A Parameterization for the Effects of Ozone on the ... - AMS Journals

DECEMBER 2012

GROGAN ET AL.

3715

A Parameterization for the Effects of Ozone on the Wave Driving Exerted by Equatorial Waves in the Stratosphere DUSTIN F. P. GROGAN AND TERRENCE R. NATHAN Atmospheric Science Program, Department of Land, Air, and Water Resources, University of California, Davis, Davis, California

ROBERT S. ECHOLS Physics Department, California Polytechnic State University, San Luis Obispo, California

EUGENE C. CORDERO Department of Meteorology and Climate Science, San Jose´ State University, San Jose´, California (Manuscript received 22 December 2011, in final form 29 May 2012) ABSTRACT An equatorial b-plane model of the tropical stratosphere is used to examine the effects of ozone on Kelvin, Rossby–gravity, equatorial Rossby, inertia–gravity, and smaller-scale gravity waves. The model is composed of coupled equations for wind, temperature, and ozone volume mixing ratio, which are linearized about a zonally averaged background state. Using the Wentzel–Kramers–Brillouin (WKB) formalism, equations are obtained for the vertical spatial scale, spatial damping rate, and amplitude of the waves. These equations yield an analytical expression for the ozone-modified wave driving of the zonal-mean circulation. The expression for the wave driving provides an efficient parameterization that can be implemented into models that are unable to spontaneously generate the ozone-modified, convectively coupled waves that drive the quasibiennial and semiannual oscillations of the tropical stratosphere. The effects of ozone on the wave driving, which are strongly modulated by the Doppler-shifted frequency, are maximized in the upper stratosphere, where ozone photochemistry and vertical ozone advection combine to augment Newtonian cooling. The ozone causes a contraction in spatial scale and an increase in the spatial damping rate. In the midstratosphere to lower mesosphere, the ozone-induced increase in wave driving is about 10%–30% for all wave types, but it can be as large as about 80% over narrow altitude regions and for specific wave types. In the dynamically controlled lower stratosphere, vertical ozone advection dominates over meridional ozone advection and opposes Newtonian cooling, causing, on average, a 10%–15% reduction in the damping rate.

1. Introduction The quasi-biennial oscillation (QBO) in zonal wind in the tropical lower stratosphere is among the most robust wave-driven circulations in Earth’s atmosphere (Baldwin et al. 2001). Indeed, the QBO signal extends far beyond its tropical seat of origin to affect, for example, the northern annular mode (Coughlin and Tung 2001), the Northern

Corresponding author address: Dustin Grogan, Atmospheric Science Program, Department of Land, Air, and Water Resources, University of California, Davis, 1 Shields Avenue, Davis, CA 95616-8627. E-mail: [email protected] DOI: 10.1175/JAS-D-11-0343.1 Ó 2012 American Meteorological Society

Hemisphere polar vortex (Lu et al. 2008), and the timing of stratospheric sudden warmings (Gray et al. 2004). Yet despite the robustness of the QBO, it is among the most difficult atmospheric circulations to spontaneously generate and accurately represent in general circulation models (GCMs). The difficulty in generating a realistic QBO hinges largely on two factors. First, the models must accurately spawn the broad spectrum of convectively coupled, equatorially trapped waves that drive the QBO. These waves include Kelvin, Rossby–gravity, equatorial Rossby, inertia–gravity, and smaller-scale gravity waves. Second, the models must accurately represent the mechanical and radiative-photochemical damping mechanisms

3716

JOURNAL OF THE ATMOSPHERIC SCIENCES

that operate on the waves in the stratosphere. Meeting these two model requirements would produce, in principle, the necessary momentum fluxes needed to generate a QBO consistent with observations. The above-mentioned modeling challenges notwithstanding, studies have made progress in spontaneously generating, without gravity wave parameterizations, ‘‘QBO-like’’ circulations in GCMs (Takahashi 1999; Hamilton et al. 2001; Watanabe et al. 2008; Kawatani et al. 2009; Kawatani et al. 2010). Perhaps the most successful spontaneously generated QBO simulation is that of Kawatani et al. (2010), who used an atmospheric GCM with horizontal and vertical resolutions of 60 km and 300 m, respectively. These spatial resolutions produced a broad spectrum of convectively coupled waves spanning zonal wavenumbers s 5 1–213 (waves s . ;100 were shown to be crucial to the westerly shear phase of the QBO). Based on a 3-yr model simulation, the amplitude and structure of the modeled QBO was in reasonable agreement with observations. The average period of the modeled QBO, however, differed sharply from the observed QBO: about 15 months (model) versus about 27 months (observations). Kawatani et al. attributed the difference between the modeled and observed QBO periods to the absence of interactive ozone in their model, which Shibata and Deushi (2005) have shown can increase the period of the QBO by a factor of about 1.8. The importance of ozone to the QBO, as well as to the semiannual oscillation (SAO) of zonal-mean wind in the tropical upper stratosphere and lower mesosphere, is well established (Leovy 1966; Echols and Nathan 1996; Cordero and Nathan 2000; Shibata and Deushi 2005). Yet, our understanding of how the distribution, transport, and chemistry of ozone combine to affect the QBO and SAO remains incomplete. Moreover, of the relatively few models that can spontaneously generate a QBO without gravity wave parameterizations, even fewer include three-dimensional, interactive ozone. There are two central challenges associated with ozone and its effects on the QBO. The first challenge hinges on understanding the causal relationships between the wave and zonal-mean fields, and the interactions involving ozone transport, ozone photochemistry, and radiative transfer. The second challenge hinges on balancing the need for sufficient model resolution to spontaneously generate and accurately reproduce a QBO, with the need to reduce computational time in order to obtain reliable statistics, especially in climate studies where multidecadal simulations may be required. Although the ability to combine long-time integrations with high spatial resolution continues to improve with increasing computer resources, challenges remain. For this reason, mechanistic

VOLUME 69

models continue as important tools for easing exposure of causal relationships, for guiding experiments, and for interpreting results obtained from high-resolution, coupled chemistry models of the tropical stratosphere. Mechanistic models have indeed advanced our understanding of how ozone transport and ozone photochemistry affect the zonal-mean circulation of the tropical stratosphere (Echols and Nathan 1996; Cordero et al. 1998; Cordero and Nathan 2000, 2005). Cordero et al., for example, used a one-dimensional (in height) model to examine how the ozone-modified Kelvin and Rossby– gravity waves together affected the model QBO. They found that the ozone increased the magnitude of the zonal-mean wind by about 1–2 m s21 and increased the period of the QBO by about 2 months. The ozoneinduced increase in the period of the QBO is consistent with Shibata and Deushi (2005), although the increase in period is considerably longer (;21 months) in Shibata and Deushi. These findings, however, differ from those of Cordero and Nathan (2000), who used a 2.5-dimensional model of the equatorial stratosphere that accounted for zonal-mean– and wave–ozone feedbacks involving Kelvin and Rossby–gravity waves. Cordero and Nathan found that the ozone feedbacks had only a small effect on the period of the QBO, though the zonal wind and temperature QBOs were 10%–20% larger than the simulations without the ozone feedbacks. Because the models are formulated differently, it is difficult to determine if the agreement between the Cordero et al. (1998) and Shibata and Deushi (2005) findings is fortuitous, or if the difference between the Cordero and Nathan (2000) and Shibata and Deushi findings is an artifact of the model differences. For example, Cordero et al. used a one-dimensional model with only two waves, with each wave coupled to ozone, whereas Shibata and Deushi used a three-dimensional, coupled chemistry model with resolved zonal waves s 5 1–42, with the smaller-scale gravity waves parameterized following Hines (1997). The gravity wave parameterization, however, did not include feedbacks with the wave–ozone field. How Shibata and Deushi’s results might change with an ozone-modified gravity wave parameterization is unclear. Cordero and Nathan’s (2000) model, however, only accounted for Kelvin and Rossby– gravity waves, though the forcing amplitudes at the lower boundary were chosen to produce a net momentum driving consistent with observations. How Cordero and Nathan’s results might change with the inclusion of an ozone-modified, high-frequency gravity spectrum also is unclear. Despite the various model limitations, there is agreement among the models and other studies regarding two features. First, models that seek to produce a realistic QBO must have a broad wave spectrum that

DECEMBER 2012

3717

GROGAN ET AL. TABLE 1. List of variables.

Variables P x, y, z 5 2H ln P0 P0 H z r(z) 5 r0 exp 2 H r0 b u(z), T(y, z), g(y, z) N2 u9(x, y, z, t), y9(x, y, z, t) w9(x, y, z, t) F9(x, y, z, t) g9(x, y, z, t) aNC (m0 ; z) A(z; T, g) B(z; T, g), C(z; T, g) G(z; T, g), j(z; T, g)

Comments Eastward, northward, and vertical direction Sea level reference pressure (51000 hPa) Scale height (57 km) Basic-state density Global mean sea level density (51.225 kg m23) Northward gradient of the Coriolis parameter evaluated at the equator (52.29 3 10211 m21 s21) Basic-state zonal-mean wind, temperature, and ozone volume mixing ratio Brunt–Vaïsa¨la¨ frequency squared (54.65 3 1024 s22) Perturbation zonal and meridional wind components Perturbation vertical wind component Perturbation geopotential Perturbation ozone volume mixing ratio Newtonian cooling coefficient Ozone heating coefficient Radiative–photochemical coefficients Shielding coefficients [as defined in Nathan and Li (1991)]

includes high-frequency gravity waves (Dunkerton 1997). Second, wave–ozone feedbacks make an important contribution to the wave damping, which affects the zonalmean wave fluxes and thus the driving of the zonal-mean circulation (Cordero et al. 1998). Motivated largely by the studies of Dunkerton (1997) and Cordero et al. (1998), we combine and extend their studies to examine the effects of ozone on a broad spectrum of tropical waves. In so doing, we analytically derive expressions for the spatial scale, spatial damping rate, and amplitude of the waves, which include the effects of ozone transport and ozone photochemistry. These expressions clearly illuminate the physics of the coupled wave–ozone feedbacks. Moreover, we show that the equations for the vertical spatial scale, spatial damping rate, and amplitude combine to yield an expression for the ozone-modified wave driving on the zonal-mean flow. The ozone-modified wave driving provides an efficient parameterization that can be implemented into models that are unable to spontaneously generate the ozone-modified, convectively coupled waves that drive the quasi-biennial and semiannual oscillations of the tropical stratosphere.

2. The model We use a model atmosphere centered on an equatorial b plane (Cordero et al. 1998). The model consists of wave equations that describe zonal and meridional wind, mass, temperature, ozone volume mixing ratio, and hydrostatic balance. We write these equations in log-pressure coordinates and linearize about a vertically sheared, basic-state zonal flow that is in radiative–photochemical equilibrium:

› › ›u ›F9 1u u9 1 w9 2 byy9 5 2 , ›t ›x ›z ›x

› › ›F9 1u y9 1 byu9 5 2 , ›t ›x ›y

›u9 ›y9 1 ›(rw9) 1 1 5 0, ›x ›y r ›z

(1)

(2)

(3)

› › ›T H 2 T9 1 y9 1u 1 N w9 ›t ›x ›y R ð‘ H r(z9) 5 2aNC T9 1 Ag9 2 G g9(x, y, z9, t) dz9 , (4) R z r0

› › ›g ›g 1u g9 1 y9 1 w9 ›t ›x ›y ›z ð‘ R r(z9) g9(x, y, z9, t) dz9 , 5 2Bg9 2 CT9 1 j H z r0 ›F9 RT9 5 . ›z H

(5)

(6)

We denote perturbation and zonal-mean quantities by primes and overbars, respectively. All of the variables in (1)–(6) are defined in Table 1. Equations (1)–(6) form a closed set that describe the linear response of a tropical wave field to coupled interactions involving the dynamical circulation, longwave radiative transfer, ozone transport, and ozone photochemistry.1

1

We use a linear representation of the ozone photochemistry, which, as Hitchcock et al. (2010) notes, compares well to the nonlinear representation of the ozone photochemistry.

3718


The radiative–photochemical portion of the model is based on Nathan and Li (1991) and Nathan and Cordero (2007). The radiative–photochemical feedbacks appear on the right-hand side (rhs) of the thermodynamic energy equation (4) and ozone continuity equation (5). The first term on the rhs of (4) represents longwave radiational cooling, which we model as Newtonian cooling based on the scale-dependent parameterization of Fels (1982). The second term on the rhs of (4) represents the local ozone heating. The first two terms on the rhs of (5) represent the temperature-dependent ozone production and destruction rate, which includes heating and reaction rates for ozone and oxygen (Lindzen and Goody 1965), and catalytic ozone loss cycles involving odd chlorine, odd nitrogen, and odd hydrogen (Hartmann 1978). Following Strobel (1978), we have incorporated the effects of multiple scattering and ground reflection into the ozone heating rate. The last terms on the rhs of (4) and (5)—called shielding effects—arise from variations in ozone above a given level. Studies have shown (Haigh 1985; Nathan and Li 1991) that the shielding effects can have an important influence on the spatial damping rates for certain wave scales. For the tropical waves considered here, however, the shielding effects are inconsequential to the vertical spatial scales and spatial damping rates. This can be shown by noting, as in Echols and Nathan (1996), that g9 is approximately oscillatory in the vertical, so that we can choose a harmonic solution for g9 such that the integrals in the shielding terms can be written as x9 5

ð‘ z

r(z9) g9(x, y, z9, t) dz9 r0

! 1 1 im0 H , 5 g9H exp(2z/H) 1 1 m20 H 2

(7)

where m0 is the (real) vertical wavenumber. We calculated the ratio of the ozone shielding effects to the local ozone effects in (4) and (5) and found the ratio is less than 1% over most of the altitude range. Thus, we neglect the shielding effects in (4) and (5). Figure 1a shows the distribution of the Newtonian cooling coefficient aNC for a vertical wavelength of 9 km. For the numerical calculations we present later, however, we calculate aNC based on the local vertical wavelength for the wave considered. Figures 1b–d show the photochemical coefficients A, B, and C, which we calculated for 158N using a September climatological zonal-mean ozone profile based on McPeters et al. (1984) and Keating and Young (1985), and a zonal-mean temperature profile based on Fleming et al. (1988); 158N is chosen because g y is largest at this latitude in the tropical stratosphere.

VOLUME 69

3. Spatial scale, spatial damping rate, and amplitude To derive the local, ozone-modified spatial scale, spatial damping rate, and amplitude for each wave type, we assume that the background fields of wind, temperature, and ozone are slowly varying in the vertical. We also assume that the radiative–photochemical forcing terms are small. As in Cordero et al. (1998), these two assumptions are made explicit by introducing the slowly varying vertical coordinate z 5 mz, where m 1, and scaling the radiative–photochemical forcing such that the righthand side of (4) and (5) are O(m). With this rescaling, the spatial variations of the background fields are such that u(z), T( y, z), and g( y, z). The wave fields in (1)– (6) are chosen to be Wentzel–Kramers–Brillouin (WKB) in form (Bender and Orszag 1978): R9(x, t; z) 5 Re R( y, z) exp i(kx 2 st) ð z : (8) 1 i [m21 m0 (z) 1 m1 (z)] dz 1 2H In (8), x 5 (x, y, z) is the position vector; R9 5 fu9, y9, w9, T9, F9, g9g represents the wave fields in wind, temperature, geopotential, and ozone, while the amplitude of the fields is denoted by R(y, z); s is the groundbased frequency; and k 5 s(ae cosu)21 , where s is the quantized zonal wavenumber, ae is the radius of Earth, and u is latitude. As in (7), m0 is the local, real, vertical wavenumber, while m1 5 m1,r 1 im1,i is the local, ozonemodified, complex correction to m0 . More specifically, the real part of m1 is an O(m) correction to the vertical spatial scale, which is proportional to m21 1,r ; the imaginary part of m1 controls the vertical spatial amplification or damping. Insertion of (8) into (1)–(6) yields an asymptotically expanded system about orders of m. The O(1) system, which is identical to Matsuno (1966), yields the dispersion relation and the meridional and vertical structures of the waves. The dispersion relation is W 2 m0 v bmW bk 2 k2 2 5 (2n 1 1) 0 . v N N

(9)

The superscript W distinguishes the wave type: Kelvin wave (W 5 KW), Rossby–gravity wave (W 5 RG), equatorial Rossby wave (W 5 ER), inertia–gravity wave (W 5 IG), and high-frequency gravity wave (W 5 GW). In deriving the dispersion relation (9), we have neglected a term arising from exp(z/2H) in (8) because it provides only a small linear correction (e.g., Andrews et al. 1987). In (9), v 5 s 2 ku is the Doppler-shifted

DECEMBER 2012

3719

GROGAN ET AL.

FIG. 1. (a) Profile of the Newtonian cooling coefficient for a wave with a vertical wavelength of 9 km. (b)–(d) Profiles of the photochemical coefficients (b) A, (c) B, and (d) C at 158N for the September climatology.

frequency, and n is a positive integer that corresponds to a meridional wave mode. In appendix A we provide the dispersion relation and the meridional and vertical structures for each wave type. The radiative–photochemical forcing enters the system at O(m). Applying a solvability condition at this order yields the local vertical wavenumber m1,r , which is inversely proportional to the vertical scale, the local spatial damping rate m1,i , and the slowly varying amplitude R(y, z) as shown: (10)

mW W W 0 (aW 1 aW mW OP 2 aVOA 2 aMOA ) , (11) 1,i 5 2 2v NC

(12)

where aW OP 5 aW VOA 5

aW MOA 5

W

m0 v W B W v W W a 2 a , m1,r 5 2 a 2v B OP v VOA B MOA

1/4 RW (y, z) 5 a0 jmW exp[P(y, z; mW 0 j 0 )] ,

ABC (B2 1 v2 )

(13)

ð ‘ Av2 W g (y, z)D(y; m0 ) dy N 2 (B2 1 v2 ) 2‘ z

(14)

AB 1 /2 2 2 (N 3 bmW 0 ) (B 1v )

ð ‘ 2‘

gy (y, z)E(y; mW )dy . 0 (15)

In (10), m1,r depends on three ozone effects: ozone photochemistry aOP , vertical ozone advection aVOA , and

3720


meridional ozone advection aMOA . In (11), m1,i depends on the same three ozone effects as m1,r , plus Newtonian cooling. In (12), a0 is a constant, which can be determined by application of a lower boundary condition (e.g., vertical velocity). In (12), (14), and (15), P(y, z; mW 0 ), W ) and E(y; m ), which are defined in appendix B, D(y; mW 0 0 are functions of the meridional and vertical structures for a given wave type. The radiative–photochemical coefficients in (10) and (11) modify a wave’s vertical spatial scale and spatial damping rate, respectively. If m1,r opposes (augments) m0 , then the vertical scale expands (contracts); if m1,i is negative (positive), then the wave spatially amplifies (damps). Newtonian cooling always damps a wave, irrespective of wave type. Thus, ozone either augments or opposes the Newtonian cooling. The central goal of our study is to determine how the effects of ozone on the equatorial waves in the tropical stratosphere drive the zonal-mean circulation. The ozonemodified wave driving on the zonal-mean circulation is measured by divergence of the Eliassen–Palm (EP) flux, which can be written as (Echols and Nathan 1996): ! Ruz › › Rby y9T92u9y9 1 r y9T92u9w9 . $ F5 r ›y HN 2 ›z HN 2 (16) Because we have obtained expressions for the ozonemodified wave fields in wind and temperature in (8), which depend on the spatial scale in (10), spatial damping rate in (11), and wave amplitude in (12), the wave driving on the zonal-mean circulation in (16) can be calculated. All that is required is knowledge of the background distributions of wind, temperature, and ozone, and the characteristics of a specific wave type: zonal wavenumber, meridional mode, and frequency. Thus the ozone-modified wave driving in (16) provides an efficient parameterization that can be implemented into models that are unable to spontaneously generate the ozone-modified, convectively coupled waves that drive the quasi-biennial and semiannual oscillations of the tropical stratosphere.

a. Basic states and radiative–photochemical coefficients We calculate the spatial scale, spatial damping rate, and amplitude based on analytical expressions for the background distributions of wind, temperature, and ozone that are consistent with observations. For the basic-state wind, we use two profiles that are representative of the descending easterly and descending westerly phases of the QBO:

VOLUME 69

u(z) 5 620 tanh[(z 2 zm )/3] ,

(17)

where z is the altitude above mean sea level and zm 5 27 km is the level of maximum wind shear (see Fig. 2a). We choose the following expression for the basic-state ozone distribution to ease the integrations in (14) and (15): 1 ^ ^ g(yL , z) 5 ag (y , z)y2 1 g(y EQ , z): 2 y 15

(18)

The corresponding meridional and vertical ozone gradients are 1 ^ ^ (y , z) , gz (yL , z) 5 ag (y , z)y2 1 g z EQ 2 yz 15

(19)

^ (y , z)y, gy (yL , z) 5 ag y 15

(20)

where L denotes the latitude where (18)–(20) are evaluated, yL 5 aeuL, and a 5 1.0 3 1028 m21 is the inverse distance from the equator (yEQ) to 158N (y15). The terms with a caret on the rhs of (18)–(20) represent the vertical distributions of zonal-mean ozone obtained from climatology. The analytical expressions for the zonal-mean ozone distribution and its gradients2 agree well with the observed distributions (see Figs. 2b–d), which we evaluate at 158N based on a September climatology using data tabulated by McPeters et al. (1984) and Keating and Young (1985). Other seasons yield similar results. Insertion of (19) and (20) into (14) and (15) yields the following radiative–photochemical feedback terms: aW OP 5

ABC , (B2 1 v2 )

(21)

3 W ^ ^ ^ (y , z) agyz (y15 , z)D 7 6g Av2 7, 6 z EQ 1 aW VOA 5 5 4 2 2 N b N(B 1 v ) 2

(22) aW MOA 5

^ (y , z)E^W ABag y 15 Nb(B2 1 v2 )

.

(23)

Appendix C lists, for each wave type, the explicit forms ^ yz 5 0 in (22), then our (11) reof (21)–(23). If we set g duces to (14) in Cordero et al. (1998), which is valid only for the Kelvin and Rossby–gravity waves.

2

The form of the vertical mean ozone gradient in (19) differs slightly from Cordero et al. (1998) because we allow the meridional ozone gradient, the first term on the rhs of (18), to vary with height.

DECEMBER 2012

GROGAN ET AL.

3721

FIG. 2. (a) The basic-state zonal flows from (17) for the descending westerly QBO (dotted) and descending easterly QBO (solid). (b)–(d) Observational (solid) and derived (dotted) vertical distributions of (b) g(z), (c) gz (z), and (d) g y (z) at 158N. The observational and derived g y (z) at 158N are indistinguishable.

b. Limiting cases We consider several limiting cases in order to clarify the differences and similarities in the wave responses to ozone. The three limits are (i) critical layers [v(k) / 0], (ii) reflecting layers [m0 (v, k) / 0], and (iii) short waves (k / ‘). Although the WKB approximation may be problematic in limits (i) and (ii), these limits nonetheless provide qualitative guidance in the interpretation of the results that we present later. We list the leading-order ozone physics for each limit in Table 2, where, to ease interpretation, we note that aOP } C, bz or g byz , and aMOA } g by . aVOA } g For all wave types, in the vicinity of a critical layer (v / 0), m1,r is controlled by aOP and aVOA , which

together cause a contraction in spatial scale; m1,i is controlled by aNC , which causes an increase in spatial damping rate. The controlling physics, however, shows different dependencies on v as v/0. For the Rossby– gravity, inertia–gravity, and gravity waves, where m0 } v22 , we obtain m1,r } v22 and m1,i } v27/2 as v/0; for the Kelvin and equatorial Rossby waves, where m0 } v21 , we obtain m1,r } v21 and m1,i } v25/2 as v/0. The fractional dependencies in the exponents for m1,i arise from the Fels (1982) parameterization for Newtonian cooling, where aNC } v21/2 as v/0. For reflecting layers, which occur where m0 (v, k)/0, the physics-controlling m1,r and m1,i depend on v and wave type (see Table 2). The Kelvin, Rossby–gravity, inertia–gravity, and gravity waves all have reflecting

3722


VOLUME 69

TABLE 2. The controlling physics for several limiting cases for the vertical spatial scale in (10) and spatial damping rate in (11). The various wave types include KW 5 Kelvin; RG 5 Rossby–gravity; ER 5 n $ 1 equatorial Rossby; and IG/GW 5 n $ 0 inertia–gravity/ gravity waves. Vertical scale m21 1,r Leading-order terms

Controlling physics

Leading-order terms

^z ) 2k(AB22 C 2 AB21 g

Photochemistry and vertical ozone advection Photochemistry and vertical ozone advection Photochemistry and vertical ozone advection

kaNC

Newtonian cooling

aNC

Newtonian cooling

aNC

Newtonian cooling

Limiting cases v/06 KW(1)

ER(2)

^z AB22 C 2 AB21 g

RG(6), IG/GW(6)

^z ) 7(AB22 C 2 AB21 g

6 mw 0 /0 KW(2), RG(2)

RG(1)

^ yz ABg

^ ) ^ 2 Bg A(k3 g y yz (B2 + k2 ) ^ yz ABg (B2 + k2 )

ER(1)

2

IG/GW(6)

^ yz 7ABg

k/‘ IG/GW

Spatial damping rate m1,i

2

^y Ag u4

Vertical ozone advection Meridional and vertical ozone advection Vertical ozone advection Vertical ozone advection Meridional ozone advection

layers where v / ‘. The inertia–gravity and gravity waves have additional reflecting layers where v/ 2 ‘; the Rossby–gravity waves have an additional reflecting layer where v / 2b/k, which is also where the equatorial Rossby waves possess their only reflecting layer. For all the wave types, except the Rossby–gravity waves, m1,r and m1,i are controlled solely by aVOA . For the Rossby–gravity waves, m1,r and m1,i are controlled by aVOA where v / ‘, and by aVOA and aMOA where v / 2 b/k. In the short wave limit (k / ‘), which is most relevant to the gravity waves, m1,r is controlled by aMOA , while m1,i is controlled by aNC and aVOA . This latter result shows that for gravity waves, the ozone feedback due to vertical ozone advection is the most important ozone contribution to the spatial damping rate and thus wave driving of the zonal-mean circulation.

4. Results a. Preliminaries In this section, we show for each wave type the vertical distributions of the ozone-modified vertical scale m21 1,r , spatial damping rate m1,i , and Eliassen–Palm flux

^ yz 2Ag ^ y 2 k21 g ^ ) A(k2 Bg yz (B2 + k2 )

Controlling physics

Vertical ozone advection Meridional and vertical ozone advection

^ yz Ak21 g 2 2 (B + k2 )

Vertical ozone advection

^ yz 6Ag

Vertical ozone advection

aNC A ^ yz ) 2 2 (g z 1 ju j g u2 u

Newtonian cooling and vertical ozone advection

divergence $ F. The distributions are calculated using observed values of zonal-mean ozone, zonal-mean wind, wavenumber, and frequency. For all of our calculations, we use the zonal-mean ozone distribution shown in Fig. 2b and three general zonal-mean wind profiles: the easterly phase of the QBO, the westerly phase of the QBO, and a motionless atmosphere. We discuss our choice of wavenumbers and frequencies next. Figure 3 shows dispersion curves (thin solid) for the Kelvin, Rossby–gravity, equatorial Rossby, and inertia– gravity waves. The curves in the figure correspond to equivalent depths of 8 (bottom curve) and 90 m (top curve) or, equivalently, vertical wavelengths of 2.8 and 9.2 km, respectively. Overlaying the dispersion curves are boxes (thick solid) that enclose the peak wave activity detected in 15-yr European Centre for MediumRange Weather Forecasts Re-Analysis (ERA-15) data (Tindall et al. 2006a) and satellite observations (Ern et al. 2008). According to Tindall et al., peak wave activity for the Kelvin, Rossby–gravity, equatorial Rossby, and n 5 0 inertia–gravity wave spectrums (circles) is concentrated in wavenumber–frequency space, whereas peak wave activity for the n 5 1 and n 5 2 eastward- and westward-propagating inertia–gravity wave spectrums are more uniformly distributed. Thus, the inertia–gravity

DECEMBER 2012

3723

GROGAN ET AL.

To illustrate the effects of ozone on the waves, we calculate m1,r , m1,i , and $ F for the waves circled in Fig. 3, and for two waves from the inertia–gravity and gravity wave spectrums (see Table 3).

b. Ozone-modified vertical scale, spatial damping rate, and EP flux divergence

FIG. 3. Dispersion curves for (a) symmetric and (b) antisymmetric equatorial waves. Equivalent depths of 8 (bottom curve) and 90 m (top curve), corresponding to a vertical wavelength of 2.8 and 9.2 km, respectively, are plotted for the KW, RG, ER, and IG waves. Overlaid are the regions of interest (thick solid, three in each figure) and peak wave power (open circle). See text for additional details.

wave spectrum contains a wide range of equivalent depths in comparison to the other wave types. We also consider small-scale gravity waves spanning quantized zonal wavenumbers 50 , s , 150, with corresponding frequencies of j3.0j , s , j6.0j cycles per day (cpd) (4–8-h periods). These wavenumbers not only fall within the observed gravity wave spectrum (Ern et al. 2004, 2011) but more importantly, they have been shown by Kawatani et al. (2010) to be mostly responsible for driving the zonal-mean circulation (as measured by the EP flux divergence). We do not show the dispersion curves for the gravity waves, however, since the wavenumbers that we consider are outside the wavenumber and frequency range in Fig. 3.

Consider first m21 1,r . For our m 1 ordering of the radiative–photochemical feedbacks (see section 3), changes in vertical scale are solely due to ozone; Newtonian cooling has no effect to order m. In the upper atmosphere, irrespective of wave type and zonal-mean wind, ozone causes the vertical scale to contract. In the lower stratosphere, depending on wave type and zonalmean wind, ozone may cause the vertical scale to expand or contract. To see this, refer to Fig. 4, which shows m1,r for each wave type for a motionless atmosphere [u(z) 5 0]. For ease of comparison, each profile in Fig. 4 is normalized to its maximum value. Positive (negative) values in Fig. 4 correspond to a contraction (expansion) of the vertical scale. For each wave type, the largest contraction in vertical scale occurs between 35 and 50 km. In this region, aOP and aVOA make similar contributions to the spatial scale. For a motionless atmosphere, as well as for the wind distributions in Fig. 2a, ozone causes the vertical wave scale to change by about 1%–7% between about 35 and 50 km. Consider next m1,i . Echols and Nathan (1996) and Cordero et al. (1998) have examined the spatial damping rate of Kelvin waves and Rossby–gravity waves for highly select wavenumbers and frequencies. Here we summarize their results, consider a broader range of wavenumbers and frequencies for these waves, and consider the other prominent tropical wave types that they did not consider. To aid in the interpretation of the effects of ozone on the wave damping, it is instructive to introduce the ratio of the dynamical to photochemical time scales B/v. In the dynamically controlled, transition, and photochemically controlled regions, which extend from the lower stratosphere to the lower mesosphere, respectively, we have B/v 1, B/v ’ 1, and B/v 1. In the dynamically controlled lower stratosphere (;20–30 km), where B/v 1, variations in m1,i are mostly due to aNC and aVOA , irrespective of wave type or zonalmean wind. For B/v 1, the expression for the spatial damping rate (11) reduces to W 21 mW i,1 ’ 20:5m0 v (aNC 2 aVOA ) .

(24)

^ z throughout the stratosphere, which ^ yz g Because g we have confirmed by numerical calculation, we are able to obtain the simplified expression aVOA ’ ^ z . 0 below about 34 km, aVOA op^ z N 22 . Because g Ag poses aNC in this region. This means that in the lower

3724


TABLE 3. Zonal wavenumber s, ground-based frequency s, zonal phase speed cx, and meridional wave mode n for the waves used to calculate m1,r and m1,i in Figs. 4 and 5. Because jcx j . juj , none of the waves possess critical levels. Because the n 5 1 and n 5 2 IGs have very similar wave characteristics, we only list below the results for n 5 1. The gravity wave mode n 5 80 has a meridional wavelength that is similar to its zonal wavelength. Wave type

s

s (cpd)

cx (m s21)

n

KW RG ER n 5 0 IG n 5 1 IG GW

1 4 1 5 12 80

0.1 20.25 20.1 0.3 60.8 65.0

46.3 229.0 246.3 27.8 630.9 629.0

21 0 1 0 1 80

stratosphere, ozone is spatially destabilizing, that is, ozone alone causes the wave to amplify in space. This is consistent with other wave–ozone studies in the tropics and extratropics (Leovy 1966; Zhu and Holton 1986; Nathan and Li 1991; Echols and Nathan 1996; Hitchcock et al. 2010). For the wind distributions shown in Fig. 2a, the vertical ozone advection reduces the spatial damping rate by about 10%–15%. In the transition region, B/v ’ 1, both ozone advection and ozone photochemistry make significant contributions to the spatial damping rate. In contrast to the dynamically controlled region, the vertical extent of the transition region depends on the wave type and wind distribution. To illustrate this, we show in Fig. 5 the relative contributions of aNC , aOP , aVOA , and aMOA to the spatial damping rate for each wave type. We show the region ;30–50 km and consider first the case where the waves are counter propagating to the zonal-mean wind; we consider the case of waves propagating with the flow later in this section. For westward- (counter) propagating waves, we use the descending westerly phase of the QBO, and for the eastward- (counter) propagating waves, we use the descending easterly phase (see Fig. 2a). In Figs. 5e,f, we show the results for the eastward-propagating inertia– gravity (EIG) and gravity waves (the westward-propagating waves yield essentially the same results). Below 32 km, the spatial damping rate increases as v decreases, consistent with the limiting cases discussed in section 3b. For the Kelvin wave, the transition region is narrow. Figure 5a shows that aOP is the primarily ozone feedback, though aVOA also makes a small but significant contribution in the midstratosphere (;34–37 km). At 47 km aOP is maximized and monotonically decreases above and below. For this midstratosphere region, the ozone feedbacks augment Newtonian cooling by about 20%–50%, consistent with previous studies (e.g., Echols and Nathan 1996; Cordero et al. 1998). For the Rossby–gravity wave, the transition region is broad. Figure 5b shows that all three ozone

VOLUME 69

feedbacks—aOP , aVOA , and aMOA —contribute significantly to the spatial damping rate in the transition region. The importance of aMOA to the damping of the Rossby– gravity waves makes them unique among all the wave types. Above about 35 km, the three ozone feedbacks together increase the spatial damping rate by about 30%– 80%. Below about 35 km, two ozone feedbacks, aVOA and aMOA , predominate and together decrease the spatial damping rate by about 0%–10%. As expected, these results agree with Cordero et al. (1998). We examine next the equatorial Rossby, inertia–gravity, and gravity waves, which were not considered by either Echols and Nathan (1996) or Cordero et al. (1998). We consider first the equatorial Rossby wave, which, like the Kelvin wave, is characterized by a narrow transition in which aOP . aVOA (see Fig. 5c). Within this narrow region, aOP and aVOA augment aNC to increase the spatial damping rate by about 5%–15% (cf. with the ;20%–50% for the Kelvin wave). For the equatorial Rossby wave, which has a relatively large vertical wavenumber, Newtonian cooling strongly dominates over ozone in the spatial damping rate. This is because the ratio of Newtonian cooling to the ozone heating is O(m10/2 ) as m0 becomes large. For the inertia–gravity and gravity waves, the transition region expands with frequency. The effects of ozone on the spatial damping rate in the transition region depends only on aOP and aVOA ; aMOA is inconsequential. Because ^ z throughout the stratosphere, the relative impor^ yz g g tance of vertical ozone advection and ozone photochemistry is measured by the ratio aVOA /aOP 5 v2 gz /BCN 2 . This ratio shows a quadratic dependence on v, meaning that near a critical level, photochemistry dominates over advection, and that away from a critical level, advection dominates over photochemistry. Similar to the ratio B/v, the ratio aVOA /aOP 5 v2 g z /BCN 2 is, in slightly disguised form, another way to distinguish the dynamical, transition, and photochemical control regions. Because v increases as we move from the n 5 0 inertia–gravity wave, to the n 5 1 inertia–gravity wave, to the gravity wave, vertical ozone advection becomes increasingly dominant over photochemistry in the spatial damping rate (see Figs. 5d–f). Between about 30 and 50 km, aVOA and aOP both augment aNC to increase the spatial damping rate by about 10%–30%. In the photochemically controlled region, B/v 1, the expression for the spatial damping rate (11) is approximated by W 21 mW i,1 ’ 20:5m0 v (aNC 1 aOP ) .

(25)

Consistent with other studies, ozone photochemistry always augments Newtonian cooling (Leovy 1966; Nathan and Li 1991; Echols and Nathan 1996; Cordero

DECEMBER 2012

GROGAN ET AL.

3725

FIG. 4. Ozone-modified vertical wavenumber normalized by its maximum value for (a) westward-propagating and (b) eastward-propagating waves in a motionless atmosphere. The wave types include KW, RG, ER, IG (n 5 0 or n 5 1), and GW.

et al. 1998; Nathan and Cordero 2007; Nathan et al. 2011; Albers and Nathan 2012). Within the region near 30–65 km, (25) controls the damping rate for waves that propagate with the zonal-mean flow. Recall from section 3b, in the critical layer limit v / 0, we have 2p mW 0 } v , where p 5 1 for the Kelvin and equatorial Rossby waves, and p 5 2 for the Rossby–gravity, inertia– 2p in gravity, and small-scale gravity waves. Using mW 0 } v 2(p11) } v (25), we have for the spatial damping rate mW i,1 (aNC 1 aOP ) as v/0. From this expression, we see that as the Doppler-shifted frequency v decreases, the spatial damping rate increases, and because aNC } v21/2 as v / 0 (see section 3b), aNC becomes increasingly

dominant over aOP . Thus, compared to waves that propagate counter to the zonal-mean flow (Fig. 5), waves that propagate with the zonal-mean flow are more effectively damped, with the effects of Newtonian cooling tending to dominate over ozone heating. For the wind distributions that we have considered, irrespective of wave type, in the photochemically controlled region, ozone feedbacks augment Newtonian cooling to increase the spatial damping rate by about 10%–20% (not shown). To calculate the effects of ozone on the wave driving exerted by the waves on the zonal-mean flow, we follow Cordero et al. (1998) and compute the latitudinally integrated EP flux divergence as

3726


VOLUME 69

FIG. 5. Contributions to the spatial damping rate due to aNC (thick solid), aNC 1 aOP (solid), aNC 1 aVOA (dotted), and aNC 1 aMOA (dashed) in (11) for (a) KW, (b) RG, (c) ER, (d) IG (n 5 0), (e) EIG (n 5 1), and (f) eastward GW.

ðz dhF z i W w 5 A(z0 )m1,i exp 22 m1,i dz9 , h$ Fi 5 (26) dz 0 Ð‘ where h i 5L21 2‘( ) dy, F z 5r[(R/H)byN 22 y9T92 u9w9], and A(z0 ) is the zonal-mean momentum flux at the lower boundary, located at z0 5 16 km (;100 hPa). Values for A(z0 ) were obtained from Tindall et al. (2006a). Because we have chosen the diabatic effects due to Newtonian cooling and ozone heating to be O(m) (see section 3), m21 1,r is also O(m) and thus does not enter the

expression for the latitudinally averaged EP flux divergence (26). The effects of m21 1,r on the EP flux divergence appear at O(m2), which we do not consider. For all of the waves, the EP flux divergence shows a significant response to ozone. As an example, we consider the n 5 1 EIG wave in both westerly and easterly phases of the zonal-mean flow. For the westerly phase, the maximum shear is at 35 km, while for the easterly phase, the maximum shear is at 18 km. These altitudes of maximum shear are consistent with the QBO and yield among the largest responses in h$ Fi for the EIG wave.

DECEMBER 2012

GROGAN ET AL.

3727

Figure 6a shows, for the EIG wave propagating with the flow in a descending westerly phase of the QBO, the vertical distribution of h$ Fi for Newtonian cooling alone (solid) and for Newtonian cooling and ozone heating combined (dotted). Ozone causes h$ Fi to increase by as much as 15% above 33 km and to decrease by as much as 8% below 33 km. For the EIG wave propagating counter to the flow in a descending easterly phase of the QBO, Fig. 6b shows that ozone causes h$ Fi to increase by as much as 60% above 32 km and to decrease by as much as 12% below 32 km. As expected, the response of h$ Fi to ozone is consistent with the response of the spatial damping rate to ozone (cf. Figs. 6b and 5e).

5. Conclusions The tropical stratosphere contains a variety of wave types that span a wide range of spatial and temporal scales. These waves include Kelvin, Rossby–gravity, equatorial Rossby, inertia–gravity, and smaller-scale gravity waves. Here we have derived expressions for the vertical spatial scale and spatial damping rate of these waves that account for feedbacks with ozone. These expressions clearly illuminate how ozone—both its photochemistry and its transport—spatially modulates the waves. Moreover, we show that the equations for the spatial damping rate and amplitude combine to yield an expression for the ozone-modified wave driving on the zonal-mean flow. The wave driving provides an efficient parameterization that can be implemented into models that are unable to spontaneously generate ozone-modified, convectively coupled waves that drive the zonal-mean circulation in the tropical stratosphere, including the QBO and SAO. To implement the parameterization only requires the following: background distributions of wind, temperature, and ozone; and, for a given wave type, the zonal wavenumber, meridional mode, frequency, and wave amplitude at the lower boundary. We have built our analysis on an equatorial b-plane model of the tropical stratosphere, which describes the linear response of a tropical wave field to coupled interactions involving the dynamical circulation, longwave radiative transfer, ozone transport, and ozone photochemistry. We show that the effects of ozone on the spatial scale, spatial damping rate, and EP flux divergence (wave driving) are strongly modulated by the Doppler-shifted frequency. The ozone effects are maximized in the upper stratosphere, where ozone photochemistry and vertical ozone advection combine to augment Newtonian cooling. This causes a contraction in vertical spatial scale and an increase in spatial damping rate.

FIG. 6. Profiles of the latitudinally integrated EP flux divergence (26) due to aNC (solid) and aNC 1 aOP 1 aVOA 1 aMOA (dotted) for the n 5 1 EIG wave propagating in a (a) descending westerly QBO and (b) a descending easterly QBO. A momentum flux of 5 3 1025 m2 s22 was imposed at the lower boundary [based on Tindall et al. (2006a)].

We have shown using observed background states of wind, temperature, and ozone that, in the mid- to upper stratosphere, the ozone-induced increase in damping rate and EP flux divergence averages about 10%–30% for all wave types, but that it can be as large as 80% over narrow altitude regions and for specific wave types. In the dynamically controlled lower stratosphere, vertical ozone advection dominates over meridional ozone advection and opposes Newtonian cooling, causing, on average, a 10%–15% reduction in the damping rate and EP flux divergence. The altitude regions where stratospheric ozone is most effective in modifying the wave, the spatial damping rate

3728


and thus wave driving are precisely the regions where modeling and observational studies show ozone is undergoing changes due to human activities. Chemistry– Climate Models (CCM) show, for example, that for various climate change scenarios, zonal-mean ozone is projected to increase in the tropical upper stratosphere and decrease in the tropical lower stratosphere (Eyring et al. 2007; Butchart et al. 2010). The increase in zonal-mean ozone in the tropical upper stratosphere is attributed to the reduction of ozone-depleting substances (Eyring et al. 2007). The decrease in zonalmean ozone in the tropical lower stratosphere, however, is attributed to increases in zonal-mean tropical upwelling caused by increases in both CO2 and sea surface temperatures (Lamarque and Solomon 2010). Consistent with the CCM studies, observations show a positive trend in ozone in the upper stratosphere (Randel and Wu 2007). In the lower stratosphere, however, the observations are inconclusive (Jones et al. 2009). As we discussed in the introduction, modeling studies have made progress in spontaneously generating, without gravity wave parameterizations, convectively coupled waves that drive ‘‘QBO-like’’ oscillations in GCMs. The complexity of these models, however, often poses challenges to understanding the causal relationships between the wave, zonal-mean, and ozone fields. Moreover, using GCMs requires balancing the need for sufficient model resolution to spontaneously generate and accurately reproduce a QBO with the need to reduce computational time in order to obtain reliable statistics, especially in climate studies where multidecadal simulations may be required. For these reasons, the scaledependent parameterization for the ozone-modified wave driving that we have derived in this study will likely aid both mechanistic and GCM studies of the tropical stratosphere. For mechanistic studies, the parameterizations are computationally efficient and easy to implement. For the GCM studies, the parameterizations can serve as an interpretive tool to understand the role of the ozone-modified wave physics operating in the models. Acknowledgments. We thank John Albers and Stephen Santilena for their insightful comments on this work. We also thank three reviewers for their constructive comments. Support for this work was provided in part by NSF Grant ATM-0733698 (TRN), NASA Living with a Star (LWS) Program Grant NNG05 GM55G (TRN and ECC) and NASA/NRL Grant NNH08AI67I (TRN), and NSF Faculty Early Career Development (CAREER) Program Grant ATM-0449996 (ECC).

VOLUME 69

APPENDIX A Dispersion Relations and Spatial Structures at O(1) The dispersion relation for each wave type is written as (Andrews et al. 1987) Nk 52 , mKW 0 v

(A1)

mRG 0 5 2sgn(v) mER 0 5

Nb v2

n1

N (b 1 vk), v2

(A2)

1 1 2 vk vk 1/2 2 n1 11 , 1 2 2 b b (A3)

Nb 1 ,GW n 1 mIG 5 2sgn(v) 0 2 v2 2 1 vk vk 1/2 1 1 n1 11 , 2 b b

(A4)

where the superscripts denote wave type: KW 5 Kelvin wave, RG 5 Rossby–gravity, ER 5 equatorial Rossby, IG 5 inertia–gravity,and GW 5 small-scale gravity wave. The dispersion relation for the Kelvin wave is a special case, corresponding to n 5 21 in (9). The lowest-order spatial structures of the wave fields are 8 9 1, > > > > > > > > > > > > > > 0, > > 8 9KW > > > > u9, > > > > > > v > > > > > > > y9, > > , > > > > > > > N > > > > > > > > < = < = w9, KW HN , (A5) 5S ,> 2i > > T9, > > > > > R > > > > > > > > > > > > > > > F9 , > > > > > v > > : ; > > , > > g9 > > > > k > > > > > > > > > > 1 > : i g > ; z N

9 8 u9, >RG > > > > > > y9, > > > > > > = < w9, > > T9 , > > > > > > > > > > F9 , > > > ; : g9

5 SRG

8 > > > > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > > > > =

> > jmRG > > 0 jHvy > > , > > > > > > R > > > > > > > > > > ivy, > > > > > > > > ! > > > > RG 2 > > ijm0 jv y > > i > > : ; gy 2 g z 2 v N

,

(A6)

DECEMBER 2012

3729

GROGAN ET AL.

8 9 > > > > ER,IG,GW > > 1 / 2 ER , IG , GW ^ > > i(Nbjm0 j) H , > > 1 > > > > > > > > > > Hn , > > 9ER,IG,GW 8 > > > > > > u9 , > > > > > > > > ER , IG , GW > > > > ER , IG , GW ER , IG , GW 1 / 2 ^ > > 2ivm > > [b/(Njm j )] , H y9, > > > > 2 0 0 > > > > = < = < w9, ER,IG,GW ER,IG,GW , 5S Hm ER , IG , GW 1 / 2 ER , IG , GW 3 0 T9 , > > > > ^ (N b/jm0 2 j) H , > > > > 2 > > > > > > > R F9 , > > > > > > > > > ; > > : > > ER , IG , GW > > g9 1 / 2 ER , IG , GW 3 ^ > > i(N b/jm0 j ) H2 , > > > > > > > > > > > > i ER,IG,GW > > ER , IG , GW ER , IG , GW 1 / 2 > > ^ fH g 2 ivm [b/(Njm j )] g g H > > 2 > > 0 0 n y z :v ;

where, for any wave type W, we define

P2 5 2

2 h W W S 5 R exp 2 . 2

(A8) P3 5

In (A7) the Hn represent Hermite polynomials; H0 5 1, H1 5 2h, H3 5 4h2 2 2, etc., where h(z, y) 5 2 1 1/2 ) y, (bjmW 0 jN

^ W 5 1 Hn11 (h) 2 nHn21 (h) , H 2 2 b2 b1 ^ W 5 1 Hn11 (h) 1 nHn21 (h) , H 1 2 b2 b1 b2 5 jmW 0 j v 2 Nk,

and

b1 5 jmW 0 j v 1 Nk.

(A9a)

ð‘ z0

c1 J 21

ð‘ 2‘

^W H ^ W ) dy dz , (H 2 2 z

z0

2jmW 0 j

2 1 nb2 (n 1 1)b1 2

^ W , b2 , and b1 are defined in appendix A. In (B3)–(B5),H 2 In (B4) c1 and J are defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi bjmW 0 j c1 5 n21 , pN 2 n! 2 (n 1 1) 1 b2 n b1 2 . b22 b21

(A10a)

J5

(A10b)

For the Kelvin wave, P is defined as P 5 2lnjvj 1/4 .

The slowly varying amplitude R for all wave types is represented as 1/4 exp[P(y, z; mW RW (y, z) 5 a0 jmW 0 j 0 )] .

(B1)

For the Rossby–gravity, equatorial Rossby, inertia–gravity, and gravity waves P 5 P1 1 P2 1 P3 ,

(B2)

n/2 , P1 5 lnjmW 0 j

(B3)

(B6)

(B7)

(B8)

The functions D and E that appear in the radiative– photochemical terms aVOA and aMOA in (14) and (15) arise from the inner products fw9T9*g and fy9T9*g, which for the Rossby–gravity, equatorial Rossby, inertia– gravity, and gravity waves are defined as W iAv W 21 m0 2 ^W H ^W (H D(y; m0 )5c1 J 2 2 ) exp(2h ), 2v N 2 (B 2 iv) (B9) 21 E(y; mW 0 ) 5 2c1 J

3

where

dz:

(B5)

(A9b)

Wave Amplitudes

(B4)

ð ‘ (jmW j) 2 2 0 z (n 1 1)b 1 2 n(n 1 1)b2 b1 2 n b2

1

APPENDIX B

(A7)

mW 21/2 0 (BNjmW 0 j) 2v

A ^ W H ) exp(2h2 ) , (H N(B 2 iv) 2 n

(B10)

^ W , Hn , and h are defined in appendix A. where c1 , J 21 , H 2

3730


VOLUME 69

APPENDIX C

The functions D and E for the Kelvin wave are defined as D(y; mKW 0 ) 5 c2

mKW iAv 0 exp(2h2 ) , 2v N 2 (B 2 iv)

E(y; mKW 0 ) 5 0,

Ozone Feedback Terms for Each Wave Type (B11) (B12)

The analytical expressions of the zonal-mean ozone gradients in (22) and (23) yield the following expressions for the ozone feedback terms: faOP , aVOA , aMOA gKW ! # " ^ ^ vag A v2 g yz z ,0 , BC, 5 2 1 N N 2bNk B 1 v2

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bjmKW 0 j c2 5 . pN

faOP , aVOA , aMOA gRG 5

(B13)

A

8 > >
> :

2 BC,

3 ^ v2 ag

^ vBag y

9 > > =

^ 7 g v2 6 yz 6 z 13 7, , 4 N N 4 bN(b 1 vk) 5 N(b 1 vk) > > ;

1 3 ER ER ^ ^ ^ ^ ag D C Bagy E 7 A 6 B z 1 yz C, 7, 6BC, faOP , aVOA , aMOA gER 5 2 A 5 N@ N b Nb B 1 v2 4 2

faOP , aVOA , aMOA gIG/GW 5

A

1 2jmW 0 j

3 2 1 2 (n11) n1 b1 2n(n11)b2 b11n n2 b2 2 2 3 , 2 2 (n11)b1 1nb2 (C5) 2 2 nb2 b 1 (n 1 1)b2 b1 2 1 . E^W 5 W 2 1 nb2 (n 1 1)b1 m0 2

1

^ D ^ 2B g ag 6 6BC, v B z 1 yz @ N N b

where for the ER waves, IG waves, and GWs, ^W 5 D

0 ^ IG,GW

B2 1 v2 4

(C6)

The b2 and b1 are defined in appendix A.

REFERENCES Albers, J. R., and T. R. Nathan, 2012: Pathways for communicating the effects of stratospheric ozone to the polar vortex: Role of zonally asymmetric ozone. J. Atmos. Sci., 69, 785–801. Andrews, D. G., J. R. Holton, and C. B. Loevy, 1987: Middle Atmosphere Dynamics. Academic Press, 489 pp.

(C2)

0

^ v2 B g

2

(C1)

(C3) 3

^ E C Bag y C, A Nb

ÎG,GW

7 7, 5

(C4)

Baldwin, M. P., and Coauthors, 2001: The quasi-biennial oscillation. Rev. Geophys., 39, 179–229. Bender, C. M., and S. A. Orszag, 1978: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. International Series in Pure and Applied Mathematics, Vol. 1, McGraw-Hill, 593 pp. Butchart, N., and Coauthors, 2010: Chemistry–Climate Model simulations of twenty-first century stratospheric climate and circulations changes. J. Climate, 23, 5349–5374. Cordero, E. C., and T. R. Nathan, 2000: The influence of wave– and zonal mean–ozone feedbacks on the quasi-biennial oscillation. J. Atmos. Sci., 57, 3426–3442. ——, and ——, 2005: A new pathway for communicating the 11-year solar cycle to the QBO. Geophys. Res. Lett., 32, L18805, doi:10.1029/2005GL023696. ——, ——, and R. S. Echols, 1998: An analytical study of ozone feedbacks on Kelvin and Rossby–gravity waves: Effects on the QBO. J. Atmos. Sci., 55, 1051–1062. Coughlin, K., and K.-K. Tung, 2001: QBO signal found at the extratropical surface through northern annular modes. Geophys. Res. Lett., 28, 4563–4566. Dunkerton, T. J., 1997: The role of gravity waves in the quasibiennial oscillations. J. Geophys. Res., 102 (D22), 26 053– 26 076. Echols, R. S., and T. R. Nathan, 1996: Effect of ozone heating on forced equatorial Kelvin waves. J. Atmos. Sci., 53, 263–275.

DECEMBER 2012

GROGAN ET AL.

Ern, M., P. Preusse, M. J. Alexander, and C. D. Warner, 2004: Absolute values of gravity wave momentum flux derived from satellite data. J. Geophys. Res., 109, D20103, doi:10.1029/ 2004JD004752. ——, ——, M. Krebsbach, M. G. Mlynczak, and J. M. Russell III, 2008: Equatorial wave analysis from SABER and ECMWF temperatures. Atmos. Chem. Phys., 8, 846–869. ——, ——, J. C. Collie, C. L. Hepplewhite, M. G. Mlynczak, J. M. Russell III, and M. Riese, 2011: Implications for atmospheric dynamics derived from global observations of gravity wave momentum flux in the stratosphere and mesosphere. J. Geophys. Res., 116, D19107, doi:10.1029/2011JD015821. Eyring, V., and Coauthors, 2007: Multimodel projections of stratospheric ozone in the 21st century. J. Geophys. Res., 112, D16303, doi:10.1029/2006JD008332. Fels, S. B., 1982: A parameterization of scale-dependent radiative damping rates in the middle atmosphere. J. Atmos. Sci., 39, 1141–1152. Fleming, E. L., S. Chandra, M. R. Schoeberl, and J. J. Barnett, 1988: Monthly mean global climatology of temperature, wind, geopotential height, and pressure for 0-120 km. NASA Tech. Memo. NASA-TM-100697, 91 pp. Gray, L. J., S. Crooks, C. Pascoe, S. Sparrow, and M. Palmer, 2004: Solar and QBO influences on the timing of stratospheric sudden warmings. J. Atmos. Sci., 61, 2777–2796. Haigh, J. D., 1985: A fast method for calculating scale-dependent photochemical acceleration in dynamical models of the stratosphere. Quart. J. Roy. Meteor. Soc., 111, 1027–1038. Hamilton, K., R. J. Wilson, and R. S. Hemler, 2001: Spontaneous stratospheric QBO-like oscillations simulated by the GFDL SKYHI general circulation model. J. Atmos. Sci., 58, 3271–3292. Hartmann, D. L., 1978: A note concerning the effect of varying extinction on radiative-photochemical relaxation. J. Atmos. Sci., 35, 1125–1130. Hines, C. O., 1997: Doppler-spread parameterization of gravitywave momentum deposition in the middle atmosphere. Part 2: Broad and quasi monochromatic spectra, and implementation. J. Atmos. Sol. Terr. Phys., 59, 387–400. Hitchcock, P., T. G. Shepherd, and S. Yoden, 2010: On the approximation of local and linear radiative damping in the middle atmosphere. J. Atmos. Sci., 67, 2070–2085. Jones, A., and Coauthors, 2009: Evolution of stratospheric ozone and water vapour time series studied with satellite measurements. Atmos. Chem. Phys., 9, 6055–6075. Kawatani, Y., M. Takahashi, K. Sato, and S. P. Alexander, 2009: Global distribution of atmospheric waves in the equatorial upper troposphere and lower stratosphere: AGCM simulation of sources and propagation. J. Geophys. Res., 114, D01102, doi:10.1029/2008JD010374. ——, K. Sato, T. J. Dunkerton, S. Watanabe, S. Miyahara, and M. Takahashi, 2010: The roles of equatorial trapped waves and internal inertia–gravity waves in driving the quasi-biennial oscillation. Part I: Zonal mean wave forcing. J. Atmos. Sci., 67, 963–380.

3731

Keating, G. M., and D. F. Young, 1985: Interim reference ozone models for the middle atmosphere. Atmospheric Structure and Its Variation in the Region 20 to 120 km: Draft of a New Reference Middle Atmosphere, K. Labitzk, J. J. Barnett, and B. Edwards, Eds., Vol. 16, Middle Atmosphere Program: Handbook for MAP, SCOSTEP, University of Illinois, 318 pp. Lamarque, J.-F., and S. Solomon, 2010: Impact of changes in climate and halocarbons on recent lower stratosphere ozone and temperature trends. J. Climate, 23, 2599–2611. Leovy, C. B., 1966: Photochemical destabilization of gravity waves near the mesosphere. J. Atmos. Sci., 23, 223–232. Lindzen, R., and R. Goody, 1965: Radiative and photochemical processes in mesospheric dynamics: Part I, models for radiative and photochemical processes. J. Atmos. Sci., 22, 341–348. Lu, H., M. P. Baldwin, L. J. Gray, and M. J. Jarvis, 2008: Decadalscale changes in the effect of the QBO on the northern stratospheric polar vortex. J. Geophys. Res., 113, D101114, doi:10.1029/2007JD009647. Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan, 44, 25–42. McPeters, R. D., D. F. Heather, and P. K. Bhartia, 1984: Average ozone profiles for 1979 from the NIMBUS 7 SBUV instrument. J. Geophys. Res., 89 (D4), 5199–5214. Nathan, T. R., and L. Li, 1991: Linear stability of free planetary waves in the presence of radiative-photochemical feedbacks. J. Atmos Sci., 48, 1837–1855. ——, and E. C. Cordero, 2007: An ozone-modified refractive index for vertically propagating planetary waves. J. Geophys. Res., 112, D02105, doi:10.1029/2006JD007357. ——, J. R. Albers, and E. C. Cordero, 2011: Role of wave-mean flow interaction in sun-climate connections: Historical overview and some new interpretations and results. J. Atmos. Sol. Terr. Phys., 73, 1594–1605. Randel, W. J., and F. Wu, 2007: A stratospheric ozone profile data set for 1979–2005: Variability, trends, and comparisons with column ozone data. J. Geophys. Res., 112, D06313, doi:10.1029/2006JD007339. Shibata, K., and M. Deushi, 2005: Radiative effect of ozone on the quasi-biennial oscillation in the equatorial stratosphere. Geophys. Res. Lett., 32, L24802, doi:10.1029/2005GL023433. Strobel, D., 1978: Parameterization of the atmospheric heating rate from 15 to 120 km due to O2 and O3 absorption of solar radiation. J. Geophys. Res., 83, 6225–6230. Takahashi, M., 1999: Simulation of the quasi-biennial oscillation in a general circulation model. Geophys. Res. Lett., 26, 1307–1310. Tindall, J. C., J. Thuburn, and E. J. Highwood, 2006a: Equatorial waves in the lower stratosphere. I: A novel detection method. Quart. J. Roy. Meteor. Soc., 132, 177–194. Watanabe, S., Y. Kawatani, Y. Tomikawa, K. Miyazaki, M. Takahashi, and K. Sato, 2008: General aspects of a T213L256 middle atmosphere general circulation model. J. Geophys. Res., 113, D12110, doi:10.1029/2008JD010026. Zhu, X., and J. R. Holton, 1986: Photochemical damping of inertio– gravity waves. J. Atmos. Sci., 43, 2578–2584.