What causes the atmospheric DMs that have a broader (narrower) zonal scale dipolar structure to possess a longer (shorter) persistence? In this section, we conduct a detailed vorticity budget diagnosis based on Eq. (2) for the composite events of DMs which prove useful in revealing the key physical processes responsible for the different persistence of DMs. To simplify our discussions, the delivered results are solely based on the model’s DMs and we only focus on the two extreme scenarios of a 1/1 DM and 1/8 DM hereafter
4.
First, we calculate the composite results of each term in Eq. (2) at 300 and 850 hPa for the 1/1 and 1/8 DM events. Then, for the purpose of a convenient display, those composite results are conducted a zonal mean (0°–360°) calculation for the 1/1 DM events and a sectoral mean (0°–45°E) calculation for the 1/8 DM events. To avoid unnecessary repetition, we only discuss the results of their positive events in the following paragraphs.
Figures 6 and 7 respectively show the 300- and 850-hPa zonal mean results of the composite tendency term, term1, term2 …, term6, TEVF, HFEVF, LFEVF, the sum of terms 1−5 $\left(\sum\nolimits_{i = 1}^5 {{\text{term}}(i)}\right)$ and the sum of term6 and TEVF $ \left({\text{term}}6 + {\text{TEVF}}\right) $ for 1/1 DM positive events from –10 to 10 days.
In the upper troposphere, the 300-hPa composite tendency term of 1/1 DM positive events depicts a longitudinal dipolar structure with positive (negative) values at around 55°–70°N and negative (positive) values near 40°–55°N during negative (positive) lag days representing the growth (decay) of 1/1 DM positive events (see Fig. 6a). Obviously, the first five linear terms (terms 1–5) play a negligible role on the formation of the tendency term of 1/1 DM positive events (see Figs. 6b–f and Fig. 6k). This should be due to the zonally symmetric feature of a 1/1 DM.
TEVF produces a meridional dipolar structure of tendency with positive values near 55°–75°N and negative values near 35°–55°N during negative lag days; while TEVF produces a tilting meridional dipolar pattern of tendency with negative values at around 40°–60°N and positive values at around 25°–50°N during positive lag days (see Fig. 6h). The effects of TEVF are greatly neutralized by term6 (see Fig. 6g). Generally speaking, the tendency term for 1/1 DM positive events can be well depicted by using the sum of term6 and TEVF only (see Fig. 6l). HFEVF tends to produce a “positive-negative-positive” tripolar structure of tendency from 25°–70°N during the entire lifetime of 1/1 DM positive events (see Fig. 6i). While the contributions of LFEVF on the formation of 1/1 DM positive events are less important and somewhat complicated (see Fig. 6j). Therefore, HFEVF is a major contributor to TEVF.
Figure 7 shows that term6 assumes an overwhelming position in the 850-hPa vorticity budget results. Term6 at 850 hPa produces positive vorticity tendencies at high latitudes and negative vorticity tendencies at mid-latitudes from lag –10 to 10 days indicating that 1/1 DM positive events in the lower troposphere are dominantly driven and maintained by term6 against dissipation. We also notice an inverse association between the positive (negative) tendency of term6 at 850 hPa to the negative (positive) tendency of term6 at 300 hPa.
The results shown in Figs. 6 and 7 indicate that a 1/1 DM, from the upper troposphere to the lower troposphere, is primarily driven by term6 and nonlinear TEVF. Note that the cyclonic (anticyclonic) circulations aroused by the convergence (divergence) of eddy vorticity flux at the upper troposphere will spontaneously induce a horizontal divergence (convergence) of mass due to the Coriolis force. To balance the divergence (convergence) of mass in the upper troposphere, an upward (downward) vertical velocity at the middle troposphere is required, which naturally induces mass convergence (divergence) at the lower troposphere. Therefore, as observed in Figs. 6 and 7, the TEVF and term6 tend to balance each other at 300 hPa, and term6 at 850 hPa and 300 hPa are closely related to each other. Of course, eddy heat fluxes can also affect the 300- and 850-hPa divergence, noting that their contributions are implicitly included in Eq. (2). Thus, term6 in the upper and lower troposphere are actually slaved by eddies. Therefore, from the upper to lower troposphere, a 1/1 DM is entirely a nonlinear eddy-driven mode although a linear term (term6) also plays a non-negligible role in its tendency term. Therefore, the persistence of 1/1 DM events relies on the persistence of the nonlinear terms TEVF or HFEVF.
Similar to Figs. 6 and 7 but for the 1/8 DM positive events, the 300- and 850-hPa sectoral mean results of the diagnosed vorticity budget are shown in Figs. 8 and 9. As in the case of 1/1 DM positive events, the nonlinear term TEVF of 1/8 DM positive events also produces a similar meridional dipolar structure of tendency but with a much stronger amplitude during negative lag days (see Fig. 8h); while the signs of the meridional dipolar structure of tendency produced by TEVF are immediately reversed during positive lag days (see Fig. 8h). Still, the effects of TEVF are strongly balanced by term6 (see Fig. 8g), but in this case, the net contributions of the sum of term6 and TEVF on the tendency of 1/8 DM positive events are rather noisy and less important compared to the linear terms (see Fig. 8l). In fact, unlike 1/1 DM positive events, the pattern and temporal evolution of 300-hPa tendency term of 1/8 DM positive events are primarily determined by the linear terms (see Fig. 8k), especially terms 1 and 4 (see Figs. 8b, e). These results are consistent with Athanasiadis and Ambaum (2010) who found that the potential vorticity (PV) variability of the NAO at the 315 K isentropic level is dominantly driven by advection and the eddy forcing only explains a small amount of the tendency variance.
Figure 9 shows that in the lower troposphere, 1/8 DM positive events, similar to 1/1 DM positive events, are also primarily driven by term6 at 850 hPa. Note that this term is closely related to the eddy forcing in the upper troposphere as discussed in the previous paragraphs. Thus, the results shown in Figs. 8 and 9 indicate that 1/8 DM positive events are mainly driven by the linear terms in the upper troposphere, while, in the lower troposphere, 1/8 DM positive events are indirectly driven by the nonlinear eddy forcing. Therefore, unlike the 1/1 DM, the 1/8 DM is a linear and nonlinear mixed-driven mode. Still, the persistence of a 1/8 DM should largely depend on the persistence of the nonlinear terms because the persistence of the nonlinear eddy forcing is still the key factor that determines the persistence of a 1/8 DM in the lower troposphere.
Compared to 1/1 DM positive events, the persistence of the nonlinear eddy forcing in 1/8 DM positive events is apparently shorter. Therefore, the persistence of a 1/8 DM is naturally shorter than that of a 1/1 DM. Now, we need to understand why the persistence of the nonlinear eddy forcing is shorter in a 1/8 DM. Figure 10 shows the composite anomalous 300-hPa storm tracks (V variance), 300-hPa divergence, 850-hPa divergence, and 500-hPa vertical velocity (omega) zonal mean results for 1/1 DM positive events and sectoral mean results for 1/8 DM positive events from lag –10 to 10 days. Both 1/1 and 1/8 DM positive events are associated with a northward displacement of storm tracks. Still, the persistence of the northward displacement of storm tracks is better (worse) for 1/1 (1/8) DM positive events, which naturally produce a more (less) persistent anomalous nonlinear eddy forcing.
A baroclinic positive eddy feedback mechanism proposed by some previous studies underscores that the adiabatic cooling or heating due to the anomalous vertical motion associated with the anomalous secondary circulation driven by the anomalous eddy forcing serves to enhance the baroclinicity, which acts to sustain the anomalous displacement of the storm tracks, and thus, maintain the anomalous eddy forcing aloft (Robinson, 2006; Barnes and Hartmann, 2010). In Fig. 10, we observe anomalous vertical downward motions at around 35°–55°N and anomalous vertical upward motions at around 55°–70°N at the most of lag days of 1/1 DM positive events (see Fig. 10d). Obviously, this kind of distribution of anomalous vertical motions will enhance the baroclinicity at around 55°N which favors the northward displacement of storm tracks shown in Fig. 10a. Such a distribution of anomalous vertical motions is closely related to the anomalous divergence or convergence in the upper and lower troposphere (see Figs. 10b, c). However, for 1/8 DM positive events, this kind of distribution of anomalous vertical motions only presents itself during negative lag days (see Fig. 10h). From lag 0 to 4 days, anomalous vertical downward motions in the vicinity of 35°–55°N abruptly change to anomalous vertical upward motions (see Fig. 10h), which reduce the baroclinicity at around 55°N, and thus oppose the surviving northward displacement of storm tracks as well as the anomalous nonlinear eddy forcing there. Therefore, the persistence of the nonlinear eddy forcing is shorter for a 1/8 DM.
For 1/8 DM positive events, the abrupt change of vertical motion after lag 0 day is closely related to the sudden emergence of anomalous divergence in the upper troposphere and convergence in the lower troposphere at around 35°– 55°N from lag 0 to 4 days (see Figs. 10f, g). The consequent sudden emergence of three-dimensional circulation anomalies cannot be initiated by the anomalous eddy forcing during those lag days, because HFEVF at those lag days still tends to generate negative vorticity tendency in the mid-latitudes at the upper troposphere (see Fig. 8i), which arouses anomalous convergence in the upper troposphere and divergence in the lower troposphere at those locations. We notice that the linear terms produce a positive vorticity tendency in the mid-latitudes right after lag 0 day (see Fig. 8k). To account for this positive vorticity tendency, there should be divergence in the upper troposphere and convergence in the lower troposphere in the mid-latitudes. Therefore, we argue that the sudden emergence of anomalous vertical upward motion near 35°–55°N from lag 0 to 4 days, as shown in Fig. 10h, is indirectly caused by the linear terms. Thus, it is the linear terms that reduce the persistence of the northward displacement of storm tracks and consequently the persistence of the anomalous nonlinear eddy forcing there.
Note that the broader the zonal scale of the atmospheric DM, the greater the zonal symmetry, the less influential the linear terms, and the longer the persistence of anomalous nonlinear eddy forcing. Therefore, the atmospheric DMs with a broader (narrower) zonal scale naturally possess a longer (shorter) persistence.