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Pergamon

and Solar-Terre~trral

PII: Sl364-6826(97)00048-5

Phyws,

Vol. 59, No. II, pp. 2185 2202, 1997 G 1997 Elsevirr Science Ltd All rights reserved. Punted in Great Britam 136&6826/97 $17 OO+OOO

Natural oscillations of the ionosphere-thermosphere-mesosphere (ITM) system C. K. Meyer and J. M. Forbes Department

of Aerospace

Engineering University

Sciences, Program in Atmospheric of Colorado, Boulder, U.S.A.

and Oceanic

Sciences,

(Receired 19 February 1991; accepted 4 June 1997) Abstract-Previous investigations of the natural resonances of the atmosphere (‘normal modes’) focused on oscillations whose energies are confined to tropospheric/stratospheric altitudes. Given that a broad spectrum of forcing is also present in the mesosphere and lower thermosphere (MLT) because of gravity wave dissipation, solar radiation absorption, Joule heating, etc., one might ask what type of resonance characteristics the upper atmosphere possesses. The present work seeks to identify and characterize those natural oscillations present in the ITM, and to compare with those realized in the lower atmosphere. Such information is potentially useful for interpreting observations of the ITM and first-principles model calculations. A 2D linearized perturbation model (Global Scale Wave Model-GSWM) is used here to determine the spectral response of the ITM. In a manner similar to that of Salby (1981a), the atmosphere is forced over a range of periods, and the total response (kinetic and potential) is examined. The forcing is placed at the base of the thermosphere (approx. 100-130 km) and so is representative of UV solar radiation absorption, the deposition of horizontal momentum by vertically propagating gravity waves and joule heating. It is found that the ITM region does not exhibit the same response characteristics as the lower atmosphere; the usual peaks at 2, 5, 10 and 16 days are not present. Instead the natural responses occur near periods 14.9, and 7 h for westward symmetric waves; 11, 9 and 5 h for westward asymmetric waves; 38, 20, and I3 h for eastward symmetric waves; and 15, 12, and 9 h for eastward asymmetric waves for zonal wave numbers 1. 2 and 3 respectively. These periods and dependencies on zonal wave number are very similar to those of gravitational normal modes in an isothermal atmosphere without mean winds. Observational data which provide evidence for the realizations of these theoretical results in the atmosphere are reviewed. ‘$3 1997

Elsevier Science Ltd. All rights reserved

1. INTRODUCTION In an isothermal atmosphere without dissipation, the linearized equations governing atmospheric motions reduce to an eigenfunction-eigenvalue problem in latitude and height. The eigenfunctions which specify the latitude structure are the solutions to LaPlace’s tidal equation called Hough functions. The vertical structure equation, when the atmospheric forcing can be specified, determines the vertical structure of each Hough mode. The atmospheric response corresponding to each mode depends on its eigenvalue and on how well its latitudinal shape matches that of the forcing. In the absence of forcing, the specification of a rigid lower boundary (vertical velocity = 0) leads to a single solution to LaPlace’s tidal equation. The corresponding eigenfunctions are the ‘normal modes’ of the atmosphere and fall into two classes of solutions: gravitational and rotational (Longuet-Higgins, 1968). The extent to which normal modes are realized in 2185

a non-isothermal atmosphere with mean winds and dissipation has been pursued by numerous investigations. The concept of atmospheric resonance has been invoked in an attempt to explain the unexpectedly large semidiurnal variations in surface pressure (Siebert, 1961). Salby (1981a, 1981b) performed a series of numerical investigations to show that elevated responses or quasi-resonance’s could occur for realistic atmospheric configurations, finding that the gravest modes should be realized for westward zonal wave numbers S = 1, 2 and 3. Varying the frequency at which the lower boundary of the atmosphere is forced, he showed that elevated responses occurred at periods of observed wave motions in the atmosphere; see Fig. 1 for results at S = 1. The clustering of peaks in the figure near 5, 10 and 16 days can be associated with specific atmospheric normal modes (eigenfunction of LaPlace’s Tidal Equation). The manifestation of quasi-resonance’s in a realistic atmosphere is determined by a number of factors.

2186

C. K. Meyer and J. M. Forbes PERIOD (days) 106

g!

10

5

perature) are provided for the preferred periods of oscillation (Section 4). Possible relationships between the present results and observed short period features in the ITM are discussed in Section 5.

16

105

% 6

104

;

103

2.

102

DESCRIPTION

OF THE MODEL

2.1. Overview -0.26

-0.24

-0.20

NORMALIZED

-0.16

-0.12

FREQUENCY,

-0.06

-0.04

r~ I Q

Fig. 1. Spectral response for westward zonal wave number one for equinox (solid) and solstice (dashed) conditions. The antisymmetric mode (lOday) is excited by an odd step in surface forcing (reverses sign at equator), while symmetric modes (5, 16 day) were driven by surface forcing which was constant with latitude. (adapted from Salby, 1981b).

Salby (1981a, 1981b) illustrated sensitivity to background mean winds in the lowest three scale heights, as would be expected with waves with relatively slow phase speeds. Wind shears also contributed to a broadening, shifting, and damping of the spectral response. Lindzen and Blake (1972) showed that dissipation mechanisms have the effect of broadening the atmospheric frequency response and diminishing the amplitude, and that the effect increases with the period of the wave. Using a global circulation model, Norton and Thuburn (1996) showed that unstable regions of the atmosphere can serve to amplify resonant behavior. Finally, it is implicitly assumed in all of these studies that the atmospheric forcing (internal or external/broad or otherwise) must exist to ‘trigger’ a resonant response. The atmospheric region above 80 km (the ionosphere-thermosphere-mesosphere or ‘ITM’ system) is subject to forcing by solar radiation absorption, joule heating, energetic particle precipitation and gravity wave momentum deposition over a wide range of spatial and temporal scales. The region is also characterized by large horizontal winds and dissipation in comparison to the lower atmosphere. However the resonance characteristics of the ITM system are unknown. By analogy with Salby’s (1981a,b) approach, the purpose of the present work is to establish the natural oscillations of the ITM system through a series of numerical simulations wherein the atmospheric response is measured as a function of the location and frequency of the forcing. In the following sections we will describe the atmospheric model (Section 2) and the effect of changing the altitude of the forcing (Section 3). Frequency response curves for thermospheric forcing and perturbation fields (winds and tem-

The GSWM (Hagan et al., 1993, 1995) is a 2D linearized steady-state numerical model of global scale atmospheric waves based upon the tidal model of Forbes (1982). The period and zonal wave number are specified and the latitude height structure of the wave is calculated subject to specified forcing. It assumes that the atmosphere is a shallow, compressible, hydrostatic, perfect gas and solves for the perturbations upon a mean zonal background state which includes winds and realistic distributions of eddy and molecular dissipation. 2.2. Numericalformulation The model is based upon a system of six primitive equations: the zonal and meridional momentum equations, the vertical momentum equation, the continuity equation, the ideal gas law and the thermal energy equation. Solutions of the form e(“‘21fei(~t+s3,) are assumed for all state variables (x = stretched vertical coordinate, t = time) where s = zonal wave number, and cr = frequency are specified. These six equations are combined algebraically to eliminate the pressure and density terms. The result is four coupled (in height and latitude) partial differential equations in terms of u’, v’, w’, and bT’. These equations are discritized with respect to height and latitude and placed in finite difference form, yielding a matrix set of coupled ordinary differential equations. The solution is found by applying a gaussian elimination algorithm developed by Lindzen and Kuo (1969). The result is a complex vector [u’, v’, w’, 6Y] at each model grid point. 2.3. Background atmosphere A climatological background atmosphere is specified. Zonal winds used in this study consist of three components. Between 0 and 80 km temperature and density fields are taken from MSISE90 (Hedin, 1991) and used to calculate the pressure at each point. Geostrophic balance is then applied to determine the zonal winds. A second order P-plane approximation (Fleming and Chandra, 1989) is used at the equator. Above 80 km an empirical model based upon data from a global array of radars measuring winds between 70 and 110 km winds (Portnyagin, 1986) is used. Above 100 km the Hedin et al. (1996) fields are used. The

Natural oscillations of the ITM system wind fields are smoothly merged in the transition regions. Vernal equinox and winter (NH) solstice background atmospheres are used for thermospheric forcing studies and a September 1st atmosphere is used for lower atmospheric forcing analyses. A low/ moderate set of geomagnetic and solar indices are used, i.e. Ap = 4 and F,07 = 15. 2.4. Boundury conditions and grid spacing At the poles ~2’and 6T’ are set to zero. At the bottom boundary of the model u” is assumed to be zero. At the top it is assumed that molecular diffusion dominates and that there is no flux of horizontal heat and momentum from infinity. The lower boundary is set at 0 km for lower boundary forcing, 20 km for stratospheric, 50 km for mesospheric and 90 km for thermospheric forcing. The upper boundary is 300 km in all cases. The model uses a stretched vertical coordinate system yielding a step size of approximately 0.4 km in the troposphere, 1.5 km at 90 km and 8 km at 300 km. With the lower boundary at 90 km 48 vertical steps are used. The model utilizes 59 grid points (3”) in latitude and runs from pole to pole. 2.5. Dissipation The model includes realistic dissipation from the surface through the thermosphere. Mechanisms include surface friction, height dependent radiative damping caused by 0, (Salby, 1981a) and CO, (Zhu and Strobel, 1991) eddy thermal conductivity and kinematic eddy viscosity, molecular thermal conductivity and molecular viscosity, and ion drag (Forbes, 1982). 2.6. Forcing Two types of forcing are used: in-situ heat sources (located at either 35, 70 or 110 km) and vertical velocity forcing (placed at the surface for lower atmospheric analyses, and at 90, 100 and 110 km when examining effect of forcing location). The GSWM model response is found to be generally independent of the method of forcing. Forbes et al. (1995~) showed that forcing by vertical velocities at the surface and insitu heating in the troposphere produced very similar response characteristics in the mesosphere and lower thermosphere. Our numerical experiments confirm this for similar distributions of momentum, heat excitation and vertical velocity forcing at 90 km. The heat sources used for results presented here are characterized by a gaussian distribution in height with a IO km half width. Heating rates are assumed constant with latitude for the ‘symmetric’ cases and are constant with latitude except for a 180” phase shift

2187

between hemispheres for the ‘asymmetric’ cases. The peak heating rate is arbitrarily set to be consistent with upper mesospheric and lower thermospheric UV absorption under quiet solar conditions: 2x 1O-7 J m3 s (Forbes and Garret, 1979). 2.1. Analysis method The zonal wave number and frequency are specified, with the frequency (period) of forcing varied from 0.05 to 6 cycles per day (4 h to 20 days). Lower boundary, stratospheric and mesospheric forcing cases only consider westward zonal wave number one and uniform symmetric forcing. Thermospheric energy source cases consider eastward and westward zonal wave numbers 1, 2 and 3, and both symmetric and asymmetric forcing. 2.8. Measuring response The response is a measure of the total energy (kinetic and potential). The kinetic energy is found directly from the zonal and meridional perturbation velocities. The potential energy (geopotential height) is calculated based upon perturbation pressure found using the linearized equation of state, where the density perturbation is derived from the velocity and temperature perturbations and the continuity equation. The contribution at each point is scaled by the volume of the atmosphere it represents and is scaled to account for the exponential decrease (increase) in density (perturbation quantities) with increasing altitude. These two factors combine to form a weighting function that depends upon the altitude and latitude of the grid point. An integrated atmospheric response can then be calculated for any desired latitudinal or height regions by summating the appropriate points.

3. RESPONSES

TO SURFACE

ATMOSPHERIC

AND LOWER

FORCING

The intent of the first portion of the study is twofold: to provide a comparison of the GSWM spectral response with that predicted by normal mode theory and to examine the changes in this response as the forcing region is raised from the surface to the lower thermosphere. All model runs in this section of study are made for westward zonal wave number one using a September 1st chmatological atmosphere and uniform (symmetric) forcing, with the exception of the thermospheric case (see later). Figure 2 shows the GSWM integrated response to uniform vertical velocity forcing at the lower boundary. The solid curve indicates the response of the atmosphere below 90 km. Below 90 km there is a sharp

2188

C. K. Meyer and J. M. Forbes

’

105,

Lower Boundary Forcing m

1

7

1

1

*

3

3

Westward, S=l

,oo

ij 0

Solid - Response below 90 km I I I,,,, 5 Period (days)

* 10

15

Fig. 2. Spectral response to symmetric lower boundary forcing for westward zonal wave number one (S = 1). Response below 90 km (above 90 km) shown as solid (dashed) line. Each curve normalized to peak value at 4.15 days.

at 4.75 days, a broader peak at 14days and a smaller peak at 9 days. The 4.75 day peak appears to be a realization of the 1st symmetric Rossby normal mode commonly referred to as the Sday wave (Madden and Julian, 1972) or (1 ,l) mode (s, n--s: s = zonal wave number, n = meridional index, notation after Salby, 198 1b and Longuet-Higgins, 1968). The peak at 9days falls within the observed 8lOday range of the 2nd asymmetric Rossby normal or (1,2) mode (Forbes et al., 199.5~). Its realization in response to uniform symmetric forcing may be a result of uniform energy projecting onto a slightly asymmetric mean wind field in the stratosphere and mesosphere. Perturbation fields (not shown) of the 4.75 and 9 day waves are consistent with the 1st symmetric and 2nd asymmetric Hough mode structure predicted by normal mode theory. The elevated response at 14 days appears to be the realization of the 2nd symmetric normal mode referred to as the quasi- 16 day wave (Forbes et al., 199%) or (1,3) mode. The character (period and structure) of these long period features may depend on the background atmosphere used here. Results may change slightly if winds from different seasons are used or if future wind models (UARS, TIMED) are incorporated into the background atmosphere. The dotted curve in Fig. 2 shows the spectral peak

response of the atmosphere above 90 km to lower boundary forcing. The largest response is again seen at 4.75 days; however, a secondary peak is present at 6.0 days with elevated response between the two. The smaller 9 day peak is still evident whereas the 14 day feature is no longer seen. It appears that the (1,l) and (1,2) Rossby normal modes are capable of penetrating into the thermosphere but the (1,3) mode is not. The lack of penetration of this wave above 100 km was verified for a variety of wind conditions in the Forbes et al. (1995~) study of the quasi 16 day oscillation. The response of the atmosphere above 90 km is distinctly different at shorter periods where an elevated response is seen at 12 and 24 h. This feature will be addressed in more detail later. As the location of the forcing is raised the spectral response character of the atmosphere changes. Figure 3 show the GSWM response to a uniform stratospheric heat source centered at 35 km. The solid line, indicating response below 90 km, has the largest peak at 6.0 days, and secondary peaks at 4.75 days and 1224 h. The lack of elevated response at longer periods indicates that the 8-10 day and 16 day waves originate primarily in the troposphere and lower stratosphere. The response above 90 km (dotted line Fig. 3) is dominated by short-period waves. In response to mesospheric forcing at 70 km (not shown) the largest

Natural

oscillations

2189

of the ITM system

Stratospheric Heat Source

: 100 0

Westward,

.. ..,_, .’ __ ,:’ ‘..

S=l I

I,,

,

5 Period(days)

/, 10

':. ;._,

_...' /.:’ 15

Fig. 3. Spectral response to stratospheric heat source for westward zonal wave number one. Response below 90 km (above 90 km) shown as solid (dashed) line. Each curve normalized to its peak value at 6 days.

response both above and below 90 km is seen at 12 h, with a smaller broad response between 4 and 9 days. The two peaks near 5days in response to stratospheric forcing and the broad 4-9day mesospheric forcing feature at first appear to be the Sday wave broadened and shifted to longer periods by the mean winds. However the spectral response character of the Sday wave was shown to be relatively insensitive to the mean wind field (Geisler and Dickinson, 1976; Salby, 198 1b). In addition, the perturbation fields for the stratospherically forced 6day wave are not consistent with the (1,l) Hough mode structure. Previous analyses (Manney and Randel, 1993; Meyer and Forbes, 1997; Norton and Thuburn, 1996) indicated the importance of barotropic and baroclinic instability in the realization of stratospheric and mesospheric waves. Unstable regions, owing their existence primarily to vertical shears in the mean winds maintained by momentum deposition caused by dissipating gravity waves, provide a source of energy for the enhancement of waves at preferred periods. As shown by Meyer and Forbes (1997) the quasi 6 day oscillation that dominates the spectral response in Fig. 3 is primarily an instability-driven oscillation. The reader is referred to that work for more details. The varying nature of the response described above indicates two things. The realization of longer period waves diminishes as the altitude of the source is raised,

indicating that the longer period waves originate at lower levels in the atmosphere. The preferred response of the thermosphere (above 90 km) occurs at the shortest periods studied (less than 24 h), a result that increases in significance as the location of forcing is raised (i.e. the 12 h feature dominates in response to mesospheric source). These trends continue when the forcing is raised to the lower thermosphere as shown in Fig. 4. The solid line depicts the GSWM response for westward S = 1, uniform forcing under equinox conditions. There is a significant peak near 12 h and no significant response is seen at periods greater than 24 h. Recall from Fig. 2 that the 5 day wave was present above 90 km when the atmosphere is forced either at the surface or in the stratosphere. The absence of a 5day peak in response to thermospheric forcing indicates that a realistic thermosphere cannot support a distinct 5 day wave (first symmetric Rossby wave) without energy propagation from below. Short-period waves are relatively insensitive to dissipation due to their large phase speeds. However, longer period waves, with slower phase speeds, are more likely to be affected by dissipation. Three dissipation mechanisms act to suppress longer period waves in the upper mesosphere and lower thermosphere: Newtonian cooling, molecular viscosity and ion drag. Model runs were made with each of these mechanisms removed and it was found that ion

2190

C. K. Meyer and J. M. Forbes

Thermospheric Forcing I’ II I i ’ 1 II ’ 1 , 1 ” Westward, S=l

1Ol

6 E: $

l(

1 0

I

I

/

2

I

I

I

I

/

I,,

4 6 Period (days)

,

I

8

/

,

,

10

Fig. 4. Spectral response to thermospheric forcing for westward zonal wave number one. Realistic dissipation with Ion drag (solid) and with ion drag removed (dashed).

drag is the dominant dissipation mechanism. The dashed line in Fig. 4 represents results when the ion drag terms in the equations of motion were set to zero. An elevated response is seen between three and 6 days, peaking near 4.5days, as well as a sharp peak near 12 h. It is clear that the effect of ion drag is to suppress longer-period waves to the level of the background response, and to slightly depress and broaden the response at shorter periods.

4. THERMOSPHERIC

4.1. General character

FORCING

qf response

All results reported in this section are for a gaussian heat source centered at 110 km with a 10 km half width. As noted previously, the main features of the thermospheric response are unaffected by the particular type of forcing employed (i.e., heat source, momentum source, vertical velocity at 90 km). The response to both symmetric and asymmetric heating distributions was examined for eastward and westward zonal wave numbers 1,2 and 3. Runs were made

for both equinox and solstice background atmosphere configurations, under quiet solar conditions. The magnitude of the forcing chosen is equivalent to that of absorbed UV solar radiation N, and 0, under solar minimum conditions. Frequencies (periods) between 0.1 and 6 cycles per day (4 h to 10 days) are specified as input. The solid curve in Fig. 4 is representative of the character of the response for all cases studied. For all wave numbers, forcing shapes and atmospheric configurations the only significant response is seen at periods less than 24 h (48 h) for westward (eastward) waves. The longer-period Rossby waves evident in the lower and middle atmosphere are not observed. Hence, the focus of the remaining analysis is restricted to the short-period features. In the lower and middle atmosphere, seasonal wind changes can have significant effects on the spectral character of longer period waves, which can undergo large Doppler shifts (Salby, 198lb). In our thermospheric simulations, however, seasonal changes in the background atmosphere produced minimal changes in the spectral response. The phase speeds of the short periods waves are quite large (790m SC’ for a 14 h period S = 1 wave at the equator) and the mean winds will not significantly alter the intrinsic frequency of the waves. Further verification of the lack of influence of the background winds on these short period features was found by making model runs with mean winds tapered to zero above 110 km. The character of the spectral response was unchanged from the full wind-field runs. 4.2. Westward

waves

Figure 5a (5b) shows the spectral response curves for westward zonal wave number 1, 2 and 3 in response to symmetric and asymmetric heat sources. The period of the peak response decreases with increasing zonal wave number. Table 1 summarizes the GSWM results for westward waves and classifies the observed features. Table 1 also shows the theoretical periods of the gravitational normal-mode solutions of LaPlace’s tidal equation for a 10 km equivalent depth atmosphere. It can be seen that the model results approximate the theoretically-predicted periods of the 1st symmetric and asymmetric gravitational normal modes. Further evidence that the peak responses seen in model results are realizations of gravitational normal modes can be found in the perturbation field structure of the waves. Figure 6 shows the horizontal velocity and temperature perturbation fields at the peak periods for the westward, symmetrically forced waves.

2191

Natural oscillations of the ITM system

a) Symmetric 35

“1

’I

’

b) Asymmetric

Forcing 1 “‘I

“1

I ”

Forcing

1’

Westward Waves

Westward Waves i ‘I ‘I

s=3

s=2 S=l

S=l

0

5

10 Period

15

20

0

5

10

Period

(hours)

15

20

(hours)

Fig. 5. Spectra1 response spectra for westward zonal wave numbers 1 (solid), 2 (dotted), and 3 (dashed) to lower thermospheric a) symmetric and b) asymmetric forcing.

Table 1. Periods of westward waves as predicted by the GSWM, classification of solutions and periods of Class I normal mode solutions (gravity) to LaPlace’s Tidal Equation for a 10 km equivalent depth Westward modes Period (h)

GSWM Peak

10 km hn

12-14 8-12 68

13.24 11.08 8.93 8.57 7.07 6.01 10.43 8.63 7.17 7.09 6.02 5.19

lo-11 610 5

Classification 1stsymmetric

1st symmetric 1st symmetric 2nd symmetric 2nd symmetric 2nd symmetric 1st asymmetric 1st asymmetric 1st asymmetric 2nd asymmetric 2nd asymmetric 2nd asymmetric

gravity gravity gravity gravity gravity gravity gravity gravity gravity gravity gravity gravity

s

For zonal wave number 1 the zonal and meridional components of the wave maximize in the polar regions in the lower thermosphere (125-130 km). For zonal wave numbers 2 and 3, the largest zonal (meridional) response is seen at the equator (mid latitudes) and maximizes above 250 km. Meridional winds peak near f30” with S = 3 fields more compact in latitudinal extent than S = 2. The temperature perturbations maximize at the equator for S = 1,2 and 3. The magnitude of the perturbations increases slightly with increasing zonal wave number (the forcing magnitude is kept independent of zonal wave number). Zonal and meridional wind expansion functions can be found from the Hough modes of LaPlace’s tidal equation. These represent the where the winds should maximize and where they should change sign. The thick lines in Fig. 8a represent the symmetric zonal wind profiles taken from these theoretical normal

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90

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for all zonal wave numbers.

1(14 h), 2 (9 h) and 3 (7 h) in response to uniform symmetric forcing. Contour interval is 5 m s-l

-60

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100 60

100 -90

160

30

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250

300

150

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5

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Westward Waves - Uniform Symmetric Forcing

Natural

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of the ITM system

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2193

2194

C. K. Meyer and J. M. Forbes

modes. For S = 1 (solid) the absolute maximum occurs at the pole. For S = 2 (dotted) and S = 3 (dashed) the absolute maxima occur at or near the equator. The latitudinal shape of these normal modes closely matches the behavior of the zonal wind perturbation in Fig. 6 (Note: the model perturbations are always shown as a positive magnitude and the sign is controlled by the phase of the response). A similar match is also evident in the meridional winds, shown in the thick lines in Fig. 8b. Absolute maxima are evident at the poles for S = 1, at 40’ for S = 2 and at 30” for S = 3 again closely matching the latitudinal structure of the perturbation fields shown in Fig. 6. The phase structure of the response fields (not shown) show the appropriate phase reversal at latitudes close to those indicated by zero crossings in Fig. 8. The geopotential height components of Hough mode expansions (not shown), which are closely associated with temperature perturbations, are all maximum at the equator, just like the GSWM temperature perturbations. Responses to asymmetric forcing also suggest a realization of gravitational normal modes in the model results. Figure 7 contains the perturbation fields at the peak periods of the westward asymmetrically-forced waves. As with the symmetric forcing the largest horizontal velocities are seen in the polar regions near 125 km for S = 1. For S = 2 and 3 the zonal winds maximize at mid latitudes above 250 km. The meridional winds for S = 2 (S = 3) show three maxima, at the equator and at f 50” (k 40”). The temperature perturbations are largest at f30’ latitude between 150 and 170 km. As with symmetric forcing the intensity of the response increases slightly as the zonal wave number is increased. Phase profiles of the peak oscillations (at the latitude of the maxima) are given in Fig. 9 as a reference for comparison with future observations of these short period features. In all cases the oscillations have vertical wavelengths in excess of 100 km. The thin lines in Fig. 8a and 8b represent the expansion profiles of the zonal (meridional) winds for the 1st asymmetric normal mode. Zonal and meridional profiles exhibit absolute maxima at the poles for S = 1. Zonal S = 2 (3) profiles are largest at 50” (40”) and meridional profiles have absolute maxima at 50’60” and again at the equator. The height expansions (not shown) for S = I, 2 and 3 all have the largest magnitude at 30”. As with the symmetric modes, the theoretical asymmetric profiles show excellent agreement with the structure of the perturbation fields at peak periods found in the numerical simulations. The phase structure of perturbation fields also contain

phase reversals at the approximate by the theoretical profiles.

location

predicted

4.3. Eastward waves Figure 10a and 10(b) shows the spectral response for eastward zonal wave numbers 1, 2 and 3 in response to a symmetric (asymmetric) heat sources. A broad response is seen for symmetric S = 1 with all other cases exhibiting a sharper, and in most cases double-peaked structure. Table 2 lists the modelderived periods and classification for the eastward waves as well as the associated theoretical periods of a 10 km equivalent depth atmosphere. For eastward waves the first symmetric modes are identified as Kelvin waves, and the second symmetric modes are gravity waves The longer period asymmetric modes are the eastward propagating Rossby-gravity waves, and the shorter periods waves are asymmetric gravity waves. As with the westward waves trends of the peak periods with respect to zonal wave number are in reasonable agreement with those predicted by normal mode theory. The Hough mode structure and the modeled perturbation fields again show excellent agreement. Figure 11 contains the perturbation fields corresponding to the peak periods eastward symmetrically forced waves shown in Fig. 10a. For all zonal wave numbers the largest zonal wind and temperature perturbations are seen at the equator. The zonal winds peak at 120-l 30 km and again above 200 km for S = 3. The temperature response maximizes at 150 km. Meridional winds are much smaller than zonal winds, and are zero at the equator. These features are consistent with the structure of equatorial

Table 2. As Table I except for eastward waves Eastward Modes Period (h) GSWM Peak 25-38 20 13 9-12 6 13-17 11-13 9 9-l 1 6-8 5

lOkm hn

Classification

32.52 16.43 11.05 9.31 7.62 6.32 13.48 10.05 1.97 7.39 6.23 5.37

Kelvin (1st symmetric gravity) Kelvin (1st symmetric gravity) Kelvin (1st symmetric gravity) 2nd symmetric gravity 2nd symmetric gravity 2nd symmetric gravity Rossby-Gravity/asymmetric Rossby-Gravity/asymmetric Rossby-Gravity/asymmetric 1st asymmetric gravity 1st asymmetric gravity 1st asymmetric gravity

s

1 2 3 I 2 3

1 2 3 1 2 3

Natural

2195

of the ITM system

oscillations

a) Zonal Wind Components

-1 .o 0

I 30

I 60

1 90

60

90

Latitude (deg) b)Meridional

.,.I 0

Wind Component

30 Latitude (deg)

Fig. 8. Hough mode structures for symmetric modes (thick lines) and asymmetric modes (thin lines) [S = (solid), S = 2 (dotted), S = 3 (dashed)] ( a ) westward zonal wind and (b) westward meridional wind.

Kelvin waves. Figure 12 contains the perturbation fields at the peak periods of the eastward propagating asymmetrically-forced waves. Zonal winds exhibit two maxima that move equatorward with increasing zonal wave number. The meridional winds are largest at the equator with secondary maxima at high/mid latitudes. The largest winds are seen above 200 km. The temperature fields show two maxima centered at mid-latitudes and exhibit decreasing latitudinal extent with increasing wave number. The horizontal wind and height expansion functions for eastward waves (not shown) are in excellent agreement with the per-

1

turbation fields for both symmetric and asymmetric responses and again the phase structure of the response is consistent with that predicted by theory. Figure 9(c) and 9(d) indicates the phase structure of the eastward symmetric (asymmetric) oscillations. Again, long vertical wavelengths ( > 100 km) are indicated. 4.4. Ejyect oj’source size, type and location It is known that the atmospheric response can be dependent on the size, shape and location of forcing.

C. K. Meyer and J. M Forbes

2196

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Fig. 9. Selected phase profiles [S = 1 (solid) S = 2 (dotted), S = 3 (dashed)] at latitude of maxima. (a) meridional wind. westward wave numbers, symmetric forcing: 78”s (S = I), 30”N (S = 2,3) (b) meridional winds, westward wave numbers, asymmetric forcing: 78”s (S = I), 50”N (S = 2,3) (c) temperature, eastward wave numbers, symmetric forcing: equator (S = 1,2,3) (d) temperature, eastward wave numbers, asymmetric forcing: equator (5’= 1,2,3).

To isolate the effects of source scale, type and location on the results described above, several additional runs are made. Responses are compared for gaussian shaped thermospheric heat sources of 5, 10 and 20 km half widths all centered at 110 km. The shape of the response vs. period curves is unchanged, only the magnitude varied, increasing with increasing source half width. Figure 13 shows results for heat sources centered at 80, 90, 100 and 110 km (S = 1, westward, uniform, IO km half width). The period of the maximum response varies slightly with source location. The peak response is seen at 11h for 80, 90 and 100 km sources and at 12 h for a 110 km source. However the structure of the perturbation fields (not shown) is unchanged. The structure of the fields at 11h in response to 80 km forcing are virtually identical to the S = 1 14 h fields shown in Fig. 6 and are still suggestive of the 1st symmetric normal mode discussed above. A similar effect, slight variations in spec-

tral character but identical field structure, is seen when the bottom of the mode1 was placed at 90, 100 and 110 km and vertical velocity forcing is applied. Response when the lower boundary forcing is placed at 110 km is virtually identical to a 110 km heat source. The effect of latitudinal variations in the source is briefly investigated by centering a heat source, gaussian in latitude, at 45”S, 0” and 45”N. In each case the source has a 20” half width and is located at 110 km (10 km half width). For all three source locations the peak response is found at 12-14 h as in the case where forcing is uniform in latitude. For the equatorial source the response is global and the perturbation fields are nearly identical to those shown for S = 1 in Fig. 6. For the source centered at 45”s (45”N) a larger repsonse in seen in the southern (northern) hemisphere. The structure of the response fields in the same hemisphere as the source closely resemble those shown in Fig. 6, in the hemisphere opposite the source a

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b) Asymmetric Forcing

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weaker response is seen. The magnitude of the response to forcing at 0” was nearly identical to that of a source uniform in latitude whereas the mid-latitude sources yielded smaller response. From this small subset of runs it appears that even an energy source with limited latitudinal extent yields the short period gravitational normal modes discussed above. If you excite the atmosphere at a normal mode frequency it will have a response with a normal mode structure regardless of the shape of the forcing. 5. OBSERVATIONS

OF SHORT PERIOD WAVES IN THE ITM

Given that forcing is present throughout the range of periods of the oscillations identified by our numerical simulations, it would seem likely that such features exist in the atmosphere and perhaps even in the (albeit sparse) observational records. Hernandez et a/. (1993) reported that 10.1 h S = 1 and 12.2 h, S = 1 oscillations are found in both South Pole optical OH emis-

sion and Scott Base (78” S) MF radar observations. The authors interpreted these waves as normal modes. Forbes (1995b) observed a large (f 20m s’) 12 h, S = 1 oscillation of the northward wind in South Pole meteor radar data (approx. 95 km). This feature is also seen in the Thermosphere-Ionosphere-Mesosphere-Electrodynamics General Circulation Model simulations (Palo and Roble - private communication). A variety of mechanisms have been proposed to explain these observed short period features in the upper atmosphere including realization of normal nonlinear interactions between multiple modes, waves, and the effects of gravity wave momentum deposition. Forbes (1995b) attributed the S = 1, 12 h oscillation to a nonlinear interaction between the migrating semidiurnal tide and a stationary wave with et al. (1986) have proposed a S = 1. Waltersheid gravity-wave driven ‘pseudo-tide’ theory to explain the large 12 h oscillation seen at high latitudes. Our numerical simulations provide strong evidence for a normal mode interpretation: the observed 12 and 10 h

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C. K. Meyer and J. M. Forbes

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Fig. 13. Response spectra above 150km showing the effects of varying the location of the heat source: 80km (dotted), 90km (dashed), 1OOkm (dot-dash), IlOkm (solid). All sources are uniform in latitude, 10 km half width for westward S = 1.

oscillations are indeed gravitational normal modes of the lower thermosphere. The 12 h oscillation falls within the our region of elevated response, 12-14 h, for the 1st symmetric normal mode (theoretical period of 13.24 h), and the 10.1 oscillation matches with our 1st asymmetric normal mode (theoretical period of 10.43 h). Numerical simulations presented here indicate both of the westward S = 1 waves exhibit peak horizontal wind amplitudes in the polar regions, accompanied by an absence of temperature variations, consistent with structure observed by Forbes (1995b) and Hernandez et al. (1996). The long vertical wavelength (approx., Fig. 9) for the S = 1 westward wave is also consistent with observations (Hernandez et al., 1996). Hamilton and Garcia (1986) identified both the S = 1 and S = 2 Kelvin waves in long-term surface pressure records. Our analyses indicate that these two waves have the potential to exist in the ITM region as well. Manson and Meek (1990), Manson et al. (1987) report that 16, 10, 8 and 6 h waves regularly occur in the mesopause region and attributed to tidal harmonics and non-linear interactions of the 2 day wave and the semi-diurnal tide. A 6 h feature was seen over Arecibo in tidal ion layer motions between 125% 150 km (Morton et al., 1993). The authors interpret this as a quaterdiurnal tidal feature arising because of a non-linear frequency doubling of the semi-diurnal are satellite observations tide. Unfortunately

incapable of resolving waves at the preferred periods found in this study. Our results do not indicate significant eastward waves at period greater than 2 days thus the 2-4 day eastward propagating waves reported by Fraser et al. (1993) and Hernandez et al. (1996) do not appear to be normal modes of the upper atmosphere.

6. CONCLUSIONS

Model simulations performed with an atmosphere that contains realistic dissipation and September 1st climatological winds indicate that the first two symmetric and first asymmetric Rossby normal modes are realized in response to lower boundary forcing. Above 90 km only the first symmetric mode (5 day wave) is evident. As the altitude of forcing is raised the longerperiod waves no longer appear in GSWM results, indicating that the S-10 day and quasi-16 day waves originate primarily in the troposphere and lower stratosphere and that mean winds in the middle atmosphere act to prevent vertical propagation into the ITM region. Again the Sday wave is evident above 90 km; however the dominant response is seen at periods less than 24 h. As the forcing is raised into the mesosphere the total atmospheric response is now dominated by responses at periods shorter than 24 h. A broad response is seen near five days but the character of the response is no longer a clear manifestation

Natural

oscillations

of the 5 day wave. Instead results suggest that atmospheric instability plays an important role. For a thermospheric heat source the model predicts only a short period response. Dissipation in the form of ion drag is responsible for the suppression of any significant response at longer periods. Thermospheric heat sources yield features that closely resemble the theoretical normal modes or eigenfunctions of LaPlace’s Tidal Equation. The periods and perturbation field structure of the westward zonal wave number 1, 2, and 3 model responses are consistent with the I st symmetric and asymmetric gravity waves. The longer period mixed Rossbygravity waves and Rossby waves (planetary waves) are not present in the model response. Eastward zonal wave number responses are consistent with the symmetric Kelvin and gravity waves, and the asymmetric Rossby-gravity and gravity waves. Our numerical simulations indicate that short period normal modes can be realized in an atmosphere with realistic winds and dissipation mechanisms if forcing is present at the correct frequency. These features do not vary significantly with seasonal changes in the background winds. With thermospheric forcing available over a broad range of temporal scales it is likely that waves exist at these preferred periods. Acknowledgements---The

work was sponsored by Grant NAGW-4451 from NASA Headquarters to the University of Colorado. Acknowledgment is made to the National Center for Atmospheric Research, which is sponsored by the National Science Foundation, for the computing time used in this research. The authors would like to thank Dr. Scott Palo for his assistance in calculating the eigenfunctions of LaPlace’s Tidal Equation.

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and results for the solar diurnal component. J. Geophys. Res. 87, 5222-5240. Forbes, J. M. and Garret, H. B. (1979) Theoretical studies in atmospheric tides. Ret). Geophys. Spnce Phys. 17, 19511981. Fraser, G. J., Hernandez, G. and Smith, R. W. (1993) Eastward-moving 2-4 day waves in the winter Antarctic mesosphere. Geophys. Res. Lett. 20, 154771550. Geisler, J. E., Dickinson, R. E. (1976) The 5-day wave on a sphere with realistic zonal winds. J. Atmos. Sci. 33, 632641. Hagan, M. E.. Forbes, J. M. and Vial, F. (1993) Numerical investigation of the propagating of the quasi 2-day wave into the lower thermosphere. J. Geoph_vs. Res. 98, 231933 23205. Hagan, M. E., Forbes, J. M. and Vial, F. (1995) On modeling migrating solar tides. Geophys. Res. Lert. 22, 8933896. Hamilton, K. and Garcia, R. R. (1986) Theory and Observations of the Short-Period Normal Mode Oscillations of the Atmosphere. J. geophys. Res. 91, 11867711875. Hedin, A. E., Fleming, E. L., Manson, A. H.. Schmidlin, F. J., Avery, S. K., Clark, R. R., Fancke, S. J., Franser. G. J.. Tsuda, T., Vial, F. andvincent, R. A. (1996) Empirical wind model for the middle and lower atmosphere. J. utmos. terr. Phys. 58, 1421-1447. Hedin, A. E. (1991) Extension of MSIS thermospheric model into the middle and lower atmosphere. J. geophys. Res. 96, 1139-1172. Hernandez. G., Fraser, G. J. and Smith, R. W. (1993) Mesospheric 12 hour oscillation near south pole Antarctica. Geophvs. Res. Lett. 20, 178771790. Hernandez, G., Forbes, J. M., Smith, R. W., Portnyagin, Y.. Booth. J. F. and Markarov. N. (1996) Simultaneous mesospheric wind measurements near south pole by optical and meteor radar winds. Geophys. Res. Le/t. 23, 107991082. Lindzen, R. S. and Blake, D. (1972) Lamb waves in the presence of realistic distribution of temperature and dissipation. J. geophys. Res. 17, 216&2176. Lindzen, R. S. and Kuo, H. L. (1969) A reliable method for the numerical integration of a large class of ordinary and partial differential equations. Monthly Weather Review 97, 732--734.

REFERENCES Chapman and Lindren, (1970) Atmospheric Tides. pp. 200. Gordon and Breach Publishers. Geisler, J. E. and Dickinson, R. E. (1976) The 5-day wave on a sphere with realistic zonal Winds. /. Atmos. Sci. 33, 632-641. Fleming, E. L. and Chandra, S. (1989) Equatorial zonal winds in the middle atmosphere derived from geopotential height and temperature data. J. Atmos. Sri. 46, 86&866. Forbes, J. M. (1995a) Tidal and planetary waves: The upper mesosphere and lower thermosphere: A review of experiment and theory. Geophysical Monograph, 87, American Geophysical Union. Forbes, J. M. (1995) First results from the meteor radar at south pole: A large 12-hour oscillation with zonal wave number one. Geophys. Res. Letr. 22, 324773250. Forbes, J. M., Hagan, M. E., Miyahara, S., Vial, F., Manson, A. H., Meek, C. E. and Portnyagin, Y. I. (1995) Quasi 16day oscillation in the mesosphere and lower thermosphere. J. Geophys. Res. 100,914999163. Forbes, J. M. (1982) Atmospheric tides 1. Model description

Longuet-Higgins, M. S. (1968) The eigenfunctions of Laplace’s tidal equation over a sphere. Philosophical transactions of the Royal Meterological Society, - London A262, 51 l-607. Madden, R. A. and Julian, P. A. (1972) Further evidence of global-scale 5-day pressure waves. J. atmos. Sci. 29, 1464 1469. Manney, G. L. and Randel, W. J. (1993) Instability at the winter stratopause: A mechanism for the 4-day wave. J. atmos. Sci. 50, 3928-3938. Manson, A. H. and Meek, C. E. (1990) Long Period (-8 20 h) wind oscillations in the upper and middle atmosphere at Saskatoon (52”N): Evidence for non-linear tidal effects. Planet. Space Sci. 38, 1431~1441. Manson, A. H., Meek, C. E., Fellous, J. L. and Massebeuf, M. (1987) Wind oscillations (- 6 h-6d) in the upper and middle atmosphere at Monpazier (France, 45’N, 1’E) and Saskatoon (52’N, 107’W) in (1979.(1980). J. atmos. terr. Phys. 49, 1059.

Meyer, C. K., Forbes, J. M. (1997) A 6.5.day westward propagating planetary wave. J. Geophys. Res., accepted. Morton, Yu Tong, Mathews, J. D. and Zhou. Q. (1993)

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Further evidence for a 6 h tide above Arecibo. J. amnx. terr. Phys. S&459465. Norton, W. A. and Thuburn, J. (1996) The two-day wave in a middle atmosphere GCM. Geophys. Res. Lett. 23,21132116. Portnyagin, Y. (1986) The climatic wind regime in the lower thermosphere from meteor radar observations. J. ammos. Ierr. Phys. 48, 109991109. Salby, M. L. (1981) Rossby normal modes in non uniform background condition, Part I: Simple fields. J. atmos. Sci. 38, 1803-1826.

Salby, M. L. (1981) Rossby normal modes in non uniform background condition, Part II: Equinox and solstice conditions .I. atmos. Sci. 38, 1827-1840. Siebert, M. (1961) Atmos. tides, advances in geophys. 7, 1055 187. Waltersheid, R. L., Sivjee, G. G., Schubert, C. and Hamwey, R. M. (1986) Large-amplitude semi-diurnal temperature variations in the polar mesopause: evidence of a pseudo tide. Nature 324, 347-349. Zhu, X. and Strobe], D. F. (1991) Radiative damping in the upper mesosphere. J. atmos. Sci. 48, 184199.

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