Gravity Waves in the Middle and Upper Atmosphere

7 Gravity Waves in the Middle and Upper Atmosphere 7.1 7.2 7.3 7.4

Introduction Background Interia-Gravity Waves in the Middle Atmosphere Planetary Waves in the Middle Atmosphere 7.4.1 Rossby Wave 7.4.2 Tropical Atmosphere 7.5 Midlatitude Wave Spectra 7.5.1 High-Frequency Waves: f 7.5.2 Mid-Frequency Waves: f  N 7.5.3 Low-Frequency Range:  ∼f 7.6 Modeling the Gravity Wave Fluxes in the MUA 7.6.1 Hydrodynamic Models 7.6.2 Ray Tracing

7.1

INTRODUCTION

It is now well established through observations and theory that gravity waves play an essential role in the global circulation of the atmosphere. Although the horizontal scales of the most active gravity waves are much smaller than the planetary and Rossby waves which dominate the general circulation, gravity waves transport energy and momentum from the troposphere and deposit it in the mesosphere and thermosphere. These transports have a large impact on the spatial and temporal characteristics of the middle and upper atmosphere. In almost all cases, the horizontal resolutions of general circulation models and global climate models are too course to resolve the scales of the important part of the gravity wave 181

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spectrum. Thus, their necessary contributions to global simulations are missed. As discussed by McLandres (1998), the modeler is left with two choices; either increase the horizontal resolution of the models or parameterize the effects of the gravity waves. To date, the highest model resolution is 20 km (Murakami et al., 2012), but this is still greater than the wavelengths of important waves. Thus, for the forceable future modelers will be using parameterizations of the effects of gravity wave, and this requires a comprehensive understanding of the complex links between gravity wave fluxes of energy and momentum and the dynamics of the middle and upper atmosphere. Until now we have dealt mostly with theoretical issues and simple examples. However, when we enter the middle and upper atmosphere (MUA) it becomes difficult to satisfy the requirements for linear theory. Background flows cannot be assumed constant and horizontally homogeneous. Compressibility is generally not an issue in the lower atmosphere; however, in the MUA decreasing air density becomes important, and terms involving the isothermal scale height, Hs , cannot be ignored. We have previously applied linear theory to cases involving single waves or wave packets described by a single dominant wave. However, in the MUA we are confronted with a spectrum of gravity waves with frequencies ranging from the Coriolis frequency f to the local Brunt–Väisälä frequency N. Disturbances have horizontal scales ranging from tens to thousands of kilometers and vertical scales ranging from a few to several tens of kilometers. Because of ez/2Hs growth, the large amplitude waves can break. Wave breaking almost always occurs in the upper and middle atmosphere. Yet, within the face of these complexities, the linear theory is robust and useful.

7.2

BACKGROUND

In the early days of long-distance radio transmission, variations and interruptions of over-the-horizon transmissions initiated the study of traveling ionospheric disturbances (TID). In his classic paper, Hines (1960) proposed that TIDs were caused by upward propagating gravity wave disturbances originating in the troposphere. Gossard (1962) calculated the gravity wave spectrum of vertical energy flow out of the troposphere. He found that the parts of the spectrum corresponding to propagating gravity waves have periods between about 10 min to 2 h, and can at times transport large amounts of energy through most of the MUA. His results were in agreement with data at D-layer heights (80–100 km) taken by Hargreaves (1961) using radio wave reflections in the ionosphere, and by Greenhow and Neufeld (1959) using meteor trails. These and the great number of papers published since agree that most of the energy driving the general circulation originates in the troposphere, and that the majority of this energy is transported upward by gravity waves. One of the major problems faced by global-scale and synoptic-scale modelers is the incorporation of gravity wave fluxes into their models. These models require relatively large horizontal grid

BACKGROUND

183

cells; however, because these cells are generally larger than the wavelengths of the energy-carrying gravity waves, the wave fluxes cannot be directly calculated. They must be parameterized, which will be discussed in Chapter 8. Gravity wave transport of energy throughout the atmospheric layers results in a coupling between these layers. This coupling is the essence of the general circulation. Vincent (2009) reviewed the coupling mechanisms and concluded: 1. The majority of the waves observed in the lower stratosphere couple energy and momentum upward into the middle and upper atmosphere. 2. Largest momentum fluxes are observed over regions of high topography, but these regions have the greatest wave variability. 3. On a zonally averaged basis, momentum fluxes over mountains and oceans are approximately equal. 4. Thermospheric gravity waves are selectively filtered by the kinematic dissipation. Only the high frequency, long vertical wavelength components penetrate to the highest altitudes. 5. There is a strong solar cycle effect in gravity wave propagation into the thermosphere. Gravity waves propagate to higher altitudes during high sunspot conditions than during solar minimum conditions. An interesting example of atmosphere–earth coupling is given by Artu et al. (2005) who found a correlation between observed gravity waves in the ionosphere and a tsunami. This was the first observation of a coupled tsunami-gravity wave. The study of synoptic and global scale motions is far beyond the scope of this book, and it is impossible to give a comprehensive understanding of gravity waves above the troposphere in a single chapter. Indeed, text books and extensive reviews are devoted to this study (see for example, Hines, 1974; Lilly and Gal-Chen, 1983; Fritts, 1984; Andrews, Holton, and Leovy, 1987; Randall, 2000; Fritts and Alexander, 2003; Holton, 2004; Brassure and Solomon, 2005; Mohankumar, 2008; Alexander, 2010; Warner, 2011 and references therein). In light of this, we present here only those details that pertain to gravity waves keeping in mind that these details are still under active investigation, and many questions remain unanswered. Figure 7.1 illustrates the climatological vertical temperature structure of the atmosphere which is divided on the bases of dynamics and chemistry into three regions, the lower, middle, and upper atmosphere. The lower atmosphere contains the troposphere and tropopause; the middle atmosphere extends from the base of the stratosphere to the top of the mesosphere; the upper atmosphere includes the thermosphere and the ionosphere; it extends from the top of the mesopause to the exosphere and outer space. Each region experiences different dynamics and different responses to gravity waves. Figure 7.2 shows the profile of Brunt–Väisälä frequency from the ground surface to the thermosphere. We see that N2 increases with height in the troposphere, mesosphere, and lower thermosphere; however, in the thermosphere N 2 decreases rapidly with height above about 115–120 km. In the stratosphere, N 2 decreases with height. The wave stability of the various layers

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Gravity Waves in the Middle and Upper Atmosphere

FIGURE 7.1 Midlatitude yearly average temperature profile in the middle and upper atmosphere. Also showing atmospheric zones based on temperature profile.

FIGURE 7.2 Vertical profiles of Brunt–Väisälä frequency for the middle and upper atmosphere.

can be simply scaled as suggested by Fritts (personal communication). We make the following assumption: horizontal wavelengths are much greater than vertical

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BACKGROUND

wavelengths so that k  m, and the vertical scale for changes in the background variables is greater than a vertical wavelength, i.e., the WKB assumption holds. We now write the Richardson number in terms of the perturbations, Ri =

N2 1 ρ0 u 1 w1 ∂u ∂z

.

(7.1)

From the polarization equation u1 =

m w1 . k

(7.2)

Multiplying (7.2) by w1 gives m 2 w . k 1 The magnitude of the vertical derivative of 7.2 is u 1 w1 =

∂u 1 m2 ∼ w1 . ∂z k Using (7.3) and (7.4) in (7.1) gives Ri ∼

N2 3

ρ0 mk 2 w13

.

(7.3)

(7.4)

(7.5)

Under WKB, m = Nk/ where  is the intrinsic frequency. Then Ri ∼

N 2 3 . N 3 kw13

(7.6)

We assume that  and k are constants or slowly varying with height. We also make the ad hock assumption that w1 is slowly varying when no critical levels are present. We then get the result that 1 . (7.7) N Relation (7.7) indicates that in regions where N is decreasing with height Ri increases, i.e., the upward propagating gravity wave becomes more stable, but in regions of increasing N, Ri decreases indicating less gravity wave stability and possible wave breaking. Thus, in the troposphere where N increases with height wave breaking is to be expected, which indeed is the case for the smaller (horizontal wavelength) waves. However, in the stratosphere wave breaking is suppressed by the increasing stability, and waves tend to propagate through it with little dissipation, but wave reflection is possible. In the mesosphere, wave stability decreases and because of the large horizontal wind shears, wave instability and transfers of pseudomomentum and drag occur. It is this deposition of momentum that drives the mesosphere winds and leads to seasonal wind direction reversals. To first order, the atmospheric general circulation is driven by differential absorption of solar heating at the ground surface. In response, an upward Ri ∼

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Gravity Waves in the Middle and Upper Atmosphere

FIGURE 7.3 (a) The calculated MUA radiative equilibrium temperature distribution at solstice with winter hemisphere on the right. (b) The observed zonal mean MUA temperature distribution at solstice with the winter hemisphere on the right. Taken from Wehrbein and Levoy (1982). heat flux lies above the warm equatorial regions and a downward heat flux is found above the cold polar regions. In the absence of eddy motions, the zonal mean temperatures in the middle atmosphere would be determined by a seasonally varying radiative equilibrium. Because of a balance between the Coriolis force and the meridional temperature gradient, the global circulation would be an averaged zonal wind with no meridional circulation. In such a situation, there would be no coupling between the middle and lower atmosphere. As an illustration consider Fig. 7.3 taken from Wehrbein and Levoy (1982). Figure 7.3a shows the calculated radiative equilibrium MUA temperature distribution at solstice with winter hemisphere on the right. Figure 7.3b shows the observed zonal mean temperature distribution in the MUA at solstice with the winter hemisphere on the right. The difference in temperatures between the radiative equilibrium winter hemisphere and the observed temperatures is quite large, the latter being much warmer. Mesosphere temperatures in the equilibrium summer hemisphere are much lower than the observed temperatures. Thus, we see that coupling exists between the lower and MUA such that energy and momentum are transported up from the lower atmosphere. The

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major transfers of momentum and energy into the middle atmosphere are through gravity waves in the equatorial regions. Additional transfers by gravity waves occur in midlatitudes. Fritts and Alexander (2003) describe several gravity wave sources in the troposphere including topography, convection, shear instability, geostrophic adjustment, and wave–wave interactions. Each of these generation mechanisms produce waves with individual characteristics. For example, hydrostatic topographic waves propagate upward almost directly above the terrain feature and have a narrow spectrum of horizontal wavelengths, but convection waves propagate vertically and horizontally with a wide spectrum of wavelengths. Globally, the troposphere is a ‘soup’ of nonlinear gravity waves with wide-ranging frequencies, wavelengths, and amplitudes such that individual waves and their sources are at best difficult to identify by observations.

7.3

INTERIA-GRAVITY WAVES IN THE MIDDLE ATMOSPHERE

Because of the wide spectrum of gravity waves launched from the lower atmosphere, modeling in terms of monochromatic waves is inappropriate. We cannot say a priori which waves will be reflected, absorbed at critical levels, or become dynamically unstable. Thus, we must consider a spectrum of gravity waves with intrinsic frequencies  such that f <  < N. However, linear inviscid wave theory can be useful in isolating mechanisms and interpreting observations. Because we are dealing with large waves with low frequencies, the Coriolis force will have an effect. Thus we must solve for inertia-gravity waves. Following Holton (2004) the equations are: 1 ∂p du − fv+ = X, dt ρ ∂x dv 1 ∂p + fu + = Y, dt ρ ∂y dw 1 ∂p + + g = 0, dt ρ ∂z 1 ∂u ∂v ∂w + + + = 0, ρ ∂x ∂y ∂z dθ = Q, dt  κ p p0 θ= , ρR p

(7.8) (7.9) (7.10) (7.11) (7.12) (7.13)

where d/dt = ∂/∂t + u∂/∂ x + v∂/∂y + w∂/∂w, X, Y , and Q represent generalized forcings, θ is potential temperature (1.42) and the other symbols have their usual meanings. Linearizing (7.8)–(7.13) where the slowly varying

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Gravity Waves in the Middle and Upper Atmosphere

background variables have subscript 0 and first-order perturbations have subscript 1 gives   ∂ Du 1 ∂u 0 p1 + w1 − f v1 + = 0, (7.14) Dt ∂z ∂ x p0   p1 ∂v0 Dv1 ∂ + w1 + f u1 + = 0, (7.15) Dt ∂z ∂y p0     ∂ p1 1 p1 ρ1 Dw1 + − +g = 0, (7.16) Dt ∂z p0 Hs p 0 ρ0   N2 D θ1 = 0, (7.17) + w1 Dt θ0 g   D ρ1 ∂v1 ∂w1 w1 ∂u 1 + + − + = 0, (7.18) Dt ρ0 ∂x ∂y ∂z Hs   1 p1 θ1 ρ1 = 2 (7.19) − , θ0 cs p 0 ρ0 where D ∂ ∂ ∂ = + u0 + v0 , Dt ∂t ∂x ∂y

(7.20)

and we have dropped the forcing terms and the background velocity shears. This is consistent with the WKB approximation. We now assume wave solutions of the form   z  A1 = A exp i (kx + ly + mz − ωt) + . (7.21) 2Hz Using these wave solutions in (7.14) to (7.19) gives the polarization wave equations −i u˜ − f v˜ + i k p˜ = 0 − i v˜ + f u˜ + il p˜ = 0,   1 −i w˜ + i m − = −g ρ, ˜ 2Hs −i θ˜ +

N2 w˜ = 0, g

  1 −i ρ˜ + i k u˜ + il v˜ + i m − = 0, 2Hs p˜ θ˜ = 2 − ρ. ˜ cs

(7.22) (7.23) (7.24) (7.25) (7.26)

Recall that  is the intrinsic frequency defined in (2.18). Solving (7.22) to (7.26) for w˜ and equating real and imaginary parts we get two equations. One equation is g 1 N2 . = − cs2 Hs g

(7.27)

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189

The second equation will be similar to (2.36), the Taylor–Goldstein equation, with dispersion relation     1 (2 − f 2 ) 1 2 2 2 2 = N m , (7.28) − K + f + 2 K 2 + m 2 + 4Hs2 cs2 4Hs2 where K 2 = k 2 + l 2 . In the Boussinesq approximation, cs → ∞, (7.27) reduces to N2 1 . = Hs g

(7.29)

In Chapter 1, we defined an exponentially decreasing atmospheric density in (1.58). The vertical derivative of this density is ∂ρ0 ρ0 =− , ∂z Hs

(7.30)

hence, −

1 ∂ρ0 N2 1 = = . ρ0 ∂z g Hs

(7.31)

Also, in the Boussinesq approximation the dispersion relation reduces to   N 2 K 2 + f 2 m 2 + 4H1 2 s 2 = . (7.32) K 2 + m 2 + 4H1 2 s

From (7.32) the vertical wavenumber is m2 =

1 K 2 (N 2 − 2 ) − . 2 2 ( − f ) 4Hs2

(7.33)

If the wave is propagating, then m must be real, and so f <  < N. In the notation of Fritts and Alexander (2003), the group velocities are [k(N 2 − 2 ), l(N 2 − 2 ), m(2 − f 2 )]   .  K 2 + m 2 + 4H1 2 s (7.34) Using the polarization (7.22) to (7.26) we can develop the relations   2   2  − f2  − f2 u˜ = v, ˜ (7.35) p˜ = k + i f l l − i f k (cgx , cgy , cgz ) = (u 0 , v0 , 0) +

w˜ =

 m−





N 2 − 2 

u˜ =

i 2Hs

i k − f l i l + f k

p, ˜

(7.36)

v. ˜

(7.37)



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Gravity Waves in the Middle and Upper Atmosphere

It is of interest to note the phase relation between u˜ and v˜ in (7.37). For a zonally propagating wave, the meridional velocity perturbation is f u, ˜ (7.38)  and we see that the horizontal perturbation velocities are 90◦ out of phase. As the wave propagates upward the horizontal wave vector rotates cyclonical, in the northern hemisphere, resulting in a spiral hodograph (see, for example, Gill, 1982; Tsuda et al., 1994; Sato, 1994; Guest et al., 2000; Walterscheid, Schubert, and Brinkman, 2001). This spiraling behavior will be explained in Chapter 8 where we discuss observations in the MUA. v˜ = −i

7.4

PLANETARY WAVES IN THE MIDDLE ATMOSPHERE 7.4.1

ROSSBY WAVE

As Holton (2004) points out, the predominant eddy motions in the midlatitude stratosphere are vertically propagating Rossby waves. However, Rossby waves can propagate only in the winter stratosphere. In this book, we cannot develop the theory from basic principles, and so we present here the essential features taken from Holton (2004). The geostrophic potential vorticity equation on a midlatitude β-plane in log-pressure coordinates, z = −Hs ln( p/ ps) is   ∂ + Vg • ∇ q = 0, (7.39) ∂t where q = ∇2ψ + f +

f 02 ∂ ρ0 N 2 ∂z

  ∂ψ ρ0 , ∂z

(7.40)

where ψ = / f 0 is the geostrophic stream function, f 0 is a constant midlatitude reference value of the Coriolis parameter, and is the geopotential defined by d = gdz. Linearizing, we assume that the motion consists of smallamplitude disturbances superimposed on a constant zonal-mean flow. Then letting ψ = −U y + ψ and q = q0 + q in (7.40) we get   ∂ ∂ψ

∂ + u0 = 0, (7.41) q + β ∂t ∂x ∂x where U is a constant zonal wind and q = ∇ 2ψ +

f02 ∂ ρ0 N 2 ∂z

  ∂ψ

ρ0 . ∂z

(7.42)

We now take wave solutions of the form ψ (x, y, z, t) = (z)ei(kx+ly−kc I x t )+z/2Hs ,

(7.43)

PLANETARY WAVES IN THE MIDDLE ATMOSPHERE

191

where c I x = U − cx is the intrinsic zonal phase speed. Using (7.43) in (7.41) gives the vertically-propagating midlatitude Rossby wave, d2  + m 2  = 0, dt 2 where m2 =

N2 f02



 1 β − K2 − , U 4Hs2

(7.44)

(7.45)

where we have dropped cx because for Rossby waves it is generally small compared with U, and K 2 = k 2 +l 2 . For propagating waves m must be real. This requires that U > 0, i.e., only eastward moving waves are allowed. A Rossby critical velocity, Uc is found by setting m = 0 in (7.45), i.e., Uc =

β , K 2 + α2

(7.46)

where α 2 = f 02 /(4N 2 Hs2). For the stratosphere, N 2 is typically 4 × 10−4 s−1 and the temperature is about 220 K. The scale height, Hs is about 6.4 km and the term f /2N Hs corresponds to zonal wavenumber 2 in midlatitudes. At midlatitudes, β ∼ 1.6 × 10−11 , K ∼ 4.4 × 10−7 , and α ∼ 4 × 10−7 . Using these values Uc ∼ 46 ms−1 . Thus, midlatitude Rossby wave will not propagate into the MUA if the average zonal wind is greater than about 50 ms−1 . In the winter northern hemisphere Rossby waves can propagate upward to about 60–70 km, but they cannot move out of the summer troposphere. 7.4.2

TROPICAL ATMOSPHERE

The equatorial region is of great interest. In the tropics, upwelling of energy and momentum into the upper troposphere and stratosphere occurs. How this happens is a major scientific question since these details greatly impact global scale models. In the tropics, significant topography does not exist; therefore terraingenerated gravity waves need not be considered. The remaining mechanisms are convectively generated gravity waves and upward propagating planetary waves. Convective gravity waves have been studied by, for example, Karoly, Roff, and Reed, 1996; Alexander and Holton, 1997; Walterscheid, Schubert, and Brinkman, 2001; Nakamura et al., 2003; Chun et al., 2004; Chun et al., 2011, and references therein. Fritts and Alexander (2003) give a comprehensive review of middle atmospheric gravity wave research since 1984. The planetary wave contributions to the upwelling have been studied by, for example, Matsuno, 1966; Holton and Lindzen, 1968; Andrews and McIntyre, 1976; Dunkerton, 1993; Alexander and Holton, 1997; Nakamura et al., 2003, and references therein. In the equatorial regions, f → 0, and as seen in (7.32) in the absence of Coriolis forces very low-frequency atmospheric oscillations are possible. Now the equations for midlatitude inertia-gravity waves apply, but with f = 0. The

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Gravity Waves in the Middle and Upper Atmosphere

linearized perturbations on an equatorial β-plane are given by Holton (2004). Using log-pressure coordinates, these equations are ∂ 1 ∂u 1 − βyv1 = − , ∂t ∂x ∂v1 ∂ 1 + βyu 1 = , ∂t ∂y ∂u 1 ∂v1 ∂(ρ0 w1 ) + + ρ0−1 = 0, ∂x ∂y ∂z ∂ 2 1 + w1 N 2 = 0. ∂t∂z

(7.47) (7.48) (7.49) (7.50)

Now substituting wave solutions of the form ˆ i(kx+mz−ωt )+z/2Hs q1 = qe

(7.51)

gives the differential equations for the meridional structure ˆ −i ωuˆ − βy vˆ = −i k , ˆ ∂

, −i ωvˆ + βy uˆ = − ∂y     ∂ vˆ i −k uˆ + wˆ = 0, +i m + ∂y 2Hs   i ˆ + wN ω m−

ˆ 2 = 0. 2Hs 7.4.2.1

(7.52) (7.53) (7.54) (7.55)

Vertically Propagating Rossby-Gravity Waves

Equations (7.52) to (7.55) can be combined to get    2 2 β 2 y 2m 2 ∂ 2 vˆ ω m k 2 β − vˆ = 0. + − k − ∂y 2 N2 ω N2

(7.56)

The boundary conditions for (7.56) are vˆ → 0 as |y| → ∞. Thus, (7.56) is an eigenvalue problem with modal solutions. As shown by Matsuno (1966), these solutions will exist if   m ω2 N 2 k 2 = 2n + 1, n = 0, 1, 2 . . . (7.57) − β −k + Nβ ω m2 where n is the modal index, i.e., n = 0 is the gravest mode. For n = 0 N (β + ωk). (7.58) ω2 When β = 0 we recover the dispersion relation for hydrostatic propagating gravity waves. For eastward propagating waves, ω > 0, and for westward propagating waves, ω < 0. Thus, eastward propagating Rossby waves have shorter vertical |m| =

PLANETARY WAVES IN THE MIDDLE ATMOSPHERE

193

wavelengths than westward propagating Rossby waves. The n = 0 mode exists only if c = ω/k > −β/k 2 .

(7.59)

Because k = s/a where the zonal wavenumber s is the number of wavelengths around a latitude circle and a is the mean radius of the earth, (7.59) will hold only for frequencies |ω| <

2 E , s

(7.60)

where E is the earth’s angular speed of rotation. For frequencies that do not satisfy (7.60), the waves will not decay away from the equator and the boundary conditions will not be satisfied. As always, for upward propagation m < 0. Observations by Yanai and Maruyama (1966) show that for the Rossby-gravity wave, the periods are 4–5 days; the zonal wavenumber is 4; the vertical wavelength is 4–5 km, and the average phase speed relative to the ground is −23 ms−1 . 7.4.2.2

The Kelvin Wave

For the Kelvin wave, the motions are zonal, i.e., vˆ = 0. The equations take the form ˆ −i ωuˆ = −i k , ˆ ∂

, βy uˆ = ∂y   1 2 ˆ + uk −ω m +

ˆ N 2 = 0. 4Hs2

(7.61) (7.62) (7.63)

Using (7.63) to solve for and using this in (7.51) and (7.52) gives two solutions. The first solution is βy 1 ∂ uˆ =− , uˆ ∂y c

(7.64)

and integration gives uˆ = uˆ0 e−0.5β/cy . 2

(7.65)

The horizontal wind perturbations are in the zonal direction and are Gaussian function of y where uˆ 0 is the perturbation speed at the equator. The second solution is   1 2 2 c m + (7.66) − N 2 = 0. 4Hs2 Observations show that for stratospheric Kelvin waves the vertical wavelength is between 6–10 km. Assuming a wavelength of 8 km and letting Hs = 6 km we see

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Gravity Waves in the Middle and Upper Atmosphere

that m 2 ∼ 6 × 10−7 m−2 and 1/4Hs2 ∼ 6 × 10−9 m−2 , and so we can drop the scale-height term to get m2 =

N2 , c2

(7.67)

but this is the same as (7.33) in the case of long horizontal waves. Observations by Wallace and Kousky (1968) show that for Kelvin waves the wave period is 15 days; the zonal wavenumber is 1–2; the vertical wavelength is 6–10 km, and the average phase speed is 25 ms−1 .

7.5

MIDLATITUDE WAVE SPECTRA

Gravity waves entering the middle atmosphere from below extend over a wide range of frequencies and wavelengths (see, for example, Dunkerton and Butchart, 1984; Dewan and Good, 1986; Fritts and VanZandt, 1987; Fritts et al., 1988; Marks and Eckermann, 1995; Alexander and Dunkerton, 1999; Fritts and Alexander, 2003; Preusse and Ern, 2008) The inertia-gravity wave approaches a singularity as 2 → f 2 and so the range of frequencies of internal waves is f <  < N, where we are reminded that  is the intrinsic frequency, ω − u 0 k. We see that the frequency range of middle atmospheric gravity waves is limited at the lower end, i.e., f = 2E sin φ, where φ is the latitude. 7.5.1

HIGH-FREQUENCY WAVES:   f

When   f we can drop f from the equations in Section 7.3. From the basic definition,  = K c I = 2π/λh c I where c I is the intrinsic phase speed, and we see that high-frequency waves will have short horizontal wavelengths, λh . From Chapter 2, we saw that in the non-hydrostatic case short horizontal waves, i.e., large K, will become evanescent or reflected more readily than longer waves. In the high frequency regime under the Boussinesq approximation (7.28) takes the form 2 = 

N2 K 2 K 2 + m2 +

1 4Hs2

.

(7.68)

From (7.68) the vertical wavenumber is m2 =

K 2 N2 1 − K2 − . 2 4Hs2

(7.69)

Setting m = 0 we can solve (7.69) for the maximum intrinsic phase speed, c I,max = ±

N K2 +

1 4Hs2

.

(7.70)

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MIDLATITUDE WAVE SPECTRA

Fritts and Alexander (2003) point out that only waves with horizontal wavelength > 10 km are important in the middle atmosphere. Using λh = 10 km, Hs = 7 km, and N = 0.02 s−1 , c I,max ≈ 33 ms−1 . Neglecting f and for m 2  1/(4Hs2) the dispersion relation is N2 K 2 = N 2 cos2 β, (7.71) K 2 + m2 where β is the angle between the wave vector and the horizontal plane. Note that (7.71) is identical to (2.57) but for three dimensions. Under these same assumptions the group velocities and other wave characteristics are the same as in Chapter 2. Fritts and Alexander (2003) point out that for Hs ∼ 7 km, the compressibility term becomes significant for waves with vertical wavelengths greater than about 30 km. They also point out that the compressibility effect is important in the study of airglow near the mesopause (Swenson, Alexander, and Haque, 2000). For long vertical wavelengths, the vertical scale of variations of background variables, especially N, will not be large in comparison to λz . Since this goes against the conditions for the WKB method, the applicability of the WKB become problematical. As the vertical wavelength becomes great m → 0 which marks a turning point for the upward propagating wave, i.e., wave reflection. Using (7.32) with m = 0 and ignoring rotation effects, the maximum horizontal intrinsic phase speed |c I,max | at the reflection point is   1 −1/2 |c I,max | = N K 2 + , (7.72) 4Hs2 2 =

which will be small at the turning point. Thus |c I,max | will be small for small horizontal wavelengths. Shorter wavelength waves will be either reflected or ducted in the stratosphere. However,  cannot be so large or K be so small as to make m complex. As Fritts and Alexander (2003) point out, a turning point for the wave occurs where m → 0 and this will occur where  → K N/2Hs . Thus, in the stratosphere where N decreases with height the high frequency long horizontal wavelength waves are likely to be reflected downward. As we have seen in Section 7.4.2, in the equatorial zones very low wave frequencies can exist. Then the vertical wavenumber (7.33) becomes m≈ 7.5.2

KN . 

(7.73)

MID-FREQUENCY WAVES: f    N

In the middle frequency range, the effects of f and 1/2Hs will not be significant, although this may not be true in limiting cases. Following Fritts and Alexander (2003), we √ define the horizontal phase speed as ch and the horizontal wavenumber as kh = k 2 + l 2 . The dispersion then simplifies to kh  = N , (7.74) m

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and the vertical wavenumber becomes N . (7.75) cI h Following our convention in Chapter 3, if kh > 0 then c I h = (ch − u 0 ) > 0 and for k < 0 c I h < 0. At a critical level c I h = 0. Using (7.74) and (7.75) it can be shown that the group velocities have the simple forms  (7.76) u g = u 0h + kh and  (7.77) wg = − . m The polarizations equations have the form p1 = wg (7.78) w1 and kh c I h  kh w1 = . (7.79) =− = u 1h m N N In the previous chapters, we almost always considered the horizontal winds constant leaving numerics to the more general condition. Now we note that as the wave moves upshear, i.e., ∂c I /∂z > 0, group velocity, vertical wavelength, and intrinsic frequency increase. The revers holds in regions of downshear. Equations (7.79) show that as the wave moves upshear, the vertical wave speed perturbation becomes increasingly greater than the horizontal perturbation speed. We also see from (7.78) that the pressure perturbation becomes large for upshear propagation. |m| =

7.5.3

LOW-FREQUENCY RANGE:  ∼ f

For low-frequency waves where 2  N 2 , the vertical wavenumber (7.33) takes the form K 2 N2 1 − 2. (7.80) 2 2  − f 4h s These waves are the inertia-gravity waves and are associated with large-scale topographic features and mesoscale to synoptic scale weather systems. The singularity where  → f is important; however, as f decreases, as in the tropics, very low frequency gravity waves are possible. In the tropics where there is little significant topography low frequency gravity waves are generated by large convective systems. The compressibility term 1/2Hs becomes important as m becomes small. We can find a critical frequency, c by following Marks and Eckermann (1995). We set m = 0 in (7.80) and solve for c to get m2 =

2c = 

K 2 N2 K2 +

1 4Hs2

.

(7.81)

MODELING THE GRAVITY WAVE FLUXES IN THE MUA

197

Now the range of allowed frequencies is f <  < c  N. Marks and Eckermann (1995) note that the compressibility term is small and is often dropped; however, this can lead to calculated frequencies lower than real frequencies. In this case, m can become large and the vertical wavelength λz can become small.

7.6

MODELING THE GRAVITY WAVE FLUXES IN THE MUA

As mentioned at the beginning of this chapter, global-scale atmospheric models require realistic parameterizations of the gravity wave fluxes into the MUA (McLandres, 1998). This requires a comprehensive understanding of gravity wave fluxes of energy and momentum. A vast literature on these subjects exists. Fritts and Alexander (2003) give a recent review of research into middle atmosphere gravity waves, and Fritts and Lund (2011) review recent research on gravity waves in the thermosphere and ionosphere. Here, we can summarize, only crudely, the fundamentals of this research. Basically, analyses proceed on two paths, hydrodynamic models and ray tracing. 7.6.1

HYDRODYNAMIC MODELS

In hydrodynamic models, differential Equations (7.8) to (7.12) are solved, and the body forces X, Y , and Q represent divergences of gravity fluxes. The temporal and spatial scales of these problems are arbitrary. Theoretical expressions or parameterizations of the forcing terms are proposed and used to match observations or develop further theories. Some of these parameterizations will be described in Chapter 8. Examples of these approaches are found, for example, in Andrews and McIntyre (1976), Lindzen (1981), Holton (1982), Tanaka (1996), Lott and Teitelbaum (1993), Alexander and Dunkerton (1999), Walterscheid, Schubert, and Brinkman (2001), Chen, Durran, and Hakim (2005), and Chun et al. (2011). 7.6.2

RAY TRACING

We have seen in Section 2.5.2 that when a wave packet moves along a path that is tangent to the group velocity vector, the wave action, ρ0 E/ is constant. This forms the basis for ray tracing. Marks and Eckermann (1995) give an excellent description of the method based on Lighthill (1978). We assume a wave packet with mid-frequency waves (Section 7.5.2) moving within a slowly varying flow without gradients in the wind field. Then with the Boussinesq approximation 2 =

N 2 K 2 + f 2 (m 2 + α 2 ) , K 2 + m 2 + α2

(7.82)

where α = 1/4Hs2. The vertical wavenumber is given by m2 =

(2c − 2 )(K 2 + α 2 ) , 2 − f 2

(7.83)

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where c is the critical or cutoff frequency (7.81). The dispersion relation is ω = ω(ki , x i ), i = 1, 2, 3, and we assume that the background flow and the wave parameters do not vary with time so that ω is constant along the ray path. The equations for the ray path are then dx ∂ω = = ug, (7.84) dt ∂k ∂ω dy = = vg , (7.85) dt ∂l ∂ω dz = = wg . (7.86) dt ∂m The refraction equations for the ray path are dk ∂ω =− , (7.87) dt ∂x ∂ω dl =− , (7.88) dt ∂y dm ∂ω =− . (7.89) dt ∂z These equations are in their simplest form. In application, there can be several rays each representing different origins or initial conditions. For each ray, initial longitudes and latitudes, elevations, horizontal wavenumbers (k0 , l0 ), and initial ground-based frequency ω0 are specified. Using these values, (7.83) is evaluated for m 0 . The sign of m 0 is taken opposite to  so that wg > 0, i.e., upward propagating energy and momentum. The amplitude of the wave packet is calculated by the wave-action equation ∂A 2A + ∇ • ( vg A) = − , (7.90) ∂t τ where A = E/ is the wave action and τ is a dampening time scale. The initial value of A is the rms horizontal velocity perturbations at the starting point of the ray, i.e., A0 = (u 21 + v12 )1/2 . The position and amplitude of the wave packet along the ray path are calculated forward in time and space noting that the background wind speeds can slowly vary in time and location. Examples of ray tracing calculations can be found in Dunkerton and Butchart (1984), Dunkerton (1984), Miyhara (1985), Schoeberl (1985), Hines (1988), Marks and Eckermann (1995), and Preusse and Ern (2008).

PROBLEMS

1. What is the frequency range for hydrostatic inertia-gravity waves in the middle and upper atmosphere?

PROBLEMS

199

2. An upward propagating inertia-gravity wave in the upper stratosphere at 50◦ north has a period of 1 h and a zonal wave vector. Calculate the major and minor axis of the hodograph formed by the velocity perturbations. What is the direction of rotation of the perturbation velocity vector. 3. What is a β-plane, and explain its purpose. 4. Explain the observation that inertia-gravity waves with long horizontal wavelengths pass through the stratosphere but those with short horizontal wavelengths do not. 5. Account for the fact that internal hydrostatic gravity waves can propagate to greater altitudes in the tropics than in midlatitudes. 6. What is the vertical wavelength at the point of wave reflection? 7. What is the wave property that allows ray tracing? 8. In the absence of wave breaking, what two wave quantities are conserved. 9. In going from (7.82) to (7.83) what assumption has been made? 10. What distinguishes mid-frequency inertia-gravity waves from those wave with high or low frequencies?