Zonal scale and temporal variability of the Asian monsoon anticyclone in an idealised numerical model

The upper-level monsoon anticyclone is studied in a 3D dry dynamical model as the response of a background circulation without any imposed zonal structure to a steady imposed zonally confined heat source. The characteristics of the background circulation are determined by thermal relaxation towards a simple meridionally varying state, which gives rise to baroclinic instability if meridional gradients are sufficiently large. This model configuration allows the study of the dependence of the monsoon anticyclone response on characteristics of both the imposed heating and the background state, in particular including interactions between the anticyclone and the active dynamics on its poleward side in the form of a jet and baroclinic eddies. As characteristics of forcing and background state are varied a range of different behaviours emerges, many of which strongly resemble phenomena and features associated with the monsoon anticyclone as observed in re-analysis data. For a resting background state the time-mean anticyclone is highly extended in longitude to the west of the forcing region. When the active mid-latitude dynamics is included the zonal extent of the time-mean anticyclone is limited, without any need for the explicit upper-level momentum dissipation which is often included in simple theoretical models but difficult to justify physically. We further describe in detail the spontaneous emergence of temporal variability in the form of westward eddy shedding from the monsoon anticyclone for varying strength of the imposed heating. By varying the strength of the background mid-latitude dynamics we observe a transition of the system from a state with periodic westward eddy shedding to a state dominated by eastward shedding. The details of the time-mean structure and temporal evolution depend on the structure of the background flow, and for certain flows the monsoon anticyclone shows signs of both westward and eastward shedding.

One of the major features of the atmospheric general circulation is the Asian monsoon. This seasonal large scale circulation pattern is active during the boreal summer months, peaking during June-August (JJA), and spans large parts of south and south-east Asia. The monsoon can strongly influence the weather and climate of the northern hemisphere (NH) via various mechanisms, e.g. locally through strong convective rainfall in the monsoon region or globally since it potentially forms an transport is of lesser importance. They concluded that the dynamics in the tropical upper troposphere is essentially non-linear and nearly inviscid. Studies such as these are often cited as important, but as yet no clear consensus has emerged on what physical processes balance the negative forcing of potential vorticity above the monsoon heating and hence control the zonal structure of the anticyclone.
Other studies used a slightly different approach to obtain a (quasi-)realistic monsoon response in three-dimensional numer-70 ical simulations of the tropical middle atmosphere, e.g. Hoskins and Rodwell (1995) or Liu et al. (2007). The general idea in these studies was to relax the zonal mean zonal wind component towards climatological or idealised profiles, while keeping the wave part freely evolving. This approach obtained tropical circulations that were localised in longitude. However, fixing the zonal mean zonal wind tends to suppress non-linear processes as it does not allow for full wave-mean-flow interactions and it is further not clear whether such an approach is in practise equivalent to some form of large scale friction. 75 Another approach to modelling the upper level monsoon response to thermal forcing in a quasi-realistic setting has been to suppress the development of baroclinic instabilities via an imposed damping or to avoid their occurrence by restricting model integrations in time (usually to about 20 days). An example of this is the work of Hoskins and Jin (1991), who performed initial value experiments for tropical circulations. Among other cases they perturbed a resting atmosphere and a climatological zonal mean zonal wind profile with a Gill-like stream function pattern to find the climatological wind profile to significantly 80 suppress the equatorial flow and causing a polewards propagation of Rossby waves. This study, however, can only give limited insights into a potential (steady) zonal confinement of the response since the model is not constantly forced and the initial flow becomes strongly modified due to baroclinicity after about 14 days. Hoskins and Rodwell (1995) modelled the monsoon circulation as response to thermal or orographic forcing and investigated the importance of non-linear effects and mountains. They found the general structure of the monsoon anticyclone to be well 85 reproduced by a linear, thermally forced model. However, again Hoskins and Rodwell (1995) restricted their experiments to early times in order to avoid baroclinic instabilities developing in their system. Such short integration lengths might simply bypass the problem of unrealistic zonal extension without the system actually reaching a steady state. Jin and Hoskins (1995) used a very similar approach and modelled zonally confined Gill-like flow structures as response to an equatorial heating, but also limited considerations to early times. 90 Ting and Yu (1998) investigated the steady response of a baroclinic atmosphere with climatological basic state to a localised equatorial heating, but suppressed the occurrence of baroclinic instability by adding a 15-day mechanical and thermal damping to the system. Such added friction processes directly affects the explicitly forced flow, which potentially explains the zonal localisation of the response in their experiments. Hendon (1986) performed Gill-like tropical heating experiments in a 2-layer model with baroclinically unstable background and an inviscid upper layer to study the relative position of the upper level 95 anticyclones without restricting the experiments to early times or suppressing the instability of the basic state. However, the problem of zonal localisation was not addressed in this study in any detail. In Section 3 below we show that baroclinic instability (which is ignored or suppressed in many of these investigations) can in fact play an important localising effect on the monsoon circulation.
The problem of zonal confinement of a flow induced by a local heat source was recently investigated by Amemiya and Sato 100 (2018) in a single-layer model, who found that a meridionally varying depth can have a localising effect. Such a meridional depth gradient does introduce a zonal mean wind (a westerly jet in their case) and does therefore correspond to a simple representation of certain dynamic mid-latitude characteristics. However, the basic state used includes regions with negative meridional PV gradient and thus its relevance for describing the monsoon system is not clear.

Eddy Shedding by the monsoon anticyclone 105
Alongside characteristics of the time-mean monsoon anticyclone, such as its zonal localisation, the temporal variation of the anticyclone is also not yet fully understood. The time variation is potentially important because of its meteorological implications and also, as noted previously, for its implications for transport of chemical species from the upper troposphere, including from within the centre of the anticyclone, into the extratropical stratosphere. Different authors have observed and described westward and eastward shedding of vortices from the (bulk) monsoon anticyclone. These vortices (or eddies) are 110 often defined via localised, coherent PV or geopotential height structures. Fluctuations in flow associated with these vortices can transport trace gases or water vapour out of the monsoon region (Garny and Randel, 2013;Vogel et al., 2014).
The mechanisms for the two types of time dependence, i.e. westward shedding and eastward shedding, are potentially very different and the two phenomena have been previously investigated by various other authors. Hsu and Plumb (2000) studied a particular westward shedding event of the monsoon anticyclone and showed that the response of a single-layer model to a local 115 steady forcing can become unstable for certain parameter ranges, giving a potential explanation for the westward shedding process. Indeed Davey and Killworth (1989) had previously analysed the response to a steady mass source in a single layer beta-plane model intending to describe and explain oceanographic phenomena (e.g. Mediterranean outflow). They observed a transition from a steady state, a westward extending patch of low PV (sometimes referred to as beta-plume), to an eddyshedding state as the strength of the forcing increases, similar to the transition shown later in Figure 11. Davey and Killworth 120 (1989) further argued that eddy shedding occurs when the meridional PV gradient within the beta-plume of the steady linear solution becomes negative and the system therefore unstable. Popovic and Plumb (2001) demonstrated with re-analysis data that such westward shedding events occur regularly and about 2-4 times each monsoon season. Analysing Fourier spectra of ERA-I PV data Fadnavis et al. (2018) concluded that the periodic eddy shedding events from the monsoon anticyclone happen with a frequency on synoptic time scales (≈10 days).

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Their estimate for the frequency of the almost-periodic shedding process seems to be in rough agreement with the findings of Popovic and Plumb (2001).
The Liu et al. (2007) work noted previously studied the response of to idealised and semi-realistic monsoon thermal forcing distributions and observed a periodic shedding of eddies from the main PV low of the monsoon anticyclone for sufficiently strong heating magnitudes. The analysis and model setup used by Liu et al. (2007) mainly differs from the study reported in this 130 paper in two aspects. For one, Liu et al. (2007) emphasise the importance of 'Tibetan heating' in creating the phenomenon of westward eddy shedding, while we will mostly study in detail the changes in temporal variability of the response as the forcing magnitude increases and, in addition, perform a set of experiments in a basic state with temporally varying background flow (Section 4). Second Liu et al. (2007) relaxed their system towards a prescribed steady zonal flow profile which, as mentioned earlier, can potentially alter the dynamics and restrict the time-mean response. The model configuration used in this study (see 135 Section 2 for details) does not restrict the winds of the forced response. Rupp and Haynes (2020) found that westward shedding of eddies from a localised region of steady negative PV forcing in a single-layer beta-plane model can be interpreted as result of a spatio-temporal instability of the system. They further analysed the dependence of the underlying instability and the resulting qualitative state of the system on different characteristics of the forcing and (also) found a transition from a stable/steady to an unsteady/eddy-shedding state for increasing strength and/or 140 decreasing length scale of the forcing, which is not fully captured by a simple reversal of the PV gradient.
The different phenomenon of eastward shedding has emerged more gradually through independent studies focusing on meteorological implications on the one hand and on chemical transport on the other. Postel and Hitchman (1999) noted that there was frequent breaking of Rossby waves over the North Pacific in the upper troposphere in NH summer and suggested that that the low PV part of the characteristic pattern of wave breaking originated in the Asian anticyclone to the west. Enomoto 145 et al. (2003) studied the dynamics of the Bonin high which is a characteristic feature of the western North Pacific circulation in late NH summer and occurs near Japan. They showed in a modelling study that the anticyclone could result from the roll-up of a low PV filament that had been drawn out of the monsoon anticyclone by fluctuations on the mid-latitude jet, leading to the formation of isolated anticyclones to the (north-)east of the bulk anticyclone. Note, however, that Enomoto et al. (2003) emphasised the importance of an external Rossby wave source (the 'Silk Road cooling' to the west of the monsoon) in order 150 to form these isolated vortices in their simulations. As we show in Section 3 we can reproduce a similar anticyclonic feature to the north-east of the main monsoon without any extra local wave source.
Later papers such as Garny and Randel (2013) and Vogel et al. (2014) identified the chemical signatures of these dynamical events forming isolated anticyclones via an interaction of the monsoon anticyclone and perturbations on the mid-latitude jet and referred to them as 'eastward-shedding' events. Note that eastward shedding is rather distinct from a further consequence 155 of time variation of the anticyclone identified by Dethof et al. (1999) where the time variation of the anticyclone and the jet often leads to a filament of low PV extending eastward and poleward into the extratropical stratosphere and subsequently being absorbed into the stratospheric air mass through mixing, perhaps also forming a coherent anticyclonic vortex during this process.
Although different types of shedding events associated with perturbations to the main anticyclone have been identified by 160 monsoon researchers, there is still uncertainty about precise mechanisms. Furthermore the connection between shedding events and other phenomena such as potential bimodality of the monsoon anticyclone Zhang et al. (2002) is not always made clear.
Indeed some of the features that have been previously identified by other authors in their studies of the monsoon anticyclone can, in retrospect, be identified as signatures of eddy shedding. For example, Hoskins and Rodwell (1995) analyse the shortterm response to three-dimensional semi-realistic heating distribution and obtain some agreement between the instantaneous 165 model stream function field and the climatological JJA-averaged stream function obtained from re-analysis data. However they identify a 'major defect' of their (instantaneous) model response in the form of a tendency of the anticyclone to 'split into three separate centres, with additional maxima near east Africa and Japan' that is not clearly visible in the time-averaged re-analysis fields. Enomoto et al. (2003) later identified the the 'defect' anticyclone over Japan as a representation of the Bonin high. Ren et al. (2015) used a composite analysis and PV budget calculations to investigate mechanisms corresponding to eastward 170 extension-phases of the monsoon anticyclone. They find a strong correlation between the eastward extension and anomalous heat and rainfall patterns over east Asia, indicating a potential meteorological impact of eastward shedding events. Similarly, Luo et al. (2017) describe and analyse an east-west oscillation event of the monsoon anticyclone during 2016. These oscillation events can include a split of the monsoon anticyclone and a zonal shift of its geopotential height maximum. Luo et al. (2017) further show that this shift can lead to a substantial zonal mass flux and thus might have important implications for horizontal 175 transport. They further link the oscillatory behaviour to a bi-modality of the monsoon, as it was described by Zhang et al. (2002).
A unified explanation of the zonal and temporal variation seen in all these phenomenon is as a consequence of eastward and westward shedding events.

Outline
Our aim in this paper is to clarify the mechanisms that control the spatial structure of the monsoon anticyclone, particularly its 180 zonal scale, and its time variability.
In particular we address the following questions: -Can we find a mechanism that leads to a zonal localisation of the time mean response and does not rely on strong mechanical damping throughout the atmosphere?
-Does the response to a steady localised forcing in a 3D numerical model show evidence for westward eddy shedding 185 similar to the results that have previously been reported for single-layer models with a transition to an eddy-shedding state as the forcing strength increases?
-What influence does the interaction with mid-latitude dynamics have on the locally forced monsoon anticyclone and what role does it play in the phenomenon of eastward eddy shedding?
We address these questions by using a minimal model, based on three-dimensional primitive-equation dynamics, in which 190 the monsoon circulation is forced by an imposed localised heating and the background zonal flow is determined by imposing a meridional (equator-pole) temperature gradient, rather than by being determined (via initialisation or relaxation) by a meridional flow profile that approximates observations. The strength of the localised heating and of the meridional temperature gradient are varied and the implications for the monsoon anticyclone are determined.
The structure of the paper is as follows. In Section 2 we describe the numerical model used in the study and also the re-195 analysis dataset that provides examples for comparison with the modelled behaviour. Section 3 focuses on the time mean response to localised heating, with particular focus on the zonal scale of the response. In Section 4 we then investigate the different forms of temporal variability of the response and its dependence on the forcing magnitude and the basic state. Finally we give a brief summary of our findings in Section 5.' 2 Model and data used in this study 200 The present study involves numerical experiments with a three-dimensional atmospheric model, as well as various diagnostic analyses of re-analysis data. This Section outlines the specifics of the model and the dataset used.

The numerical model
Numerical experiments are performed using the dry non-linear primitive equation model IGCM1 developed at the University of Reading (Hoskins and Simmons, 1975). In the model horizontal dimensions are represented via a spherical harmonics 205 series truncation at total wave number 42, corresponding to resolution of about 2.8 • at the equator. The vertical dimension is represented by 40 σ-levels (ratio of pressure and surface pressure) equally spaced on the log-pressure scale defined via z = −H ln σ, where H = 7 km is a scale height. This gives a vertical level spacing of 0.7 km up to the model top at about 28km. Throughout this paper we will use the log-pressure height z rather than σ-coordinates.
All experiments are run using a Held-Suarez-like (HS) basic state, which is obtained by relaxing the temperature of the 210 system at a rate r (φ, σ) towards the restoration profile T r (φ, σ), both defined in Equation 1. The corresponding approach was suggested by Held and Suarez (1994) and gives an easy and reliable way to produce a simple representation of the large scale circulation of the mid-latitude atmosphere in the form of a mid-latitude mean zonal jet and a strong temporal variability due to baroclinic instability. In addition to the thermal relaxation the basic state set-up includes a simple representation of surface friction in the form of a linear damping of horizontal winds at a rate of 1/day at z = 0, gradually reducing to zero at about 215 z = 2.5 km (exact implementation follows the definition in Held and Suarez (1994)). (1) The parameters appearing in Equation 1 are defined in Table 1. For most of the parameters of the basic state we use the same values as have originally been used by Held and Suarez (1994), with the exception of the meridional temperature gradient 220 parameter ∆T , which we vary to alter the characteristics (in particular the strength) of the induced mid-latitudinal background flow. Following the proposal by Polvani and Kushner (2002), we further added a term '−T as sin φ' to the restoration profile in Equation 1, allowing us to introduce the hemispherical asymmetry associated with the summer season that is of most relevance to the Asian monsoon. The same approach was adapted by McGraw and Barnes (2016) and Chen and Plumb (2014) to investigate eddy transport in the lower atmosphere, both choosing T as = 20 K, a value twice as large as was originally used by Polvani and Kushner (2002). In this study we use both, a (hemispherically symmetric) 'annual-mean' state (T as = 0 K) and an asymmetric 'summer' state (T as = 20 K), in order to assess the significance of differences in the structure of the background state.
A tropical monsoon flow is forced by imposing a localised steady heating with structure

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where φ and λ represent latitude and longitude, respectively, Q 0 controls the magnitude of the heating, r 0 its horizontal extent and V (z) its vertical structure. The forcing is centred at φ 0 = 20 • latitude and λ 0 = 80 • longitude, with a radius of r 0 = 10 • . Note that the zonal position of the forcing is completely arbitrary and has no influence on the results, since we did 235 not impose any zonally asymmetric features other than the explicit heating. For easier visualisation of the response we defined the range of longitudes in our domain to be −120 The vertical structure V (z) is given by i.e., the heating is confined to a height below z top and maximises at z max . We chose different a structure of V (z) above 240 and below the maximum at z max in order to reduce the vertical gradient of the heating at lower levels, which weakens the corresponding PV forcing and therefore de-emphasise the cyclonic lower level part of the forced monsoon response. However, the results presented in this study do not rely on the precise structure of the forcing.
In each simulation the initial state is that of an isothermal resting atmosphere and the flow is allowed to evolve in the absence of localised forcing (Q 0 = 0) until it reaches what is effectively a statistically steady state. At least 1000 days was allowed for  Table 1 gives an overview of the parameter ranges for the imposed heating and the basic state used in our experiments. 3 Zonal scale of the time mean response 3.1 Localisation due to mid-latitude dynamics As mentioned earlier the monsoon anticyclone is associated with a pronounced PV minimum in the UTLS region. Figure 1 shows the horizontal structure of this minimum on the 370K isentropic surface, as it appears in the mean over several NH summers (JJA for 2000(JJA for -2009. Note that only a limited range of longitudes is included in the figure.  We use the numerical model configuration described in Subsection 2.1. Figure 2 shows the time mean PV structure and the corresponding anomaly from the basic state on the 340 K isentropic surface for a forcing with magnitude Q 0 = 5 K/day and a resting background atmosphere, i.e. ∆T = 0 and T as = 0. The chosen isentropic level roughly corresponds to z = 13 km at the latitudes of the heating, and hence lies between the height of maximum heating (z max = 10 km) and the top of the heating 275 (z top = 15 km) and in a region with strong negative PV forcing due to the large negative vertical gradient of the imposed heating profile. We can therefore expect to observe a relatively strong anticyclonic PV response on this level. From Figure   2 it is apparent that the PV response is zonally elongated and not confined to the vicinity of the heating. Zonally elongated structures are visible in both, the full PV field and the anomaly field (recall that the zonal position of our forcing is arbitrary due to the zonally symmetric basic state, but for ease of comparison we have centred the forcing at 80 • N ). days) 1 . This zonal non-localisation of the monsoon anticyclone is in strong contrast to what is found in re-analysis date (e.g., Figure 1) and indicates that additional physical or dynamical processes have to be included in the model in order to obtain a 285 monsoon anticyclone response with a realistic zonal scale.
As a next step we modify the basic state of our system by choosing ∆T = 60 K in Equation 1 and thus making the basic state baroclinically unstable at mid-latitudes (for now we keep T as = 0). The resulting dynamics leads to the formation of mid-latitude jet streams with associated highly variable baroclinic eddies. Figure 3 shows the horizontal structure of the time mean PV and stream function (ψ) response on the 335K isentropic surface 290 and the 13 km height plane, respectively 2 . The streamfunction ψ represents the non-divergent part of the horizontal flow. Since we are considering relatively long time scales on which the flow is close to geostrophic balance, the streamfunction is expected to be a useful representation of the actual horizontal flow, which will be non-divergent to good approximation.
The top panel of Figure 3 shows the full anomaly response to a localised steady heating in a baroclinically unstable atmosphere. Two main features can be seen:

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a zonally confined patch around the forcing region (green circle) and an elongated zonal band of anomaly to north of the forcing.
The bottom panel of Figure 3 only shows the zonally asymmetric (azonal) component of the ψ and PV response, i.e., with zonal mean subtracted. We find that the anticyclone response in form of a confined anomaly patch remains mostly unchanged as we remove the zonal mean, while the northern, elongated feature disappears almost entirely. These results suggest that the  change to the background circulation as associated with non-zero ∆T has resulted in a subtropical monsoon anticyclone that is much more zonally confined that was found with ∆T = 0, but the role of the zonally elongated response at mid-latitudes cannot immediately be dismissed and we therefore examine this further in the following subsection.

Distinction between response to zonal mean and zonally varying part of the forcing
To further study the nature and relative importance of the zonal band structure seen in Figure 4 we conduct an experiment 305 using an imposed forcing identical to the one described Section 2, but without a zonal mean component (the zonal mean of the original heating was subtracted along the entire latitude band). Figure 4c shows the response of the system to the forcing without zonal mean component. As can be seen, the zonally symmetric band-structure is not present in this case and the PV anomaly response looks almost identical to the response in the case with full heating (including zonal mean), when subtracting the zonal mean of the final response (Figure 4b). This indicates that the response to the zonal mean part and to the azonal part 310 of the heating are almost independent and the zonally extended feature is not caused by the 'local contribution' of the forcing but only by its zonal mean component.
In particular the localisation to the west of the forcing is similar in Figures 4b and c, despite the corresponding basic states differing slightly in their zonal wind structure (due to the changes in wind corresponding to the zonal band structure, also seen later in Figure 5). This suggests that the direct advection by the mean flow is less important in localising the response and thus  Various studies (e.g., Ring and Plumb, 2007;Butler et al., 2010) have shown that simple dry GCMs with basic states similar to the HS configuration can indeed exhibit an annular-mode-like response when perturbed by a zonally symmetric forcing. As we have shown in Figure 4, the response to the zonally symmetric part of the heating is manifested in the zonally symmetric 330 part of the response, which can be separated from the zonally localised monsoon anticyclone response forced by the azonal part of the heating. meridional eddy advection of zonal wind u v (shading), where u and v describe the deviation of zonal and meridional wind from the zonal mean, respectively, and the overbar indicates the zonal mean. The mean eddy momentum transport is a measure for the strength of the baroclinic eddies and indicates their role in driving the mean flow of the system. Figure 5b shows the corresponding modification of the zonal wind and eddy flux when the system is forced by a local heating with Q 0 = 5 K/day.
A polewards shift of the jet can be seen, driven by changes in u v . Note that the mean flow and the eddies seem to respond 340 as a coupled system and the response is consistent with the typical EOF response associated with annular modes (e.g., Butler et al. (2010)). The results presented above suggest that the elongated mid-latitude feature shown in Figure 3 has no particular significance with regard to the monsoon anticyclone itself and hence that the addition of the meridional temperature gradient and the resulting baroclinic instability and jet formation is the key ingredient which results in zonal localisation of the monsoon 345 anticyclone response to the confined heating. One has to be careful when interpreting Figure 5 since the shown anomalies include the localised anticyclone structure in the forcing region that can be seen in Figure 5b. However, since the change in mean zonal wind is centred at about 40 • latitude, and hence north of the forcing, it seems to indeed mostly capture the annular-mode-like feature.
We now investigate in more detail how the monsoon anticyclone response to a confined steady heating changes as we gradually vary the meridional temperature gradient parameter ∆T . This will lead to a change in zonal jet strength, centre position and shape, as well as corresponding changes in the baroclinic eddies. Figure 5 shows the meridional profiles of the mean zonal wind at 13 km. It can be seen how the jets generally shift polewards and become stronger as ∆T is increased. Note that the zonal wind and meridional shear within the forcing region (10 − 30 • latitude) is almost the same for all shown values of ∆T 355 (the exception is the case with T as = 20 K which we will discuss in a later section). Several studies have previously investigated the effect of varying physical parameters in HS-like states, e.g., Zurita-Gotor (2008) who found a similar tendency for poleward-shift, strengthening and broadening of the jets when changing the meridional temperature gradient.
The altered background state does affect the time mean response of the flow to the localised heating, as can be seen in Figure   360 7. As we increase the meridional temperature gradient ∆T we see how the distinction between the two features discussed in Section 3.2 (localised anticyclone and zonal-band-structure) becomes more apparent. First, the response gradually becomes zonally localised at the latitudes of the heating, forming a coherent anticyclone in the forcing region. Second, the zonal mean component of the anomaly response seems to shift northward, which can be interpreted as formation of an annular-mode-like response, as shown earlier. Figure 5 shows the jet maximum tends to move polewards as ∆T increases and a similar polewards 365 shift of the corresponding EOF would not be surprising.
In the rest of this subsection we investigate the response to a steady confined heating for a summer basic state in which hemispheric asymmetry has been incorporated by setting T as = 20 K, as discussed in Section 2. The intention is not to contrast the hemispheres, but to obtain a basic state that is within the northern hemisphere (where the forcing is a located) a more realistic representation of the summer state, rather than the more annual-mean-like state of the original HS configuration with 370 T as = 0. Figure 8 illustrates the main difference between the basic state mid-latitude jet profiles for the annual-mean and summer states. Note in particular the zonal wind for the summer case at a height of 13 km ( Figure 8c) is close to zero in the centre of the forcing region at 20 • latitude.
The upper level ψ and PV response of the summer state to a steady localised forcing are shown in Figure 9. As for the symmetric case (Figure 3), we find an annular-mode response, in the form of a zonally symmetric band to the north of the 375 forcing region, and a pronounced localised anticyclone confined to the vicinity of the forcing region. The annular-mode-part of the response seems to be broader and span a larger meridional range than that observed in the case with symmetric basic state.
The structure of the localised anticyclone, however, seems to be qualitatively similar to the symmetric case. The fact that the localisation of the response in the annual-mean and summer cases are similar, while the zonal winds essentially vanish within the forcing region (see Figure 8) is further evidence that stirring by baroclinic eddies plays a significant role in localising the 380 response and it is not solely due to the direct advection by the zonal mean flow (also briefly discussed in Section 3.2).
The azonal mean response seems to show zonal extensions, with the response stretching eastward to the north and westward to the south of the main response, giving the response a a North-East/South-West tilted appearance. Similar extensions can be seen in the symmetric HS case (Figure 3) although they do not seem to extend as far out of the forcing region and are weaker in magnitude. Further examination (see Section 4) suggests that these zonal extensions arise from east-and westward eddy 385 shedding. As we explain eastward shedding is caused in part by advection by the mid-latitude jet and hence happens preferably closer to the centre of the jet (located north of the forcing), while westward shedding can only occur in the absence of strong eastward winds and hence occurs further to the south (where the zonal winds are weak or even westwards).
In the next section we investigate the time dependence of the monsoon response in a baroclinic background, with special interest in the phenomena of east-and westward shedding. clone PV structure during specific east-or westwards shedding events (e.g., Hsu and Plumb, 2000;Garny and Randel, 2013;Vogel et al., 2014). Figure 10 provides further illustration of shedding events, in this case as observed in re-analysis data during be identified on 11th of July in Figure 10a. Over the following days the northern edge of the anticyclone becomes distorted and the patch of low PV eventually breaks into two almost equally sized anticyclones on July 19th ( Figure Figure 10c). The events between July 11th and 19th are representative for a westward eddy shedding event. In the case shown the broken-off anticyclone does not propagate westwards very far, but stays rather in place. The reason might potentially be the interaction 400 with the mid-latitude jet and a pronounced baroclinic eddy, which can be seen at about -30 • longitude and 40 • latitude on 19th of July in Figure 10c . The meridional wind associated with the eddy further seems to disintegrate the shed vortex over the period until August 1st in Figure 10f.
A few days after the westward shedding event, on July 30th in Figure 10e, we can observe an example of an eastward shedding event. At about 130 • longitude one can see how a filament of low PV gets pulled out of the main anticyclone due to 405 meridional advection of a passing by baroclinic eddy. The filament then breaks off, rolls-up and subsequently gets advected eastwards by the mid-latitude jet on August 1st (Figure 10f). Note that, whilst there is significant variation between the daily PV fields shown in the different panels of Figure 10, in each case a clear monsoon anticyclone, manifested as a coherent low PV structure (albeit sometimes split into two) can be clearly identified. is entirely steady and simply consists of a westwards extending beta-plume which slowly decays with distance to the forcing, 420 probably due to the weak thermal damping of the model (as also mentioned in Section 3.1). In the case with a strong heating (Q 0 = 5 K/day) the behaviour of the model is completely different. In this case the system does exhibit a periodic creation and westward shedding of vortices from the region of (steady) heating, inducing a strong temporal and zonal variability. The eddies propagate westward at an essentially constant speed of about 12 m/s, indicated by the constant slope of the diagonal line in Figure 11d. The spontaneously emerging temporal variability of the response to a steady forcing that can be seen in Figure   425 11 is a potential explanation for the observed phenomenon of westward eddy shedding in relation with the observed monsoon anticyclone (e.g., Figure 10). Note that, as discussed in Section 3, the lack of strong dissipation in our model leads to an extreme westward elongation of the response and an eventual re-emergence of air parcels in the forcing region due to the spherical geometry of the domain.
In cases where the system is in the shedding regime, i.e. with strong heating magnitudes, this phenomenon can, in particular, 430 lead to a 'phase-locking process', where vortices that re-emerge in the forcing region can trigger a new shedding event. Such a phase-locking mechanism can potentially influence flow characteristics like the shedding frequency or the size of shed vortices and the corresponding details of the response have to be interpreted with caution. Other authors have previously investigated eddy shedding in a periodic domain, but either restricted their experiments to early time behaviour (Davey and Killworth, 1989) or did not mention the phenomenon of phase-locking (Hsu and Plumb, 2000). However, phase-locking is less 435 relevant for experiments in which the response is, via some process, zonally localised, for example experiments that include a representation of mid-latitude dynamics (as discussed in Section 3). In order to obtain a better understanding of how the westward shedding process evolves Figure 12 displays the horizontal distribution of ψ at 13 km for several consecutive days. At day 1495 (i.e., Fig. 12a) we find a localised anticyclone developing inside the forcing region and slowly strengthening and propagating westward over the next days. As it moves out of and away 440 from the forcing region the stream function inside the forcing region decreases slightly (see Fig. 12d). Once the anticyclone moved sufficiently far away from the forcing the stream function another isolated vortex starts developing inside the forcing region and correspondingly the stream function starts to increase again (Fig. 12f). Figure 12 suggests the shedding process to be characterised by the periodic production of individual vortices inside the forcing region along with a continuous westward propagation of the strengthening vortex, resulting in alternating periods of high and low equatorward flow anomalies within the 445 forcing region and therefore variations of the southward advection of high background PV. We find the evolution of the stream function response during a westward shedding event in our three-dimensional model to be similar to what has been described by other authors in relation to single-layer studies on westward shedding (Hsu and Plumb, 2000;Davey and Killworth, 1989).
For the rest of this section we focus on experiments in the shedding regime, i.e., experiments with imposed heating of magnitude and investigate the details of the intrinsically emerging temporal variability for various basic state configurations.

Transition to eastward eddy shedding via introduction of mid-latitude dynamics
Next we investigate how the (transient) behaviour of the response changes as we make the basic state baroclinically unstable by introducing a meridional temperature gradient, i.e., by increasing ∆T in Equation 2. As mentioned earlier, the baroclinically unstable basic state develops westerly jets and baroclinic eddies in mid-latitudes. The latter induce a strong spatial and temporal variability to the background flow and hence the PV and stream function fields generally show a range of quickly evolving 455 spatial structures which can correspond to anomalies of fairly large magnitude compared to the time mean field. Since the response magnitude of the explicitly forced anticyclone strongly depends on the forcing strength (as suggested by Figure   11) one can imagine situations where the response to a steady localised heating is weak relative to the varying anomalies of the background and thus it is difficult to identify a clear and pronounced anticyclone on a day-to-day basis. In situations with sufficiently strong forcing, and therefore strong response, one would, on the other hand, expect to see a well-defined anticyclone 460 in terms of a coherent low-PV structure (as is typically the case in re-analysis date; see Figure 10).  Figure 13 shows instantaneous snapshots of the PV response to heating distributions with varying magnitude Q 0 . For the case with Q 0 = 10 K/day a clear patch of low PV can be identified in the vicinity of the forcing region, relating to a clearly visible anticyclone, i.e. similar to the observed cases shown in Figure 10. For the weaker heating with Q 0 = 3 K/day, on the other hand, the anticyclone is barely visible and could easily be misinterpreted for a feature of the basic state (e.g., a baroclinic 465 eddy). This dependence of the response amplitude on the forcing magnitude is a persistent and usual characteristic of the flow.
Whilst the PV field of course varies on a day-to-day basis, these overall characteristics of the PV field (as the forcing magnitude varies) are robust and reproducible. In all three experiments shown in Figure 13 the PV anomaly of the anticyclone becomes clearly visible when taking long time averages since the azonal anomalies of the background state average out and the monsoon response creates a uniquely identifiable azonal feature in or near the heating region.

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The nature of time dependence of the response changes fundamentally as we gradually increase ∆T and thus gradually introduce active mid-latitude dynamics. Figure 14 shows the time evolution along latitude lines at the northern edge (top panel) and through the centre (bottom panel) of the forcing region for a range of values of ∆T . In the case where the basic state is at rest (Fig. 14a+b)  In the case with strong mid-latitude dynamics (Fig. 14g+h), on the other hand, almost no evidence for westward shedding can be seen, but we find clear indication of eastward propagating coherent features. These eastward shed eddies have pronounced PV signatures at 30 • , i.e., to the north of the forcing region and closer to the centre of the mid-latitude jet (see Figure 6), suggesting them to be advected eastwards by the background wind. To gain a better understanding of the dynamical processes leading to the modified temporal variability of the response in 480 experiments with strong mid-latitude dynamics (e.g. Figure 14d+h) it is useful to study in detail the evolution of a specific eastward shedding event. In order to see a clear evolution of the anticyclone on a day-to-day basis (as discussed earlier; see, e.g., Figure 13) we choose a forcing magnitude of Q 0 = 10 K/day for this experiment, although we see similar behaviour for weaker magnitudes (as also suggested by Figure 14 for Q 0 = 5 K/day). 1535. This behaviour, which in our simulations occurs repeatedly as a response to a combinations of steady monsoon heating and statistically steady baroclinic eddy dynamics in the extratropics, is very similar to what has been reported during specific events (e.g., Enomoto et al., 2003;Garny and Randel, 2013). Note that the centre location of eastward shed vortices in Figure   15 is located northward of the forcing region, in agreement with the PV signals observed in Figures 14e and g.

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A slightly different kind of behaviour seems to be happening for the weakly baroclinic background state with ∆T = 10 K. As can be seen in Figure 14d a coherent bulk anticyclone has formed at 20 • latitude and between about -20 • and 100 • longitude.
There is no pronounced time variation or preferred direction of propagation obvious and the response seems relatively steady.
However, to the north of the forcing region ( Figure 14c) patches of anomaly seem to appear to the west of the forcing region, propagate eastwards towards it and eventually disappear near the eastern edge of the forcing. In the following paragraphs we 500 discuss this state in more detail, before moving on to a (hemispherically asymmetric) summer basic state. Figure 16 shows how the PV field on the 340 K isentropic surface evolves between days 1542 and 1550 for a background state with ∆T = 10 K. The arrows highlight the position of two pronounced baroclinic eddies forming a wave-like structure on the mid-latitude PV gradient of the background. As the eddies move eastward, they start to interact with the PV minimum of the anticyclone. The meridional winds associated with the horizontal gradients in PV pull low PV out of the anticyclone northward 505 on one side and advect heigh-value PV southward on the other. A potential dynamic interaction of the monsoon anticyclone could then further intensify this baroclinic wave structure. It is also possible that the combination of strong westerlies of the mid-latitude jet and the northern part of the anticyclone form a sort of wave-guide, as pointed out by Enomoto et al. (2003), which additionally favours certain waves and hence leads to a stronger modulation of the PV field. Amemiya and Sato (2018) observed a somewhat similar behaviour in a steadily forced single-layer model with meridionally 510 varying mean depth, inducing a background westerly mean wind. They describe a statistically steady localised response, which experiences westward shedding of eddies followed by a 'circling back' of shed vortices towards the forcing region. However, it should also be noted that the basic state used by Amemiya and Sato (2018) is associated with a pronounced reversal of the meridional PV gradient at the latitudes of the imposed forcing due to the varying mean depth of the modelled fluid layer. It is questionable if such a state is a useful representations of the sub-tropical UTLS, as to some extent discussed by e.g. Kraucunas 515 and Hartmann (2007), who argue that sub-and extratropical basic state flows should not be introduced through a layer depth gradient to avoid unrealistic behaviour of the system.
The process shown in Figure 16 leads to the emergence of coherent PV anomaly structures at about -20 • longitude and subsequent eastward propagation of such, as shown in Figure 14c. However, the meridional advection does not lead to a complete separation of low-PV patches from the bulk anticyclone and subsequent eastward propagation of these patches, as we 520 observe for basic state configuration with stronger meridional temperature gradient (see Figure 15). A potential explanation is that the baroclinic eddies in the case with ∆T = 10 K are too weak to produce sufficient meridional advection of PV and/or are located to close to the forcing region to create elongated PV filaments (recall that the jet core is located farther north for larger values of ∆T ).
Different authors have presented indication for the existence of a bi-modality of the monsoon anticyclone, with respective 525 east-or westward displacement of the anticyclone centre (e.g., Zhang et al. (2002)). Similar theories suggest the occurrence of a 'split-state' of the anticyclone, showing two distinct centres (e.g., Vogel et al. (2015)). Nützel et al. (2016) found that only one (NCEP-1) out of seven analysed re-analysis datasets showed indication for a pronounced bi-modality for a range of time scales. The modulations of the PV low of the anticyclone by baroclinic eddies in Figure 16 can produce pronounced and zonally separated PV minima on certain latitudes. It is not clear if this mechanism, however, can potentially explain the observation of 530 mentioned split-phases in certain diagnostics and datasets.
Since the phenomenon described above modulates the northern edge of the anticyclone it creates a north-south asymmetry of the PV minimum in the form of a wavy northern edge and an only weakly perturbed southern edge. We find similar modulations of the northern edge in PV fields of re-analysis datasets, e.g. just before a westward shedding event on July 15th in Figure 10.
We also observe east-and westward shedding behaviour in experiments with an (hemispherically asymmetric) summer basic 535 state, i.e. when choosing T as = 20 K in Equation 1. Figure 17 shows the time evolution of PV along two latitude lines on the 335 K isentrope. Clear evidence for both, westward shedding at the latitudes of the forcing region and eastward shedding north of it, is visible. Westward shedding seems to occur much more frequently than eastward shedding and the shed vortices travel much farther. As described for the symmetric cases in Figure 14 the westward shedding seems to happen mostly on the latitude of the forcing, while eastward shedding has a clear signature to the north of the forcing region. Both characteristics, the north-540 /southward shift of east-/westward shedding and the relatively more pronounced westward events, seem to be consistent with re-analysis data observations and our general understanding of these features (e.g., Popovic and Plumb, 2001;Enomoto et al., 2003;Vogel et al., 2014).
A crucial aspect of the relative importance of east-/westward shedding is potentially given by the strength and position of the mid-latitude jet relative to the forcing region. As explained, Figure 14 shows the change from west-to predominantly eastward 545 shedding as we increase ∆T in the (symmetric) annual-mean HS state. This induces (among other things) a strengthening of the jet and baroclinic eddies, both features are essential for the occurrence of eastward shedding process. Hence we find the described transition as the mid-latitude flow becomes stronger with increasing ∆T . In the (asymmetric) summer case we find evidence for both types of shedding, although a clear dominance of westward shedding can be seen. Figure 17 indicates generally weaker and poleward shifted jet for T as = 20 K, compared to the symmetric case with T as = 0. The existence of such

Summary and conclusions
In this study we analysed the response of a three-dimensional dry dynamical model to a steady and spatially localised imposed heating distribution, aimed to model the Asian monsoon anticyclone circulation. Particular focus was given on the modification 555 of the response when the localised forcing was applied to an atmosphere that includes a simple representation of mid-latitude dynamics (mid-latitude jet and baroclinic eddies) compared to an atmosphere at rest. In a range of numerical experiments we identified a set of characteristics and behaviours that are potentially relevant when describing the three-dimensional circulation of the Asian monsoon anticyclone, including a localised zonal scale and eastward and westward eddy shedding phenomena.
As shown, in re-analysis data the time mean structure of the PV low associated with the monsoon anticyclone is zonally 560 localised and thus confined to the vicinity of the forcing region. In numerical model simulations with resting basic state (∆T = 0), however, the response to a simple monsoon heating is only very weakly localised, with the anticyclone extending far away from the forcing region and its zonal extent essentially being determined by the weak thermal damping of the upper troposphere lower stratosphere. In experiments with mid-latitude dynamics (∆T = 0) the response forms a localised anticyclone at the latitudes of the forcing and a change in zonal mean flow to north of forcing region. Further examination supports the idea that 565 these two aspects of the response are effectively independent and that the interactions with mid-latitude dynamics, in particular the baroclinic eddies, allow a time-mean response that is localised to the west and also extends to the (north-)east. The details of the spatial structure of the anticyclone change as ∆T is varied or the basic state is changed from a more annual-mean state to a state more representative of the summer time conditions in the northern hemisphere.
The dependence of the spatial structure of the time-mean response on the background is easier to understand if we look 570 in more detail at time dependence of the system. In the case with resting basic state the response shows a transition from a steady beta-plume state to a state with westward eddy shedding for sufficiently strong forcing. In cases with time-varying basic state the nature of the time dependence of the response changes significantly and the system exhibits a transition from a state with westward shedding only to a state dominated by eastward shedding as ∆T increases, and thus structure and strength of the background flow change. While in some cases (equinox flow) westward shedding seems to be inhibited, in other cases 575 westward shedding persists (summer flow) and the response shows signs of both shedding-behaviours (as observed in the real atmosphere).
The presented model exhibits a range of behaviours and reproduces various properties of the monsoon circulation. The combined simplicity of the setup and ability to simulate different monsoon characteristics provides potential conceptual explanations for many aspects of the monsoon structure and variability and gives a way to study them quantitatively.