To determine the physical mechanisms, we took a two-step approach. Step 1 is a tendency analysis involving the continuity equation for a tracer given by (Brasseur and Solomon, 2006) 2∂μ∂t+uacos⁡ϕ∂μ∂λ+va∂μ∂ϕ+w∂μ∂z-1ρ0∂∂zρ0Kzz∂μ∂z-1ρ0∂∂zρ0Dμ∂μ∂z+1ρ0∂∂zρ0μwD=P-L, where μ is CO volume mixing ratio, t is time, ϕ is latitude, λ is longitude and z is geopotential height. The variables u, v and w are the neutral zonal, meridional and vertical winds, respectively; Kzz is the eddy diffusion coefficient; Dμ is the molecular diffusion coefficient and wD is its corresponding diffusive separation velocity; P is the chemical production rate; L is the chemical loss rate; ρ0 is the atmospheric neutral density; and a is the radius of the Earth which is 6.37×106 m. Note that the eddy diffusion coefficient is calculated from a gravity wave parameterization (Richter et al., 2010; Garcia et al., 2017). The molecular diffusion coefficient is as defined in Smith et al. (2011). The DW1 component of each term is calculated by fitting the terms into the equation Xλ,t=X‾+X^n,scos⁡(πt/24-(-1)λ-ψ^n,s) using a 2D least-squares fit. This determines the contributions of zonal advection, meridional advection, vertical advection, eddy diffusion, molecular diffusion and photochemical production to SD-WACCM-X CO μ′'s DW1 component. Comparing these terms will determine the main processes behind SD-WACCM-X CO μ′. Salinas et al. (2020) recently used this to determine the mechanisms of lower thermospheric carbon dioxide's (CO2) local-time variations. The result of this analysis gives us SD-WACCM-X's suggested mechanisms for μ′. This first method is clearly only applicable with model outputs because some of the required parameters cannot currently be observed.

Figures 4a, b and B1 show the results of our tendency analysis for March equinox season. Figure 4a shows the DW1 amplitude of the time-derivative term in the continuity equation. It can be characterized by a primary equatorial peak of around 30 ppm d-1 between 85 and 92 km as well as secondary mid-latitude peaks of around 12 ppm d-1 between similar altitudes. This latitude structure is consistent with that of the (1,1) mode. Figure 4b shows the DW1 amplitude of the vertical advection term in the continuity equation. It exhibits a similar magnitude and latitude–altitude profile to Fig. 4a. Figure B1 shows the DW1 amplitudes of the other terms in the continuity equation. These figures show that in SD-WACCM-X, vertical advection in March equinox has the closest magnitude and latitude–altitude structure to the time-derivative term. Chemical production has a higher magnitude than vertical advection but because, as mentioned earlier, the chemical production timescale is slower than dynamical timescales, its latitude–altitude structure is not similar to the time-derivative term.

Now, we need to determine if the same mechanism holds for the observations. This is step 2. A tendency analysis is still currently impossible with observations because we do not have observations of all the needed parameters. However, we can use a method called the adiabatic displacement method to quantify the contributions of vertical advection on CO μ′ as observed by MLS and as simulated by SD-WACCM-X. The adiabatic displacement method involves calculating the perturbation on a tracer μw′ due to any tide-induced vertical advection using the following equation: 3μw′=T′S∂μ‾∂z. T′ is the tidal perturbation of temperature. S=∂T∂z+gcp is static stability. g is the acceleration due to gravity (9.8 m s-2). cp is the heat capacity of air at constant pressure (1004 JK-1 kg). ∂μ‾∂z is the vertical gradient of a tracer's zonal-mean profile. This equation is derived by first linearizing the continuity equation (Eq. 2). Then, we assume only the vertical advection term is important. Finally, we set all primed variables into the form ei(kx-σt), where k is the zonal wave number and σ is the tidal frequency. This gives us the following equation: 4iku‾-σμw′+∂μ‾∂zw′=0. The same can be done to a form of the thermodynamic equation that assumes all temperature changes are due to adiabatic motion. This gives us the following equation: 5iku‾-σT′+Sw′=0 Combining Eqs. (4) and (5) gives Eq. (3).

Comparing CO μ′ and CO μw′ will determine how much of CO μ′ is driven by vertical advection. If CO μ′ and CO μw′ are similar, then we can argue that vertical advection does primarily drive CO μ′. Applying this method on the model outputs will assess the consistency of this method and the previous method with regards the role of vertical advection in μ′. Forms of this method have already been applied on the analysis of the tidal or local-time variations of other tracers (Akmaev et al., 1980; Angelats I Coll and Forbes, 1998; Marsh et al., 1999; Shepherd et al., 1995, 1997; Ward, 1999; Zhang et al., 1998; Marsh and Russell, 2000; Oberheide and Forbes, 2008; Smith et al., 2010; Marsh et al., 2011; Salinas et al., 2022)

Apart from proving that vertical advection primarily drives CO μ′, Eq. (3) will also explain why the latitude and altitude structure of a tracer's DW1 component and temperature's DW1 component may be correlated. Equation (3) indicates that if vertical advection does primarily drive a tracer's DW1 component, and since Fig. 1 has shown that zonal-mean CO's vertical gradient is positive, CO μ′ and T′ are correlated. This also indicates that an increase in μ' requires T′>0, which under adiabatic conditions implies a net downwelling. Conversely, a decrease in μ′ implies T<0′ and net adiabatic upwelling.

Figure 4c shows the differences between SD-WACCM-X CO μ′ and SD-WACCM-X CO μw′. The differences are within ±0.6 ppm. Peak positive difference of around +0.6 ppm is found over the Equator, while peak negative difference of around -0.6 ppm is found over the mid-latitudes. These differences are an order of magnitude lower from the correct values which indicates that SD-WACCM-X CO μw′ is very similar to SD-WACCM-X CO μ′. Thus, Fig. 4a, b and c indicate that in the model, CO μ′ and CO μw′ are very similar in March equinox. This explains the positive correlation between the latitude structure of SD-WACCM-X CO μ′ and SD-WACCM-X T′ in March equinox. We now show MLS CO μ′-μw′ in Fig. 4d. Figure 4d shows that the largest differences are above 90 km. Below 90 km, the average difference is around -0.5 ppm. This also indicates good similarity in March equinox between MLS CO μ′ and MLS CO μw′. This explains the positive correlation between the latitude structure of MLS CO μ′ and MLS T′ in March equinox. For both MLS CO μ′ and SD-WACCM-X CO μ′, Fig. 4c and d indicate that the positive perturbations are driven by a relative downwelling due to the DW1 tide, while the negative perturbations are driven by a relative upwelling.

We now determine the physical mechanisms behind CO μ′ in June solstice. We first show the results of the tendency analysis in SD-WACCM-X. Figure 5a shows the DW1 amplitude of the time-derivative term in the continuity equation. It can be characterized by a primary low-latitude peak of around 8–12 ppm d-1 between 85 and 95 km. Amplitudes are larger over the northern low-latitude than over the southern low-latitude. Mid-latitude peaks are also present, but the northern mid-latitude peak of around 7 ppm d-1 is larger than the southern mid-latitude peak of around 3 ppm d-1. Figure 5b shows the DW1 amplitude of the vertical advection term in the continuity equation. It shows a similar magnitude and latitude–altitude profile to Fig. 5a. Figure B2 shows the DW1 amplitudes of the other terms in the continuity equation. These figures show that in SD-WACCM-X, vertical advection in June solstice has the closest magnitude and latitude–altitude structure to the time-derivative term. Like March equinox, the chemical production also has a higher magnitude than vertical advection, but its latitude–altitude structure also is not similar with the time-derivative term.

We now also use the adiabatic displacement method to quantify how much CO changes due to DW1-induced vertical advection in June solstice. Figure 5c shows the differences between SD-WACCM-X CO μ′ and SD-WACCM-X CO μw′ for June solstice. The differences are within ±0.5 ppm. Peak positive difference of around +0.5 ppm is found over the Equator above 90 km, while peak negative difference of around -0.4 ppm is found over the northern mid-latitudes. These differences are also an order of magnitude lower than the correct values, which indicates that SD-WACCM-X CO μw′ is very similar to SD-WACCM-X CO μ′. Thus, Fig. 5a, b and c indicate that in the model CO μ′ and CO μw′ are remarkably similar in June solstice. These figures indicate that for SD-WACCM-X in June solstice the positive perturbations are driven by a relative downwelling due to the DW1 tide, while the negative perturbations are driven by a relative upwelling.

We now show June solstice MLS CO μ′-μw′ in Fig. 5d. Figure 5d shows that the largest differences between latitudes 30∘ S and 60∘ N are above 90 km. Below 90 km, the average difference is around -0.5 ppm. This also indicates good similarity between MLS CO μ′ and MLS CO μw′ in June solstice for regions less than 90 km between latitudes 30∘ S and 60∘ N. On the other hand, between latitudes 30 and 60∘ S, the largest differences are found between 80 and 92 km. This also indicates that, unlike in March equinox, MLS CO μ′ and MLS CO μw′ are not similar throughout all latitudes in June solstice. They are only similar between latitudes 30∘ S and 60∘ N. These figures indicate that for MLS in June solstice not all positive perturbations are driven by a relative downwelling, and not all negative perturbations are driven by a relative upwelling.

This subsection will determine the seasonal and interannual variability of the (1,1) mode of CO. To explain it, we will first compare the (1,1) mode of CO with the (1,1) mode of temperature (hT′). Then, we will determine the role of vertical advection by projecting the latitude profiles at each altitude of CO μw′ (presented in Sect. 5) with the (1,1) Hough function profile. The corresponding projection coefficients will be denoted as CO hw′.

Figure 6 first shows the seasonality of observed and modeled CO hμ′ and hT′. Figure 6a shows the (1,1) mode. Figure 6b shows the seasonality of MLS CO hμ′. The largest coefficients are above 80 km, and they are all positive projection coefficients, indicating that there is a positive correlation between a latitude profile of MLS CO μ′ and the (1,1) Hough mode between 80 and 92 km. This is consistent with the features shown in Fig. 2a and c. Between 80 and 90 km, the seasonality is characterized by a semi-annual oscillation with primary peak of around 4 ppm in March equinox and secondary peak of around 3 ppm in September equinox. Above 90 km, it appears as though the primary peak is moving towards June solstice. Figure 6c shows the seasonality of SD-WACCM-X CO hμ′. The largest coefficients are above 85 km, and they are all also positive projection coefficients. Between 85 and 95 km, the seasonality is also characterized by a semi-annual oscillation with primary peak of around 6 ppm in March equinox and secondary peak of around 5 ppm in September equinox. Figure 6b and c shows that SD-WACCM-X does capture the observed semi-annual oscillation of the projection coefficients. However, the altitudinal variations are different from observed. This is consistent with the differences in MLS CO μ′ and SD-WACCM-X CO μ′. The coefficients of SD-WACCM-X CO hμ′ are also higher than MLS.

Figure 6d shows the seasonality of MLS hT′. Focusing on the coefficients above 80 km, we find that, like MLS CO, they are all positive projection coefficients, indicating that there is a positive correlation between MLS T′ and the (1,1) Hough mode between 80 and 92 km. Between 80 and 90 km, it also has a semi-annual oscillation with primary peak coefficients of around 25 K during March equinox and secondary peak coefficients of around 20 K during September equinox. Above 90 km, the seasonality appears to shift into an annual oscillation with peak amplitudes in June solstice. Figure 6e shows the seasonality of SABER hT′. Between 80 and 95 km, it also has a semi-annual oscillation with primary peak of around 20 K during March equinox and secondary peak of around 15 K during September equinox. Unlike MLS hT′, its seasonality does not seem to change above 90 km. Figure 6f shows the seasonality of SD-WACCM-X hT′. Between 80 and 95 km, it also has a semi-annual oscillation with primary peak of around 27 K during March equinox and secondary peak of around 20 K during September equinox. Unlike MLS hT′, its seasonality does not also seem to change above 90 km. Also, SD-WACCM-X hT′ is consistently larger than SABER hT′, but its coefficients are not too different from MLS hT′.

Figure 6b and d showed that the seasonality of MLS CO hμ′ and MLS hT′ is similar. However, MLS hT′ and SABER hT′ in Fig. 6e have differences. The seasonality of MLS hT′ changes slightly above 90 km, while the seasonality of SABER hT′ does not. Figure 6b, d and e shows that the seasonality of MLS CO hμ′ may be affected by the incomplete local-time sampling of MLS or its coarse vertical resolution. This would be consistent with the results shown in Figs. 2 and 3. Figures 2 and 3 showed that MLS CO μ′ and MLS T′ may be affected by inadequate sampling over the mid-latitudes.

Figure 6b and c showed that MLS CO hμ′ is stronger than SD-WACCM-X CO hμ′. Figure 6e and f also showed that SABER hT′ is stronger than SD-WACCM-X hT′. A larger MLS CO hμ′ than simulated is consistent with a larger realistic MLS or SABER hT′ than simulated. An underestimation of SD-WACCM-X hT′ indicates inaccuracies in the simulated background atmosphere, tidal source or tidal dissipation mechanisms.

Apart from these differences amongst the datasets, Fig. 6b and d as well as Fig. 6c and f shows that CO hμ′ and hT′ in MLS observations and SD-WACCM-X simulations have similarities in their seasonality. We now check whether this similarity is also found for all months and all years of MLS observations and SD-WACCM-X simulations. We will also check if CO hμ′ is primarily driven by vertical advection across all months and years of observations and simulations. In this subsection, we determine the importance of vertical advection by projecting the latitude profiles at each altitude of the previously calculated CO μw′ with the (1,1) Hough function profile. The corresponding projection coefficients will be denoted as CO hw′.

Figure 7a shows the MLS CO hμ′ from 2004 until 2020. Figure 7b shows MLS CO hT′ from 2004 until 2020. Their overall morphology is similar; that is, for all years, there is a semi-annual oscillation with primary peak in March equinox and secondary peak in September equinox between 80 km and 90 km. Above 90 km, their seasonality shifts into having a primary peak close to June solstice. This could suggest that the latitude structure of DW1's phase during solstice (equinox) causes maximum (minimum) values when taking the difference of values at ∼ 02:00 LT and at ∼ 14:00 LT. This consequently enhances (reduces) MLS CO hμ′. This is difficult to validate, with a very high degree of uncertainty even with SABER data, because of the differences in MLS and SABER's vertical resolution. Figure 7c shows the MLS CO hw′. Comparing Fig. 7a and c shows that most of the seasonal features are indeed captured. Figure 7d shows a scatter plot between CO hμ′ and CO hw′ at ∼90 km, which is the approximate altitude where these MLS parameters attain peak amplitudes. It shows a correlation of 0.97, indicating that the variations of hμ′ and hw′ are very similar.

Figure 8a shows SD-WACCM-X CO hμ′ from 2004 until 2020. Figure 8b shows SD-WACCM-X hT′ from 2004 until 2020. Their overall morphology is also similar; that is, for all years, there is a semi-annual oscillation with primary peak in March equinox and secondary peak in September equinox. We now calculate SD-WACCM-X CO hw′. Figure 8c shows the SD-WACCM-X CO hw′. Comparing Fig. 8a and c shows that most of the features are indeed captured. Figure 8d shows a scatter plot between SD-WACCM-X CO hμ′ and SD-WACCM-X CO hw′ at ∼92 km, which is the approximate altitude where these SD-WACCM-X parameters reach peak amplitudes. It shows a correlation of 0.996, indicating that the variations of CO hμ′ and CO hw′ are very similar. This correlation is higher than MLS CO hμ′. We suggest that this may be due to the aliasing over the mid-latitudes in MLS CO μ′ Previous studies using this adiabatic displacement method only involved the analysis of tracer observations at one instance in time (Akmaev et al., 1980; Angelats I Coll and Forbes, 1998; Marsh et al., 1999; Shepherd et al., 1995, 1997; Ward, 1999; Zhang et al., 1998; Marsh and Russell, 2000; Oberheide and Forbes, 2008; Smith et al., 2010; Marsh et al., 2011; Salinas et al., 2020, 2022). Our work adds to these studies by presenting an approach involving the adiabatic displacement method that involves proving the importance of vertical advection at both seasonal and interannual timescales.

Figures 7 and 8 indicate that, across interannual timescales, both observed and modeled CO hμ′ and CO hw′ are highly correlated. In this subsection, we identify the interannual phenomena in CO hμ′ that can apparently also be seen in CO hw′ because they are both highly correlated. We focus on interannual phenomena that are known to already affect the (1,1) mode in temperature: the quasi-biennial oscillation and the El Niño–Southern Oscillation (Lieberman, 1997; Vincent et al., 1998; McLandress, 2002a, b; Gurubaran et al., 2005, 2009; Mayr and Mengel, 2005; Liebermann et al., 2007; Wu et al., 2008; Mukhtarov et al., 2009; Pancheva et al., 2009; Xu et al., 2009; Pedatella et al., 2012, 2013; Gan et al., 2014; Liu et al., 2017; Zhou et al., 2018; Kogure et al., 2021; Pramitha et al., 2021; Cen et al., 2022). We use a cross-wavelet analysis to determine the dominant oscillations in MLS CO hμ′ or SD-WACCM-X CO hμ′ that coincide with the 30 mb QBO index and multi-variate El Niño–Southern Oscillation index (MEI).

Figure 9 identifies interannual phenomena found in CO hμ′. Figure 9a shows the time-series of MLS CO hμ′ at ∼90 km, SD-WACCM-X CO hμ′ at ∼92 km and the QBO index. Figure 9b shows the time-series of MLS CO hμ′ at ∼90 km, SD-WACCM-X CO hμ′ at ∼92 km and the MEI index. Figure 9c shows the cross-wavelet spectrum between MLS CO hμ′ and the QBO index. In this and the succeeding spectra, encircled regions with the high spectral power correspond to oscillations statistically significant in both time-series (Grinsted et al., 2004). The arrows indicate the phase relationship between the time-series. If the arrow points right, both time-series are in phase. If the arrow points left, both time-series are anti-phase. If the arrow points upward or downward, there is a lag between the time-series. In our case, an upward arrow indicates that both time-series are in phase but the QBO or ENSO index time-series peaks later than the other time-series. A downward arrow indicates that both time-series are also in phase but the QBO or ENSO index time-series peaks ahead of the other time-series. Depending on the arrows, one can deduce the correlations between CO hμ′ amplitude and QBO or ENSO. Consequently, the deduced correlation will imply whether CO hμ′ increases or decreases during, for example, westerly QBO phase.

Figure 9c reveals that both MLS CO hμ′and the QBO index have a statistically significant oscillation with periods of around 24 months. It is statistically significant between 2005 and 2018. The arrows are pointed slightly upward which indicates that MLS CO hμ′ peak slightly ahead of the QBO index. The arrows also indicate that MLS CO hμ′ amplitude increases during the westerly phase of the QBO while it decreases during the easterly phase of the QBO. Remarkably similar features are found in the cross-spectrum between MLS hT′ and the QBO index (Fig. C1c). The QBO's impacts on the (1,1) mode of temperature is well known (Lieberman, 1997; Vincent et al., 1998; Mayr and Mengel, 2005; Wu et al., 2008; Gurubaran et al., 2009; Mukhtarov et al., 2009; Pancheva et al., 2009; Xu et al., 2009; Gan et al., 2014; Pramitha et al., 2021). These studies have shown that the (1,1) mode is enhanced during the westerly phase of the QBO while it is reduced during the easterly phase of the QBO. Our work adds to these previous studies by showing that MLS CO's (1,1) mode is also affected by the QBO in the same way.

Figure 9d shows the cross-wavelet spectrum between MLS CO hμ′and the MEI index. This spectrum reveals that both MLS CO hμ′ and the MEI index have a statistically significant oscillation of around 30 months between 2008 and 2012. This does coincide with the strong 2010–2011 La Niña event. The arrows are pointed almost fully to the left, which indicates that MLS CO hμ′ and ENSO are anti-correlated during this event; that is, MLS CO hμ′ amplitude increased during this La Niña event. The anti-correlation also indicates that, if we were to solely use this event as a basis, MLS CO hμ′ should decrease during El Niño events. Very similar features are found in the cross-spectrum between MLS hT′ and the ENSO index (Fig. C1d). ENSO's impacts on the (1,1) mode of tide is well explored (Gurubaran et al., 2005; Liebermann et al., 2007; Pedatella et al., 2012, 2013; Liu et al., 2017; Zhou et al., 2018; Kogure et al., 2021; Cen et al., 2022). These studies have shown that the general response is that (1,1) mode is reduced during El Niño, although the reduction is modulated by secondary mechanisms like gravity waves (Cen et al., 2022).

Figure 9e shows the cross-wavelet spectrum between SD-WACCM-X CO hμ′ and the QBO index. This spectrum reveals that both SD-WACCM-X CO hμ′ and the QBO index have a statistically significant oscillation with periods ranging from 20 to 30 months, and the statistical significance is found throughout all years. From 2015 to 2020, both have a statistically significant oscillation with a period of around 15 to 20 months. Very similar features are found in the cross-spectrum between SD-WACCM-X hT′ and the QBO index (Fig. C1e). Like MLS CO hμ′, the arrows are pointed slightly upward, which indicates that SD-WACCM-X CO hμ′ peaks slightly ahead of the QBO index and that SD-WACCM-X CO hμ′ amplitude increases during the westerly phase of the QBO while it decreases during the easterly phase of the QBO. The differences between Fig. 9c and e could suggest that the model may be overestimating the impacts of the QBO during certain periods.

Figure 9f shows the cross-wavelet spectrum between SD-WACCM-X CO hμ′ and the MEI index. This spectrum reveals that both have a statistically significant oscillation with a period of around 24 to 36 months. This period of statistical significance lasts from 2006 to 2016. Unlike MLS CO hμ′, the statistical significance for SD-WACCM-X CO hμ′ includes both the 2010–2011 La Niña and the 2015–2016 El Niño event. The differences between Fig. 9d and f could suggest that the model may be overestimating the impacts of ENSO during certain periods. Very similar features are found in the cross-spectrum between SD-WACCM-X hT′ and the ENSO index (Fig. C1f). Like MLS hμ′, the arrows are pointed almost fully to the left between 2006 to 2013, which indicate that, for the 2010–2011 La Niña period, SD-WACCM-X CO hμ′ also increased. However, between 2013 and 2016, the arrows are pointed almost downward, which indicates that, for the 2015–2016 El Niño period, SD-WACCM-X CO hμ′ increased. In addition, the arrows also indicate that ENSO peaks ahead of SD-WACCM-X CO hμ′. Most studies have found that the (1,1) mode should decrease during El Niño events. However, our results indicate that the effect of ENSO reversed during the 2015 El Niño. Kogure et al. (2021) has explained this. Their work showed that the enhanced (1,1) tide in 2015 was a result of the overlapping occurrence of an easterly QBO phase and an El Niño event. Lieberman et al. (2007) also showed that the (1,1) mode increased during ENSO events because the climatological dry tongue disappears during the El Niño phase, leading to a more longitudinally uniform water vapor distribution and therefore a stronger (1,1) forcing by water vapor heating. Our work adds to these previous studies by showing that MLS CO's (1,1) mode is also affected by ENSO in the same way.

The previous subsection found that QBO and ENSO variabilities are present in both MLS CO hμ′ and SD-WACCM-X CO hμ′. In this section, we quantify the changes in MLS CO hμ′ or SD-WACCM-X CO hμ′ due to QBO. We do not quantify the changes due to ENSO because there were only a few events during our data span. Hence, any estimated response may be biased. We use multiple linear projection (MLR) to estimate the response of MLS CO hμ′ or SD-WACCM-X CO hμ′ to the QBO. Finally, we use the same MLR to reconstruct the time-series with the QBO index and then compare this reconstruction with a low-pass filtered version of the time-series. Note that while an MLR analysis offers an estimate of the response of one parameter to another, the reconstructed time-series using these MLR coefficients constrains the fluctuations to either be in-phase or completely anti-phase of the other parameter. It also assumes that the amplitude fluctuations are the same as that of the other time-series. For example, in the case of MLS CO hμ′ and the QBO index, the MLR reconstruction of MLS CO hμ′ can only be a time-series that is either completely in-phase with the QBO index or completely anti-phase. The overall fluctuation of the reconstruction will also only be a multiple of the QBO index time-series. With the low-pass filtered time-series, we can reconstruct the time-series that accounts for non-in-phase or non-anti-degree phase differences. The low-pass filtered time-series also accounts for the exact amplitude fluctuations as a function of time. Thus, in the case of MLS CO hμ′ and the QBO index, the low-pass filtered MLS CO hμ′ will reveal how dominant QBO periodicities are.

Figure 10 shows our MLR analysis and our low-pass filtering of MLS CO hμ′ and SD-WACCM-X CO hμ′ to quantify the responses of these parameters to QBO and ENSO. Figure 9 showed that the phase-relation between MLS CO hμ′ or SD-WACCM-X CO hμ′ and the QBO index were consistent for all years. However, the phase-relation between MLS CO hμ′ or SD-WACCM-X CO hμ′ and the QBO index changed. In this figure, we separate this analysis. Figure 10a and b focuses on the QBO response, while Fig. 10c to f focuses on the ENSO response.

Figure 10a shows the de-seasonalized MLS CO hμ′ from 2004 until 2020, reconstructed using an MLR (hereafter MLR recon MLS CO hμ′) and filtered with a fifth order low-pass filter with a cut-off period of 6 months (hereafter low-pass filtered MLS CO hμ′). The equation within the plot shows the MLR coefficients between the de-seasonalized MLS CO hμ′ and the QBO and ENSO indices. However, for this plot, we will ignore the coefficients for ENSO because, as mentioned above and as will be shown later, the phase relationship changes. Both reconstructions are overplotted with the QBO index. Figure 10b shows the same as Fig. 10a but for SD-WACCM-X CO hμ′. Figure 10a shows a QBO MLR coefficient of 0.014 for MLS CO hμ′. The MLR fit has p value = 0.002, which is less than 0.05 indicating statistical significance. With this value, MLS CO hμ′ increases by around 0.21 ppm or a +8 % variation (1000.212.49=8 % variation where 2.49 is the temporal mean included in the MLR equation) when the QBO index is at the typical peak westerly value of around +15 m s-1. On the other hand, MLS CO hμ′ decreases by around -0.42 ppm or a -16 % variation when the QBO index is at the typical peak easterly value of around -30 m s-1. Figure 10a also shows that the low-pass filtered MLS CO hμ′ is dominated by low-frequency periodicities, whose combination yields a time-series that is dominated by QBO-like fluctuations. The time-series looks very similar to MLR recon MLS CO hμ′ and the QBO index. Using the standard deviation of the low-pass filtered MLS CO hμ′ as a measure of the variation, we calculate a variation of ∼0.9 ppm which is slightly higher than the 0.63 ppm variation estimated with the MLR coefficients. Comparing the low pass filtered MLS CO hμ′ and the QBO index further shows that peak westerly phase values in the QBO index occur just after the local maximum values. This is consistent with the cross-wavelet spectrum arrows in Fig. 9c. Figure 10b shows a QBO MLR coefficient of 0.029 for SD-WACCM-X CO hμ′. The MLR fit has p value less than 0.0001, indicating statistical significance (p value <0.05). With this value, SD-WACCM-X CO hμ′ increases by around 0.44 ppm or a +10 % variation when the QBO index is at the typical peak westerly value. SD-WACCM-X CO hμ′ decreases by around 0.88 ppm or a -20 % variation when the QBO index is at the typical peak easterly value.

Figure 10b also shows that the low-pass filtered SD-WACCM-X CO hμ′ is also dominated by QBO-like fluctuations that are like the MLR recon SD-WACCM-X CO hμ′ but slightly larger in variance at around 1.52 ppm and corrected for the phase differences determined by the cross-wavelet spectrum arrows in Fig. 9d.