Clear air turbulence (CAT) has relevance for aviation but also for cross-tropopause transport of chemical constituents. This study presents a 44-year climatology (from 1958 to 2001) of indicators for CAT for the Northern Hemisphere tropopause on the basis of reanalysis data (ERA40) from the European Centre for Medium-Range Weather Forecasts. Small Richardson numbers (Ri), negative squared Brunt-Väisälä (N2) and negative potential vorticity (PV) values are used as indicators for Kelvin-Helmholtz, hydrostatic, and symmetric instability, respectively. Additionally, an empirical indicator (TI) is applied. All indicators have winter frequency maxima for CAT over the North American east and west coasts. Other local maxima are found over the western part of the North Atlantic and North Pacific, the Himalayas, central Europe, and eastern China. In summer, frequencies are smaller, except for N2, and patterns are shifted poleward. Frequencies of TI and PV depend strongly on jet position, whereas Ri has moderate dependence. N2 frequencies are largest over land and are not markedly influenced by jets. The frequency maxima relative to the jets differ with a northern maximum for TI, a southern maximum for PV, and one for Ri that is essentially along the jet axis. For the 44-year period pronounced nonlinear trends are identified with an increase of 40–90% over the North Atlantic, United States, and European sector. For the two phases of the North Atlantic Oscillation and the Pacific/North American flow pattern the interannual variability of CAT, indicated by TI and PV, is significant (in contrast to Ri and especially N2).
 Clear air turbulence (CAT) remains an unsolved problem of environmental fluid dynamics with high practical and theoretical relevance. Ellrod et al.  define CAT as “aircraft turbulence that occurs at altitudes of 5.6 km or higher, either in cloud-free conditions or within stratiform clouds.” CAT is most frequent in the tropopause region [Dutton and Panofsky, 1970] near jet streams and internal fronts associated with extratropical cyclones [Keller, 1990]. About two thirds of CAT occurrences are found near jet streams, where the environment is optimal for the production of sheared stable layers [Ellrod et al., 2003]. CAT is also found in macroscale regions of sharply curved anticyclonic flows and in regions of deep mesoscale convection and breaking gravity waves in the midlatitudes [Ellrod et al., 2003]. At present, however, the theoretical understanding of CAT remains somewhat limited.
 Since CAT is not explicitly resolvable at the grid spacing of current forecast models it is not possible to exactly forecast CAT, even if high-resolution, limited area models are used (such as the German/Swiss aLMo [Buss et al., 2005]). Therefore the investigation of CAT usually involves the determination of synoptic or mesoscale parameters that are likely conducive to turbulence such as vertical and horizontal wind shear, convergence, horizontal deformation, lapse-rate discontinuities, and strong horizontal thermal gradients [Ellrod and Knapp, 1992]. Ultra-high-resolution radar observations [Atlas et al., 1970] and numerical simulations [Lane et al., 2004] provide evidence for a relationship between Kelvin-Helmholtz instability (KHI) and the generation of CAT by wave breaking. Mancuso and Endlich  statistically related different meteorological quantities to CAT observations and found the best correlation with CAT for the product of vertical wind shear and deformation. This product and an additional indicator for CAT (the product of vertical wind shear and deformation plus convergence) have been adapted by Ellrod and Knapp  and successfully validated against a large number of observations. Turbulence diagnostics based on horizontal deformation are controversial for two situations where the conventional linkage between frontogenesis, deformation and CAT, which was used in the aforementioned studies, is not appropriate. First, deformation may be frontolytic in cases. Second, the linkage is not appropriate in strongly anticyclonic flows. In these areas, nonfrontogenetical mechanisms due to anticyclonic relative vorticity such as gravity wave activity by geostrophic adjustment and inertial instability are more appropriate [Knox, 1997]. Roach  proposed an expression for the rate of change of the logarithm of the gradient Richardson number (Ri) following the air (the Roach equation), which is a function of the deformation and subgrid-scale phenomena associated with CAT. This indicator was subsequently linked to CAT observations by Keller . Brown  used two turbulence indices based on the Roach equation. A comparison with observations indicated better results than for Ri only. Finally, Frehlich and Sharman  used statistical quantities based on turbulence theory, illustrating higher probabilities of detection than the CAT indicators presented by Ellrod and Knapp .
 In this study a novel 44-year climatology of indices for different types of fluid instability that may lead to CAT is calculated for the Northern Hemisphere tropopause region. For a selected case these indices are compared with aircraft measurements of turbulence. The advantages of this climatological approach over one based on pilot reports of turbulence is a drastically improved and uniform spatial and temporal coverage. Aircraft traffic is guided along certain flight corridors and therefore lacks this uniformity. The focus in this study is on CAT near the extratropical dynamical tropopause. The motivation for this work was to investigate CAT indicators using ERA40 reanalysis data and to create a comprehensive climatology of these indicators. In a forthcoming study we intend to assess the influence of CAT on its contribution to mixing between the stratosphere and the troposphere using these indicators.
 In section 2 the data used in this study are presented. In section 3, four conventional CAT indicators are discussed. In section 4 a case study of observed CAT over the North Atlantic is investigated using reanalysis data from the European Centre for Medium-Range Weather Forecasts (ECMWF). Given satisfactory results from the case study, a 44-year climatology of the four different turbulence indicators near the dynamical tropopause is then presented in section 5. Finally, in section 6 the variability, possible trends, and the association to major atmospheric flow patterns are discussed, and in section 7, conclusions as well as a prospectus for future work are given.
2. Methodology and Data
 All analyses in this study are based upon the 44-year ECMWF reanalysis data set (ERA40) covering the time period from September 1957 to August 2002 with a resolution of 6 h. ERA40 has a spectral resolution of T159 (∼90 km in the midlatitudes) in the horizontal and a vertical resolution of 60 levels (from the surface up to 0.1 hPa). For more details on ERA40, see Uppala et al. . The fields (e.g., horizontal and vertical wind components, temperature, and pressure) required for the calculation of the turbulence indicators used in this study were interpolated to an equal angle grid with 1° horizontal resolution. The indicators were then calculated on the original hybrid model levels, and finally, the indicators were interpolated to pressure and potential vorticity (PV) levels.
 ERA40 has a horizontal and vertical resolution which is too coarse to explicitly resolve turbulence. Therefore the assumption underlying our approach is that turbulence-generating mechanisms have their origin at resolvable scales and that energy cascades down to the smallest scales, but it is unclear what the exact cascade mechanism is [Sharman et al., 2006]. Hence it is expected that only regions of strong and extended turbulence, originating from resolvable scale features, can be located using ERA40 indicators for CAT. Indeed, Ellrod et al.  discuss the contribution of macroscale forcings (resolvable in ERA40) that cause atmospheric instabilities and subsequently CAT. Moreover, Shapiro  found that the primary contributors to turbulent heat or momentum fluxes within tropopause folds originate from the low-frequency components (corresponding to length scales of ∼10 km) of the turbulent motions. Therefore the macroscale and the rather large scale synoptic or mesoscale features in ERA40 that we associate with CAT are considered sufficient to climatologically investigate indices for CAT.
 To verify the ERA40-derived CAT fields, measurements of turbulence from the Nitrogen Oxides and Ozone Along Air Routes (NOXAR) research campaign were used [Dias-Lalcaca et al., 1998]. The measurements presented in section 4 were made by a SWISSAIR B-747 commercial aircraft flying from Zurich to Atlanta. The vertical velocity was calculated from the measured horizontal wind speed, vertical acceleration, and relative angles with approximately 1 Hz. The data output of the vertical velocity w is available as the standard deviation ()1/2 over intervals of three seconds 1/3 Hz, which allows one to resolve eddies larger than 250–500 m. For further information on the vertical wind speed derivation, see Brunner .
 The NOXAR data set may have insufficient resolution to capture the smallest scale turbulent fluxes. High values of ()1/2 nevertheless are used as a first-order indication for turbulent regions.
3. Turbulence Indicators
 Turbulent flows and their main characteristics are discussed in most text books of fluid dynamics (e.g., Pope ). For this study an important property of turbulent flows is their ability to strongly mix a fluid (as compared with molecular mixing) and hence to effectively reduce gradients (e.g., of momentum and potential temperature). Turbulent flows develop from flow instabilities via growing disturbances by extracting energy from the mean flow. The following three mechanisms for turbulence and associated indicators are considered in this work: Kelvin-Helmholtz instability (Ri), hydrostatic instability (N2), and symmetric instability (PV). Additionally, an empirical indicator (TI) for CAT is investigated. Following Sharman et al. , the relative performance of individual CAT diagnostics is dependent on the model. For their turbulence forecasting system they use several more sophisticated methods that are partly based on our indicators. Our approach is to use simple diagnostics that are directly linked to instabilities and hence are more easily interpreted. Nevertheless, it should be kept in mind that instability is a necessary but not absolutely sufficient condition for CAT. The four CAT indicators are discussed below.
 1. The gradient Richardson number is defined as the ratio of the squared stabilizing Brunt-Väisälä frequency and the squared destabilizing vertical wind shear [Stull, 1988]
where θv is the virtual potential temperature, u and v are the horizontal wind components, and g is the Earth's gravity. KHI develops for Ri smaller than 0.25 [Miles and Howard, 1964]. It can occur wherever there is a low static stability and a high vertical wind shear, for instance, in the vicinity of jets or frontal zones where vertical shear is expected to be largest. Ellrod and Knapp  say that KHI is the principal mechanism responsible for CAT.
 2. The squared Brunt-Väisälä frequency (N2) corresponds to the numerator in (1)
Vertical parcel displacements are hydrostatically stable, neutral, or unstable depending on the sign of N2. Unstable regions (where N2 < 0) in the atmosphere exist only locally and are removed by convective overturning and subsequent turbulent mixing, which tends to form neutral stratifications (for further information, see Holton  or Gill ). N2 is an important quantity for the development of hydrostatic instability and indirectly also for KHI and symmetric instability (see below). (The criterion for hydrostatic instability is similar to the one for convective instability. The only difference is that θe is used instead of θv in the latter case. Because of the small amount of water vapor in the extratropical tropopause region the difference between the two is minor.)
where ζ, ρ, f, k, and θ denote the vertical component of the relative vorticity, the air density, the Coriolis parameter, a unit vector directing toward the vertical, and the potential temperature, respectively. The approximation in equation (3) is valid for synoptic and large scale flows. Similar to the way in which vertical displacements are resisted by the buoyancy force in statically stable layers, horizontal parcel displacements are resisted by the Coriolis force in a rotating fluid. Symmetric instability (or inertial instability) results from an imbalance between the inertial forces and the pressure gradient force for a radially displaced parcel. In the Northern (Southern) Hemisphere the flow is symmetrically unstable provided that the vertical component of the absolute vorticity η = ζ + f times the stratification ∂θ/∂z is negative (positive) [Hoskins, 1974; Holton, 1992]. In such regions the fluid will begin to mix, just as convection mixes vertically, until the PV becomes positive (negative) again.
 4. The paper by Ellrod and Knapp  describes two forecast indices based on horizontal deformation, vertical wind shear and divergence. The first index is the product of vertical shear and deformation and is used in this study
The link of this quantity to CAT is based on observations and is predominantly empirical. The physical basis of its use as a turbulence measure is given by Petterssen's frontogenetic intensity equation relating frontogenesis, vertical shear, and deformation via the thermal wind relationship [Mancuso and Endlich, 1966], which is questionable in cases of anticyclonic flows [Knox, 1997] or in cases of frontolysis. TI describes CAT produced by jet streams and upper-level fronts but does not account for wave breaking and convection [Ellrod et al., 2003]. Ellrod and Knapp  validated TI with pilot reports of CAT and found considerably high (low) values of probability of detection (false alarm rate). TI has also been applied to investigate turbulence-induced STE over the eastern Mediterranean Sea [Traub and Lelieveld, 2003]. Finally, Brown  gives the turbulent energy dissipation rate by an expression which is similar to this turbulence indicator. Endlich  investigated more than 20 flights associated with turbulence in either straight jets, ridges, or troughs. For straight jets, turbulence occurred rarely, whereas for ridges (troughs) it occurred primarily on the anticyclonic (cyclonic) shear side of the jet core. According to references in Endlich's paper, most investigations show a maximum of turbulence on the cyclonic shear side. Therefore TI might be a better indicator than those discussed above if cyclonic curved jets are analyzed. On the other hand, Ellrod and Knapp  found that TI performs well in situations of upper-level ridging, which is said to be the most common type of major CAT over the North Atlantic. On the basis of observations, CAT in anticyclonic flows is most frequent, whereas cyclonic conditions produce the most intense CAT [Ellrod et al., 2003].
 It is important to mention that the instabilities discussed above can interact and occur at the same time and place. As will be discussed in section 5.1, regions of negative Richardson numbers (see equation (1)) and associated KHI are colocated with regions of hydrostatic instability. It has also been shown that there are similarities between hydrostatic and symmetric instability [Xu and Clark, 1985]. This complicates the investigation of different instabilities and can at times preclude a clear distinction of the relevant dynamical processes involved.
4. Case Study: Cross North Atlantic Flight
 In order to assess the ability of ERA40 to locate CAT and to investigate suitable thresholds for the turbulence indicators, multiple case studies were performed with NOXAR measurements. Results from a flight from Atlanta to Zurich on the 17 February 1996 are presented in Figure 1. On the 250 hPa chart (left) a trough is visible over the eastern United States. The labels W1, W2, and W3 indicate regions with strong standard deviations of the vertical wind speed ()1/2 > 1.0 m s−1. W1 and W2 are found in the vicinity of frontal systems that are often associated with strong vertical winds and not always necessarily with turbulence. In contrast, W3 (0634 UTC on 239 hPa flight altitude) is far away from fronts and will therefore be further investigated. Local minima (maxima for TI) of the CAT indicators and the wind speed are depicted on the right (Figure 1) at 0600 UTC (close to the time when the aircraft encountered turbulence near W3).
 Near W3, PV, Ri, and N2 have local minima where large values of ()1/2 were measured. At least for this event and using ERA40 data these indicators appear to identify regions of CAT. In contrast, the TI indicator does not capture the turbulent spot at W3 on the anticyclonic shear side. A possible explanation is based upon the following result by Knox : In strongly anticyclonic flows, indicators based upon the absolute vorticity (such as PV) are more appropriate than those based on deformation (such as TI). This is consistent with our case study and the criterion of PV < 0. Moreover, in strongly anticyclonic flows the deformation (and therefore also TI) is limited by inertial instability and hence is generally larger on the cyclonic shear side (visible on the right panel (Figure 1)). Unfortunately, it cannot be verified whether or not the local TI maximum in Figure 1 is associated with CAT, since only turbulence measurements from the anticyclonic shear side are available. TI was found to be an accurate indicator in additional case studies with strong cyclonic flows (not shown).
 The present case study illustrates that it is difficult to unambiguously determine which instability mechanism is responsible for the presence of turbulence at a specific region. It is also apparent that the jet structure itself might play an integral role in determining turbulent regions. Additionally, a case study over Wales presented by Pavelin  was compared to the turbulence indicators calculated with ERA40 (not shown). The results are satisfactorily consistent.
5. The 44-Year Climatology of Clear Air Turbulence Indices
 This section presents frequency distributions of the analyzed turbulence indices near the dynamical 2 potential vorticity unit (PVU) tropopause for the Northern Hemisphere winter and summer from 1958 to 2001. The section is structured as follows: First, in section 5.1, the climatological approach is described. In sections 5.2 and 5.3 the geographical distribution and the vertical structure of the climatology are discussed, respectively. Finally, in section 5.4, sensitivity analyses are presented in order to determine the robustness of this climatology. Since the spatial and temporal patterns (amplitude) of the CAT indicator frequencies turn out to be weakly (strongly) sensitive to their thresholds, the main focus in the remainder of this paper is not on the absolute values of the turbulence frequencies but rather on their climatological patterns.
5.1. Data Analysis
 Reasonable thresholds for each of the CAT indicators had to be selected for the analysis of the ERA40 data (see Table 1). These are based on previously published results, theoretical considerations and the results of case studies (see section 4) validated with aircraft measurements. Additionally, it was considered that Endlich  found frequencies for moderate (severe) turbulence within jet streams of approximately 2.5% (0.2%).
Table 1. Thresholds Used in This Study for Each CAT Indicator for Transition to Turbulent Flow, Thresholds From Theory or From Literature, and the Unit of Each Indicator
 The negative Ri pattern is always associated with negative N2 values and hence hydrostatic instability. In order to be able to separate hydrostatic from Kelvin-Helmholtz instability the Ri threshold was set to 0 < Ri < 0.75 instead of Ri < 0.25. With this choice the Ri pattern does not simply reproduce the N2 pattern. For negative PV values a similar difficulty occurs: They can also be due to negative N2 values (if the absolute vorticity is positive) and hence hydrostatic instability rather than symmetric instability. However, since negative PV values could also be due to negative absolute vorticity (if N2 is positive) a similar separation as for Ri is not possible. In the following climatology it is shown that even though hydrostatic instability, KHI, and symmetric instability have similar spatial patterns the latter two are more frequent and therefore not only the result of negative values of N2.
Figures 2 and 3are discussed in section 5.2 and indicate where and when the large scale, upper-level conditions for turbulence (see Table 1) near the 2 PVU tropopause are most favorable according to the different CAT indices. The overlaid contour lines represent the seasonal mean horizontal wind speed at the tropopause. In order to investigate the contribution of CAT, described by the four indicators, to the turbulent mixing near the tropopause, five equally spaced levels in a band around the tropopause are taken into account. (The five levels are the height of the 2 PVU tropopause in hPa +50, +25, +0, −25, and −50 hPa. See section 5.4 for a discussion of the sensitivity of the patterns on the choice of the vertical distance from the tropopause.) Whenever the value of Ri, PV, and N2 (TI) falls below (exceeds) the threshold at one or more of these five levels the grid location is considered as possibly turbulent.
5.2. Geographical Distribution
 1. TI is by definition (see equation (4)) strongly dependent on vertical wind shear and horizontal deformation. As shown in Figures 2a and 3a, the highest frequencies of TI are found on the northern side of the jets. This asymmetry with respect to the southern (anticyclonic shear) and northern (cyclonic shear) side of the jets may be for two reasons. First, as we have already mentioned, deformation (and hence also TI) are limited by inertial instability in strongly anticyclonic flows. Second, anticyclonic jets, which have a maximum in deformation on the northern side as compared to a maximum on the southern side of cyclonic jets, may be climatologically predominant. During winter (Figure 2a) the distribution of TI has two maxima, one over the United States and the western part of the North Atlantic and the other over eastern China and the western part of the North Pacific. High frequencies are found in a zonally elongated band between 30 and 45°N, except over the North Atlantic and to some extent the Rocky mountains where the mean jet reaches further to the north (up to 70°N) and hence high TI frequencies. The turbulence frequencies are generally below 10%. A local maximum with values up to 20% is found downstream of the Himalayas. In summer (Figure 3a) the frequencies are considerably reduced, with maximum values of 6%. The maxima are shifted to the north (compared with winter results), located poleward of 40°N, and more dispersed. There is an additional local maximum in the area of the Black Sea and the Caspian Sea. This region corresponds to the TI and STE maximum found by Traub and Lelieveld  and a local maximum in the frequency of tropopause folds in summer [Sprenger et al., 2003]. The main Northern Hemisphere mountain chains (the Alps, Himalayas, Appalachians, and Rocky Mountains) are reflected in the TI pattern. They can affect the large scale circulation by inducing vertically propagating gravity waves (though only poorly represented in ERA40) and, hence, influence the CAT indicators by changing the stratification or creating additional shear. Ellrod et al.  present a 9-year winter climatology of TI at 9.5–10.8 km height derived from the National Centers for Environmental Prediction Aviation Model. The distribution of the mean TI values in their climatology is similar to the distribution of the maximum TI frequencies seen in Figures 2a and 3a, as well as the strong interannual variability (see section 6.1) and the poleward shift and decrease during summer.
 2. Maximum frequencies of small but positive Ri values are also expected close to the wind maxima. Because of strong latitudinal variations of the jets and long-term averaging this characteristic is only partly discernible in the 44-year climatology (Figures 2b and 3b). If analyzed for single seasons, the wind and Ri maxima coincide (not shown). Generally, the large scale winter pattern (Figure 2b) is similar to the TI pattern discussed above. Surprisingly, the eastern China-North Pacific maximum associated with the Pacific jets is much weaker than the U.S.-North Atlantic maximum, even though the mean Pacific jet is climatologically stronger. The turbulence frequency maxima do not exceed 7% except downstream of the Himalayas (>10%). The position of the maxima relative to the jet, which is along the jet axis for Ri (particularly evident in shorter, e.g., 1-year climatologies, not shown), is slightly displaced from the TI maxima, which lie north of the jet axis. In summer (Figure 3b) the maxima are again shifted poleward and dispersed, most probably due to low stability rather than wind shear (the Ri summer climatology partly matches with the N2 summer climatology given in Figure 3c). The central Asian maximum, for example, is likely associated with the Asian summer heat low. There is no significant Ri maximum over the Black Sea, which could mean that the TI maximum in this region is predominantly determined by horizontal deformation rather than vertical wind shear. Again, at least for winter the main Northern Hemisphere mountain chains are reflected in the Ri climatology.
 3. The N2 pattern (Figures 2c and 3c) is not strongly affected by the position of the mean jet stream, a conclusion based on short-term analysis (monthly and seasonal climatologies, not shown). Maxima occur in all four seasons and are primarily located over land. There are pronounced maxima over Europe, North America, and central Asia-Himalayas. Similar to the results for the previous indicators, the maxima reveal a seasonal north-south shift, which is largest over Asia. In summer (Figure 3c) the North American maximum is weak and does not extend to the east coast, while over Asia maximum frequencies up to 4% are found. All maxima are north of the mean jet position, and there is no hydrostatic instability occurring near the (sub)tropical tropopause (in the tropics where the 380 K isentrope is lower than the 2 PVU isosurface, the dynamical tropopause is defined by the 380 K isentrope). The frequencies for hydrostatic instability indicated by negative N2 are surprisingly large and may, at least to some extent, be caused by layer inconsistencies in the ERA40 data. These areas may not always be unstable in reality but are probably regions of weak hydrostatic stability (the same holds for negative Ri and PV values since they are a function of N2 as well).
 4. Jets are associated with strong relative shear and curvature vorticity (ζ = ζcurve + ζshear; Holton ). Therefore it is expected that also (negative) PV values (a function of the relative vorticity ζ (see equation (3))) will be affected by the jets. Local maxima of negative PV are expected south of the jets, where symmetric or inertial instability is more likely than on the northern side, because of small relative vorticity on the anticyclonic shear side of jets [Knox, 1997]. At lower latitudes, negative PV values are more probable because of low values of the Coriolis parameter f. However, since we have no experience with the use of PV as a CAT indicator near the equator, the tropics will not be discussed and only frequencies north of 20° are plotted in Figures 2d and 3d. The most striking winter maxima are situated on the southern side of the subtropical jet over Africa, Asia, and the North Pacific. The remaining climatological winter and summer patterns poleward of 30° latitude are similar to those of TI, Ri, and N2, with a winter maximum of over 20% near the Himalayas. The maxima over the U.S. east and west coasts of over 10% in winter and approximately 5% in summer, as well as the European and the central Asian maxima, match fairly well with the Ri and N2 maxima. There are some similarities with an inertial instability climatology from Knox and Harvey , but direct intercomparison is not possible since the climatologies are on different vertical levels.
5.3. Vertical Structure
 Zonally and seasonally averaged cross sections of the turbulence indicators are shown in Figure 4 for three different sectors: the U.S. east coast-west Atlantic sector (Figures 4a and 4d), the European-Mideast sector (Figures 4b and 4e), and the Chinese east coast-west Pacific sector (Figures 4c and 4f). Additionally, the mean position of the 2 PVU surface, wind speed, and isentropes are included. Note that for this analysis the CAT indicator frequencies were calculated not only near the tropopause but on 11 pressure levels from 500 to 100 hPa in order to be able to produce meaningful cross sections. The absolute values of the frequencies are decreased compared to section 5.2 because of the fact that only single levels are considered, whereas in Figures 2 and 3, five levels were taken into account. In the case of PV and N2 the difference is smaller because three of the five levels (2 PVU, 2 PVU–25 hPa, and 2 PVU–50 hPa) considered in section 5.2 hardly ever contribute to symmetric (PV < 0) or hydrostatic instability (N2 < 0).
 The maximum frequencies for TI are slightly north of the jet core, as already discussed in section 5.2. Ri has its maxima in the U.S. east coast-west Atlantic and the Chinese east coast-west Pacific sectors immediately below the mean jet core where the vertical wind shear is strongest. Only in the European-Mideast sector is the mean wind field structure separated into the subtropical and the polar front jets. The former is much stronger and persistent, whereas the position of the latter varies strongly meridionally with, for instance, the phases of the North Atlantic Oscillation (NAO). Therefore the position of the turbulence frequency maximum relative to the jets is less evident. Negative N2 occur near the dynamical tropopause in the vicinity of the midlatitudinal jets. There seems to be no favored position relative to the jet. At least for summers in the European sector (Figure 4e), negative N2 and low Ri values are found in the subtropics below the tropopause. As it was shown in section 5.2 negative PV values occur predominantly south of the mean jet. At least at the tropopause height, maxima of frequencies of negative PV values extend astonishingly far to the north (especially in summer but with smaller frequencies than in winter).
 From Figures 2, 3, and 4 one could speculate that turbulence in the European sector is not typically associated with the jets. This notion is disproved via an investigation of the likelihood of turbulence in both strong and weak wind environments for Europe as well as for other regions in the Northern Hemisphere. Only turbulence according to the N2 indicator is not dependent on the wind strengths.
 Generally, the maxima of all indicators in the midlatitudes are located near the dynamical tropopause (similar to the mean jet), corresponding to a region of STE flux maximum [Sprenger and Wernli, 2003]. The climatological position of negative N2 and PV values is sometimes found above the mean tropopause, which occurs because of the averaging of strong height variations of the tropopause (especially in the midlatitudes). Similarly, because of long-term averaging, negative N2 frequencies exist in regions where the mean stratification is stable.
 In summary, the following results can be drawn: (1) The empirical CAT indicator TI is largest north of the jets; (2) KHI, indicated by small but positive Richardson numbers, is most likely near the jet cores; (3) symmetric instability is most frequent south of the jets (especially in anticyclonic jets); and (4) hydrostatic instability is only slightly dependent on the jet position and is most frequent over land where convection and gravity wave activity are most probable (triggered by mountains or the former also by heating of land surfaces in summer).
5.4. Sensitivity Studies
 1. A sensitivity analysis shows that the amplitude of the CAT indicator frequencies is strongly dependent on their chosen thresholds: TI frequencies are doubled/tripled if the threshold is lowered from 12 × 10−7 to 8 × 10−7 s−1, Ri frequencies are almost doubled if the upper threshold is increased from 0.75 to 1.0, N2 frequencies more than triple if the threshold is increased from 0 to 2 × 10−5 s−2, and the PV frequencies double if the threshold is 0.1 PVU instead of 0 PVU. Therefore a direct intercomparison of frequency amplitudes is not possible. In contrast, the seasonal and spatial patterns of the frequency distributions are independent of the thresholds (figures are not shown).
 2. The following three different spacings for the vertical distance of the five levels (used for the climatology shown in section 5.2) are investigated (figures are not shown): 12.5 hPa, 25 hPa, and 50 hPa. All indices containing stratification terms (PV, Ri, and N2) have strongly decreasing frequencies if the vertical distance is decreased because of the increase in stratification toward the tropopause (and the stratosphere). The result is geographical patterns that are only similar in the position of their maxima but not in the frequency amplitudes. TI, on the other hand, has similar frequencies and the same geographical patterns for all three spacings (slightly smaller values for the closest spacing).
6. Aspects of Variability
 The 44-year climatology in the tropopause region allows for a thorough investigation of the interannual and intra-annual variability of the CAT indicators. First, in section 6.1, seasonal cycles and possible trends of the CAT indicators are discussed. In section 6.2 the geographical variability of the CAT indicators in the North Atlantic and European sector is analyzed in association with positive and negative phases of the NAO, and some remarks on the Pacific/North American flow pattern (PNA) are given.
6.1. Seasonal Cycle and Trends
 To examine the seasonal cycle and trends over the 44-year period, a descriptive statistical analysis (seasonal trend decomposition procedure based on loess, STL) was used [Cleveland et al., 1990]. It is a filtering procedure for decomposing a time series into trend, seasonal, and remainder components. It repeatedly applies a Loess smoother to the data. By adding the three components, one ends up with the original time series. (For the STL algorithm the following parameters were chosen: period of the seasonal component (12), length of the seasonal smoother (209), and length of the trend smoother (119).)
Figure 5 (left) shows 44-year time series of CAT indicator frequencies near the dynamical tropopause (grey lines; the same approach as in section 5.2 is applied to calculate the CAT frequencies) with overlaid nonlinear trend estimates from STL analysis (bold black lines) for the North Atlantic sector (90°W–10°E; 30–70°N). Figure 5 (left) illustrates the year-to-year variability, whereas Figure 5 (right) emphasizes the mean seasonal cycle component from STL. In addition to the North Atlantic, the North Pacific (110°E–150°W; 25–65°N), U.S. (135–75°W; 30–70°N), and European (10°W–30°E; 35–60°N) sectors were considered (not shown).
 The amplitude of the annual cycle for all turbulence indicators exhibits significant interannual variability. The amplitude between the years with the smallest and largest intra-annual variability varies by a factor of ∼2.1 for TI, ∼3.2 for PV, ∼5.0 for N2, and ∼5.5 for Ri (Figure 5 (left)).
 The STL analysis provides positive trend components from 1958 to 2001 for all indicators. It corresponds to an increase of roughly 70% for TI, 90% for Ri, 40% for N2, and 60% for PV. For the U.S. and the European sector the trends are similar. In order to preclude the possibility that the trends are due to the choice of the thresholds of the indices a sensitivity study was performed using the same thresholds as in section 5.4. The analysis reveals that the trends are not dependent on the thresholds. It is important to keep in mind that changes in the amount and type of assimilated data over the 44-year period of study (e.g., radiosondes and satellite data [Bengtsson et al., 2004b; Uppala et al., 2005]) may have a bearing on the presented trends. With the ability to resolve more finely structured features may come increased shearing and hence increased probability for CAT. Therefore the actual trends are possibly smaller than those presented in this study. Bengtsson et al. [2004a] show that additional assimilated satellite data produces a kink in the time series (approximately 1979) of total kinetic energy calculated with ERA40, which then leads to an artificial positive trend component. Since the calculated trends are quite constant throughout the whole ERA40 period, at least for PV, Ri, and TI for the North Atlantic and the U.S., we cannot neglect their existence. Also, the positive trends for the North Atlantic are in good agreement with an increase of the positive phase of the NAO since the 1970s [Hurrell, 1995, 1996]. As it is shown in the following section, the NAO index (NAOI) correlates with all the CAT indices, especially with PV and TI. Note that several other studies have found trends (both positive and negative) of dynamical features near the extratropical tropopause (see F. Isotta et al. (Long-term trends of synoptic-scale breaking Rossby waves in the Northern Hemisphere between 1958 and 2001, submitted to International Journal of Climatology, 2007) for PV streamers and wind speed and Croci-Maspoli et al.  for blocking occurrences and associated tropopause heights). The trends for the North Pacific, on the other hand, look somewhat different; that is, for all indicators the frequencies are strongly increasing toward 1976 and then slowly decreasing.
 The mean seasonal cycle component from the STL analysis reveals pronounced seasonal variability for all indicators (Figure 5 (right)). The winter maximum and the summer minimum of TI, Ri, and PV, which was already discussed in sections 5.2 and 5.3, are quite distinctive. The N2 seasonal cycle, with a late winter maximum and an early summer minimum, is less pronounced (also evident in Figures 2 and 3). These findings are consistent with those for the U.S. and European sectors. Over the North Pacific, there is a rather constant TI seasonal cycle with a striking summer minimum. Ri and PV reveal a late spring maximum, whereas N2 shows a winter minimum and a summer maximum. Because of the seasonal smoothing inherent in the STL analysis the amplitude of the seasonal cycle is generally smaller than would be expected from the raw data depicted in Figure 5 (left).
6.2. Geographical Variability and NAO
 The NAO describes a major atmospheric flow variability pattern in the winter Northern Hemisphere [Wallace and Gutzler, 1981]. It is used in this study to investigate the interannual variations of winter CAT indicator frequencies. CAT indicators and mean wind speeds in the tropopause region (the same approach as in section 5.2 is applied to calculate the CAT frequencies) are shown for the 10 winter months between 1958 and 2001 with the largest (Figure 6 left) and the smallest (Figure 6 middle) values of the NAOI (taken from the Climate Prediction Center (CPC)). (The 10 winter months with the largest NAOI (>1.5) within the ERA40 time period are December 1982 and 1999; January 1984 and 1986; and February 1973, 1981, 1989, 1997, 1999, and 2000. The ones with the smallest index (<−1.4) are December 1961, 1963, 1978, 1987, 1989, and 1995; January 1970 and 1977; and February 1958 and 1978.) Figure 6 (right) shows the correlation of the monthly mean CAT indicator frequencies of the whole ERA40 period with the monthly NAOI. Correlations that are statistically significant on the 5% level are framed by the bold black contour lines.
 Dynamically, the NAO+ phase corresponds to a stronger pressure difference between the subtropical high and the Icelandic low and is associated with a more northern position of the jet stream (vice versa for NAO−). This characteristic is visible in Figure 6 (left and middle). The mean composite patterns of the turbulence indicators TI and PV follows the mean jet position. For Ri and, especially, N2 the latitudinal displacements during the different phases of the NAO are smaller. For higher Ri thresholds, and therefore less influence by small N2 values, the displacement would be larger. In Figure 6 (right) there are significant correlations between TI and the monthly NAOI, both positive (from Newfoundland to northern Europe, tropical North Atlantic, and north Africa) and negative (Greenland, east Canada, and central North Atlantic). Because of the maximum frequencies of PV that are south of the mean jets compared to the northern TI maximum the regions with significant positive and negative correlations are also shifted to the south (except for the positive correlation over northern Europe). Ri and N2 have significant positive correlations over northern Europe and sporadic ones across the North Atlantic similar to PV, whereas significant negative correlations are rare. Again, there is a slight tendency for a TI maximum on the cyclonic (northern) shear side of the mean polar front and subtropical jets, a PV maximum on the anticyclonic (southern) side, and a Ri maximum along the jets axis.
 Additionally, the PNA [Wallace and Gutzler, 1981] was used to analyze the interannual variability of the CAT indicator frequencies in the Pacific region (not shown). For TI and PV, there are extended regions with significant positive correlations (along the jet in the North Pacific for both and the North American West Coast for PV) and negative correlations (north and south of the North Pacific jet) with the PNA index (also taken from CPC). Similar to the NAO patterns discussed above, the regions of significant correlation with PV are shifted to the south compared to those of TI. Ri and N2 have locally isolated negative correlations over the North Pacific and positive ones along the North American West Coast.
 This study presents a novel 44-year climatology of different clear air turbulence indicators in the tropopause region. The climatological distribution of CAT indicators and their interannual variability and trends are discussed. The climatology is calculated from the ERA40 reanalysis from ECMWF [Uppala et al., 2005] and some of the results are directly compared with aircraft measurements from the NOXAR research campaign [Dias-Lalcaca et al., 1998].
 Kelvin-Helmholtz instability was diagnosed via small Richardson numbers, hydrostatic instability via negative squared Brunt-Väisälä frequencies, and symmetric instability via negative potential vorticity values. In addition, an empirical turbulence indicator called TI [Ellrod and Knapp, 1992] was calculated. Sensitivity studies indicate that the frequency patterns of the CAT indicators are quite robust with respect to their thresholds. In contrast, the absolute values of the turbulence frequencies are strongly dependent on the chosen thresholds. Key results of this study are as follows:
 1. For all indicators, there is a winter frequency maximum for CAT over the North American east and west coasts. Other local maxima are found over the Himalayas, central Europe, eastern China, and the western part of the North Atlantic and North Pacific.
 2. In summer the frequencies are reduced in amplitude (only slightly for N2) and shifted to the north. The seasonal variability, with a winter maximum and a summer minimum, is quite robust for TI, Ri, and PV. For N2 the seasonal cycle is less pronounced.
 4. There are significant differences in the spatial climatological patterns of the individual indicators. The N2 frequencies turn out to be largest over land and are rather independent of the position of the jet streams, whereas TI, PV, and, to a lesser extent also, Ri are associated with the jets. For Ri, TI, and PV, there are significant differences in the frequency maxima relative to the jet positions, with a maximum to the north for TI, to the south for PV, and along the jet axis for Ri. The asymmetrical distribution of turbulence indicators containing deformation (such as TI) or relative vorticity (such as PV) agree with the results described by Knox .
 5. Over the 44-year period, pronounced trends are identified with an increase of roughly 70% for TI, 90% for Ri, 40% for N2, and 60% for PV over the North Atlantic, U.S. and European sector. The North Pacific trends strongly increase until 1976 and subsequently decrease. Changes in the amount and type of assimilated data used for ERA40 were not taken into account and may have affected the absolute values of the calculated trends.
 6. The interannual variability of CAT is significant as indicated by the CAT indicators and can be correlated with the two phases of the North Atlantic Oscillation as well as with the Pacific/North American flow pattern. The interannual variations of the TI and PV patterns are consistent with the variation of the jet position associated with the NAO, whereas the Ri and, especially, the N2 patterns are not markedly influenced by the jet stream position. During positive phases of the NAO, generally larger turbulence frequencies occur, which might be due to stronger jets, and associated with that, more frequent instabilities.
 Because of the relevance of CAT for the mixing of air masses and chemical constituents between the troposphere and the stratosphere the link between stratosphere-troposphere exchange and CAT indicators deserves further investigation. As a caveat, it should, nevertheless, be kept in mind that other diabatic processes (such as condensational or radiative heating) are also associated with PV changes which can lead to STE. The relevance of these diabatic processes will be discussed in a forthcoming study.
 We would like to thank MeteoSwiss and ECMWF for access to the ERA40 data set. Special thanks to Dominik Brunner for providing the NOXAR data that was used for validation. Discussions with Heini Wernli, Markus Jonas, Mischa Croci-Maspoli, Francesco Isotta, Olivia Martius, Sandro Buss, and Huw Davies were highly appreciated. Richard Moore's proofreading is gratefully acknowledged. We would also like to thank the reviewers for their useful and critical comments that certainly helped to increase the quality of this paper.