POP Modes from Model Data

Figures 6-2 and 6-3 show the real and imaginary parts of the V and W> matrices for the first two POP modes (filled contours). The bold line separates positive from negative patterns, while the sign of the patterns is arbitrary, but consistent between V and the corresponding W> matrix and between the POP pattern and its corre- sponding PC time series. The white contours indicate the climatological zonal mean zonal wind. The V modes in Figure 6-2 show a dominantly stratospheric pattern, describing a latitudinal wobbling of the polar vortex around its climatological mean position. In the troposphere, the dominant pattern is a latitudinal wobbling of the tropospheric jet around its climatological mean position, but with a much weaker signature. According to these patterns, a poleward shift of the polar vortex is accom- panied by an equatorward shift of the tropospheric jet and vice versa. The W>modes show a similar variability, but with a dominant tropospheric pattern. The imaginary parts of these patterns (Figure 6-3) exhibit a similar stratospheric variability, but the opposite connection to the troposphere. The remainder of this chapter will focus on the V patterns, since the W> patterns turned out to be not as robust to lag and retained variance. This will have an impact on the ability of the patterns to provide the modes for a forcing to project on, as will be discussed in the last section of this chapter. However, since only the internal variability of the model data is consid- ered and analyzed here, this limitation will not affect the results obtained by the V patterns.

V1 real, decorr=63, lag=40, unweighted, notopo −75 −60 −45 −30 3 10 30 100 300 900 W1 real, decorr=63 −75 −60 −45 −30 3 10 30 100 300 900

V2 real, decorr=63, lag=40, unweighted, notopo

−75 −60 −45 −30 3 10 30 100 300 900 W2 real, decorr=63 −75 −60 −45 −30 3 10 30 100 300 900

Figure 6-2: Filled contours: Real part of the POP patterns V (top) and W>(bottom) for the first (left) and second (right) mode. White contours: Zonal mean zonal wind averaged over the entire run, with a contour interval of 10 ms−1. The zero-wind line is bolded, negative contours are dashed.

V1 imag, decorr=63, lag=40, unweighted, notopo −75 −60 −45 −30 3 10 30 100 300 900 W1 imag, decorr=63 −75 −60 −45 −30 3 10 30 100 300 900

V2 imag, decorr=63, lag=40, unweighted, notopo

−75 −60 −45 −30 3 10 30 100 300 900 W2 imag, decorr=63 −75 −60 −45 −30 3 10 30 100 300 900

Since the patterns are complex, they exhibit an oscillating structure between the real and the imaginary patterns. Verifying the response in the oscillation of the real and imaginary parts of the principal component time series, it can be shown that the combination at times exhibits an oscillatory pattern with periods on the order of a hundred days, i.e. in the general form of V(q, n) cos(t) +V(q, n) sin(t), corresponding to a progression as described in Section 6.2.2. At other times the time series stays in one phase for a while and intermittently oscillates back and forth between two dominant patterns. The phases of oscillatory motion correspond to times of stronger wind variability in the wind time series. Figure 6-4 shows an example of an oscillatory pattern corresponding to a 90 day excerpt of the control run. All observed oscillatory patterns exhibit the same direction of rotation.

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Figure 6-4: Days 40 to 130 of the principal component time series in the complex plane. Every dot represents the daily value of the principal component time series, with the annotated dots represented in red. The circular shape of the time series in the complex plane is indicative of an oscillatory behavior of the modes.

More specifically, the complex patterns can be reconstructed to the full dataset using Equation (6.9). Since the modes are complex, the POP modes as well as the time

series can be separated into real and imaginary parts (vr, vi) and (pr, pi), respectively,

using V(q, 1) = vr+ivi, V(q, 2) = vr−ivi, and PC(1, t) = pr+ipi, PC(2, t) = pr−ipi.

For including the first and second mode only, this yields

U0(q, t) = V(q, n) PC(n, t) where n = 1, 2 = V(q, 1) PC(1, t) + V(q, 2) PC(2, t) = (vr+ ivi)(pr+ ipi) + (vr− ivi)(pr− ipi)

= 2vrpr− 2vipi. (6.12)

The reconstruction of the data from Equation (6.12) is shown in Figure 6-5 for the same time period as Figure 6-4. The data weighted by pressure (see Section 6.5.4) turns out to be more revealing for the connection to the tropospheric structure than the unweighted data, which is why the weighted pattern is shown here, while the unweighted pattern shows the same stratospheric pattern and the same tropospheric connection, but due to the weaker tropospheric magnitude the connection to the tro- posphere is not as obvious there. The progression of the patterns shows a strong emphasis on the mode describing the poleward shift of the polar vortex along with an equatorward shift of the tropospheric jet. Between days 40 to 60, the stratospheric pattern weakens along with an equatorward shift of the tropospheric pattern, leading to a reversal of the tropospheric jet shift. This reversal precedes the subsequent shift in the stratospheric pattern by a couple of days, yielding a pattern emphasizing an equatorward shift of the polar vortex along with a poleward shift of the tropospheric jet. Overall, the pattern corresponding to the poleward vortex shift and the equa- torward jet shift dominates the time series of the oscillatory pattern, indicating that this pattern may be present most of the time, with a faster run through the oppo- site pattern. The analysis of other episodes of oscillatory motion between the first two POP modes indicates very similar results, with the longest-lived and dominant pattern being the one of a poleward vortex shift and the equatorward jet shift. This type of variability strongly resembles annular mode variability which is the dominant

pattern in the extratropical atmosphere (see Section 1.4.2). It is therefore not sur- prising that this pattern is found using POP analysis, it is however illuminating to see the progression of the pattern along with the corresponding time scales.

POP Modes from Forced Model Data

The modes for the control run showed a distinct latitudinal wobbling of the polar vortex along with a corresponding latitudinal movement of the tropospheric jet. This does not correspond to the annular mode pattern described in Section 1.4.2, which de- scribes the dominant wintertime extratropical variability as a weakening/strengthening pattern of the stratospheric vortex along with a latitudinal wobbling of the tropo- spheric jet. Since stratospheric variability is considerably smaller in the control run as compared to the real atmosphere, as shown in 4, the dominant variability in the control run is not the characteristic weakening/strengthening pattern, but a latitudi- nal wobbling. This is an important finding for defining the internal variability of the stratosphere - troposphere system for weak tropospheric wave forcing.

In order to verify that the POP patterns pick up the variability which is char- acteristic to the winter stratosphere, POP and EOF patterns were also computed for the topography run described in Section 2.2.2 (not shown). Both the POP and EOF patterns for the topography run pick up the characteristic winter variability corresponding to a weakening (strengthening) of the stratospheric vortex along with an equatorward (poleward) shift of the tropospheric jet (not shown). This pattern corresponds to the mode observed in the topography run as shown in Section 5.4.1.

In order to correctly compute the projection of the forcing onto the POP modes for the forced system, an effective torque will have to be computed as described in Ring and Plumb (2008). This will be a matter of future research for this study.