Parameterization of the inertial gravity waves and generation of the quasi-biennial oscillation

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jd016778, 2012 Parameterization of the inertial gravity waves and generation of the quasi-biennial oscillation X.-H. Xue, 1,2 H.-L. Liu, 3 and X.-K. Dou 1,2 Received 25 August 2011; revised 18 January 2012; accepted 27 January 2012; published 22 March [1] In this work we extend the gravity wave parameterization scheme currently used in the Whole Atmosphere Community Climate Model (WACCM), which is based upon Lindzen s linear saturation theory, by including the Coriolis effect to better describe the inertia-gravity waves (IGW). We perform WACCM simulations to study the generation of equatorial oscillations of the zonal mean zonal winds by including a spectrum of IGWs, and the parametric dependence of the wind oscillation on the IGWs and the effect of the new scheme. These simulations demonstrate that the parameterized IGW forcing from the standard and the new scheme are both capable of generating equatorial wind oscillations with a downward phase progression in the stratosphere using the standard spatial resolution settings in the current model. The period of the oscillation is dependent on the strength of the IGW forcing, and the magnitude of the oscillation is dependent on the width of the wave spectrum. The new parameterization affects the wave breaking level and acceleration rates mainly through changing the critical level. The quasi-biennial oscillations (QBO) can be internally generated with the proper selection of the parameters of the scheme. The characteristics of the wind oscillations thus generated are compared with the observed QBO. These experiments demonstrate the need to parameterize IGWs for generating the QBO in General Circulation Models (GCMs). Citation: Xue, X.-H., H.-L. Liu, and X.-K. Dou (2012), Parameterization of the inertial gravity waves and generation of the quasi-biennial oscillation, J. Geophys. Res., 117,, doi: /2011jd Introduction [2] The quasi-biennial oscillation (QBO) exists in the equatorial stratosphere and consists of an alternating pattern of eastward and westward zonal-mean zonal winds with periods from about 22 to 34 months. It was first reported by Ebdon [1960] and Reed et al. [1961], and a comprehensive review of the subject is provided by Baldwin et al. [2001]. The features of the QBO are clearly seen in the zonal-mean zonal winds over the equator for from ERA-40 data set (Figure 1, The alternating easterly and westerly winds are most apparent between hpa, with downward propagation of the westward and eastward regimes at an average rate of 1 km/month, and nearly constant amplitudes between hpa. [3] It is well known that the QBO is driven by the momentum transfer of waves propagating upward from the troposphere. Many previous studies confirmed that the 1 CAS Key Laboratory of Geospace Environment, Department of Geophysics and Planetary Sciences, University of Science and Technology of China, Hefei, China. 2 Mengcheng National Geophysical Observatory, School of Earth and Space Sciences, University of Science and Technology of China, Hefei, China. 3 High Altitude Observatory, National Center for Atmospheric Research, Boulder, Colorado, USA. Copyright 2012 by the American Geophysical Union /12/2011JD waves with different horizontal/vertical scales, i.e., the Kelvin waves, Rossby-gravity waves, inertial gravity waves, and mesoscale gravity waves, contribute to the formation of the QBO [e.g., Lindzen and Holton, 1968; Holton and Lindzen, 1972; Takahashi and Boville, 1992; Dunkerton, 1997], but the relative contribution of each wave is not very clear yet. Probably because all the waves are not properly represented, it is still a challenge to correctly simulate the QBO in general circulation models (GCM). [4] A simple way is using an artificial forcing, in which, the QBO can be imposed based on cyclic, fixed-phase, or observed winds. Some works have successively reproduced the QBO in GCMs and studied the corresponding effects in a global region [e.g., Hamilton, 1998; Pascoe et al., 2006; Richter et al., 2011; Yamashita et al., 2011]. Ideally, the waves driving the QBO can be resolved in GCMs with sufficient resolution, so the QBO can be simulated selfconsistently and realistically. A series of works have focused on this after Takahashi [1996], but the period of the QBOlike oscillations were usually less than 2 years [e.g., Takahashi, 1999; Horinouchi and Yoden, 1998; Hamilton et al., 1999, 2001; Kawatani et al., 2010]. [5] The fact that the resolved waves do not produce the correct QBO period may indicate the problems with the wave excitation in the model, which is most likely due to the parameterized convection. It is also impractical to conduct the long-term climate simulations using hyperfine spatial resolutions with the current computing resources. Given 1of14

2 Figure 1. Time-height cross section of monthly mean zonal-mean zonal winds averaged between 2.5 N and 2.5 S of ERA-40 data set during Winds are in units of ms 1 and are plotted in intervals of 10 ms 1. The solid lines show the westerly winds and the dotted lines show the easterly winds. the importance of the QBO in studying climate and climate variability, there is a need to properly parameterize the effects of equatorial waves to produce a QBO. In many GCMs, the parameterized GWs in the model are the mesoscale waves, with a horizontal wavelength 100 km. Most of these waves break in the mesosphere and lower-thermosphere (MLT). For example, in the Whole Atmosphere Community Climate Model (WACCM), the maximum forcing in the MLT region due the mesoscale GWs is 100 ms 1 day 1 [Richter et al., 2008, 2010]. If the GWs carrying the same momentum flux (0.001 Pa in WACCM) were to break in the stratosphere ( hpa), they would cause an acceleration rate similar to that required by the QBO. According to the linear theory [Holton, 1982], the gravity wave breaking altitude z b 2H ln(2p/l h A), where z b, H, l h and A represent the breaking level, scale height, horizontal wavelength and amplitude of vertical winds at a launching level respectively. It is thus clear that an order of magnitude increase in horizontal wavelength or, equivalently, in wave amplitude will lead to a decrease of z b by 4.6H 32 km. This is consistent with the finding by Kawatani et al. [2010] that the IGWs with zonal wavelength 1000 km were the main contributors to the easterly wind shear phase of the simulated QBO and contributed 50 70% of total wave forcing during the westerly wind shear phase of the QBO in their high-resolution atmospheric GCM simulations. Therefore, to account for the stratospheric forcing while keeping the mesospheric forcing, the IGWs should be properly parameterized. In fact, some GCMs have already produced quite realistic QBOs by incorporating a parameterization of the IGWs with the Hines [1997] scheme or the Warner and McIntyre [1999] scheme [e.g., Scaife et al., 2000; Giorgetta et al., 2002; Shibata and Deushi, 2005; Giorgetta et al., 2006; Scinocca et al., 2008; Naoe and Shibata, 2010]. [6] The goal of this work is to develop a way to parameterize the IGWs, building on the scheme of Lindzen [1981] but taking the Coriolis effect into account, and to test the capability of such a scheme to drive the QBO in a GCM with the standard model horizontal and vertical resolutions. The paper is organized as follows: Section 2 gives a description of the model used in this work. Section 3 examines the controlling of the equatorial oscillations by the parameterized GWs, based on the conventional Lindzen parameterization scheme. Section 4 formulates a new parameterization scheme that includes the Coriolis effect on the equatorial stratospheric wind oscillation. Based on the numerical tests in section 3 and section 4, section 5 presents the QBO structure generated by the parameterized IGWs and with a detailed discussion of the zonal-mean zonal winds structures, and section 6 is the summary. 2. Model Description [7] The WACCM (version3.5) is one of a few GCMs that encompass the atmosphere from the Earth s surface to the lower thermosphere, which allows for studies of chemical, dynamical, and radiative coupling processes between the lower and upper atmosphere. [8] WACCM is based on the software framework of the National Center for Atmosphere Research s Community Atmospheric Model (CAM). The version used in this study, WACCM3.5, is a superset of CAM, version 3.5 (CAM3.5) with the vertical model domain extending to 145 km. WACCM 3.5 uses the finite volume dynamical core of Lin [2004] with 66 vertical levels, with variable vertical resolution: 1.1 km in the troposphere above the boundary layer, km in the lower stratosphere, 1.75 km around the stratopause (50 km) and 3.5 km above 65 km. The horizontal resolution used for this study is (latitude longitude). 2of14

3 [9] WACCM3.5 uses all of CAM3.5 s physical parameterizations (details are provided by Collins et al. [2006]), with more detailed GW drag and vertical diffusion parameterizations. In addition to the orographic GW parameterization based on work by McFarlane [1987], the mesoscale GWs from deep convection and frontal systems are also parameterized in WACCM3.5 [Garcia et al., 2007; Richter et al., 2010]. The parameterization of gravity wave forcing in these schemes are all based on the linear saturation theory [Lindzen, 1981]. [10] The current WACCM does not produce a QBO internally. This is likely because the resolved equatorial waves in WACCM (and in CAM), including the Kelvin waves, Rossby-Gravity waves, and IGWs are weak compared to those extracted from reanalysis, according to earlier work by Richter et al. [2008], which also showed that wave forcing by the resolved waves comes mostly from the wave number 1 and 2 components. The causes for the weak waves are not clear, but are possibly related to the convective parameterization used and to the coarse spatial resolution. Thus, the current WACCM has an imposed QBO by nudging the tropical winds to observations based on work by Balachandran and Rind [1995], which has been previously tested in the Freie Universität Berlin Climate Middle Atmosphere Model (FUB-CMAM) [Matthes et al., 2004]. The vertical range of the nudging is between 86 to 4 hpa with a time constant of 10 days, and its latitudinal distribution is Gaussian with a half width of 10 degrees centered at the equator between 22 N and 22 S [Richter et al., 2010; Matthes et al., 2010]. Please note in this work, the imposed QBO forcing is turned off to avoid the ambiguity. 3. The Effects of the Parameterized GW Spectrum Characteristics on the Equatorial Stratospheric Wind Oscillation [11] We introduce a spectrum of the GWs with horizontal wavelength l h = 1000 km (the horizontal wave number 2p 1000km k ¼. Please note that in this section we do not consider the Coriolis effect, thus what is parameterized is the GW with large horizontal scale, rather than an IGW). The wave components are specified by phase velocity from c 0 to c 0 with a uniform spacing of dc ms 1. c 0 is the cutoff phase speed and its setting is discussed below. For simplicity, the momentum fluxes are assumed to be uniform across these components ( i.e., momentum flux t(c) = constant for c 0 c c 0 ), and the source spectrum is thus symmetric. For convenience, we also refer c 0 as the spectral width in our discussion. [12] These GWs are launched uniformly from 30 Nto 30 S at model level 200 hpa with an initial momentum flux t = Pa for each spectral element. These conditions will be kept for all cases in this paper. [13] We first started the numerical experiment without consideration of the Coriolis effect on the parameterized GW, that is, we employed the GW spectrum with the conventional Lindzen scheme [Garcia et al., 2007]. According to the Lindzen scheme, when a GW becomes convectively unstable its amplitude saturates and thus the wave momentum flux is reduced, which leads to mean flow acceleration/ deceleration. It can have multiple saturation levels until it meets the critical level, where the wave momentum flux is reduced to zero. To facilitate a better comparison, a summary of the main features of the conventional GW parameterization scheme is given here: [14] The dispersion relation for the vertical propagating GWs: m ¼ N U c Here, m, N, U, and c are the vertical wave number, buoyancy frequency, the background zonal wind speed, and the gravity wave phase speed, respectively. [15] The critical level (when a wave disappears according to the dispersion relation): c ¼ U [16] The saturation momentum flux (when a wave becomes saturated and begins to break): t ¼ r 0 kju cj 3 2N Here, k and r 0 are the horizontal wave number and atmospheric density, respectively. [17] The momentum tendency within a saturated region determined from the convergence of t : U t ¼ 1 t r 0 Z ku ð c 2NH Here, is an efficiency factor, which represents the temporal and spatial intermittency in the wave sources. [18] We now perform WACCM simulations with this GW parameterization, and investigate the dependence of the zonal wind oscillation in the equatorial stratosphere on the spectral characteristics of the parameterized GWs The Spectral Width of the GW and the Amplitude of the Zonal Wind Oscillation [19] Figure 2a shows the daily equatorial zonal-mean zonal winds (averaged over 2.5 N to 2.5 S) between 100 hpa to 1 hpa for a 500-day, standard WACCM3.5 simulation with the imposed QBO turned off in the model. It is clear that the easterly winds are dominant above 100 hpa, and the semiannual oscillations (SAOs) appear above 3 hpa, and no QBO develops. [20] Now we set up the first case with the GW parameterization (equations (1) (4)) and choose the parameters for GWs as c 0 =30ms 1, dc =1ms 1, and = 1.0 (Case 1). [21] The equatorial zonal-mean zonal winds from this simulation are shown in Figure 2b. The most notable change is that the winds display oscillations between 70 hpa and 3 hpa with a downward phase progression. The SAO structure above 3 hpa is still present by examining the periodogram of zonal wind at 1 hpa, but has become more variable compared with Figure 2a. [22] The zonal-mean zonal wind at 10 hpa is shown in Figure 3a. It is seen from the plot that the wind oscillation is nearly symmetric (in positive v.s. negative magnitude, but not in duration. This will be discussed later.) with the Þ3 ð1þ ð2þ ð3þ ð4þ 3of14

4 Figure 2. Time-height cross section of the daily zonal-mean zonal winds averaged from 2.5 N to 2.5 S of a 500-day WACCM outputs, Winds are in units of ms 1 and are plotted in intervals of 10 ms 1. The solid lines show the westerly winds and the dotted lines show the easterly winds. (a) The standard WACCM with the imposed QBO winds turned off in the model. (b) Case 1 (c 0 =30ms 1, dc = 1ms 1, and = 1.0). maximum easterly speed near 45 ms 1 and westerly speed near 50 ms 1. From a Lomb-Scarge periodogram analysis (Figure 3b), the most prominent period of the wind oscillation is around 70 d, much shorter than a typical QBO period (800 d). [23] The result of this case demonstrates that the prescribed GWs do break and deposit their momentum in the stratosphere according to Lindzen s parameterization scheme. Although the breaking levels agree with the observed QBO, the disagreements with the observations are also obvious. Most importantly, the period of the wind oscillation is too short, the magnitude of the oscillation is too strong, and the positive and negative peaks of the oscillation are nearly equal in magnitude. The short period of the zonal wind oscillation indicates that the forcing, and thus the acceleration rate from the parameterization, are too strong. The large magnitude and the symmetry of the oscillation are likely associated closely with the width and the symmetry of the GW spectrum, respectively. In Case 2, we test the change of the zonal wind oscillation when the spectral width is reduced, and the corresponding parameters are selected as c 0 =20ms 1, dc = 1ms 1, and = of14

5 Figure 3. The equatorial zonal-mean zonal winds and their Lomb-Scarge periodograms near 10 hpa for (a and b) Case 1 (c 0 =30ms 1, dc =1ms 1, and = 1.0), (c and d) Case 2 (c 0 =20ms 1, dc =1ms 1, and = 1.0), (e and f) Case 3 (c 0 =20ms 1, dc =2ms 1, and = 1.0), (g and h) Case 4 (c 0 =20ms 1, dc =2ms 1, = 1.0, and Dc =5ms 1 ) and (i and j) Case 5 (same as Case 4, but with Coriolis effect). The red dashed lines in the equatorial zonal-mean zonal wind plots show the zero wind speed. The black dashed lines in Lomb-Scarge periodograms represent the significances for the confidence level of 95%, and the green dot-dash lines indicate the periods of 80 d and 160 d. [24] In this case, we reduce the cutoff phase speed c 0 of GW to 20 ms 1 and keep the other parameters the same as Case 1. The equatorial zonal-mean zonal wind near 10 hpa from 500 days of this simulation is shown in Figure 3c. Comparing with Case 1, one can find that both cases exhibit symmetric wind oscillations, but with different amplitudes and periodicity. The maximum amplitudes of the wind oscillation for this case become 35 ms 1, about 10 ms 1 smaller than Case 1. This is consistent with the reduction of the width of the GW source spectrum, because the acceleration rate from a breaking wave is proportional to the cube of its intrinsic phase speed (equation (4)). The decrease of the acceleration rate also increases the period of the wind oscillation, to 84 d near 10 hpa (this will also be discussed in next subsection), as shown in Figure 3d. The GW spectral width thus affects both the magnitude and the period of the zonal winds oscillation. [25] Since the spectral width 20 ms 1 of GWs gives amplitudes for the equatorial zonal-mean zonal winds in the stratosphere comparable to that of the QBO ( 35 ms 1 at 10 hpa, 30 ms 1 at 30 hpa, and 25 ms 1 at 50 hpa), we choose c 0 =20 ms 1 to be the cutoff phase speed of the GW spectrum for our simulations hereafter Total Momentum Flux of the GW and the Period of the Zonal Wind Oscillation [26] According to Lindzen and Holton [1968], the period of the wind oscillation should be inversely proportional to the total forcing, and thus the total momentum flux of the wave. In the parameterization, the total momentum flux is 5of14

6 determined by the spectral width and the spectral spacing dc. The total forcing is also controlled by the efficiency factor of the parameterized GWs in WACCM simulation. [27] Here we run another WACCM simulation, Case 3, with c 0 =20ms 1, dc =2ms 1, and = 1.0, in which we reduce the total momentum flux of the parameterized GWs to a half by increasing the spectral spacing dc from 1 ms 1 to 2ms 1 with other conditions unchanged. Again, the equatorial zonal-mean zonal wind near 10 hpa and its Lomb- Scarge periodogram from 500 days of the simulation is shown in Figures 3e and 3f, respectively. It is not surprising that the symmetry and the amplitude of the zonal wind are very similar to those in Case 2, but the oscillation period is now longer. From the periodogram, the dominant period is 153 d, about twice the length of that in Case 2, which is the result of the reduced total momentum flux carried by GWs. Equivalently (though computationally more expensive), the total number of the launched GWs is kept unchanged (that is dc =1ms 1 ) but the efficiency factor is reduced to 0.5 (results not shown here). [28] If we look back to the Case 2, the period of the wind oscillation is 84 d near 10 hpa, longer than that in Case 1. This also reflects the changing in the total momentum flux, that is, since the GW spectrum is flat, reducing the cutoff phase speed in this case by 1/3 (from 30 ms 1 to 20 ms 1 ) should reduce the total momentum flux to 2/3 of its original value in Case 1. It is more clear if we look into the acceleration of the zonal wind near 10 hpa, which is evaluated by dividing the total wind change over a zonal wind oscillation cycle by the period in each case: (1) Case 1: 160 ms 1 /70 d = 2.3 ms 1 day 1 ; (2) Case 2: 120 ms 1 /84 d = 1.4 ms 1 day 1. The zonal wind acceleration of 1.4 ms 1 day 1 in Case 2 is very close to 2/3 of that in Case 1 (2.3 ms 1 day 1 2/3 = 1.5 ms 1 day 1 ) The Symmetry of the GW Spectrum and the Symmetry of the Zonal Wind Oscillation [29] In previous cases, the simulated easterly and westerly winds are symmetric in amplitude. It is also seen that the average duration of the westerly wind is longer than the easterly wind. In the observed QBO, on the other hand, the easterly wind is stronger and its duration is usually longer (Figure 1). This discrepancy between the observations and the simulation suggests that the eastward forcing by the GW parameterization is too strong. One way to mitigate this problem in the model is to specify a zonally asymmetric GW spectrum that has stronger westward components. The observational evidence supporting this spectral asymmetry is that disturbances from many tropical mesoscale convective systems (especially over Africa) having the dispersion properties of Matsuno s equatorially-trapped IGWs with equivalent depths in the range m and mostly propagate westward at speeds of 18 ms 1 (S. Tulich, personal communication, 2011). The model study by S. Evan et al. (WRF simulations of convectively-generated gravity waves in opposite QBO phases, submitted to Journal of Geophysical Research, 2012) also found an east-west asymmetry with the vertical EP flux carried by the westward propagating gravity waves being larger than that carried the eastward propagating waves, which is considered as the effect of the wind above the convective source and the source motion. If we assume that convective systems are the main source of IGWs in the equatorial region, it is plausible to expect that the westward propagations of these systems lead to the westward shift of the source spectrum. [30] We now introduce an asymmetric phase speed spectrum of the GW in WACCM simulation Case 4, with c 0 = 20 ms 1, dc =2ms 1, = 1.0, and Dc =5ms 1 (westward shift), i.e., the specified phase speeds are shifted westward by Dc =5ms 1, so that the phase speed varying between c 0 Dc to c 0 Dc. [31] From Figure 3g, the wind now oscillates with stronger easterly amplitudes (maximum near 40 ms 1 ) and weaker westerly amplitudes (maximum near 25 ms 1 ), resulting from the enhanced westward forcing due to the westward shift of the GW spectrum. The corresponding periodogram is shown in Figure 3h. The period is 152 d, the same as that of Case 3 (153 d). Therefore, the shift of the GW spectrum only changes the symmetry and does not affect the period of the zonal wind oscillation. The asymmetric equatorial zonal-mean zonal winds simulated in this case are comparable with the QBO wind observations, and the shifted spectrum is used in later WACCM cases. [32] A potential downside of adding an asymmetric wave source is the possibility of introducing climate shift. We compared the global mean longwave flux and the net shortwave flux at the model top of the Cases 3 and 4, and found the differences are minimal. It should be noted that in this study our main focus is the dependence of the equatorial wind oscillation on various gravity wave parameters. 4. Coriolis Effect of the Parameterized IGWs on the Equatorial Stratospheric Wind Oscillation [33] In Section 3, the GWs (l h = 1000 km) carrying the momentum flux in general agreement with the observations (i.e., 10 3 Pa) can break in the stratospheric region and accelerate the mean flow, testifying our motivation as proposed in Section 1. However, these GWs with large horizontal wavelengths are most likely to have the inertial periods, i.e., the IGWs. Since the linear saturation formulation by Lindzen [1981] ignores the Coriolis effect, we need to develop a new scheme and extend the Lindzen s theory for the IGW with a period comparable to the inertial period. The details of the IGW parameterization are derived in the Appendix, here we only list the main differences compared with the conventional GW parameterization scheme as illustrated in Section 3. [34] The dispersion relation for the vertical propagating IGW with the Coriolis effect: m 2 ¼ N 2 ðc UÞ 2 ð5þ f 2 =k 2 where f =2Wsinf is the Coriolis parameter (where W is the Earth rotation rate and f is latitude). [35] Equation (5) shows that when (c U) 2 f 2 /k 2 =0, the vertical wavelength of the IGW will tend to zero (vertical wave number m ). Therefore the new critical level with the Coriolis term added in is: c ¼ U jf j=k ð6þ 6of14

7 [36] The momentum flux at the saturation level applying the new dispersion relation of the IGW: t ¼ kr 0 2N h i 1=2 ðc UÞ2 f 2 =k 2 ðc UÞ 2 [37] The acceleration rate within a saturated region determined from the divergence of the momentum flux is obtained: U t h ¼ 1 t k r 0 z ¼ ðc UÞ 2 f 2 =k 2 2NH i 1=2 ðc UÞ 2 It can be seen that the Lindzen formulation is recovered when f =0. [38] At the latitudes other than the equator, the critical level and the saturation level of a given wave will both be changed when the Coriolis effect is added in the parameterization scheme. The typical magnitude of f/k for the IGW with the wavelength l h =1000 km near the equator is about 2ms 1 at latitude 5, and 4 ms 1 at latitude 10. For a wind shear of 2ms 1 km 1, for example, the critical level will be vertically shifted by 1 km at latitude 5, and 2 km at latitude 10. [39] The change of the saturation level can be determined according to its definition equation [see also Lindzen, 1981, equation (13)]. In a background atmosphere with density r 0 ðþ¼r z 0 ðz 0 Þexpð ðz z 0 Þ=HÞand adiabatic lapse rate G, a wave with phase speed c, wave number (k, m), and momentum flux t 0 at the launch level will saturate when sffiffiffiffiffiffiffiffiffiffiffiffiffiffi Az ðþ¼jdt =dzj=g ¼ ð7þ ð8þ 2jmj Þ jt 0j 1=2 kr 0 ðz 0 Þ jc Uj e ðz z0þ=2h ¼ 1 ð9þ Without consideration of the Coriolis effect (substitute m with equation (1)), equation (9) has the same form as the equation (3.1) of Campbell and Shepherd [2005]. Because the vertical wave number m will become larger when the Coriolis effect is taken into account (assuming other conditions remain unchanged), the saturation level will decrease. [40] To understand the implications of the new parameterizations, we now performed WACCM simulations by parameterizing IGW forcing using the new parameterization (equation (8)) instead of the Lindzen scheme as used in previous section, and named it Case 5 (same as Case 4, but using equation (8)). [41] The daily equatorial zonal-mean zonal winds (averaged from 2.5 N to 2.5 S) between 100 hpa to 1 hpa again oscillate between the westerly and easterly winds. The wind near 10 hpa is shown in Figure 3i. As that in Case 4 (Figure 3g), the wind oscillates with stronger easterly amplitudes 40 ms 1 and weaker westerly amplitudes 25 ms 1. The corresponding dominant period in its periodogram shown in Figure 3j is 169 d, more than 10% longer than that of Case 4 (153 d). This is caused by the Coriolis effect included in the new parameterization. According to previous analysis, the critical level and the saturation level both drop to the lower altitudes with the introduction of the Coriolis term ( f /k) compared to that without. Thus, the higher atmospheric density at/above the new critical/ saturated layer reduces the acceleration rate of zonal winds (equation (8)) and leads to oscillations with longer period. [42] Now we inspect the latitude dependence of the Coriolis effect. Figure 4 shows the zonal-mean zonal wind at 10 hpa versus the latitudes and time for the standard WACCM run (Figure 4a) Case 4 (Figure 4b) and Case 5 (Figure 4c), respectively. In the standard WACCM run, the easterlies are dominant at 10 hpa between 20 Nto20 S all the time. For Case 4, the dominant easterlies in the equatorial region in standard WACCM run have been split by the periodic occurrence of westerlies. When the new parameterized IGW scheme is employed with the consideration of the Coriolis effect (Case 5), the main difference in the zonalmean zonal wind at 10 hpa is that the period of the east-west wind oscillation becomes longer (note that the weaker westerlies and stronger easterlies in Figures 4b and 4c are due to the westward IGW spectrum shift). The wind structures in the mid- to high latitudes (mainly the seasonal variations), on the other hand, do not change too much, especially in the southern hemisphere. It should be noted that the specified IGW source is uniform from 30 Nto30 S, so the latitude structure of the zonal wind oscillation is selfconsistently resolved by IGW and mean flow interactions in WACCM. 5. Generation of the Quasi-Biennial Oscillation in WACCM [43] With the knowledge gained from the above case studies, we proceed with numerical experiment of driving the QBO in WACCM using the new parameterization developed in section 4, i.e., Case 6, with c 0 =20ms 1, dc = 2ms 1, = 0.2 and Dc =5ms 1 (westward shift). [44] Here we first set the spectral spacing dc to 2 ms 1, and according to the discussion of the cases in section 3, the period of the wind oscillation will be only dependent on the efficiency factor used in the parameterization. Considering the enhancement of the period due to the introduction of Coriolis effect, we set to 0.2 in this case to achieve an oscillation period close to that of the QBO. Because the asymmetric spectrum yields wind oscillation with a strong easterly phase as shown in Case 4 and Case 5, the same spectrum is used in this case. [45] Figure 5 illustrates the monthly mean of the equatorial zonal-mean zonal winds between 100 hpa to 1 hpa from a ten-year simulation. The following features are most notable (1) an alternating pattern of easterly and westerly winds in the stratosphere with a period near 2 years; (2) the westerly winds are weaker, typically between 10 and 20 ms 1 from 40 hpa to 5 hpa, but the easterly winds are stronger and between 25 ms 1 to 35 ms 1 above 50 hpa; (3) the downward propagation of successive regimes at the average rates of 1.5 km and 1.2 km per month for the easterly shear and the westerly shear, respectively; (4) the amplitudes of zonal winds nearly constant in height between 40 hpa and 10 hpa, but decreasing rapidly below 40 hpa level. There are also features that differ from the observations. Compared to the monthly mean of the zonal-mean zonal winds from ERA-40 (Figure 1), the westerly winds of Case 6 decrease quickly below 40 hpa and there is no significant signal below 50 hpa. This is due to the reduction of the westerly forcing by the asymmetric wave spectrum, and 7of14

8 Figure 4. Time-latitude cross section of the zonal-mean zonal winds near 10 hpa for (a) standard WACCM run, (b) Case 3 (c 0 =20ms 1, dc =2ms 1, and = 1.0) and (c) Case 5 (c 0 =20ms 1, dc =2ms 1, = 1.0, and Dc =5ms 1 ). The thick black solid lines show the zero wind speed. the westerly below 40 hpa cannot be supported in the absence of the westerly forcing. At higher altitudes, the simulated QBO westerly phases extend to 2 3 hpa, higher than ERA-40, where the westerly phases starts typically at 5 hpa. [46] A more quantitative measure of the periodicity of the wind oscillation at 30 hpa using a wavelet analysis is shown in Figure 6. The period is centered near 24 months with the 99% confidence levels extending to 16 and 30 months. This indicates that, although the IGW sources are assumed to be constant in time, the model is able to produce an internal modulation of the QBO period. The period range of this case is consistent with the observations, but the dominant period of nearly 24 months is shorter than the typical QBO period of months. 8of14

9 Figure 5. Time-height cross section of the monthly mean zonal-mean zonal winds and the inertial gravity wave forcings averaged from 2.5 N to 2.5 S for ten year run of Case 6 (c 0 = 20 ms 1, dc =2ms 1, = 0.2, and Dc =5ms 1 ). Winds are plotted in intervals of 10 ms 1. Red and blue colors correspond to eastward and westward wave forcing, respectively Zonal Forcings in WACCM [47] Figure 5 also shows the acceleration rates due to the eastward and westward IGW forcings, which correspond well to the westerly wind shear and the easterly wind shear, respectively. The maximum eastward and westward wave forcings both occur around the zero wind line, signifying its role in reversing the wind. The westward forcing in plot is stronger above 10 hpa. Below that altitude, the westward forcing in Case 6 becomes stronger than the eastward forcing, and penetrates to lower altitudes, as noted in earlier discussions. Figure 7a compares the acceleration rates due to various forcing sources, i.e., the resolved waves (divergence of the Eliassen-Palm (E-P) flux (r F), red lines), the vertical advection of the mean flow ( w U z, with w being the vertical component of the residual mean circulation, green lines) and the parameterized IGWs (with k ¼ 2p 1000km, added WACCM in this study, blue lines) together with the zonal mean zonal wind (black dashed lines) and its vertical shear at 30 hpa for Case 6. We also calculate the parameterized mesoscale GWs forcing (generated from orography, deep 2p 100km convections and fronts, with k ¼, parameterized in the standard WACCM), since these waves usually break in the MLT, they are normally one order less in magnitude compared with the above forcings in the stratosphere. [48] From Figure 7a, the acceleration rates due to the resolved waves and the vertical advection are comparable though somewhat smaller compared with those of the IGWs. 2p Thus, the IGWs (k ¼ 1000km ) are the main contributor to the simulated QBO. At this altitude, the westward forcing by Figure 6. Wavelet spectrum of the equatorial zonal-mean zonal winds of Case 6 (c 0 =20ms 1, dc =2ms 1, = 0.2, and Dc =5ms 1 ). The black line indicates the 99% confidence level. The dashed line shows the cone of influence, out of which the edge effects become important. The dashdotted line shows the typical QBO period of 28 months. 9of14

10 Figure 7. (a) Wave forcing due to the resolved waves (red line), the vertical advection (green line), and IGWs (blue line), together with zonal-mean zonal wind (black dashed line) and its shear (black line) at 30 hpa for Case 6 (c 0 =20ms 1, dc =2ms 1, = 0.2, and Dc =5ms 1 ). (b) The upward E-P flux over the equatorial upper troposphere, the longest arrow represents the maximum upward E-P flux near kg s 2. IGWs is about ms 1 day 1 at peaks and the eastward forcing is 0.4 ms 1 day 1. The westward forcing by resolved waves, on the other hand, is rather insignificant, and the eastward forcing by these waves maximizes between ms 1 day 1. The relative contributions to the eastward forcing by IGWs and the resolved waves are in agreement with those derived from high-resolution GCM simulation by Kawatani et al. [2010]. Kawatani et al. [2010] also found that the equatorially trapped waves (EQW) contribute up to 10% and Rossby waves from winter hemisphere to 10% 25% (larger in the upper levels) of the total QBO forcing. The very weak westward forcing by resolved waves thus indicates that both EQW and Rossby waves from winter hemisphere are too weak in WACCM. This is probably responsible for the unrealistically long westerly and short easterly QBO phases in the middle stratosphere in WACCM. Furthermore, in Figure 7a one can find that the acceleration due to the vertical advection always acts to offset the strong westward acceleration by IGWs. This is followed by an eastward acceleration by resolved waves. The opposite, however, is not always true: the vertical advection offsets the eastward IGW acceleration sometimes (5 10 months and months in Figure 7a), but is either weak or even enhances the IGW acceleration at other times; and the resolved waves do not yield any significant westward forcing as mentioned earlier. The small westward forcing by resolved waves is probably also responsible for the differences of residual circulation, as well as the descending speed of the QBO phases between WACCM and reanalysis. For example, the average downward propagation of the QBO westerly (1.2 km/month) is slower than that of the QBO easterly (1.5 km/month) in WACCM, while ERA-40 shows a faster downward motion of the QBO westerly (1.3 km/month) than the QBO easterly (1.0 km/month). [49] As mentioned earlier, the resolved waves have an maximum eastward acceleration rates (rf > 0) about 0.2 ms 1 day 1, and the strong eastward forcing associated with the resolved waves synchronize with the westerly wind shear with the period also of 24 months. Figure 7b shows 10 of 14

11 Figure 8. The frequency-height cross section of the Fourier analysis for the equatorial zonal-mean zonal winds of (a) ERA-40 and (b) Case 6 (c 0 =20ms 1, dc =2ms 1, = 0.2, and Dc =5ms 1 ). Contours are drawn at Fourier amplitude intervals of 5 ms 1 and the confident levels larger than 95% are shaded with red color. the upward E-P flux at 5 levels near the tropopause region from 12 km to 18 km (150 hpa to 80 hpa). The flux shows a clear seasonal dependence, with the maximum upward wave activity appears in the Northern winter (below 14 km, there is another peak of the upward wave activity in summer). Moreover, each episode of strong eastward forcing by resolved waves (rf > 0) corresponds to an episode of maximum upward E-P flux (except for the second one, which has a relative small eastward forcing). The significant E-P flux divergence (rf > 0) requires both the upward wave momentum flux as well as the background westerly shear. Therefore, in this model the biennial period of the forcing by resolved waves seems to result from a combination of the annual variation of wave flux and modulation by the QBO of the wind shear Spatial and Temporal Structures [50] The frequency analyses of the equatorial zonal-mean zonal winds from ERA-40 and Case 6 (Figure 8) using a Lomb-Scarge spectrum illustrate the amplitudes of the QBO, the annual oscillation (AO), and the SAO in the stratosphere/ stratopause, in which, the confident levels larger than 95% are shaded with red color. The comparison shows that the period of the simulated QBO in WACCM is shorter than in ERA-40, and that the peak amplitude of the simulated QBO is smaller and extends to a higher altitude. We also identified a QBO spectrum peak at a higher level near the mesopause, but it is very weak and appears more like a QBO modulation of the stronger AO and SAO signals at this altitude. The AO and SAO shown in Figure 8b are qualitatively similar to ERA-40 results, though the AO in WACCM is stronger while the SAO is weaker. In addition to these large amplitude spectral components, there are other distinct frequency peaks (the confident levels are usually less than 60%) with a smaller amplitude in Figure 8b, which might be caused by the interactions between the main oscillation components. For example, the interaction between the QBO and the SAO might be responsible for a beat frequency of eight months as shown in Figure 8b in the model. [51] Figure 9 shows the latitude-height cross sections of the QBO amplitudes (u QBO ) for the ERA-40 (Figure 9a) and Case 6 (Figure 9b) in ms 1, respectively, which have been Figure 9. The latitude-height cross section of the QBO amplitudes obtained by a least squares fit of zonal-mean zonal winds at each latitude and altitude to equation (10). The contours are drawn at the intervals of 5 ms 1 and the red color regions illustrate the correlation coefficients are larger than 0.5. (a) The ERA-40 QBO amplitudes, and (b) the WACCM QBO amplitudes. 11 of 14

12 determined by a least squares fit of the zonal-mean zonal winds at each altitude and latitude to a sinusoid, using the following equation. Ut ðþ z;f ¼ u z;f 0 þ u z;f QBO cos 2p QBO þ u z;f AO cos t þ j z;f T QBO 2p t þ j z;f AO T AO þ u z;f SAO cos 2p t þ j z;f SAO T SAO ð10þ Here, z and f is the altitude and latitude. u i z,f, j i z,f, i =(QBO, AO, SAO) represent the amplitudes and phases of the QBO, AO and SAO at each altitude and latitude, respectively. T QBO, T AO and T SAO are the typical periods of the QBO, AO (12 months) and SAO (6 months). Note that the periods of the QBO in ERA-40 and Case 6 are different, for simplicity, we set T QBO = 28 months for the ERA-40 and T QBO = 24 months for Case 6. The contour regions with red color indicate the correlation coefficients are larger than 0.5. From Figure 9, the meridional structures of the QBO in the ERA-40 and Case 6 are approximately Gaussian and centered directly on the equator, with a width of about 20 latitude. As mentioned in previous section, although the specified IGW source is uniform from 30 Nto30 S, the latitude structure of the QBO is self-consistently resolved by IGW and mean flow interactions in WACCM. This is consistent with the fact, which has been pointed out by Haynes [1998], that the latitudinal width of the QBO is controlled not by the width of the momentum fluxes due to the IGWs, but by the internal dynamics of the meridional circulation itself. The differences between the QBO amplitudes in Figure 9a and 9b are (1) the QBO amplitudes in Case 6 are a little smaller than those in ERA-40; (2) there is a weak QBO amplitude peak appears in the southern polar upper stratosphere for Case Conclusion and Summary [52] The QBO is a dominant mode of variability in tropical stratosphere that is critical for studying climate variability. Based on linear saturation theory and straight forward scale analysis, we argue for the need of inertial gravity wave (IGW) in driving a stratospheric QBO. We have demonstrated that a reasonable equatorial stratospheric QBO is internally generated in WACCM3.5 using a new inertial gravity wave (IGW) parameterization scheme that extends Lindzen s linear saturation theory to include the Coriolis effect. [53] The simulated QBO in WACCM agrees with the observations in many respects: it is located in the stratospheric region over the equator with a latitudinal span of 20 ; and the easterly phase is stronger than the westerly phase, maximizes at 30 ms 1 and 15 ms 1, respectively, with the easterly phase extending to lower altitudes. The main differences of the QBO simulated in WACCM compared with that of the observation are: (1) The duration of the QBO westerly in WACCM are much longer than that of the QBO easterly in the middle and upper stratosphere; (2) The average downward propagation of the QBO westerly is slower than that of the QBO easterly; (3) The QBO westerly does not extend to the lower altitude and stops near 50 hpa. Detailed analysis shows that in the WACCM simulations the forcing of the QBO mainly comes from the parameterized IGWs. The westward IGW forcing is partially offset by the vertical advection of the mean flow, although the eastward IGW forcing is not always offset, and is sometime even enhanced, by the vertical advection. The eastward forcing by resolved waves is about 50% of the maximum eastward forcing by the parameterized IGWs, while the westward forcing by resolved waves is small. These are likely the direct causes for the first main difference mentioned above. [54] Further, to summarize the current work above, there are two main points we want to emphasize as follows The New Lindzen s Parameterized Scheme [55] In previous GCMs, the Hines or Warner & McIntyre Schemes have been widely used [e.g., Scaife et al., 2000; Giorgetta et al., 2002; Shibata and Deushi, 2005; Giorgetta et al., 2006; Scinocca et al., 2008; Naoe and Shibata, 2010]. Given the physical insights the linear saturation theory has lent to GCM studies, we have updated the traditional Lindzen s scheme to an IGW parameterization formulation including the Coriolis effect (see Appendix A), that can be applied more generally. Our numerical tests clearly demonstrate the dependence of the QBO-like oscillations on various wave parameters: (1) The QBO wind amplitudes are determined by the spectral width of the parameterized IGWs; (2) the periods of the simulated equatorial zonal-mean zonal wind oscillation are approximately linearly dependent on the total wave momentum flux, so that the period of the QBO can be conveniently controlled by adjustment of the total number or the efficiency factor (representing wave intermittency) of the IGWs (when the spectral width is determined); (3) the symmetry of the IGW spectrum controls the symmetry of the equatorial zonal-mean zonal winds; (4) with other conditions kept the same, the inclusion of the Coriolis effect in the parameterization decreases the altitude of the wave saturation/critical layers, decreases the acceleration rate due to the increasing atmosphere density, and increases the oscillation period of the equatorial zonal-mean zonal wind. The points (1), (2), and (4) are in line with the theory proposed by Lindzen and Holton [1968], and some of them have been confirmed by previous studies [e.g., Giorgetta et al., 2006; Scinocca et al., 2008] The Importance of the Parameterized IGWs in Generation of the QBO [56] With the increasing resolution, GCMs start to produce the QBO-like oscillations driven by resolved GWs, however, in the recent work of Kawatani et al. [2010], the period of the QBO-like oscillation is still shorter than 2 years. The fact that the resolved waves do not produce the correct QBO period may indicate problems with the wave excitation and/ or propagation in the model. Even in the current operational numerical weather prediction (NWP) models, which have sufficient spatial and temporal resolution to resolve much of the GW spectrum, Shutts and Vosper [2011] found that little GW energy flux by waves with horizontal wavelengths less than 200 km. On the other hand, the important role of the IGW is confirmed by Evan et al. (submitted manuscript, 2012). Using a channel version of the Weather Research and 12 of 14

13 Forecast (WRF) model with 37 km horizontal resolution, the authors found that the QBO forcing by the GWs with horizontal wavelength larger than 1000 km is much larger than or comparable to those by the waves with smaller horizontal wavelengths, depending on the QBO phase. GCMs with 1 2 deg horizontal grid and 2 km vertical resolution, however, could only resolve a portion (about 50%) of the inertial waves with horizontal scales larger than 1000 km. Thus the IGWs parameterization is necessary for GCMs with such commonly used resolutions. Overall, it is still a long-way to perform the realistic cloud resolving global climate simulations, and it is also impractical to conduct the long-term climate simulations with hyperfine spatial resolutions with the current computing resources, then a physical parameterization like the one studied here is still useful for climate simulations. Appendix A: IGW Parameterization With the Coriolis Effect [57] To consider the Coriolis effect, we start with the primitive equations [e.g., Andrews et al., 1987], linearized about a horizontally uniform basic state with background winds (U, V, 0), geopotential F, potential temperature q and density r 0 varying only in z. The disturbances of the winds (u, v, w ), geopotential F, potential temperature q and density r due to the gravity waves have the forms ðu ; v ; w ; F ; q ; r Þ ¼ e2h z ~u; ~v; ~w; ~F; ~ q; ~r expi½kxþ ly þ mz wtš These describe a monochromatic wave perturbation with wave number components (k, l, m) and ground-relative frequency w. Ignoring the background shear term of winds (U, V), one can get the gravity wave dispersion relation m 2 ¼ N 2 ðk 2 þ l 2 Þ ^w 2 f 2 1 4H 2 Where ^w ¼ w ku lv is the intrinsic frequency (the frequency that would be observed in a frame of reference moving with the background wind U, V), and f =2Wsinf is the Coriolis parameter (where W is the Earth rotation rate and f is latitude). We further assume that l = 0 (so the wave is propagating in the x direction, and ignore the rotation of the wave as it propagates upward) and m 1/2H, to simplify the dispersion relation to m 2 ¼ N 2 ðc UÞ 2 ða1þ f 2 =k 2 [58] Thus, when (c U) 2 f 2 /k 2 = 0, the vertical wavelength of the IGW will tend to zero (vertical wave number m ). This level is defined as the critical level, where a wave stops propagating and is assumed to deposit all the momentum in the background flow. Therefore the new critical level with the Coriolis effect added is reached when c = U f /k. [59] Below the critical level, a wave can also interact with the background flow when the wave becomes convectively unstable. The linearized first law of thermodynamics for the IGWs, under the assumption that T =T q =q and the ignoring the difference between log-pressure height and geographical height, may be expressed as ikðc UÞT þ w G 0 ða2þ Here G ¼ g=c p þ T= z. This equation is the same as the Equation (17) used by Holton [1982]. [60] The wave becomes convectively unstable when the total lapse rate reaches the dry adiabatic lapse rate. The wave amplitude is assumed to cease growing due to turbulence mixing when the wave becomes unstable (saturation assumption [Lindzen, 1981]), and this level is referred as the Saturation Level. Above the saturation level, the wave amplitude (and thus the momentum flux) is reduced to the extent that the wave remains statically stable. A given wave can have multiple breaking levels according to this theory, before it deposits all the momentum at a critical level, as noted by McFarlane [1987] and illustrated by Campbell and Shepherd [2005]. Following Lindzen [1981] and Holton [1982] dt =dz mt ¼ G ða3þ [61] Then the disturbed vertical wind will satisfy the following equation when the IGW is saturated by combining equations (12) and (13): ik c U w * ¼ ð Þ ða4þ m [62] At the same time, the disturbed zonal wind u can also be determined through the linearized equation by considering m 1/2H u * ¼ ic ð UÞ ða5þ [63] The momentum flux at saturation level can be derived from equations (14), (15) and (11) t ¼ r 0 u *w * ¼ kr 0 2N h i 1=2 ðc UÞ2 f 2 =k 2 ðc UÞ 2 ða6þ and the acceleration rate within a saturated region determined from the divergence of the momentum flux [Garcia et al., 2007] is obtained U t h ¼ 1 t k r 0 z ¼ ðc UÞ 2 f 2 =k 2 2NH i 1=2 ðc UÞ 2 ða7þ It can be seen that the Lindzen formulation is recovered when f =0. [64] Although the above equations are derived for the zonal wind direction (we set l = 0), it is straightforward to show the results hold true for any horizontal direction. This is because the primitive equations have ignored the zonal component for the Coriolis force for the vertical motion, and thus invariant under rotation around the vertical axis. In WACCM, the IGW sources are specified in the direction of the horizontal wind at the launch level. So k is the horizontal wave number, not necessarily zonal wave number. [65] Acknowledgments. We would like to thank ECMWF for providing the ERA-40 data set. The WACCM simulations have been carried out at the National Center for Atmospheric Research (NCAR). The authors thank Rolando Garcia and Jadwiga Richter for helpful discussions. The authors at the University of Science and Technology of China are supported 13 of 14