Tropospheric eddy feedback to different stratospheric conditions in idealised baroclinic life cycles

A pronounced signature of stratosphere-troposphere coupling is a robust negative anomaly in the surface northern annular mode (NAM) following major sudden stratospheric warming (SSW) events, consistent with an equatorward shift of the tropospheric jet. It has previously been pointed out that tropospheric eddy feedbacks, mainly induced by anomalies in the lowermost extratropical stratosphere, play an important role in creating this surface NAM-signal. We use the basic setup of idealised 5 baroclinic life cycles to investigate the influence of stratospheric conditions on the behaviour of tropospheric synoptic-scale eddies. Particular focus is hereby given on the enhancement of the tropospheric eddy response by surface friction, as well as the sensitivity to wind anomalies in the lower stratosphere. We find systems that include a tropospheric jet only (modelling post-SSW conditions) to be characterised by an equatorward shift of the tropospheric jet in the final state of the life cycle, relative to systems that include a representation of the polar vortex (mimicking more undisturbed winter-time conditions), 10 consistent with the observed NAM-response after SSWs. The corresponding surface NAM-signal is increased if the system includes surface friction, presumably associated with a direct coupling of the eddy field at tropopause level to the surface winds. We further show that the jet shift signal observed in our experiments is mainly caused by changes in the zonal wind structure of the lowermost stratosphere, while changes in the wind structure of the middle and upper stratosphere have almost no influence.


Potential influence of surface friction
The influence of surface friction onto the evolution of baroclinic eddies is potentially crucial to understand the surface signal observed after SSWs, as it can be argued that the inclusion of surface friction increases the potential for the mid-and uppertropospheric eddy field to couple to the surface winds. This can be illustrated using the evolution equation of the vertically averaged zonal mean zonal wind, given in Equation 1 (see, e.g., chapter 10 of Vallis (2017)).
where u and v are zonal and meridional wind,ū sf c the zonal mean zonal surface wind, τ the surface friction time scale, square brackets and overbars denote vertical and zonal averages, respectively, and primed quantities describe deviations from the zonal mean (note that we neglected the mean flux term as it tends to be small in our system, consistent with quasigeostrophic scaling). Here we used a linear damping of surface winds as simple parametrisation of surface friction. In the 100 case with vanishing friction (τ → ∞), only the meridional momentum fluxes can act as source for (vertically averaged) zonal momentum and changes inū tend to occur in regions of non-zero momentum flux, i.e., around tropopause level for baroclinic life cycle experiments. For finite values of τ , on the other hand, the atmosphere can 'exchange' momentum with the surface, allowing for a non-local coupling between surface winds and the eddy field. This additional coupling mechanism suggests that a dynamic modification of the eddy field (due to the presence of a stratospheric jet) can lead to an enhanced change of the 105 corresponding surface winds (in terms of the difference between final and initial state) in cases where surface friction is active in the system.

Structure of this study
In the present paper we further investigate what impact the presence of a stratospheric polar vortex has on the idealised tropospheric baroclinic life cycle. In particular we are interested in the sensitivity of the life cycle evolution to changes in wind 110 structure in the lower stratosphere, compared to changes in the middle and upper stratosphere, and the influence of surface friction onto the surface signal of the life cycle induced by the presence of a stratospheric jet. We hereby mostly focus on the modification of the equilibrated 'final' state of the system, as opposed to the details of the (linear) growth stage or the (non-linear) decay stage of the baroclinic wave.
Section 2 introduces the model setup used in this study and lays out the specifics of the different sets of initial conditions. In 115 Section 3 we discuss in detail various changes of the evolution of the baroclinic life cycle due to the presence of stratospheric jet, with particular focus on the NAM-like response of the troposphere in the final state of a life cycle when there is no stratospheric jet present, compared to when there is. Additionally we show that we only find a strong signature in the corresponding surface signal when the system is subject to surface friction. We then provide evidence, in Section 4, to show that this NAM-like signal is mainly caused by the modification of winds in the (extra-tropical) lower stratosphere and the inclusion of winds in the middle 120 and upper stratosphere have almost no influence on the final state of the life cycle. In Section 5 we further discuss and interpret some of our findings before, in Section 6, we summarise the main conclusions of this paper.
wave number 85. The discrete vertical levels are distributed with constant spacing ∆z = 250 m up to a height of z = 60 km, where z = −H ln(p/p 0 ) is a log-pressure coordinate with scale height H = 7.5 km and reference pressure p 0 = 1000 hPa. To minimise upper boundary effects we add 10 additional model levels between z = 60 km and z = 82 km, equally spaced in pressure. Note that we are using a substantially higher vertical resolution than has typically been used in similar studies, since we found in particular the details of the non-linear decay phase of the baroclinic life cycles to be sensitive to changes in ∆z for 130 values larger than about ∆z = 250 m, as also further explained in Section 3.  The basic state is analytically defined via a given zonal wind field and is chosen to represent two general situations, depending 135 on the choice of parameters: either a system with a tropospheric jet only (modelling post-SSW conditions), or a system that contains a tropospheric and a stratospheric jet (mimicking more undisturbed winter-time conditions). In order to also study the sensitivity of changes in the wind structure of different regions in atmosphere we further use a set of basic states which include the tropospheric jet and only the upper or lower part of the stratospheric jet, respectively. Table 1 summarises the different in Figure 1 (note that only a part of the domain is shown). The temperature distribution of to the respective initial state is calculated to be in thermal-wind balance with the prescribed wind field. Note that the resulting meridional PV gradient (thick blue contours in Figure 1) strongly depends on the vertical curvature of the underlying wind field and therefore produces a pronounced local maximum near the tropospheric jet core.
Further note that both configuration displayed in Figure 1, due to the strong dependency on the wind field structure, include 145 regions with (slightly) negative PV gradient, which could potentially influence the evolution of the life cycle. However, the corresponding initial states follow the typical setup used in this type of idealised life cycle experiment. We further performed a series of sensitivity experiments and concluded the regions of negative PV gradient to have no significant influence on the qualitative results presented in this paper. Magnusdottir and Haynes (1996) also raised the question of the effect of negative PV gradients in typical life cycle setups on the evolution of the baroclinic wave and concluded that these regions can have an 150 effect on certain details of the non-linear phase (e.g., details of the energetics), but seem to have no impact on most aspects of the qualitative behaviour.
To trigger the growth of a baroclinic wave the initial state is perturbed by super-imposing a zonally periodic near-surface temperature perturbation of fixed zonal wave number 6, centred around 45 • latitude. We found our results to be qualitatively similar for perturbations with wave number 7, but the stratospheric jet to have almost no influence on the life cycle for wave 155 numbers 5 and 8 (in these cases the purely tropospheric life cycle is generally weaker than for perturbations with wave numbers 6 and 7).
More details on how the basic state is constructed are given in the Appendix. Starting from the described initial conditions the experiments are then either run freely (without any external forcing) or including a linear Rayleigh surface friction, following the friction profile specified by Held and Suarez (1994) with a maximum friction coefficient of k f = 1 day −1 at the surface, 160 gradually reducing to zero at 700 hPa (z ≈ 3 km).

Modification of the life cycle by a stratospheric jet
We start our study by investigating in what way the general evolution of an idealised baroclinic life cycle is altered when the initial conditions include a tropospheric and a stratospheric jet, the latter representing the winter time polar vortex, compared to when they include a tropospheric jet only, as is usually the case after a SSW and is the conventional life cycle setup. In the rest of this section we therefore analyse a set of life cycle experiments with varying values of the stratospheric jet strength parameter u Smax (see Equation A2 in the Appendix) and thus varying strength of the stratospheric jet that is added onto the system with tropospheric jet only.

Modification of the baroclinic wave breaking
The evolution of idealised baroclinic life cycles is often described in terms of the distribution of potential vorticity (PV) on an 170 isentropic surface close to the jet core (or equivalently close to the tropopause). Zonal modulations in PV contours in this region of sharp PV gradient (also seen in Figure 1) give insights into the growth and decay of the eddy field, while any change in the position of the maximum in zonally averaged PV gradient represents a meridional shift of the jet. The top and middle rows in Figure 2 show the horizontal PV distribution on the 350 K isentrope at selected days for the two initial state configurations with tropospheric jet only (experiment T) and tropospheric and stratospheric jet (experiment TS).

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The general evolution of both experiments is similar to each other in the sense that the baroclinic wave grows gradually until about day 6. At that point the wave becomes non-linear, breaks and eventually decays. However, especially the non-linear decay phase shows substantial differences in the specific evolution of the PV field when a stratospheric jet is present. The wave breaking is still characterised by filaments of high PV that stretch out on the equatorward side of the jet core, break off and eventually roll up anticyclonically, but the timing of events and the details of the small scale structures are altered considerably To highlight the modification in PV evolution induced by the presence of a stratospheric jet the bottom panel of Figure   2 shows the difference in the PV field of a simulation with and without stratospheric jet. Overlaid are the corresponding 8 185 PVU contours of the two respective experiments. It can be seen that at day 6, i.e., at the end of the linear growth phase, the two baroclinic waves have a similar magnitude and structure, but are slightly phase shifted with respect to each other. This shift can potentially be explained by a minor increase in phase speed in the case with a stratospheric jet. This might either be due to a minor increase in wind speed near the tropopause (also further discussed in Sections 4 and 5) or a change of the corresponding PV gradient in that region. While a pure zonal phase shift of the wave should not have any influence on the 190 subsequent behaviour of the wave-breaking due to the zonal symmetry of the system, it does indicate a change in the dispersion relation.
At days 7 to 9, i.e., during the non-linear phase, the evolution of the system is strongly influenced by the stratospheric jet and Figure 2 shows a large difference in PV distribution. Especially at days 8 and 9 the baroclinic wave in experiment TS, including a stratospheric jet, seems to have entered a second growth phase, while the wave in experiment T still seems to 195 be decaying. As mentioned in Section 1 these 'secondary life cycles' during the non-linear decay phase have been discussed previously by Barnes and Young (1992). We find the details of the non-linear phase, like the occurrence, timing or apparent flavour (in a LC1/LC2 sense) of 'secondary cycles', to be very sensitive to small changes of the initial conditions or the details of the physical processes involved, as can also be seen in Figure 2. Recall that, as mentioned in Section 2, the occurrence and strength of these secondary cycles varied in a set of sensitivity experiments with lower vertical resolution. For the purpose of 200 this study we therefore focus primarily on the evolution of the entire life cycle, e.g., in terms of the difference between initial and some final state.

Dependency on stratospheric jet magnitude
In addition to the evolution of the PV field baroclinic life cycles can be quantified in terms of the global energetics of the system, typically with a strong focus on eddy kinetic energy (EKE), which describes the growth and decay of the baroclinic wave in 205 the region of large meridional PV gradient near the jet core (see Figure 1). In particular the decay of EKE is associated with an energy transfer to the zonal mean state, i.e., an increase of the mean kinetic energy (MKE). This increase in MKE can be associated with a poleward shift, and a corresponding acceleration, of the tropospheric jet due to wave-mean-flow interactions and poleward eddy momentum fluxes during the decay phase of the life cycle.
The way the evolution of the life cycle is altered by a stratospheric jet can be seen in terms of EKE and MKE time series, 210 shown in Figure 3 for experiments with different values for the stratospheric jet magnitude u Smax (see Appendix for details).
Note that here we use ∆MKE, which is simply the change in MKE with respect to the initial conditions and that both, EKE and ∆MKE, are displayed as vertically integrated and horizontally (over the northern hemisphere) averaged energy densities.
In agreement with Figure 2, which suggests only a phase shift in the baroclinic waves during the linear phase, but no difference in magnitudes, Figure 3 shows essentially no sensitivity to introducing a stratospheric jet before day 6, in particular 215 we do not find any significant change in growth rate, as has been reported by other authors, e.g., Wittman et al. (2007). A potential explanation for the strong change in growth rate found by Wittman et al. (2007) could be a substantial difference in meridional PV gradient (due to the substantial modification of the vertical curvature of zonal wind at the tropopause) between their different experimental setups. The basic states used in the present study, on the other hand, do only slightly differ in terms of their tropopause level PV gradients (see Figure 1). However, during the non-linear-phase, so from day 7 onwards, the 220 stratospheric jet seems to extensively alter the evolution of the life cycle. Especially the onset of a secondary phase of wave growth (with EKE peaking again at about day 10) seems to happen about a day earlier when a stratospheric jet is present in the system, and leads to a much stronger and more persistent secondary peak. The persistently elevated EKE of the secondary cycles during the non-linear phase (with EKE reducing again towards the final state) is consistent with the idea of a stronger LC2 flavour (which is often characterised by persistently increased EKE in the decay phase) of the secondary cycles, as is also 225 suggested by Figure 2 and is further discussed in Section 5.
The alteration of the system as we increase u Smax does not only manifest as changes in the details of how the wave breaking evolves, but also leads to a change of the final state (here defined as average over days 20-30), in particular a systematic increase of ∆MKE. (compared to the experiment T, with tropospheric jet only) can be seen in Figure 4, which shows in all subplots as black contours the evolution of the zonal mean zonal wind field at 10 km. Figure 4a furthermore shows the zonal wind anomaly of experiment T with respect to the initial conditions. One can clearly see a dipole pattern developing around the initial jet core   However, several differences can be observed. First, an overall weakening of the jet in the final state (black contours) can be observed when surface friction is included, which is easily explained by the direct dissipation of kinetic energy over the course of the life cycle due to the added friction process. The same argument holds for the disappearance of the strong wind anomaly patterns close to the surface at about 30 • and 40 • latitude in the case without friction. These patterns develop due to strong temperature fluxes in this region arising from the large meridional surface temperature gradient (see Figure 1) and they are likely not influential on the standard baroclinic life cycle evolution. More importantly, however, the vertical structure of the dipole pattern around the final jet core at 60 • latitude is drastically different between the experiments with and without surface friction displayed in Figure 5. When the system is subject to surface friction during the life cycle the corresponding dipole 260 pattern is more barotropic, thus it extends much further down and shows much stronger anomalies at the surface.

Figures 4 and 5 indicate a tendency of the tropospheric jet to exhibit a weaker poleward shift during the baroclinic life
cycle if there is no stratospheric jet present compared to when there is. This behaviour is consistent with the negative NAM response, associated with an equatorward shift of the tropospheric jet, observed during periods following SSWs (see Baldwin and Dunkerton, 2001). It further provides a simple way to quantify the eddy feedback processes potentially involved in creating 265 the corresponding jet shift signal. Figure 5 shows the shift signal only to have a significant surface contribution if the system is subject to surface friction.
To further illustrate the surface signal observed in our model experiments Figure 6 shows the geostrophic geopotential height field Z, calculated by solving the equation    In the rest of this paper we investigate the influence of a stratospheric jet onto the final state of the baroclinic life cycle and the resulting NAM-like signature in more detail. In particular we identify a region in the lower stratosphere which is highly sensitive to changes in the zonal wind that are induced by the inclusion of a stratospheric jet. In the previous section we established that introducing a stratospheric jet can modify the evolution of the system in an idealised baroclinic life experiment, as has also been shown by other authors (e.g., Wittman et al., 2004). In this section we show that the system is particularly sensitive to changes in wind structure in the extratropical lower stratosphere (heights below about 25 km), while changes in the middle and upper stratosphere have almost no influence on the final state. In order to investigate this 295 sensitivity we analyse a set of experiments with initial conditions that include a tropospheric jet, as well as a stratospheric jet with modified vertical structure.
We modify the structure by multiplying the profile of the stratospheric jet used in experiment TS by a function η(z) (see

Equation A2
). We choose η(z) to follow a tanh-profile, which allows us to smoothly set the winds of the stratospheric jet component to zero below or above a set transition height z η and thus investigate which part of the stratospheric jet has the 300 strongest influence on the life cycle. We hereby refer to the experiments where we only include the part of the stratospheric jet below height z η as 'T S <zη ', and correspondingly refer to the experiments where we keep the part above z η as 'T S >zη ' (for simplicity we drop the units of z η within this notation and set it to be kilometres). See the Appendix for details on how the basic state is defined. Details of the vertical structure of the various initial wind fields can also be seen in Figure 9, displaying the zonal wind at 310 60 • latitude, i.e., at the northern flank of the tropospheric jet and through the core of the stratospheric jet. A very prominent difference is that profiles where the stratospheric jet reaches into the lower stratosphere have substantially increased wind speeds in that region (roughly between 10 and 25km), compared to profiles where the contribution of the jet is mostly confined to the troposphere or the middle and upper stratosphere. This criterion divides the six profiles into two groups, 'Set 1' consisting of profiles T, TS >25 and TS <10 with weak winds in the lower stratosphere, and 'Set 2' consisting of profiles TS, TS >10 and 315 TS <25 with strong winds in the lower stratosphere. In most of the rest of this section we analyse the experiments with different initial conditions keeping in mind the grouping into these two sets.
To visualise the NAM-like jet shift signature of the final state, and to investigate which contribution to this jet shift can be associated the different parts of the stratospheric jet, Figure 10 shows the zonal mean zonal wind averaged over days 20-30 and the corresponding anomaly from experiment T (with tropospheric jet only). We first look at the experiments of Set 1. The final state zonal wind field of experiment TS >25 (Figure 10c) does not show any substantial deviation from experiment T, indicating that winds in the middle and upper stratosphere have virtually no influence on the life cycle. Experiment TS <10 , with superimposed winds confined to the troposphere, shows a dipole pattern, which could potentially be attributed to the projection of the wind modification on e.g., the increase in tropospheric jet magnitude or the vertical shear, also further discusses in Section 5. However, also note that the superimposed winds of the stratospheric jet 325 do not abruptly vanish at the given cut-off height (e.g., above 10 km for TS <10 ), but follow a smooth transition over the course of about 4 km and therefore still reach into the lower stratosphere region.
The experiments of Set 2 (bottom panel of Figure 10 The significance of the lower stratospheric wind anomalies are discussed further in Section 5.

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The surface signal of the NAM-like response discussed above can be seen in Figure  In Section 4 we showed that the NAM-response observed in the final state of our life cycle experiments is mostly caused by the change in wind structure in the lower stratosphere when including the stratospheric jet, rather than wind anomalies in 400 the middle and upper stratosphere (see Figure 10). However, it should be noted that changing the wind structure in the lower stratosphere does also introduce changes in various other characteristics of the corresponding initial conditions, like the height of maximum wind speed, the vertical wind shear in the upper troposphere (roughly up to 10km) and the magnitude of the tropospheric jet (especially obvious for profiles T and TS in Figure 9). However, these three characteristics are intrinsically not completely independent and can potentially all affect the evolution of the life cycle. This can be seen, e.g., since the vertical 405 wind shear is (via thermal wind balance) related to the horizontal temperature gradient, which drives the growth of baroclinic waves and can, among other things, modify their (linear) growth rate (although note that the near surface shear is almost identical in the different experiments).
We performed a set of sensitivity experiments (not shown) with tropospheric jet only and varying tropospheric jet magnitude (and therefore increased vertical shear in the troposphere). We found that an increase in tropospheric jet strength also leads to 410 a increased poleward shift during the life cycle (i.e., an equatorward shift of the jet in the final state of experiment T relative to a case with stronger tropospheric jet), similar to the shift observed in Figure 5a. In order to achieve a jet shift signal of similar magnitude as the one shown in Figure 5, however, it was necessary to increase the jet magnitude by order of 10 m/s (the difference in tropospheric jet magnitude between experiments T and TS is only of the order of 1 m/s.), indicating that other characteristics of the initial state need to contribute and the observed jet shift cannot purely be a result of a strengthened 415 tropospheric jet. The inclusion of the stratospheric jet does to some extend project onto the mentioned characteristics (e.g., height of the jet core and tropospheric shear) of the total zonal wind profile and the resulting jet shift can potentially be interpreted as the result of a combination of factors. Figure 11 further suggested that we essentially recover the surface geopotential height signal of experiment TS (with full stratospheric jet), when adding the corresponding signals of experiments T <10 and T >10 . Such additivity of responses might 420 be another indication that the stratospheric jet projects onto various other structures and characteristics (e.g., tropospheric shear and jet core height) and the corresponding jet shift response forms as the result of a combination of responses to those modifications. However, while the anomalies of the respective experiments seem to be additive when it comes to the surface geopotential height, the middle tropospheric jet shift response in Figure 10 does not appear to follow the same additive behaviour.
As discussed, Figure 10 suggests the NAM-like jet shift signature of the life cycle due to the inclusion of a stratospheric jet 425 to be mainly caused by the corresponding change in winds in the lower stratosphere, rather than the winds in the middle and upper stratosphere, where the stratospheric jet itself is strongest. A similar conclusion can be drawn from the energetics of the system (provided as supplementary material), which shows a consistent increase in MKE of the final state for the experiments in Set 2 (as defined in Section 4), compared to the experiments in Set 1, in a system that does not include surface friction. As also explained in Section 3, this increase in MKE is caused by the relative meridional shift of the final tropospheric jet. Note The corresponding NAM-like jet-shift response has an increased surface signal if the system includes surface fiction, which 445 might seem counter-intuitive, but is consistent with the idea of an increased coupling of surface winds to the eddy momentum transport at tropopause level due to the friction.
We further showed that the system is mainly sensitive to changes of the wind structure in the lower stratosphere (heights between 10 km and 25 km), rather than to zonal wind anomalies in the middle and upper stratosphere.
The findings of this paper improve our basic understanding of the weather and climate system in the mid-latitude tropo-450 sphere and lower stratosphere. In particular they provide a potential explanation for the downward propagation of zonal wind anomalies from the stratosphere into the troposphere and the related negative NAM signal observed after SSWs. The idealised lice cycle setting further provides a quantitative way to analyse the importance of the tropospheric eddy feedback during and after SSWs.
Appendix A: Appendix: Construction of initial state

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The basic state used to initialise our experiments is defined via a zonally symmetric zonal wind field, consisting of two individual components: a tropospheric jet U T (representing the mid-latitude jet) and a stratospheric jet U S (representing the polar vortex). The total wind field is then given by the sum of both components U = U T + U S , with the tropospheric jet profile being given by U T = u T max (z/z T mid ) exp ((1 − (z/z T mid ) α ) /α) sin 3 π sin 2 (φ) ,