The 2-day wave is a planetary-scale wave recurrently observed in the summer mesosphere. It is predominantly zonal wave number 3 (though wave numbers 2 and 4 are also observed) and has a period of approximately 2 days. It is generally accepted that the 2-day wave has characteristics of both global-scale Rossby-gravity normal mode and baroclinic/barotropic instability. Here we examine how local instability amplifies the global Rossby-gravity mode. We perform a two-dimensional instability calculation to calculate unstable modes. The fastest growing modes have wave number 3, a 35-hour Rossby mode localized in the summer hemisphere at high latitudes, and a Rossby-gravity mode with period 42 hours with similar characteristics to the observed 2-day wave. The Rossby-gravity mode was examined as the background state was changed from an isothermal atmosphere to realistic conditions. The wave was slowly Doppler shifted toward shorter periods and its spatial structure became distorted. Approaching realistic conditions the mode becomes weakly unstable. Near this point a localized Rossby mode also emerges which exists on the reversed potential vorticity gradients of the summer jet. This has the necessary characteristics to phase lock with the Rossby-gravity mode. Hence we propose that the 2-day wave amplifies from the interaction of global-scale Rossby-gravity and local modes. A nonlinear model was used to examine the unstable modes as they grow to finite amplitude. It was found that even though the 35-hour Rossby wave had the largest growth rates, it did not reach large amplitude. It was the slower growing 42-hour Rossby-gravity mode which dominated the long time evolution.
 The quasi 2-day wave (or just the 2-day wave) is a planetary-scale wave observed recurrently in the summer mesosphere. First reports were by Muller  who analyzed meteor radar wind measurements at Sheffield, UK, for August 1968. Subsequently it has been extensively observed by radar and satellite [e.g., Glass et al., 1975; Muller and Nelson, 1978; Craig et al., 1981; Rodgers and Prata, 1981; Wu et al., 1993; Meek et al., 1996; Tsutsumi et al., 1996; Nozawa et al., 2003; Garcia et al., 2005; Limpasuvan et al., 2005]. The wave appears with large amplitude after the summer solstice (July and August in the Northern Hemisphere and January and February in the Southern Hemisphere) but also has been observed at other times of year particularly at low latitudes [e.g., Harris and Vincent, 1993]. The signature is largest in meridional wind which peaks near the equator, and both zonal wind and temperature show maxima at midlatitudes with larger values in the summer hemisphere. The wave is westward propagating with predominantly zonal wave number 3 but zonal wave numbers ranging from 2 to 4 have also been observed. The period of the wave ranges from 43 to 55 hours with larger frequency variations in July and August.
 The 2-day wave has also been found in simulations of middle atmosphere General Circulation Models (GCMs), e.g., Norton and Thuburn [1996, 1997] in the Extended version of the UGAMP (UK Universities Global Atmospheric Modelling Programme) GCM (EUGCM), and McLandress et al. , in the Canadian Middle Atmosphere Model.
 There were two original theories as to how the 2-day wave is produced. One theory is that the wave is a manifestation of the gravest zonal wave number-3 Rossby-gravity normal mode [Salby and Roper, 1980]. This is the (3, 0) mode in the solution of Laplace's tidal equation [Longuet-Higgins, 1968], where for (k, n − k), k and n are the zonal and total wave numbers, respectively. The (3, 0) Rossby-gravity mode for an isothermal atmosphere at rest has a period of approximately 50 hours, and a spatial structure that has a meridional wind maxima at the equator and zonal wind and temperature minima at the equator with maxima in both midlatitudes (see Figure 7 in section 4). The normal mode theory can account for the global structure as well as the approximate period of the wave.
Salby  used a linearized primitive equation model with realistic background fields with forcing at the surface and found a resonant response at 54 hours for solstice conditions with somewhat enhanced amplitude in the summer mesosphere. Hagan et al.  confirmed these results with a global-scale wave model.
 A second theory is that the 2-day wave is produced from baroclinic instability of the summer jet [Plumb, 1983]. Plumb performed a one-dimensional instability analysis of a summer mesospheric vertical profile using the quasi-geostrophic equations and found an unstable mode with a zonal wave number 3 and period approximately 2 days. Pfister  extended the quasi-geostrophic instability analysis to two dimensions for one hemisphere. Even though he found unstable modes with the correct wave numbers and period range, their spatial structure was confined to mid and high latitudes. A related theory to baroclinic instability is that the wave is produced from barotropic instability in the summer subtropics near the stratopause [Orsolini et al., 1997; Limpasuvan et al., 2000; Schröder and Schmitz, 2004].
 Instability of the summer jet could account for the observed seasonal amplification of the wave, since regions of reversed potential vorticity gradients (a necessary condition for instability) have been diagnosed from NCEP analysis [Randel, 1994] and in GCM integrations [Norton and Thuburn, 1996]. Furthermore, the eddy momentum and temperature fluxes associated with the unstable modes are consistent with the observed 2-day wave [Lieberman, 1999; Fritts et al., 1999].
 Consequently it was suggested [e.g., Randel, 1994; Norton and Thuburn, 1996; Wu et al., 1996] that the 2-day wave has characteristics of both the global Rossby-gravity mode and an instability. This was explored by Salby and Callaghan  who performed calculations with the linearized primitive equations. They found unstable modes with e-folding times as short as 5 days that had global structure similar to the observed 2-day wave. They concluded that the global Rossby-gravity mode could amplify could amplify through extracting energy from the mean flow. Wave activity generated at the unstable region near the wave's critical line then disperses globally onto the Rossby-gravity mode structure. They demonstrated that the growth rate of the mode is very sensitive to details of the zonal mean state, relatively small changes in zonal mean wind are sufficient to remove instability.
 Despite the recent advances in understanding of the 2-day wave, there remain unanswered questions. Given the instability is due to local regions of reversed PV gradients, why have the local unstable modes predicted by Plumb  and Pfister  not been diagnosed from observations or from GCMs? In addition, how does the global Rossby-gravity normal mode amplify from a local instability? Addressing these questions is the aim of this study.
 The outline of the paper is as follows. To illustrate the seasonal amplification of the 2-day wave, section 2 presents wave spectra from meteor radar data and the EUCGM. In section 3 we perform global instability analysis using the linearized primitive equations and a background state from the EUGCM to isolate the unstable modes. This identifies an unstable global mode which is similar to the Rossby-gravity mode. Surprisingly this is not the fastest growing mode, the fastest growing mode is a local mode confined to high latitudes of the summer hemisphere. To examine how the global unstable mode arises, section 4 shows how the Rossby-gravity mode is modified by the background state by changing the zonal mean temperature and winds from isothermal at rest toward realistic values. It demonstrates that when the unstable global mode first appears, it has characteristics of previously existing stable global and local modes. We suggest the hypothesis that mutual interaction between these two modes leads to the linear instability and growth. To understand why the unstable local mode identified in section 3 is not readily observed, section 5 describes a model experiment that examines the nonlinear evolution of the unstable modes. Finally section 6 discusses the results of the paper.
2. Wave Spectra From Meteor Radar Data and EUGCM
 As an illustration of the seasonal evolution and frequencies exhibited in the Northern Hemisphere mesosphere, Figure 1 shows frequency spectra of zonal wind for 3 years of meteor radar measurements at Sheffield, UK. The data are representative of the winds at 53°25′N, 3°53′W and at a height between 85 and 95 km, the data were acquired between June 1989 and October 1994 [Muller et al., 1995]. At most times of year (apart from the diurnal signal) the frequency of fluctuations in the zonal wind are mostly less than 0.2 cycles per day. However in June, and particularly July and August, there is a strong burst of wave activity near 0.5 cycles per day. Closer inspection of Figure 1 shows that there are in fact several frequencies present, especially in 1992. For July and August, in all 3 years there is a strong peak near 0.53 cycles per day (period 45 hours), in 1991 and 1992 there are peaks at 0.47 cycles per day (period 51 hours) and 0.33 cycles per day (period 72 hours), there are also weaker peaks at 0.7 cycles per day (period 34 hours) and at higher frequencies. Figure 1 shows that there is considerable interannual variability in the exact timing and amplitude of the different wave components.
 To compare with the meteor radar data, Figure 2 shows frequency spectra of zonal wind at 54°N, 3°W and 0.003 hPa (about 90 km) for 3 years of simulation of the T42 (∼3°) resolution version of the EUGCM (for more details of the model, see Norton and Thuburn [1996, 1997]). There is good overall agreement between the spectra in Figures 1 and 2. The seasonal evolution is well captured by the EUGCM. Apart from the diurnal tide signal, only for June, July and August are there frequencies above 0.2 cycles per day. For these months the EUGCM shows a dominant frequency of 0.53–0.55 cycles per day (periods 44–45 hours) with other peaks at 0.45–0.47 cycles per day (periods 51–53 hours), 0.33 cycles per day (period 72 hours), and above 0.6 cycles per day (period 40 hours). There is also significant interannual variability in the timing and amplitude of the wave components, though in year 1 and 2 the EUGCM shows slightly larger amplitudes than in the meteor radar data.
 The good simulation of the 2-day by the EUGCM [see also Norton and Thuburn, 1996, 1997] indicates we can use zonal mean wind and temperature distribution from the model as the background state to investigate the linearly unstable modes at times of large 2-day wave amplitude. This will be done in the next section.
3. Linear Two-Dimensional Instability Analysis
 The linearized equations for deviations from a zonal mean background state are used to perform a two-dimensional instability analysis on an evenly spaced grid of latitudes and pressure levels (see Appendix A for details). Assuming an exponential time dependence for the unknowns variables (winds and temperature), the problem reduces to an eigenvalue problem with complex eigenvalues and eigenvectors. The real part of the eigenvalue gives the growth rate and the imaginary part the period or frequency of the eigenvector.
Figure 3 shows the background state (zonal mean zonal wind and temperature) used in the instability calculations. It is a mean of 1–5 July taken from the T42 integration of the EUGCM. This is a time when the 2-day wave has large amplitude in the EUGCM (see Figure 2). The shaded regions in Figure 3 are where the horizontal gradient in quasi-geostrophic potential vorticity (y) are less than −1 × 10−11 m−1 s−1 indicating the potential for instability.
 Initial calculations of the instability analysis were performed to evaluate the influence of resolution on the resulting unstable modes. Results were insensitive to levels below 30 km. So 23 levels were used in the instability calculation with a domain from 36 km to 94 km. In the horizontal 35 latitudes were used (from 85°S to 85°N, with 5° spacing).
 The instability calculation was performed with zonal wave numbers k = 1, 2, 3 and 4. The fastest growing modes at all zonal wave numbers had structure with very small vertical wavelength near the equator (figure not shown). These are inertially unstable modes near the equatorial stratopause where the criteria for inertial instability is satisfied (potential vorticity times the Coriolis parameter is less than zero). These inertially unstable modes will not be considered further in this study (though inertially unstable modes have been shown to important in other studies [e.g., Orsolini et al., 1997; Limpasuvan et al., 2000; Schröder and Schmitz, 2004], here the vertical discretization of the EUGCM means these modes are probably spurious).
 The periods and e-folding times of the most unstable modes (disregarding the inertially unstable modes) for all wave numbers are given in Table 1. For k = 1, there is only one unstable mode. The mode is an eastward propagating mode, with amplitude in the Southern Hemisphere, with period 109 hours. This mode may be related to the 4-day wave which has been observed at high latitudes in the wintertime hemisphere [e.g., Allen et al., 1997; McLandress et al., 2006]. For k = 2, again there is only one unstable mode. It is a westward propagating mode with period 49 hours and e-folding time of 94 hours. Its spatial structure is shown in Figure 4. Its maximum amplitude is in the Northern Hemisphere near 60°N, though the zonal wind has a smaller peak in the Southern Hemisphere, and the meridional wind also has a peak at the equator.
Table 1. Unstable Modes From the Linearized Instability Calculation With a Background State of 1–5 July From the EUGCMa
e-Folding Time, hours
C60, m s−1
Rows show zonal wave number, growth rate, period, and zonal phase speed at 60°N for each unstable mode.
 For k = 3 there are two westward propagating unstable modes. The fastest growing mode has a period of 33 hours and an e-folding time of 79 hours (this is the fastest growth rate for all wave numbers). Its structure is shown in Figure 5. In all the fields, the amplitude is localized around 60°N and 80 km. This is the region where y is negative and there is the strongest vertical shear zone of the summer jet (see Figure 3). The second fastest growing mode for k = 3 has a period of 42 hours and an e-folding time of 98 hours. Its structure is shown in Figure 6. It has a more global structure with amplitude in the Southern Hemisphere in both the zonal and meridional winds. This structure is similar to the 2-day wave diagnosed directly from EUGCM integrations [Norton and Thuburn, 1996, 1997] and also resembles the observed 2-day wave (though in observations and the EUGCM the maximum in meridional wind amplitude is closer to the equator).
 Finally for k = 4 there are also two westward propagating unstable modes. The fastest growing mode has an e-folding time of 85 hours and a very similar spatial structure to the 33 hours, k = 3 mode (figure not shown). There is also a slower growing mode with k = 4 with an e-folding time of 126 hours which has more global structure. Table 1 shows that the k = 2 mode, the fastest growing k = 3 mode, and the slower growing k = 4 mode all have very similar phase speeds (a common phase speed across wave numbers has also been seen in the analysis of the 2-day wave by Rodgers and Prata  and Garcia et al. ).
 The results of this section show that for k = 3, 4 the fastest growing modes have amplitude localized near the strong shear of the summer jet, yet the observed 2-day wave resembles a slower growing mode. To understand the reason for this, section 5 examines the nonlinear evolution of the unstable modes. The next section examines how the slower growing k = 3 mode arises.
4. Rossby-Gravity Mode With Varying Background States
 This section examines the mechanism for local linear instability of the mean flow which amplifies the global k = 3 mode. We will examine what happens to the structure and phase speed of the gravest zonal wave number-3 Rossby-gravity mode as the background state is slowly changed from isothermal at rest, to the full EUGCM background state used in section 3.
Figure 7 shows the structure of the Rossby-gravity mode for an isothermal at rest background state ( = 240 K and = 0). This is a stable mode (real part of eigenvalue is zero) with period of 50 hours and phase speed at 60°N, C60, of −37 m s−1. It is symmetric about the equator and has an amplitude maxima in both hemispheres in zonal wind and temperature, and a maximum on the equator in meridional wind.
 The background wind and temperature were then changed as follows,
where E and E are the EUGCM zonal mean fields, and are the background fields used in the instability calculations, and α ∈ [0, 1]. Hence α = 0 corresponds to a 240 K isothermal atmosphere with no background wind, and α = 1 corresponds to the full EUGCM temperature and wind fields.
 For increasing α, stable eigenvectors with periods between 40 hours and 58 hours were calculated. For α = 0.2 the Rossby-gravity mode was found to have a period of 49 hours (C60 = −38 m s−1), for α = 0.4 the period was 47 hours (C60 = −39 m s−1). At α = 0.55 the Rossby-gravity mode is still stable, with a period of 46 hours (C60 = −40 m s−1). Figure 8 shows the structure of this mode with α = 0.55. Compared to the mode about a state of rest (Figure 7), the structure has become distorted with increased amplitude in the Northern Hemisphere, especially in the temperature field. The meridional wind is the least altered field.
 Interestingly, for the same background state, one can identify a stable mode which has a local structure confined to high latitudes in the Northern Hemisphere (see Figure 9). This has a period of 53 hours (C60 = −35 m s−1).
 When α is increased still further to α = 0.75, a stable wave number 3 mode with global structure cannot be found. However an unstable mode can be identified with a growth rate of 20 days and a period of 45 hours (C60 = −41 m s−1). This mode has a critical line near the summer easterly jet maximum. The structure of this mode is shown in Figure 10. This mode has large amplitude at high latitudes of the Northern Hemisphere, however it also has more global structure particularly in the meridional wind field. Hence it resembles a combination of the stable Rossby-gravity (Figure 8) and local (Figure 9) modes at α = 0.55. It also resembles the structure of the second fastest growing wave number 3 mode with the full EUGCM background state (Figure 6, this has a period of 42 hours and C60 = −44 m s−1). We now interpret how this instability of the global Rossby-gravity mode arises.
 It is well established that shear instability (e.g., baroclinic instability) can be understood as the mutual interaction of two counterpropagating waves. The induced velocity field of each wave can result in phase locking and mutual growth [see Hoskins et al., 1985]. This has been demonstrated, for example, by Bretherton  for the Eady problem where two boundary Rossby waves interact, by Heifetz and Methven  in examining optimal growth in shear instability, and by Sakai  who showed Rossby and Kelvin waves can interact to produce an ageostrophic instability. Here we propose this mechanism can also be used to understand the growth of the 2-day wave.
 We showed that the westward propagating Rossby-gravity mode is weakly Doppler shifted by the background winds, its phase speed changing from C60 = −37 m s−1 with α = 0 to C60 = −40 m s−1 with α = 0.55. The local Rossby mode shown in Figure 9 has an intrinsic eastward phase speed because it is propagating on the negative PV gradients near the peak of the easterly jet (for α = 0.55, the maximum background easterly wind is approximately −40 m s−1, while the wave has C60 = −35 m s−1, hence the intrinsic phase speed is approximately +5 m s−1). With α increased still further, it could be expected that the Rossby-gravity mode is again only weakly Doppler shifted while the local Rossby mode is strongly Doppler shifted so that their phase speeds nearly match. At this point the waves can strongly interact with the induced velocity field of each wave resulting in phase locking and mutual growth. This is mechanism whereby the global Rossby mode can amplify through local instability.
5. Nonlinear Model Experiments
 The instability calculations in section 3 are based on linear theory and so are only valid for small amplitudes. In this section we employ a mechanistic GCM to perform initial value experiments that investigate the nonlinear evolution of the unstable modes. Initial conditions are the same zonally symmetric flow as used in the instability calculations. Small amplitude noise is added to excite the unstable modes and the model is integrated for several weeks to study the subsequent evolution.
 The model used is the UK Met Office Stratosphere Mesosphere Model (SMM [see Fisher, 1985]). The SMM is a global primitive equation model used for process studies in the middle atmosphere (it does not have a representation of the troposphere). The SMM grid used here has 72 longitudes and 36 latitudes, i.e., a resolution of 5° by 5°, and 35 levels in the vertical from 28 km to 96 km at 2 km spacing. Here the model is run without a radiation scheme or any representation of gravity wave drag.
 When the SMM is run with the zonal mean initial conditions taken from the EUGCM, large values of the meridional wind develop near the top of the model as a result of the initial conditions not being in balance (in the EUGCM there are other body forces, e.g., from the parameterized gravity wave drag that are important in the momentum balance). Hence a short initial run is carried out where the zonal mean winds throughout the whole domain are strongly relaxed back to the initial conditions (E, E) with a damping coefficient of 1.15 × 10−5 s−1 at every time step (i.e., an e-folding time of approximately 1 day). Note that in the initial condition E = 0. The SMM fields come into balance with a very small meridional wind field after a few days, the zonal wind and temperature fields have changed slightly to achieve balance (but all fields remain zonally symmetric). These fields are then used as initial conditions for a subsequent SMM integration where small amplitude white noise is added to the zonal wind and temperature at all grid boxes (maximum amplitude of 0.1 m s−1 and 0.1 K respectively). This white noise seeds the growth of the unstable waves. This integration starts nominally on 7 July (labeled as day 0) and is 100 days long.
5.1. Growth of Wave Number 3
 After around 10 days in the SMM integration, waves of different zonal wave numbers start growing from the initial conditions. Figure 11 shows time evolution versus latitude at 70 km of the temperature amplitude for zonal wave number 3. Small amplitudes are also present for wave numbers 2 and 4 in the Northern Hemisphere (not shown) and wave numbers 1 and 2 are found at high latitudes in the Southern Hemisphere (this could be associated with instability of the Southern Hemisphere polar night jet).
 Three periods of wave number 3 activity can clearly be distinguished in Figure 11. At first, a small amplitude wave is present between day 20 and 35 at around 55°N. Then two periods with large amplitude occur at days 35–58 and 58–74 which have their peak at 40°N.
 Examination of a zonal wave number versus frequency spectrum calculated from the zonal wind at 70 km and 52.5°N for days 10–35 (not shown) shows a large peak at wave number 3 and frequency −0.7 cycles per day, which corresponds to a period of 35 hours (the 35 hour wave hereafter). A second smaller peak is also present at wave number 3 can be seen at frequency −0.58 cycles per day, which corresponds to a period of 42 hours (the 42 hour wave hereafter).
Figure 12 shows the spatial structure of the 35 hour wave (calculated from the amplitude of the 35 hour frequency component at each point in u, v and T for days 10–35). The structure is very similar to the fastest growing wave number 3 mode found in the instability calculation (compare with Figure 5). It is localized to the Northern Hemisphere with largest amplitude at around 60°N and 80 km.
 During the period between days 35 to 55 wave number 3 activity reaches its largest amplitude. A zonal wave number versus frequency spectrum at 70 km and 52.5°N for this period (not shown) indicates the 42 hour wave now has the largest amplitude, i.e., there is a change in period of the dominant wave at the same time as there is a shift in latitude where the maximum amplitude is located (from 55°N at day 30, to 40°N at day 50). Figure 13 shows the spatial structure of the 42 hour wave for days 35–55. The wave is more global than the 35 hour wave and has a similar structure to the second fastest growing mode of the instability calculation (compare with Figure 6).
 No significant changes in the characteristics of wave number 3 are found after the amplitude minimum around day 58, the period continues to be approximately 42 hours with a spatial structure as found in days 35–55. Beyond day 70, the SMM integration shows repeated pulses of wave number 3 activity with smaller amplitude (not shown).
5.2. Baroclinic Life Cycle
 The evolution of the waves in the SMM integration can also be seen in Figure 14 which shows the distribution of Northern Hemisphere potential vorticity (PV) at 4 day intervals on the 5000 K isentrope (about 70 km). By day 36, a clear wave number 3 pattern has developed from the initial axisymmetric flow but this is confined to high latitudes, presumably this is the PV signature of the 35 hour wave.
 At day 40 the wave number 3 starts to extend equatorward so by day 44, even PV contours near the equator are strongly displaced. This is the period when the 42 hour wave emerges as the dominant wave. Note at day 44, filaments of PV extending back to the pole indicating that the waves are breaking. At day 52, three distinct vortices have formed which partially merge at day 56, this coincides with the amplitude minimum seen in Figure 11. The wave 3 structure then emerges again and remains until the end of the integration. This “life cycle” of growing, breaking and merging of vortices resembles the life cycle of baroclinic waves in the troposphere [e.g., Thorncroft et al., 1993].
 The evolution of y at 70 km and the zonal mean wind at 57°N are shown in Figures 15a and 15b. During the first wave event (days 25–35) changes to the mean flow are small. However when the 42 hour wave starts to grow in amplitude (around day 40) the summer jet is decelerated. These changes are consistent with the influence of the waves on the mean flow as indicated by the divergence of Eliassen and Palm (EP) flux on day 45 (see Figure 15c). The source of wave activity (arrows) is in the negative region of y near 70 km (shaded), the force per unit mass on the mean flow (contours) is such to reduce the vertical shear in the summer jet. Hence the waves stabilize the baroclinically unstable initial conditions with after day 45 the strongly negative regions of y having been removed (see Figure 15a).
5.4. Growth Rates
 The growth rates of the two k = 3 unstable waves in the SMM integration is now examined to compare with the linear instability calculation of section 3. We assume the meridional component wind can be represented by
where, for the nth wave, an is the complex nonsinusoidal part of the time variation (which will represent the growth and decay of the wave), An is the latitude-height structure (which is assumed constant over the run and is taken to be the structures in Figures 12 and 13), and ωn is the frequency. The meridional wind was chosen because this is the field in which the two unstable waves differ the most. The an in equation (3) are determined by assuming they are constant over a 4-day period and a best fit to v is obtained by multiple regression over the two waves at all latitudes and heights. A time series of an is determined by examining successive 4-day periods. Figure 16 shows the result for the 35 hour (solid line) and 42 hour (dotted line) waves.
 The solid curve shows exponential growth up to day 30, this is the time of the first peak in wave amplitude in Figure 11. The 42 hour wave has a much slower growth rate, however by day 40 it has grown to large amplitude, this is the time of the second peak in wave amplitude in Figure 11. The amplitude of the 42 hour wave then decays sharply and then reemerges between days 55 and 70. The initial exponential growth of both k = 3 waves indicate that both waves are independent unstable modes. To obtain a value for the growth rate, γ, a least-squares fit is performed to the logarithm of the initial part of the curves. The growth rates compare well with those obtained by the linear instability calculation. For the 35 hour wave the e-folding time from the SMM is 3.8 days compared to 3.2 days from the instability calculation. The values for the 42 hour wave are 5 days from the SMM and 4 days from the instability calculation. These growth rates are significantly faster than was found by Salby and Callaghan  who performed a similar experiment. However, with a different initial state one could expect a sensitivity of the growth rates [Salby and Callaghan, 2001].
 It could be that the 42 hour wave nonlinearly interacts with the 35 hour wave, e.g., Figure 16 shows some indication of compensating changes. This could help the 42 hour wave reach large amplitude, however they may just exist as two largely independent modes. This should be tested in future work.
 The linear instability analysis of the July background state in section 3 found a number of unstable modes with the fastest growing modes having zonal wave number 3. This is in agreement with observations which show wave number 3 as the predominant wave number of the 2-day wave. However the fastest growing mode had a period of 35 hours and its spatial structure is localized in the Northern Hemisphere at high latitudes, with no resemblance to the global structure of the 2-day wave. In the EUGCM waves with periods around 35 hours and localized spatial structure are also diagnosed. A 35 hour wave has also been diagnosed in meteor radar data from Sheffield (G. Beard, personal communication, 1999). Possibly these waves have not been reported before because they are only seen north of 50°N and have small latitudinal extent. Hence the local unstable modes predicted by Plumb  and Pfister  do appear to exist.
 A slower growing wave number 3 mode was also found in the instability analysis, with a spatial structure that resembles the most important features of the observed wave. This wave also resembles the dominant 2-day wave found in the EUGCM [Norton and Thuburn, 1996, 1997]. However the period of this wave, 42 hours, is slightly shorter than typically observed. It could be that, even though the background fields came from the EUGCM, the summer easterly jet on the days chosen (1–5 July of one year) was stronger than the July-August average.
 The structure and phase speed of the wave number-3 Rossby-gravity mode was examined in section 4 as the background state was slowly changed from isothermal at rest, to the full EUGCM background state. For an intermediate background state, we identified a stable local counter propagating Rossby mode which exists on the reversed PV gradients of the summer easterly jet. We propose that instability, and hence the seasonal amplification of 2-day wave, comes from the interaction of the global-scale Rossby-gravity mode with such a local mode. It would be interesting to try more numerical experiments to test this hypothesis. For instance one could examine the unstable modes with background states of an idealized easterly jet with constant maximum zonal wind speed but with varying amounts of vertical shear (hence reversed PV gradient) near 80 km. In this way it may be possible to more cleanly distinguish the impact on the Rossby-gravity mode of Doppler shifting by the background state compared to the interaction with the counter propagating Rossby mode.
 An integration with a mechanistic GCM was used in section 5 to examine the nonlinear evolution of the unstable waves. All the waves that emerged could be identified with unstable modes from the instability analysis with similar e-folding times over the linear growth phase. The fastest growing mode, as in the instability calculation, is the local 35 hour wave. However this wave saturates early and does not reach large amplitude. It is the slower growing, more global, 42 hour wave which reaches largest amplitude and interacts with the background flow to remove the unstable region. Hence even though local unstable modes exist, the global Rossby-gravity mode is more readily observed.
 The linearized equations for deviations from a zonal mean background state are used to perform the instability analysis in spherical and pressure coordinates. The equations are:
where (u, v, w) are the horizontal and vertical (pressure coordinate) components of the wind, T is temperature, Φ is geopotential, a is the Earth radius, f the Coriolis parameter, k is zonal wave number and ϕ is latitude. The barred quantities are zonal means (i.e., the background state), the primed quantities are departures from the zonal mean (i.e., wave quantities).
 The five unknown fields (u′, v′, w′, T′, Φ′) can be reduced to three (u′, v′, T′) by integrating the hydrostatic equation (equation (A3)) and the mass continuity equation (equation (A4)):
 As the domain of interest lies in the stratosphere and mesosphere, a flat lower boundary is considered at pressure po (6 hPa) below the region of interest [see Andrews et al., 1987] which linearized is
The upper boundary condition is:
at the top of the domain.
Equations (A1), (A2), (A5) and (A8) are discretized for a grid of J evenly spaced latitudes and K pressure levels, using second-order finite differences. Equations (A6) and (A7) are evaluated using trapezoidal rule integration. The linear model can be written as:
where the vector X(ϕ, p, t) is:
 The matrix M is constructed as follows. The first element of the vector X is set to one with all the other elements of X set to zero. Equations (A1), (A2), (A5), (A6), (A7) and (A8) are evaluated, and the result, ∂X/∂t, gives the first column of the matrix M. Then the second element of the vector X is set to one, with the other elements zero, the equations are evaluated and the result gives the second column of the matrix. The process is repeated for all the elements of X. This process is much less prone to error than trying to determine analytically the individual elements of M from the finite difference equations themselves. Note that M is complex and for this study has size ∼2450 × 2450.
 If X is assumed to have a time dependence such that:
where v are complex eigenvalues and are complex eigenvectors. The real part of the eigenvalue v gives the growth rate of the eigenvector mode, the imaginary part of v gives the frequency of oscillation of the mode. In particular, if the real part of the eigenvalue is positive, then the mode is unstable.
 A standard numerical package is used to first calculate all the eigenvalues, and then to calculate the eigenvectors of particular eigenvalues. For unstable modes the eigenvectors of interest are those with eigenvalues that have the largest positive real part and so represent the fastest growing modes. Whereas for stable modes, eigenvectors of eigenvalues with zero real part are calculated. (The classification of solutions of eigenvalue problems in the context of the atmosphere has only been done for the barotropic vorticity equation [Drazin et al., 1982; Held, 1985]. They found that the solutions belonged to 8 different classes, which might or might not be empty. Although for the primitive equations this classification has not been done, most of the solutions will belong to the same classes. Some of the classes are described here: (1) a finite number of nonsingular unstable modes, and an equal number of nonsingular damped stable modes, whose eigenvalues and eigenvectors are complex conjugates; (2) a finite number of marginally stable modes; (3) a countable number of nonsingular stable modes modified by the basic shear (these are Rossby waves); and (4) a continuum of singular neutrally stable modes (these may not be properly treated in a numerical calculation with finite truncation).
 M.R.C. acknowledges support from the Chilean government and MIDEPLAN through a “Presidente de la República” studentship and FONDECYT through grant 1050416 and project ACT19, and W.A.N. acknowledges support from the UK Natural Environmental Research Council. We would like to thank Gary Beard for supplying the meteor radar data and John Thuburn for several useful discussions. We would also like to thank John Methven, David Andrews, and three anonymous referees for providing comments on the paper.