## Abstract

Prediction of long-lived anomalous behaviour in the atmosphere is fundamental to extended range and seasonal forecasting. Prediction of changes in the climatology of such anomalous behaviour is also fundamental to regional climate modelling. Anomalous atmospheric behaviour is often related to mid-latitude tropospheric ‘blocking’ patterns, where the normal westerly flow associated with the temperature difference between the Poles and the Equator is disrupted. Following recent work on stratosphere–troposphere coupling, we show that the vertical structure of the atmosphere can strongly influence the climatology of tropospheric blocking. We invoke dynamical theory to argue that the development and decay of anomalous circulations is most efficient for a preferred aspect ratio of the flow, implying that the development of large-scale anomalies requires a large vertical scale. Evidence for this link comes from the observed evolution of the geopotential height. In particular, we find that the development of the large-scale tropospheric anomalies associated with blocking requires a vertical scale extending well into the stratosphere. This process is inhibited during periods of high stratospheric activity, when the vertical scale of tropospheric developments is restricted, leading to the persistence of large horizontal scales.

## 1. Introduction

Now that numerical weather prediction for a week ahead has achieved a fair degree of accuracy, there is huge interest in identifying useful signals of predictability for longer periods. Any success in this would have very large economic benefits. A key aspect of this is improved understanding and prediction of the ‘blocking’ patterns that disrupt the normal climatological atmospheric circulation and are responsible for persistent anomalous weather [1]. A related problem concerns the possible changes to the climatology of anomalous weather patterns, and hence regional climate, as a result of global warming. The importance of blocking to weather and climate is now recognized in the mainstream media, where it is mentioned in stories on exceptional cold spells [2].

This paper seeks to advance understanding of the dynamical mechanisms underlying tropospheric blocking by examining five Northern Hemisphere winters. Using best estimates of the state of the atmosphere produced by a numerical weather prediction centre (so-called operational analyses), we show that instantaneous circulation patterns are almost always transient and baroclinic (so that the wind direction changes with height). This implies that, despite their long-lived nature, anomalies cannot be described using quasi-steady solutions of the governing equations, for such solutions would have to be barotropic (so that the wind direction does not change with height). Although most studies of long-lived anomalies are statistical in nature, typically relying on empirical orthogonal functions (EOFs), the preceding considerations are still relevant. For example, the leading EOF in the classic study of stratosphere–troposphere coupling by Baldwin & Dunkerton [3] is some way from being barotropic and does not represent a quasi-steady solution of the equations of motion.

In support of this, we note that barotropic dynamics cannot describe the development and decay of anomalous patterns on scales larger than the deformation radius (see [4] for background and [5] for numerical illustration); furthermore, weather systems occur on length scales comparable to the deformation radius, where rotational effects become dominant. The existence of large-scale blocks therefore implies that blocking development and decay must be essentially baroclinic [6]. Such developments require the conversion of available potential energy to kinetic energy.

Standard dynamical theory helps elucidate this process. The energy conversion is most efficient when the aspect ratio of the flow assumes a preferred ratio defined by the local rotation and stratification: *H*/*L*∼*f*/*N*, where *H* and *L* are characteristic vertical and horizontal scales, *f* is the Coriolis parameter and *N* is the Brunt–Vaisala frequency [7], p. 217. Thus, there is maximum variability on a horizontal scale *L*∼*L*_{d}, where *L*_{d}=*NH*/*f* is the deformation radius, whereas the dynamics are relatively stable (i.e. anomalies tend to persist) for *L*>*L*_{d} [4]. We associate changes in blocking with *L*∼*L*_{d} and persistent blocks with *L*>*L*_{d}. Changes in blocking occur via baroclinic development and decay when the horizontal scale is large but not too large: anomalies have small Doppler-shifted, Rossby-wave phase speeds and are long-lived [8], §12.3 for zonal wavenumbers *k*∼3–4 (or equivalently *L*∼1300–1700 km).

The foregoing suggests that the vertical structure of the atmosphere plays a key role in both weather and climate. Traditionally, the stratosphere, which lies above the tropopause at approximately 10 km, has been considered to be of secondary importance to numerical weather prediction. This is because the standard theory for the development of mid-latitude weather systems considers only tropospheric dynamics. However, recent work on medium- to long-range forecasting has highlighted the influence of the stratosphere on the troposphere. Although the idea that the stratosphere influences wave properties of the troposphere [9] as well as its steady state [10] is not new, the upsurge of activity has been largely inspired by Baldwin & Dunkerton [3], among others, who showed that stratospheric anomalies precede the development of tropospheric anomalies, implying downward propagation on a time scale of several weeks and predictability of the troposphere based on the state of the stratosphere. Many follow-up studies have examined implications for weather forecasts [11] and climate prediction [12]. One topic of research in this area is the connection between tropospheric blocking patterns and stratospheric anomalies, more particularly sudden warmings [13,14].

In a review of the evidence for stratospheric influence on blocking, Woollings *et al.* [15] noted that the results are somewhat contradictory. From their own study, based on a statistical analysis of the ERA-40 dataset, they concluded that there are several different mechanisms or associations between blocking and stratospheric variability. The complexity of stratosphere–troposphere coupling has been noted in many other studies.

Nevertheless, the possible effect of stratospheric structure on baroclinic development has been studied by a number of authors. Wittman *et al.* [16] examined the effect of the vertical shear of the zonal wind in the lower stratosphere. The vertical wind shear plays a key role in baroclinic instability; in the troposphere, it drives the development of mid-latitude weather systems through the transport of temperature (or mass). From a linear stability analysis and idealized life cycle simulations, Wittman *et al.* [16] show that positive vertical wind shear in the lower stratosphere increases the horizontal scale of the fastest growing mode, which favours a greater likelihood of long-lived anomalous circulations (i.e. blocking), and the vertical extent of disturbances. This is consistent with our claim that the efficiency of baroclinic growth depends on the aspect ratio: a positive vertical shear in the lower stratosphere increases the depth of the layer where there is a single-signed vertical shear. Colucci [17] shows that stratospheric contributions to the time evolution of the geopotential can be just as important as tropospheric contributions in a number of observed cases, including the development of a blocking pattern. King *et al.* [18] studied a particular case of baroclinic growth to illustrate that the baroclinic energy conversion process continues to operate into the lower stratosphere.

Our study complements these by using operational analyses to examine the importance of this mechanism over a complete season and by using global statistics rather than local diagnostics. Our diagnostics show that the baroclinic energy conversion process contributes directly to the growth and decay of geopotential anomalies into the stratosphere on the occasions when there are large changes in blocking patterns. Although the sign of the average vertical shear (and thus of local baroclinic energy conversion) typically reverses at the level of maximum wind, the effects of baroclinic instability can extend across the tropopause. As shown by Wittman *et al.* [16], positive vertical shear can be maintained locally. This direct upward effect is not captured in the standard Eady model of baroclinic growth, which lies behind much thinking about baroclinic development, because of the use of a rigid upper boundary condition at the tropopause. We characterize this effect using the vertical integral of the thermal advection, a quantity that measures poleward heat transport and lies at the heart of baroclinic instability. Baroclinic development occurs when the sign of this vertical integral is maintained over depth.

Section 2 reviews the data and diagnostic methods used in our study. In §3, we first illustrate that horizontal thermal advection is a major contributor to the amplitude and variability of blocking: this means that baroclinic processes must be important. We then show that the vertical scale of the developments is time-dependent and related to their amplitude and horizontal scale. We finally investigate the link with the basic-state baroclinicity: while thermal advection typically remains important in the stratosphere, implying a vertical scale extending across the tropopause, other contributions to the stratospheric evolution can dominate. On such occasions, the vertical structure of baroclinic development is restricted with an accompanying effect on the deformation radius and the preferred scale of baroclinic developments. The inference is that the stratosphere can affect the climatology of tropospheric blocking through control of the deformation radius.

Our results imply, in agreement with Scaife *et al.* [19], that an accurate representation of the stratospheric vortex structure (and thus the vertical wind shear in the lower stratosphere) is necessary for accurate predictions of blocking. We do not investigate the stratospheric processes responsible for establishing a particular vortex structure. This has been extensively studied in the literature; for instance, see the review by Gerber *et al.* [20].

## 2. Data and diagnostic methods

### (a) Data

The data are derived from Met Office operational analyses for the winters of 2006/7, 2007/8, 2008/9, 2009/10 and 2010/11, archived daily at 00 UTC. For brevity, the different winters will sometimes be referred to using only the year at the beginning of the season. The horizontal resolution of the analyses was 40 km from 2006/7 to 2009/10 and 25 km for 2010/11; in the former case, there are *N*_{x}=400 points in the zonal direction, in the latter *N*_{x}=640. The vertical domain extended to 40 km (with 50 vertical levels) in 2006/7 and 2007/8, whereas, in the remaining years, it extended to 80 km (with 70 vertical levels).

We analyse the geopotential height on pressure levels. Geostrophy and hydrostatic balance are sufficiently accurate on scales of interest to allow inferences to be made about winds and temperatures. A 3 day running mean is used to focus attention on large-scale synoptic changes, associated with the formation and breakdown of blocking patterns, rather than on individual baroclinic developments. This differs from the case studies of stratospheric influence on baroclinic development [17,18]. Blocking diagnostics in the literature often impose a temporal persistence condition (typically approx. 5 days; [21]), which represents a stronger constraint than the running mean.

We consider data between 30^{°}N and 60^{°}N. The geographical restriction is arbitrary but convenient: the wave-mean decomposition that underlies our spectral analysis (see §2*c*) is difficult to interpret at higher latitudes; previous studies [22] have avoided this problem by using different methods of analysis.

### (b) Baroclinic structure

We begin by demonstrating that the observed data are strongly baroclinic. Figure 1 shows maps of the geopotential height for 1 December 2010 at four levels: 500, 100, 50 and 10 hPa. On this date, which fell within a very cold period over the UK [23], there are large anomalies at 500 hPa over northwest Europe and the eastern Atlantic. However, it is clear from the geopotential at higher levels that this was not at all a barotropic pattern: the phase of the anomalies changes significantly with height. This extensive baroclinicity is bound to lead to highly transient behaviour on synoptic time scales, and even if barotropic instability contributes locally, then it is likely to be swamped by the changes induced by baroclinic effects.

Figure 2 shows corresponding maps for 1 February 2011. By this time, temperatures had increased over the UK, and the flow was much more large scale and zonal. The anomalies are more strongly correlated up to 50 hPa than in figure 1. Above 50 hPa another large anomaly develops, but it is out of phase with the lower level flow.

Further evidence for the baroclinic nature of disturbances comes from figure 3. Over a significant depth of the atmosphere (extending up to 20 hPa), the geopotential amplitude decreases almost simultaneously from day 55 to day 60, and increases almost simultaneously from day 70 to day 75. This is consistent with baroclinic development or decay over this depth but not with vertical wave propagation. Explanations of stratospheric variability often assume that baroclinic growth and decay are confined to the troposphere, tropospheric variability being transferred to the stratosphere via vertical wave propagation. Recent examples include Scaife *et al.* [19] and Woollings *et al.* [15]. This line of thinking follows the classical analysis of wave trapping given by Charney & Drazin [24], which explains the weak propagation of planetary waves into the summer hemisphere and the filtering of small-scale disturbances in the winter hemisphere [25]. But given that typical vertical group velocities of Rossby waves are approximately 5 km per day [8], p. 523, vertical propagation does not occur rapidly enough to explain the quasi-simultaneous growth and decay of amplitude observed in figure 3. A more natural explanation is that baroclinic growth will naturally extend into the stratosphere, for the tropopause is not rigid. This effect is included in the modified Eady model of baroclinic growth studied by Wittman *et al.* [16].

Figures 1 and 2, along with similar ones from other dates and years (not shown), suggest that the vertical scale, *H*, over which coherent baroclinic development takes place is not constant. This is significant because, within our framework of three-dimensional balanced dynamics, changes in the vertical scale are related to the horizontal scale and thus to blocking. As noted in §1, baroclinic instability theory shows that energy conversions between available potential energy and kinetic energy are most efficient when the aspect ratio *H*/*L* equals *f*/*N*.

The characteristic zonal wavenumber associated with baroclinic activity can be estimated for some limiting cases: taking *N*=10^{−4} and *f*=10^{−4} s^{−1}, then *k*∼5 when *H*∼10 km (250 hPa), which is appropriate when the stratosphere decouples from the troposphere. If the stratosphere and troposphere were strongly coupled and the same value of *N* retained, then we would obtain *k*∼3 when *H*∼16 km (100 hPa). However, the increased value of *N* in the stratosphere means that *H* will not need to be as large as this. Although these estimates are very rough, they do suggest that *H* must extend across the tropopause into the lower stratosphere in order for the preferred wavelength to be around 3–4 and blocking to occur. In agreement with the idea that anomalies persist for *k*<*k*_{d}, numerical weather prediction models show enhanced stratospheric and tropospheric predictability for [26], while for larger *k* (in the synoptic range of scales) predictability is rapidly lost via baroclinic activity.

We look for evidence of a link between the typical horizontal and vertical scales by diagnosing the horizontal scale of disturbances and the vertical coherence of the basic state.^{1} Because *H* depends on the stratospheric structure, it will not be surprising if it is strongly time dependent, a feature that cannot be included in the standard quasi-geostrophic analysis.

### (c) Diagnostic methods

Our diagnostics use ‘global’ (i.e. area-averaged) flow properties. This allows estimates of mean horizontal and vertical scales, which can, in turn, be related to properties of the basic state. Diagnostics of this type have been used, for instance, by Hussain & Lupo [27] and Lupo *et al.* [28]. Most previous studies of blocking have used local diagnostics, notably a persistent reversal in the westerly geostrophic wind [1] or the westerly thermal wind [21] at a specific longitude along some ‘central blocking latitude’. As blocking depends on both amplitude and scale (see §1), these aspects are diagnosed separately.

We diagnose the amplitude by calculating the mean magnitude of the meridional geostrophic wind between 30^{°}N and 60^{°}N. This is proportional to
2.1
where *ϕ*_{b}=*π*/6,*ϕ*_{t}=*π*/3, and
where (*λ*,*ϕ*) are longitude and latitude, and is a 3 day running mean of the geopotential height on a pressure level *p*. The partial derivative is approximated with a centred difference; for convenience, the zonal endpoints are excluded from the discrete representation of the integral.

The integrand in (2.1) represents the magnitude of the meridional geostrophic wind, *v*, multiplied by 2*Ωa*, where *Ω* is the angular velocity of the Earth’s rotation and *a* is the radius of the Earth. In the extratropics, *v* does not exhibit a systematic variation with latitude. Large values of correspond to situations in which the disruption of the predominantly zonal flow by a strong meridional component is maximized. A blocked circulation pattern requires a high value of and a horizontal scale around wavenumbers 3–4; taking the absolute value of means that we do not distinguish between different types of blocks, e.g. those bringing cold polar air to the extratropics (as in the case of the major blocking episodes that occurred over the UK during the winter of 2010/11) and vice versa. We also do not distinguish cases where the large amplitude occurs simultaneously in several different geographical locations from cases where the zonal flow is only disrupted in a single sector. will be used as an indirect measure of the strength of blocking. It is less suited as a blocking criterion. Not only is the scale dependence neglected, a threshold value also needs to be specified, as in Tibaldi & Molteni [1].

We first demonstrate that the time variations of (and by extension blocks) are strongly related to baroclinic energy conversion. According to quasi-geostrophic theory [8], p. 530, the rate of conversion of potential to kinetic energy is the product of the vertical velocity *ω* and the potential temperature. On scales larger than the deformation radius, which is the scale of interest here, and in the absence of radiative or dissipative effects, *ω* is determined primarily by the thermal advection, ** u**⋅

**∇**

*T*[8], p. 208. Because measures the amplitude of geopotential gradients, we calculate the magnitude of the vertically integrated thermal advection, , which contributes to the geopotential tendency, 2.2 Constant scaling factors have been omitted. As is conventional, only the meridional contribution to the thermal advection,

*vT*

_{y}, is considered; the results reported below change slightly when the zonal contribution is retained. From the definition of the geopotential, , where

*R*is the gas constant and

*g*is the gravitational acceleration, it may be seen that measures the contribution of thermal advection to the magnitude of the time tendency of the geopotential, |

*Z*

_{t}|. Here,

*p*

_{b}=900 hPa.

We then demonstrate the expected relation between horizontal and vertical length scales for the efficient conversion of potential to kinetic energy. We diagnose the horizontal length scale *L* spectrally. Decomposing into Fourier modes in the longitudinal direction and writing the mode coefficients as gives
2.3
*k*_{m}=20 is the number of retained modes. This value is appropriate for large-scale balanced dynamics; the results change slightly when the complete set of modes is used. For brevity, the dependence on pressure and time has been suppressed.

We then define *L* by averaging over latitude. Set
2.4
whence
2.5
and then define *L*=2*πa*/〈*k*〉. To facilitate, comparison between years 〈*Z*〉 is normalized by the number of horizontal grid points, 〈*z*〉=〈*Z*〉/*N*_{x}. Note that the zonal mean, *k*=0, is excluded from 〈*z*〉.

The diagnostic for the vertical scale, *H*, should reflect the depth of the baroclinic developments. An objective calculation is difficult. One approach that comes immediately to mind is projection onto vertical normal modes [29]; however, the increase in the amplitude of the pressure perturbation (by a factor of 300 from the surface to 80 km on account of the exponential decay of density) and the sensitivity of the mode structure to the position of the model top makes this method unsuitable for analysing real data.

A typical disturbance vertical scale can be inferred from time series of . This implies that the vertical scale is the depth over which geopotential changes remain well correlated. It is expected that this will extend into the stratosphere, because the maximum effect of thermal advection on the geopotential tendency will be at the tropopause, but the effect will decrease at higher levels, the rate of decrease depending on the stratospheric structure. This is quantified by the correlation, over the length of the time series *T*_{s}, between a given level *p* and a reference level *p*_{0} (which is unrelated to *p*_{b}),
2.6
The angle brackets denote an area average (see (2.4)), whereas the overbar denotes a time average; is the standard deviation of . A large value of indicates that the depth available for baroclinic development extends from *p*_{0} to *p*; henceforth, we restrict attention to . For brevity, we abbreviate this as and suppress the units from *p*.

We wish to relate the typical disturbance length scales to properties of the basic state on which the baroclinic waves develop. In order to impose a scale separation between the basic state and the important scales for blocking developments, we define the basic state, , as the restriction of the geopotential to wavenumbers 0 and 1. Using (2.4), this gives
2.7
This type of basic state has been used to study the vertical propagation of planetary waves in the stratosphere [22,30] and the forcing of the polar vortex during episodes of Euro-Atlantic blocking [14]. The scale separation is clearly somewhat arbitrary as the actual evolution will be fully nonlinear.^{2}

Such a basic state will not usually be a steady-state solution of the equations of motion. For the idea of baroclinic growth on this basic state to make sense, we need a separation of time scales between the evolution of the basic state and that of the disturbances themselves, which is typically *O*(10) days. Analogous to (2.5), we first calculate , the mean amplitude of the basic-state geopotential, from time series of ; we then calculate the standard deviation to characterize the time variability. Large values suggest that there is no time-scale separation.

For geostrophic, hydrostatic flow, the baroclinicity is measured by *u*_{z}, the vertical derivative of the zonal geostrophic wind, or equivalently *T*_{y}. The importance of the vertical shear for stratosphere–troposphere coupling has been discussed by Wittman *et al.* [16]. A basic-state vertical scale is obtained from , the vertical integral of *u*_{z},
2.8
(Constant scaling factors have again been omitted.) increases with height up to the level of maximum wind and then decreases at a rate determined by the stratospheric structure. Because the hydrostatic relation makes it difficult for the geostrophic wind (and hence *v*) to vary rapidly with height, we expect to be closely related to . Indeed, we have confirmed that *v* is strongly correlated from 500 to 50 hPa.

## 3. Results

We now demonstrate the main implications of the dynamical theory reviewed in §1 and elaborated upon in §2*b*. These are

— Thermal advection is a major contributor to the variability in the large-scale geopotential and to the amplitude and variability of blocking. This means that baroclinic processes are relevant to blocking development and decay.

— The vertical scale of the developments is time dependent and related to their amplitude and horizontal scale. In particular, the development and decay of blocking patterns requires a vertical scale extending well into the stratosphere.

— The vertical scale of the developments, which is effectively a disturbance scale, is related to properties of the basic state. In particular, the vertical scale is reduced when the stratosphere is highly transient.

In this section, we examine these predictions using seasonally averaged statistics for all five winters and detailed analyses for 2008–2010.

### (a) The importance of thermal advection

In this section, we demonstrate the relevance of thermal advection to the baroclinic development processes by comparing the time evolution of the meridional wind, , with the vertically integrated thermal advection, . Figure 4 plots time series of and for *p*=500, 300, 100 and 50 hPa. Beginning with for 2010/11 (figure 4*a*), the curves for the two tropospheric levels of 500 and 300 hPa are very similar in character, basically changing in step with each other. Between days 25 and 60, these curves show large-amplitude episodes, corresponding well to observed blocking episodes in the European sector: December 2010 was the second coldest in the 350 year time series of central England temperatures [23]. In the second half of this period, there is less variability and lower amplitudes, in agreement with the observed reduction of blocking in the European sector. In the stratosphere, is larger between days 25 and 50; after day 90 these variations are not particularly well correlated with those in the troposphere.

We expect the vertically integrated thermal advection to show more time variability than because is directly related to the time derivative of *Z*. The link between and *Z*_{t} is assessed by calculating |*ΔZ*/*Δt*|=|(*Z*(*t*+*Δt*)−*Z*(*t*))/*Δt*|. Using the omega equation to estimate the vertical velocity [32], the ratio of to |*ΔZ*/*Δt*| should be approximately 2 on scales where the effects of vorticity advection and thermal advection are comparable (i.e. for *L*∼*L*_{d}). Averaging over the season with *Δt*=24 h, the ratio is approximately 2 at 300 hPa, approximately 6 at 50 hPa and even larger in the stratosphere (presumably because other processes contribute to the geopotential tendency). These results confirm that is an important contributor to *Z*_{t}. (There are similar results for other years.)

We determine whether thermal advection and baroclinic development correspond well by assessing the degree to which persistent increases and decreases in are reflected in . Because is related to |*Z*_{t}|, whereas is related to |*Z*_{λ}|, is associated with smaller horizontal scales than . Consequently, we also compare with 〈*Z*〉, which is an alternative diagnostic for baroclinic development.

Turning to the time series of (figure 4*b*), they are well correlated at the two tropospheric levels of 500 and 300 hPa. Generally, the largest peaks occur during periods of large variation in , suggesting that tropospheric variations of are strongly influenced by thermal advection and baroclinic effects. During the second half of the season, however, the correspondence between and is weaker: although is lower on average, remains large. Time series of 〈*Z*〉 (figure 5*a*) support the claim that the correspondence between baroclinic development and thermal advection is weaker during the second half. There is a general upward trend in and 〈*Z*〉 but not .

The differences in the degree of correspondence among , and 〈*z*〉 from the first to second halves reflect differences in stratospheric structure. In the absence of significant large-scale dynamical processes in the stratosphere, time series of and 〈*z*〉 stay well correlated, because tropospheric geopotential anomalies propagate upwards via hydrostatic balance. In the first half of the season (up to day 60), changes little above 300 hPa, and stratospheric thermal advection is unimportant. Thus, is well correlated between 300 and 100 hPa (the correlation for days 0–60 is 0.815); however, the corresponding time series of 〈*z*〉 are not well correlated. This indicates that there are large temperature changes in the lower stratosphere on the large scales sampled by 〈*z*〉; because 〈*z*〉 is well correlated through the stratosphere, deep circulations may be responsible for these changes. In the second half of the season, there is significant stratospheric thermal advection and relatively little vertical correlation in ; however, stratospheric and tropospheric time series of 〈*z*〉 become fairly well correlated after day 80. This indicates that stratosphere–troposphere coupling is being established on the large scales sampled by 〈*z*〉.

Figure 4*c*,*d* shows equivalent plots for 2009/10. The most obvious difference with respect to 2010/11 is that a clear contrast between the first and second halves of the season cannot be seen. This is obvious in the tropospheric time series of . With respect to , the most striking tropospheric features are the two short-lived peaks around days 25 and 75. With respect to , there is a consistent upward trend in stratospheric values (up to day 80) which is not reflected in either or 〈*z*〉 (figure 5*b*). After day 40, increases significantly above 300 hPa, implying that stratospheric thermal advection has also increased in strength. This indicates strong decoupling of stratosphere and troposphere.

Figure 4*e*,*f* shows equivalent plots for 2008/9. As in 2009/10, a clear contrast between the two halves of the season cannot be discerned. Average values of are greater than in 2009/10, but average values of are similar. The key difference between the years is the degree of vertical coupling. In 2008/9, the time series of are very well correlated between the troposphere and stratosphere, suggesting that there is strong stratosphere–troposphere coupling at the scales measured by . For example, around day 80 there is a very large peak in both troposphere and stratosphere that corresponds to large values of and to relatively large increases in above 300 hPa; the implication is that baroclinic development extends well into the stratosphere. However, the strong stratosphere–troposphere coupling does not extend to the largest scales. 〈*z*〉 is not particularly well correlated during this period (figure 5*c*).

In all three seasons, we see a link between tropospheric variability and thermal advection. Thermal advection is maintained into the stratosphere and on occasion the baroclinic development process extends across the tropopause. If no other dynamical mechanisms play a significant role, then can have a strong influence on the stratosphere. Examples of such episodes, during which stratospheric thermal advection is significant and large values of correspond to increased tropospheric variability, are seen around days 90–100 in 2010/11, days 25 and 75 in 2009/10, and days 20 and 75 in 2008/9. Nevertheless, the overall evolution of the stratospheric geopotential is not primarily determined by thermal advection.

### (b) Time dependence of the vertical scale: relation to disturbance statistics and horizontal scales

We have already noted that baroclinic development and decay (roughly speaking, blocking) and the extent to which baroclinic processes extend into the stratosphere vary from year to year. We now seek to characterize the baroclinic processes by examining diagnostics of vertical and horizontal disturbance scales.

We begin with the former. Table 1 gives seasonal averages for 2006–2010 of , the vertical correlations with respect to 100 hPa of the overall variability at pressure *p*. An implied vertical scale can be defined by , which measures the vertical coherence between 100 and 300 hPa. is largest in 2008 and 2006 and considerably smaller in the other years. Strong troposphere coupling during 2008 was discussed in §3*a*.

Table 1 also shows seasonally averaged statistics of , namely mean and standard deviation. We regard the latter as an independent diagnostic of temporal variability. These statistics illustrate the connection between the time-dependent vertical scale and the disturbance statistics. The amplitude and variability of , in troposphere and stratosphere, are largest in 2008. This is the year when and the implied vertical disturbance scale are largest; and are also largest in 2008. Considering all 5 years, tropospheric variability is large when the vertical scale is large.

Table 2 shows the mean and standard deviation for 〈*z*〉, the area-averaged geopotential perturbation from the zonal mean. These diagnostics can also be related to the vertical scale implied by . Analogous to table 1, years with the largest implied vertical scale have the largest tropospheric variability as diagnosed by 〈*z*〉. In the stratosphere, however, variability of 〈*z*〉 cannot be clearly related to . This is not surprising as measures variability on a smaller scale than is typical in the stratosphere.

If baroclinic development is to be maximized, then this relation between amplitude and vertical scales must also carry over to horizontal scales. In order to maintain balance, perturbations with a large vertical scale require a large horizontal scale. Because a large-amplitude perturbation inevitably has a large latitudinal scale, it is also expected that the longitudinal scale will be large.

We illustrate this using time series of the mean horizontal scale. 〈*k*〉 includes both baroclinically developing and non-developing perturbations. We expect that non-developing perturbations will primarily be on a scale *L*>*L*_{D} and developing perturbations on a scale *L*∼*L*_{D}, while the deformation radius *L*_{D}∝*H*. In seasons where and *H* are large, we expect perturbations to have larger horizontal scale and developing ones to be of larger amplitude.

Figure 6 plots time series of 〈*k*〉 for various cases. For 2010, there is a clear contrast between the first and second halves: at 300 hPa, 〈*k*〉≃3.3±0.5 during the former, and ≃2.8±0.4 during the latter. This contrast, which can also be seen in figure 4*a*, is lost in a seasonal average; therefore, we supplement tables 1 and 2 with values calculated for each half. At 300 hPa, the mean of is 1346 in the first half and 1270 in the second, the respective standard deviations being 16.0 and 10.3; is 0.815 in the first half and 0.68 for the entire season; the mean of 〈*z*〉 is 120.9 in the first half and 120.6 in the second, the respective standard deviations being 11.3 and 12.5. The upshot is that, over the course of the season, there is a very large drop in variability of , the value in the first half being comparable to 2009 and that in the second half being lower than in any of the full seasons. Thus, 〈*k*〉≃2.8 in the second half probably represents non-developing perturbations on a scale greater than *L*_{D}, whereas 〈*k*〉≃3.3 in the first half suggests that there are large growing perturbations for *k*>3.3. Although these values depend on the choice of averaging period, they do highlight the contrast seen in figure 6. The behaviour of the large scales, as measured by 〈*z*〉, is similar in the two halves.

For 2009, nearly all the time, implying relatively stable dynamics with the equivalent *L* being greater than *L*_{D}. This is consistent with a smaller vertical scale and relatively low values of tropospheric variability (table 1). Although the standard deviation of is not particularly low in the troposphere, the standard deviation of 〈*z*〉 is very low at 300 hPa. This suggests that the variability is primarily on small scales (as implied by the low value of ). Thus, the episodes of increased around days 25 and 75, noted at the end of §3*a*, are not accompanied by large changes in 〈*k*〉.

For 2008, 〈*k*〉 at 300 hPa is lower than in 2010, consistent with a larger vertical scale than the first half of 2010 (table 1). The increase in 〈*k*〉 relative to 2009 reflects the increased level of activity around *L*≃*L*_{D}. Large increases in 〈*k*〉 around days 40 and 100 correspond to the large amplitudes of . Around day 80, there are also large values of ; although 〈*k*〉 does not show a peak value in the troposphere, these values do correspond to a peak of 〈*k*〉 in the stratosphere which brings the mean 〈*k*〉 in the troposphere and stratosphere to about the same value.

These results demonstrate the connection between the vertical structure and tropospheric developments. The degree of stratosphere–troposphere coupling, as measured by the correlation of between 100 and 300 hPa, appears to play a key role. By determining the vertical scale, the stratosphere determines the deformation radius and influences tropospheric variability. The data also support the expected links between vertical and horizontal scales, and between horizontal scales and tropospheric variability.

### (c) The relation between tropospheric disturbances and the basic state: the role of the vertical scale

Following Wittman *et al.* [16], we now seek to relate the changes in the vertical extent of baroclinic energy conversion and in the amplitude and variability of the disturbance (§3*b*) to properties of the basic state. We have already seen that large vertical disturbance scales, as diagnosed by , correspond to large perturbation amplitudes (tables 1 and 2); however, the connection to disturbance scales is not a simple one because the vertical correlations of and are associated with different horizontal scales. Nevertheless, given that baroclinic growth will be maximized on a scale *L*∼*L*_{D}, we expect a spatially coherent basic state to be a prerequisite for large vertical correlations in baroclinic growth. It is common to relate the intensity of baroclinic developments to the ‘Eady index’, which measures the vertical shear of the basic state [16]. In order for the growth to be maintained in time, this basic state must also be temporally coherent.

We thus compare the time series of baroclinic development, i.e. , with a diagnostic of basic-state baroclinicity, the vertically integrated thermal gradient, , equation (2.8). is proportional to the zonal-wind difference across the layer used for the calculation. Because hydrostatic coupling makes it difficult for horizontal gradients of *Z* to change rapidly in the vertical, *v* tends to exhibit strong vertical correlations (not shown) and one may expect and to be closely related.

Figure 7 shows plots of . In 2010/11 (figure 7*a*), the variations at 500 hPa are similar to the variations of (figure 4*a*) during the first half of this season. This suggests a connection between disturbance amplitude and basic-state baroclinicity. Later in the season, there is a persistent increase in , well correlated between the stratosphere and troposphere. During this period, there is also increased vertical correlation of (figure 4*b*); however, as discussed in §3*a*, these increases in probably reflect decoupling of the stratosphere from the troposphere, i.e. they are driven by large-scale stratospheric dynamics.

In 2009/10 (figure 7*b*), the values of at 300 hPa are not very different from those in 2010/11. If tropospheric disturbances were determined solely by basic-state baroclinicity, then we would expect the values of to be similar; however, in 2009/10, is much smaller at 500 and 300 hPa (table 1). Compared with 2010/11, and do not correspond as well. In 2009, the values of are lower than in 2010/11, and the differences between 300 and 100 hPa larger (figure 4). This suggests a breakdown of vertical correlation, indicative of decoupling between troposphere and stratosphere.

In 2008/9 (figure 7*c*), the values of at 300 and 100 hPa are similar to those in 2009/10 in the middle of the period, but less so early and late in the season. But, once again, the tropospheric disturbance statistics are inconsistent with : tables 1 and 2 indicate more baroclinic growth in 2008/9 than in 2009/10. Thus, we need an explanation for tropospheric amplitude differences that does not rely exclusively on the baroclinicity of the basic state.

The discussion in §3*a* suggested that smaller values of and lower amplitudes of tropospheric developments are associated with decoupling between the troposphere and stratosphere. Decoupling implies a lack of coherence in the basic state and may serve to decrease baroclinic growth below what would be expected from . We look for evidence of large changes across the tropopause as measured by the seasonally averaged mean and standard deviation of (table 3). There are significant interannual differences. The standard deviation of jumps significantly (by roughly a factor of 2) from the upper troposphere to the lower stratosphere, i.e. from 300 to 100 hPa, for 2009/10 and 2007/8. This indicates that the basic state is less coherent in space and time. During these years, the stratosphere and troposphere largely decouple; during other years, notably 2008/9 and 2006/7, the evolution of the basic state is more coherent across the tropopause. The latter are the years in which the tropospheric is largest.

The results in this section show that the relationship between basic-state baroclinicity and tropospheric disturbances is not a straightforward one. When the stratosphere and troposphere decouple, as in 2009/10 and the second half of 2010/11, blocking statistics cannot be explained in terms of a basic-state vertical scale (defined, for example, by ) extending across the tropopause: the effective vertical scale is shorter. But when stratosphere and troposphere are more strongly coupled, as in 2008/9 and the first half of 2010/11, the large basic-state vertical scale, which is associated with coherent thermal advection, provides more insight into tropospheric developments. In such periods, an Eady index, based on the local vertical shear, may be a useful indicator of tropospheric development [19].

## 4. Summary

We summarize the main points from §3. The raw geopotential data are nearly always strongly baroclinic and therefore strongly time dependent; the vertical coherence of the baroclinic developments in the troposphere is strongly time dependent; and there are large interannual and, in the case of 2010/11 and 2006/7 in particular, large intraseasonal differences in the amplitude and variability of the departures from the zonal mean. These points are consistent with standard dynamical theory, which predicts that changes in effective depth are linked to changes in the horizontal scale of the developing perturbations, whereas larger scale perturbations persist for long periods.

Diagnostic techniques were developed to support this view. While simple inspection of geopotential fields is sufficient to show that the flow is almost always baroclinic, the demonstration of the expected relation between horizontal and vertical scale requires the use of hemispheric spectral diagnostics to define the horizontal scale. Defining the typical vertical scale is much harder, and we used diagnostics of the effect of the thermal advection on the geopotential tendency: thermal advection is only present if there are baroclinic developments, whereas other contributions to the geopotential tendency can come from a variety of causes. We also explored the mechanisms by which the vertical scale is determined. Reductions in the vertical scale were identified by time series of the r.m.s. meridional wind; a loss in vertical correlation of these time series indicates that developments in the stratosphere proceed largely independently of those in the troposphere, implying that the vertical extent of tropospheric developments is reduced.

This study highlights the influence of the stratospheric structure on tropospheric developments—in particular, blocking—via control of the time-dependent deformation radius. During years in which stratosphere and troposphere are tightly coupled (as in 2008/9 and 2006/7), baroclinic developments penetrate through much of the stratosphere, whereas, at other times, there is little penetration above the tropopause and the stratosphere and troposphere essentially decouple. The vertical structure of the basic state is also linked to the mean horizontal scale and hence to the amount of tropospheric variability; for example, seasons with an active independent stratosphere, such as 2009/10 and 2007/8, have relatively large horizontal scales and less tropospheric variability. We have not attempted to study the reasons behind the changes in the basic-state structure. Our study underscores why this structure is of fundamental importance in predicting the tropospheric climatology.

## Acknowledgements

This work benefited from discussions with Adam Scaife. The constructive comments of two anonymous referees and Mike Davey were very useful in improving the paper and pointing out additional relevant citations.

## Footnotes

One contribution of 13 to a Theme Issue ‘Mathematics applied to the climate system’.

↵1 In principle, variability in the mean Brunt–Väisälä frequency could also be important; however, in the periods that we study and compare, it does not seem to be the dominant issue.

↵2 A basic state more appropriate for the nonlinear case could be obtained, for example, by rearranging potential vorticity on isentropic surfaces [31].

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.