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Pure and Applied Geophysics

, Volume 176, Issue 5, pp 1923–1958 | Cite as

Influences of Gravity Waves on Convectively Induced Turbulence (CIT): A Review

  • Robert D. SharmanEmail author
  • S. B. Trier
Article

Abstract

Thunderstorms are known to produce turbulence. Such turbulence is commonly referred to as convectively induced turbulence or CIT, and can be hazardous to aviation. Although this turbulence can occur both within and outside the convection, out-of-cloud CIT is particularly hazardous, since it occurs in clear air and cannot be seen by eye or onboard radar. Furthermore, due to its small scale and its ties to the underlying convection, it is very difficult to forecast. Guidelines for out-of-cloud CIT avoidance are available, but they are oversimplified and can be misleading. In the search for more appropriate and physically based avoidance guidelines, considerable research has been conducted in recent years on the nature of the phenomenon, and in particular, its connection to gravity waves generated by the convection. This paper reviews the advances in our understanding of out-of-cloud CIT and its relation to convective gravity waves, and provides several detailed examples of observed cases to elucidate some of the underlying dynamics.

Keywords

Convective gravity waves convectively induced turbulence Kelvin–Helmholtz instability aviation hazards 

1 Introduction

Aircraft encounters with turbulence are an important safety and operational hazard for the aviation industry (e.g., Gultepe 2018 this issue). They are responsible for hundreds of injuries each year, occasionally cause aircraft damage, are the source of tens of millions of dollars of additional operational costs for commercial airlines, and thus indirectly increase the cost of air travel. Aircraft avoid turbulence, especially the severe events, whenever possible through the use of aviation weather forecasts and en route tactical avoidance procedures. Yet, turbulence owes its existence to a variety of atmospheric phenomena (e.g., jet streams, terrain-induced flows, and thunderstorms) and each of these requires its own unique forecasting or avoidance strategy tailored to the dynamics of the underlying source or sources.

Progress in improving avoidance methods for a number of these sources is hampered by an incomplete understanding of the governing atmospheric small-scale dynamics, especially with respect to convectively induced turbulence (CIT). CIT occurs within convective clouds, as well as in the clear air above or around the cloud, sometimes at substantial distances from the cloud boundaries. Although turbulence within the cloud can often be avoided through the use of visual cues or through available radar observations, out-of-cloud CIT is difficult to avoid, because it is invisible, and is, therefore, often the cause of unexpected turbulence encounters that may be significant. These out-of-cloud CIT encounters may occur as aircraft: (1) try to circumnavigate the high radar reflectivity regions of thunderstorms to minimize deviations from planned routes; (2) encounter convection that is either unexpected or appears benign from aircraft onboard radar; (3) encounter storm tops that rapidly rise into the aircraft’s flight path; or (4) are inadvertently directed into turbulent regions by air traffic control (Hamilton and Proctor 2003).

Estimates of the relative frequency of CIT vary, depending on location, and correlation time and spatial windows used to compare a thunderstorm to a turbulence report. Pond (2015) estimates up to 7% of commercial flights over Europe are within 35 km of a thunderstorm. Meneguz et al. (2016) and Kim and Chun (2011) both estimate “moderate-or greater” CIT frequencies of about 10% over Europe and the northeastern Atlantic, and South Korea, respectively. Over the U.S., Kaplan et al. (2005) found that 86% of 44 “severe” turbulence cases examined were within 100 km of deep convection, while Cornman and Carmichael (1993) estimate CIT is responsible for as much as 60% of aircraft accidents. Wolff and Sharman (2008) estimate as much as 20% of turbulence reports issued by pilots are associated with nearby cloud–ground lightning. Figure 1 provides an alternative analysis of CIT frequencies over the U. S. based on 5 years of correlating 3D lightning stroke data with reports of turbulence from automated commercial aircraft reports (see Sect. 3) of at least “moderate” intensity. The relative frequency of CIT occurrence is  > 40% over large sections of the Great Plains and is  > 60% over parts of Texas, Florida, the Gulf of Mexico and the Atlantic. However, over the Northeast and Western half of the country, it is considerably smaller (10–20%). The lightning data in Fig. 1 were obtained from the Earth Networks lightning detection system (ENTLN) and include cloud-to-ground and in-cloud lightning pulses. The overall pattern of relative frequency of CIT occurrence is very similar to the pattern of deep convective events derived from the Atmospheric Infrared Sounder (AIRS) NASA Aqua satellite data (Hoffmann and Alexander 2010), from reanalysis data (Kang et al. 2017), and from high-resolution radiosonde data (Gong and Geller 2010; Geller and Gong 2010).
Fig. 1

Frequency of occurrence turbulence in situ EDR reports in the vicinity of convection as indicated by ENTLN detected total (cloud-to-ground and in-cloud) lightning, as a percentage of total turbulence reports. Based on 5-years comparison of EDR reports ≥ 0.22 m 2/3 s −1 to total lightning reports. Reports are binned into ~ 54-km squares and compared to lightning pulse data—that is at least one lightning pulse within ± 0.5 h and ± 50 km from the binned reports

Figure 2 shows a schematic of known out-of-cloud CIT generating regions in the upper troposphere and lower stratosphere (UTLS). The sketch in Fig. 2 provides an indication of where UTLS turbulence would most likely occur relative to the cloud boundaries, but exact locations may be transient and depend strongly on the particular storm type and its environment. This pattern is consistent with recent large eddy simulations (LES) of deep, moist convection (e.g., Lane and Sharman 2014; Verrelle et al. 2017). As shown in the diagram, one major source of out-of-cloud CIT is through convectively induced gravity waves propagating and breaking in the clear air away from the cloud (e.g., Lane et al. 2012). Enhanced shears above the cloud top (e.g., Lester 1993), cloud interfacial instabilities (e.g., Grabowski and Clark 1991; Lane et al. 2003), instabilities induced by sublimation of ice underneath anvils (Kudo 2013), and instabilities associated with cumulus turrets and overshooting tops (e.g., Bedka et al. 2010; Ahmad and Proctor 2011) can also lead to turbulence. Pseudo-obstacle effects may also generate turbulence above and in the downwind wake of the cloud (e.g., Burns et al. 1966; Lemon 1976; Pantley and Lester 1990; Bedard and Cunningham 1991; Bedard 1993; Fric and Roshko 1994). Some of these instabilities may extend well above (Prophet 1970) or below the visible anvil cloud. For more detailed discussions, see Lester (1993), Lane et al. (2012), and Sharman et al. (2012). Within the cloud, turbulence is usually most prevalent and most intense near deep convective updraft and downdraft regions, i.e., near strong vertical velocity gradients especially in the upper third of the cloud depth (Honomichl et al. 2013). Though usually lighter, turbulence also often occurs in the anvil of mature storms (e.g., Muhlbauer et al. 2014). Any of these regions may or may not coincide with the precipitation and thus strongest radar echo patterns as seen on onboard or ground-based radar. An example is shown in Fig. 3 indicating that, in least this case, the strongest turbulence is above, not within, the largest radar reflectivity regions.
Fig. 2

Schematic of common out-of-cloud CIT generation regions (blue areas) in a mature thunderstorm

Fig. 3

Comparison of vertical cross section of (b) radar reflectivity (dBZ) from a convective storm to (c) radar derived energy dissipation rate to the 1/3 power (EDR m2/3 s−1), using the algorithm described in Williams and Meymaris (2016) along the transect shown as a red line in (a). Date and time is Sept 28, 2007 at 2225 UTC

The frequency of out-of-cloud CIT is difficult to determine, since aircraft position relative to the cloud boundaries must be known to very good accuracy, but pilot reports (PIREPs), with a median position error of  ~ 40 km, have location uncertainties too large for such evaluations (Sharman et al. 2014). However, comparing more accurate automated turbulence reports from some commercial aircraft to radar derived cloud boundaries shows that although turbulence is most frequent within cloud, significant frequencies of turbulence also occur above and to the sides of the cloud as well. Lane et al. (2012) show that aircraft is 15 times more likely than average to encounter moderate-or-greater (MOG) turbulence when within 15 km of convection (Fig. 4a), and nearly 90 times more likely if flying within 800 m above the radar echo top (Fig. 4b).
Fig. 4

Relative risk of MOG turbulence derived from in situ EDR measurements as a function of: a distance to convection and (b) altitude above radar echo top in convective regions. The reports were between 7.6- and 12.8-km altitude (flight levels 25,000–42,000 ft) from May to October of 2004 and 2005. From Lane et al. (2012) c American Meteorological Society. Reprinted with permission

Because of the relatively high frequency of turbulence encounters in the vicinity of convection, the U. S. Federal Aviation Administration (FAA) has developed guidelines for thunderstorm avoidance (FAA 2014 Section 7-1-29), which read in part,
  • “Turbulence should be expected to occur near convective activity, even in clear air.”

  • “Do avoid by at least 20 miles any thunderstorm identified as severe or giving an intense radar echo. This is especially true under the anvil of a large cumulonimbus.”

  • “Don’t attempt to fly under the anvil of a thunderstorm. There is a potential for severe and extreme clear air turbulence.”

  • “Do regard as extremely hazardous any thunderstorm with tops 35,000 ft or higher whether the top is visually sighted or determined by radar.”

A strategy for turbulence avoidance above visible cloud given in earlier versions of the guidelines, viz., “Do clear the top of a known or suspected severe thunderstorm by at least 1000 ft altitude for each 10 knots of wind speed at the cloud top” has been eliminated from the current guidelines. As indicated in Lane et al. (2012) and other studies, these guidelines can be misleading, because they disregard the complexities of CIT. Furthermore, they may call for tactical avoidance that is unnecessary, wasting fuel and time. Although other visual indicators may be available [e.g., satellite imagery of overshooting cloud tops (Bedka et al. 2010; Monette and Sieglaff 2014), gravity-wave patterns embedded in the anvil, rapid anvil expansion, and banded structures along the outside edge of the anvil (Lenz et al. 2009)], these can provide only indirect inferences of the presence, location, and intensity of CIT. Improved forecasting and avoidance of turbulence near thunderstorms can only be realized through a better understanding of the conditions conducive to turbulence generation, and its extent and intensity. However, in recent decades, understanding of physical mechanisms influencing CIT has increased, mainly through the use of high-resolution numerical modeling of idealized scenarios and well-observed cases. Observations have also been improving, both in quantity and precision, so that it is now possible to correctly identify CIT cases, both within and out-of-cloud.

This article specifically reviews recent progress in understanding remote CIT, which we define as turbulence that is spatially and sometimes temporally removed from its parent deep convection. Since CIT may be remote we avoid the use of the terms “near-cloud turbulence” (NCT, Lane et al. 2012), “turbulence near thunderstorms” (TNT, Lester 1993) and “turbulence near thunderstorm tops” (TNTT, Pantley and Lester 1990). Thus, hereafter, remote CIT is referred to simply as CIT. Our examination of CIT in this review encompasses both observational and numerical modeling studies.

2 Convective Gravity Waves

In recent years, high-resolution numerical simulations have shown that in many cases, out-of-cloud CIT can be linked to gravity waves generated by convection. This is not surprising since gravity waves can be produced whenever air parcels are displaced vertically in a statically stable environment. References to early observational studies of convective gravity waves can be found in Gossard and Hooke (1975), Einaudi et al. (1978), Lu et al. (1984), Grachev et al. (1984), Alexander and Pfister (1995) and Hansen et al. (2002). The observations have been corroborated by theoretical studies (e.g., Townsend 1966, 1968; Stull 1976; Lin and Smith 1986; Chun and Baik 1994; Chun 1997; Baik and Chun 1996; Baik et al. 1999; for other early references see Lin 1994a, b; Romanova and Yakushkin 1995), laboratory experiments (e.g., Deardorff et al. 1969; Mclaren et al. 1973; Cerasoli 1978; Ansong and Sutherland 2010), and numerical simulations (Deardorff 1969; Fovell et al. 1992; Alexander et al. 1995; Gavrilov and Kshevetskii 2015). Additional references concerning results from simulations are given in Lane et al. (2012) and Lane (2016). Some of the techniques used to study internal gravity waves generated by penetrative convection in stellar interiors may also be relevant to the earth’s atmosphere (see, e.g., Pinçon et al. 2016, and references therein). These and other studies have helped to better elucidate the dynamics of the waves and have identified three possible convective gravity-wave mechanisms:
  1. 1.

    thermal forcing and latent heat released within the cloud during condensation (e.g., Lin and Smith 1986; Bretherton 1988; Nicholls et al. 1991; Pandya et al. 1993; Pandya and Durran 1996; Pandya and Alexander 1999; Alexander and Barnet 2007; Chun and Baik 1998);

     
  2. 2.

    an obstacle effect (also termed “quasi-stationary forcing” by Fovell et al. 1992), whereby the cloud acts as a barrier to impinging airflow (e.g., Clark et al. 1986; Beres et al. 2002; Wang 2003);

     
  3. 3.

    a “mechanical oscillator” effect due to rapid cumulus growth impinging on a stable layer (e.g., the tropopause), overshooting its level of neutral buoyancy (termed penetrative convection) and initiating gravity waves (e.g., Pierce and Coroniti 1966; Stull 1976; Jones 1982; Fovell et al. 1992; Lane et al. 2001; Kumar 2007; Lane 2008).

     

For detailed explanations of these effects, see, e.g., Bradbury (1973), Clark et al. (1986), Fovell et al. (1992), Fritts and Alexander (2003), Kim et al. (2003), and Song et al. (2003). Fritts and Alexander (2003) indicate that the relative importance of these generating mechanisms in explaining observations depends on the particular environmental conditions. Numerical studies by Lane et al. (2001) and Song et al. (2003) also point to the importance of nonlinear advection effects in the forcing process.

The propagation characteristics of gravity waves are described in many excellent texts (e.g., Turner 1973; Gossard and Hooke 1975; Lighthill 1978; Gill 1982; Nappo 2002; Lin 2007) and review papers (Lin 1994a,b; Fritts and Alexander 2003). Unlike mountain lee waves, convective gravity waves are both mobile and transient, depending on the trajectory and life cycle of the generating cloud. The fundamental properties of buoyancy driven gravity waves can be demonstrated by a simple 2D linear, inviscid, adiabatic, nonhydrostatic, nonrotating, non-Boussinesq system, for which the following single governing equation for the wave induced density weighted vertical velocity \(\tilde{w} = w'\exp (z/2H)\), where H is an atmospheric scale height, can be derived (e.g., Nappo 2002; Lin 2007):
$$\left( {\partial_{\text{t}} \, + \,U\partial_{\text{x}} } \right)^{2} \,(\tilde{w}_{\text{xx}} \, + \,\tilde{w}_{\text{zz}} )\, + \,N^{2} \,\tilde{w}_{\text{xx}} \, = \frac{g}{{c_{\text{p}} T_{0} U^{2} }}Q.$$
(1)
Here, cp is the specific heat at constant pressure, T0 is the mean environmental temperature, Q is the heating rate (usually dominated by latent heat release), U(z) is the mean flow velocity, and N(z) is the Brunt–Väisälä or buoyancy frequency, \(N^{2} \, = \,\left( {g/\theta_{\text{v}} } \right)\,\left( {\partial \theta_{\text{v}} /{\text{d}}z} \right)\) where \(\theta_{\text{v}}\) is the virtual potential temperature and g is gravity. Subscript notation is used for differentiation in (1), with x signifying the horizontal direction and z is the direction aligned with gravity, and t is time. If a monochromatic wave solution to (1) is assumed with \(\tilde{w} = \hat{w}(z)e^{i(kx\, - \,\omega t)} \, = \,\hat{w}(z)e^{ik(x\, - ct)}\), where \(k\, = \,2\pi /\lambda\) is the horizontal wavenumber in the x direction (typically the west–east or zonal direction), with λ the corresponding wavelength, ω is a wave frequency, and c is the horizontal wave speed in the direction of propagation, then (1) becomes
$$\hat{w}_{\text{zz}} + \left( {\ell^{2} - k^{2} } \right)\hat{w} = \frac{g}{{c_{\text{p}} T_{0} U^{2} }}Q,$$
(2)
where
$$\ell^{2} = \frac{{N^{2} }}{{\left( {U - c} \right)^{2} }} - \frac{{U_{\text{zz}} }}{{\left( {U - c} \right)}} - \frac{{U_{\text{z}} }}{{H\left( {U - c} \right)}} - \frac{1}{{4H^{2} }} \approx k^{2} \left[ {\frac{{N^{2} }}{{\varOmega^{2} }} - \frac{{U_{\text{zz}} }}{k\varOmega }} \right],$$
(3)
is the so-called Scorer parameter, where
$$\varOmega = \omega - kU,$$
(4)
is the intrinsic (or Doppler shifted) frequency, i.e., the frequency of the wave relative to the mean flow, whereas ω is the frequency in a frame of reference with respect to the ground. If Q = 0 and N and U are approximately uniform with height, then (2) and (3) admit homogeneous solutions with vertical dependence of the form:
$$\hat{w} = Ae^{ + imz} + Be^{ - imz} ,\quad m = \sqrt {\ell^{2} - k^{2} } ,\quad {\text{for}}\quad \ell^{2} > k^{2} ,$$
(5)
$$\hat{w} = Ce^{ + \mu z} + De^{ - \mu z} ,\quad \mu = \sqrt {k^{2} - \ell^{2} } ,\quad {\text{for}}\quad k^{2} > \ell^{2} .$$
(6)
In (5), m is the vertical wavenumber and the + sign in the exponent (term with the coefficient A) corresponds to waves propagating energy upwards, and vice versa for the –sign (B term) (Eliassen and Palm 1961). For vertically propagating waves of the form (5), a dispersion relation that relates the frequency to the vertical and horizontal wavenumbers and the background buoyancy frequency N [ignoring the Uzz and non-Boussinesq terms for \(\ell^{2}\) in (3)] is
$$\varOmega^{2} \, = \,\,\frac{{Nk^{2} }}{{k^{2} + m^{2} }}\, = \,N^{2} \cos^{2} \alpha ,$$
(7)
where α is the angle between lines of constant phase and the vertical. As evident from (7), for vertically propagating waves, the intrinsic frequency Ω is always less than the buoyancy frequency N.

In (6), μ represents a growth rate (C term) or decay rate (D term) with height. In general, for propagation in unlimited domains, the first term is unphysical, and the wave amplitude decays with height. These are termed evanescent waves. In cases where \(\ell^{2}\) decreases with height, for example, a wave of a given k may be propagating at low levels, where 2 > k2 but may be evanescent at higher levels, where 2 < k2. In such cases, the waves are often described as “trapped” at levels where \(\ell^{2} - k^{2} \approx 0\) (e.g., Lindzen and Tung 1976).

For the propagating solutions (5), the wave propagation or phase velocities Cp in the x and z directions are
$$\begin{aligned} C_{px} = & \frac{\omega }{k} = U \pm \frac{N}{{\sqrt {k^{2} + m^{2} } }} \\ C_{pz} = & \frac{\omega }{m} = \frac{k}{m}\left( {U \pm \frac{N}{{\sqrt {k^{2} + m^{2} } }}} \right). \\ \end{aligned}$$
(8)

These relations may be helpful in the analysis of observations or output from simulations.

In reality, since the convection source is temporally and spatially dependent, a spectrum of gravity waves is generated, and since they are dispersive, different wavelengths travel at different velocities. In this case, for trapped waves, only a discrete set of wavelengths satisfy the criterion for trapping, and the waves are very well defined. More generally, the energy of groups of waves propagate according to the group velocity:
$$\begin{aligned} Cg_{x} & = \frac{\partial \omega }{\partial k} = U \pm \frac{{Nm^{2} }}{{\left( {k^{2} + m^{2} } \right)^{3/2} }}, \\ Cg_{z} & = \frac{\partial \omega }{\partial m} = \mp \frac{Nkm}{{\left( {k^{2} + m^{2} } \right)^{3/2} }}. \\ \end{aligned}$$
(9)

For k, m > 0, the group or energy propagation is eastward and downward, so negative m corresponds to upward energy propagation. Note that the wave vector k = (k, m) is perpendicular to the group velocity (k • Cg= 0), so that the group propagation is along the phase lines.

An interesting situation occurs at the altitudes where the wave speed c equals the background wind speed U, the so-called critical level zc, where \(\ell^{2}\) is undefined (Eq. 3). As waves approach a critical level from the generation region the vertical wavelength 2π/m and vertical group velocity Cgz in (9) become small, and the wave amplitudes can become large (Bretherton 1966). In this case, nonlinear processes become important, and may lead to wave steepening, overturning, and “breaking” into turbulence patches (e.g., Dörnbrack et al. 1995). Applying this concept to convective gravity waves propagating upward, waves with a downshear horizontal component of propagation may amplify and break when they approach a critical level zc above the cloud; but upshear propagating waves do not encounter a critical level, and such waves are often found to decay with altitude (i.e., are evanescent). These two different situations are illustrated schematically in Fig. 5. Convective gravity-wave encounters with critical levels have been shown in a number of studies to lead to CIT (e.g., Teitelbaum et al. 1999; Lane et al. 2003; Moustaoui et al. 2004), so identification of critical levels is important for understanding many instances of CIT. For further details on critical levels and transient effects, see, e.g., Nappo (2002).
Fig. 5

Schematic diagram of gravity-wave propagation and breakdown above deep convection in an environment with negative above-cloud vertical wind shear in a reference frame moving with the cloud-top wind. If the change in wind speed is sufficiently large, waves with negative phase speed c will encounter a critical layer where U(zc) = c, and break down into turbulence, while waves with positive phase speed will become evanescent. Typical phase speeds and horizontal wavelengths at shown at bottom left. From Lane and Sharman (2008). c American Meteorological Society. Reprinted with permission

Wave encounters with critical levels are just one example of how gravity waves may perturb the environment such that the local gradient Richardson number \(R_i\, = \,N^{2} /U_{z}^{2}\) becomes small enough to induce Kelvin–Helmholtz instabilities (e.g., Hodges 1967; Pavelin et al. 2001). The Ri criterion for instability depends on the environment, although theoretically Ri > 1 guarantees stability (e.g., Abarbanel et al. 1984; Miles 1986). In practice, the use of Ri as an instability diagnostic is hampered by its strong dependence on vertical differencing (e.g., Reiter and Lester 1968), and in simulation experiments, identification of regions of small Ri associated with gravity waves also requires that the gravity waves are properly resolved

Environmental shear, rotation, and 3D effects considerably complicate wave propagation characteristics. The introduction of three dimensions allows wave energy to spread horizontally, producing "ripples" resembling surface waves that are generated when a rock is dropped into a pond, (e.g., Piani et al. 2000; Lane et al. 2001; Lane 2016). However, in the vertical, surfaces of constant phase for a given ω < N will still take the form of a cone of half angle α = cos−1(Ω/N) centered on the vertical axis with the apex at the gravity-wave source (Eq. 7). With no mean flow, the wave amplitudes \(\propto \,J_{1} (kr)\), where r is radial distance and J1 is the Bessel function of the first kind (Ansong and Sutherland 2010), but with substantial wind impinging on the cloud or overshooting top the generated gravity waves may produce parabolic-shaped wave patterns, which resemble 3D lee waves and ship waves (e.g. Sharman and Wurtele 1983; Lin 1986; Lin and Li 1988; Sharman and Wurtele 2004; Wang et al. 2010). Discussions of shear and rotation effects can be found, e.g., in Stobie et al. (1983) and Lin (2007). For example, studies involving topographically generated mountain waves point to important effects of wind turning with height (Broad 1995; Shutts 1998; Guarino et al. 2017), and (Rossby or Jones’) singularities introduced by rotation at altitudes where z = ± f/kUz, where f is the Coriolis frequency (Jones 1967; Wurtele et al. 1996). Also an important nonlinear effect occurs when gravity waves break to form turbulence: the resulting disruption in stability can cause “secondary waves” to be generated, which can in turn propagate away from the mixing layer and possibly break at a different location than the primary waves. Examples of secondary waves generated by breaking of primary convective gravity waves can be found in Holton and Alexander (1999), Zhou et al. (2002), Snively and Pasko (2003), Chun and Kim (2008), and Lane and Sharman (2006).

In addition to these shorter wavelength convectively induced gravity waves, large amplitude, large wavelength (10–100 s km) waves or pulses, with compensating subsidence (e.g., Bretherton and Smolarkiewicz 1989), can also be generated by mesoscale regions of thunderstorms. Ruppert and Bosart (2014) contain a thorough review of the previous (mostly observational) work on this topic. Many case studies (typically based on examination of surface barograph parameters and analyses of radar and radiosondes) have shown that these mesoscale gravity waves can initiate new or modify existing deep convection. The dependence of gravity-wave characteristics on forcing scale was first demonstrated by Queney (1948) for the mountain wave problem, who showed that for mountains of scale  ~ 10 km or greater, the hydrostatic assumption is approximately satisfied [i.e., k → 0 in Eqs. (2)–(9)]. For convectively generated mesoscale gravity waves, the resultant hydrostatic disturbance can be characterized as wave packets (e.g., Eom 1975), or solitary waves (e.g., Lin and Goff 1988) and may be long-lived, lasting for several hours or more (e.g., Trier and Sharman 2016). Since the wavelength and/or the gravity-wave lifetime are large, they may be influenced by the earth’s rotation and Eqs. (2)–(9) must be modified to include rotational effects and are consequently termed “inertia–gravity waves” (see, e.g., Nappo 2002; Lin 2007).

Perhaps the first connections of short convective gravity waves to turbulence were proposed by Haman (1962) and Townsend (1966, 1968). Early observations of turbulence above cloud, which were probably related to convectively generated gravity waves, have been reported by Burns et al. (1966), Prophet (1970), Burnham (1970), and Pantley and Lester (1990). A 2D simulation case study of CIT by Keller et al. (1983), where the cloud was replaced by an obstacle, showed good agreement with a CIT encounter (Parks et al. 1985) over Hannibal, MO. A 3D cloud simulation study by Droegemeier et al. (1997) identified a connection between convectively generated gravity waves and turbulence. 2D and 3D numerical studies of a CIT encounter over Dickinson, ND by Lane et al. (2003) and Lane and Sharman (2006, 2008) demonstrated that the convective gravity waves were most likely due to the mechanical oscillator effect, and the turbulence was induced when the convective gravity waves encountered a cloud-relative critical level. Kim and Chun (2012), and others have confirmed this connection in another case. Since then, detection of convectively induced waves and turbulence in field programs, inferences from satellite imagery or radar, and from reports of turbulence through pilot experience and have become commonplace. An example of an extreme turbulence encounter above deep convection due to breaking convective gravity waves is provided in Fig. 6 from Kim and Chun (2012). The CIT occurred at a cruising altitude of 35 kft (~ 10.7-km MSL), and was marked by a change in vertical acceleration of 2.26 g during a 3-min interval. Further case study examples will be provided in Sect. 4. Emphasis there is not on the convective gravity waves themselves, but rather on how they may lead to turbulence.
Fig. 6

a Vertical acceleration (solid line), wind speed (dashed line), and wind direction (dotted line) and b TKE (dotted line) and flight level (solid line) as a function of time from 1033:10 to 1036:10 UTC 2 Sep 2007, derived from the Digital Flight Data Recorder. The area with the vertical lines from 89 to 109 s signify the duration of the turbulence event. From Kim and Chun (2012). c American Meteorological Society. Reprinted with permission

3 Turbulence Metric and Observations

Research to better understand the nature and causes of aviation-scale turbulence in general, and CIT in particular, has been hampered in part by the lack of routinely available reliable observational data. Until recently, verbal pilot reports (PIREPs) have typically been the only routine source of information about the location and severity of turbulence. These reports are, however, incomplete (since reporting is voluntary), and highly subjective (Schwartz 1996). In particular, the turbulence intensity in PIREPs is noted only as “smooth”, “light”, “moderate”, “severe” (or sometimes “heavy”), and “extreme”, but these characterizations are aircraft dependent, and are influenced by the pilot’s experience (what one pilot views as “moderate” might be perceived as “light” or “severe” by another), see Sharman (2016) for definitions of these intensities.

The intensity of atmospheric turbulent eddies within the range of eddy sizes that affect commercial aircraft (~ 100 m to a few km) is usually small enough to be perceived by occupants as “smooth”, and Sharman et al. (2014) show that about 85–90% of PIREPs are in the smooth category. However, if the turbulent eddies within this size range are energetic enough, some level of aircraft bumpiness will be perceived by aircraft occupants. “Moderate” turbulence, defined as a vertical acceleration magnitude of at least 0.5 g, may be enough for the pilot to seek changes in altitude or airspeed, but the aircraft remains in control at all times. “Severe” turbulence is defined as at least 1 g vertical acceleration, which may cause large, abrupt changes in the flight of the aircraft, and may even cause momentary loss of control.

For CIT investigations, probably more important than the subjective turbulence intensity reported in the PIREP is the associated position and time inaccuracy. Recent investigations into the accuracy of PIREPs have indicated an average position error of about 50 km (Sharman et al. 2014), which can easily be comparable to the size of a large individual thunderstorm. Thus, it becomes difficult if not impossible to determine if a report is in or out of cloud, although PIREPs may still be useful for identification of turbulence cases. To address the accuracy problem associated with PIREPs, automated in situ reports of turbulence intensity are now being routinely generated onboard some aircraft, and this information is downlinked to the ground (Sharman et al. 2014). The strategy is to provide estimates of an atmospheric turbulence metric, the cube root of the energy dissipation ε. This quantity is commonly referred to as “EDR”, and is the ICAO (2013) standard for turbulence reporting. The EDR estimates (in units of m2/3 s−1) are provided by relating spectral levels based on analyses of time series of vertical acceleration, vertical velocity or true airspeed to EDR (MacCready 1964; Cornman et al. 1995; Sharman et al. 2014). EDR is directly related to the aircraft root-mean-square (RMS) normal acceleration σg (e.g., McCready 1964):
$$\sigma_{\text{g}} = V_{\text{T}}^{1/3} \left[ {\int {\left| {A(f)} \right|^{2} S_{\text{w}} (f)} {\text{d}}f} \right]^{1/2} \varepsilon^{1/3} ,$$
(10)
where A(f) is the aircraft specific response at frequency f and true airspeed VT, and \(S_{\text{w}}\) is a specified atmospheric spectral model (e.g., Kolomogorov or von Karman) for the vertical wind w with a unit \(\varepsilon\). Implicit in all onboard computational algorithms for EDR is the existence of an inertial subrange where \(S_{\text{w}} \, \propto \,k^{ - 5/3}\) and k is the horizontal wavenumber. This has been shown by many investigators to be a good approximation in the range of wavenumbers or wavelengths that affect aircraft (e.g., Sharman 2016).
In the vertical wind-based algorithm described in Sharman et al. (2014), a 10-s sliding time window with a 5-s overlap is used to compute Sw and EDR, providing 12 EDR estimates each minute. The mean and peak of the 12 estimates are recorded and downlinked. The 1-min resolution corresponds to a horizontal scale of about 12 km at typical aircraft cruise speeds, providing a much better position of events for comparing to cloud observations than PIREPs. Figure 7 shows two examples of in situ EDR recordings outside of deep convection. Future in situ EDR implementations will identify the location of the peak EDR, providing about 1-km resolution. Useful guidelines for turbulence intensity as indicated by EDR are provided in Sharman et al. (2014). Based on comparisons to PIREPs, they find for a medium-weight category commercial aircraft (maximum takeoff 15,500–300,000 lbs) flying under typical cruise conditions that EDRs of  ~ 0.15, 0.22, 0.35 m2/3 s−1 could be identified with “light”, “moderate”, and “severe” PIREPs, respectively, although there is significant scatter in the data. These intensity thresholds are generally expected to be smaller for light aircraft and larger for heavy aircraft.
Fig. 7

Infrared satellite images of two cases with in situ commercial aviation measurements of EDR during turbulence encounters. The flight tracks and turbulence intensities (green = smooth, yellow = light, orange = moderate, and red = severe) are overlaid on the images. Adapted from Lane et al. (2012)

Some CIT case studies have used EDR estimates based on retrieved information from the flight data recorder (e.g., Lane et al. 2003; Kim and Chun 2012). This provides a still better estimate of peak EDR and its location, but these records are frequently not available to the public, and thus, most of the CIT examples presented in the next section use the automated EDR estimates available from selected commercial aircraft. As more aircraft implement some version of the EDR estimation algorithm, additional cases of CIT will be identified.

4 Example CIT Types

4.1 Overview

EDR measurements from commercial aircraft (Sect. 3) have helped characterize CIT, but are by themselves insufficient to determine its causes. However, Lane et al. (2012) describe how current research numerical weather prediction (NWP) models configured with fine-scale nesting can be used to diagnose different possible mechanisms explaining the origins of CIT. The horizontal grid spacing in the highest resolution domains of these models is typically between a few 100 m and a kilometer, which is not sufficient to resolve the motions responsible for the aircraft turbulence, but is often adequate to resolve the larger scale mechanisms directly responsible for its onset. In this section, we present examples of possible CIT initiation mechanisms from these types of research simulations of observed cases identified by EDR measurements and PIREPs.

The EDR data from a subset of the commercial aviation fleet have been overlaid on displays of high-resolution satellite imagery to develop climatologies of turbulence near specific types of cloud patterns including banded cirrus in MCS anvils (Lenz et al. 2009) and relatively isolated overshooting deep convection (Bedka et al. 2010). Such displays have also been used to identify individual cases of observed moderate-or-greater CIT (Fig. 7), which have been simulated by high-resolution research models in an effort to better understand its different causes (e.g., Fovell et al. 2007; Trier and Sharman 2009; Lane et al. 2012; Trier et al. 2012).

These modeling studies have suggested that a variety of different mechanisms could be responsible for the onset of CIT in different situations. Though the overall mechanisms are different, a common aspect emphasized in this section is the potentially important role of gravity waves as either a direct or an indirect linkage between the deep convection and CIT onset. Supporting evidence for frequent gravity-wave activity in CIT environments has resulted from processing techniques used on imagery from new high-resolution satellite platforms (e.g., Wimmers et al. 2018).

4.2 Turbulence Occurring Above Deep Convection

Locations directly above either rapidly growing (e.g., Lane et al. 2003) or dissipating (e.g., Kim and Chun 2012) deep convection are common sites for CIT. Recall from Fig. 4 that the risk of moderate-or-greater turbulence is enhanced over the background risk by 50 times (25 times) for depths of approximately 2 km (3 km) above deep convection. In their high-resolution idealized simulation of a severe turbulence encounter above an isolated continental midlatitude thunderstorm, Lane et al. (2003) note that the turbulence at different depths above the storm could result from a variety of different mechanisms. Turbulence can result from either strong vertical motions associated with vertical displacements of statically stable air induced by the storm below, small-scale shearing instabilities including Kelvin–Helmholtz instability (KHI) associated with strong deformation near the boundary of the cloud and surrounding atmosphere (e.g., Grabowski and Clark 1991), or the breaking of vertically propagating gravity waves initiated by the tropopause-penetrating convection (Fig. 5). Lane et al. (2003) note that turbulence events more than a kilometer above deep convection are likely dominated by the latter mechanism, which we focus on in the remainder of this subsection.

Though wave breaking above deep convection is often invisible, making it a significant hazard for aviation turbulence, such wave breaking is sometimes indicated by extensive cirrus plumes occurring above the deep convection (Levizanni and Setvak 1996; Wang 2003, 2007). The formation of these cirrus plumes (Fig. 8) is influenced by vertical displacements and resulting stratospheric–tropospheric exchange (STE) of water vapor during wave breaking. Homeyer et al. (2017) found that lower stratospheric cirrus plumes above deep convection are more likely to occur when there is strong flow relative to the horizontal motion of the deep convection, which may be indicative of strong vertical shear that would increase the likelihood of a critical level that leads to wave breaking.
Fig. 8

GOES 1-km visible satellite imagery for 0115 UTC 21 May 2014. State boundaries are indicated by the solid cyan lines and the thick dashed white line encircles an observed above-anvil cirrus plume. Adapted from Homeyer et al. (2017)

In a numerical sensitivity study using 2D and 3D numerical models, Lane and Sharman (2008) found that wave breaking and turbulence is most intense and horizontally widespread in strong vertical shear conditions, but under such conditions is confined to locations relatively close to the top of the deep convection (Fig. 9b). However, Lane and Sharman also note that the depth through which wave breaking and turbulence can occur is maximized for intermediate values of vertical shear (Fig. 9c), which is germane to turbulence avoidance strategies. These two results from Lane and Sharman are consistent with the strength of the environmental vertical shear influencing the height of the critical level above the deep convection. Lane and Sharman also varied the strength of the static stability in their sensitivity experiments and found that small static stability generally facilitated wave breaking and development of turbulence. Homeyer et al. (2014) found that details in the vertical structure of static stability in different synoptic situations influence the depth of vertical mixing. Though their emphasis is on general aspects of STE, these results are also relevant to wave breaking and CIT.
Fig. 9

a Form of the idealized jet-like vertical profile of horizontal winds used in the high-resolution Lane and Sharman (2008) simulations, where \(U_{ \hbox{max} }\) is the maximum horizontal wind near-cloud top and \(\Delta U\) is the wind difference through a layer of depth \(\sigma\). b, c vertical cross sections of model output from the Lane and Sharman (2008) simulations, where the blue colors are cloud condensate with the darker shades indicating larger cloud mixing ratios, and the shading above the cloud is resolved-scale turbulence kinetic energy (TKE) with gold, brown, and red colors indicating TKE values representative of light, moderate, and severe turbulence, respectively

Deep convection is much less common in midlatitudes during the cold season (e.g., Kang et al. 2017), because conditional instability is often limited or nonexistent. However, despite its lesser frequency, several important environmental factors may contribute to wave breaking above deep convection being particularly hazardous to aviation during less frequent cold-season convective outbreaks. First, because of its association with strong synoptic disturbances in the winter, deep convection is often more widespread than in summer and, therefore, may be more difficult to avoid. Second, the height of the tropopause is lower during the cold season, particularly on the cyclonic side of jet streams, where stratospheric intrusions or tropopause folds are most common. Thus, commercial aviation cruising altitudes in the winter are more likely to be in the lower stratosphere (where vertical wave propagation is supported) than in summer. Third, the strong vertical shear above intense jets located at wintertime tropopause is more likely to lead to critical levels facilitating wave breaking above any deep convection that occurs.

Figure 10 summarizes the distribution of moderate-or-greater PIREPs and EDR reports and illustrates the mechanisms influencing CIT in a late-winter case on 9–10 March 2006 reported by Trier et al. (2012). This modeling study uses a large-domain strategy with multiple nests, which are typically employed in simulations of observed turbulence outbreaks, where it is necessary to both resolve the large-scale flow pattern conducive to turbulence and the fine-scale mechanisms directly responsible for its initiation. In these types of simulations, typical horizontal (vertical) spacings for the finest grid are less than 1 km (250 m), since accurate simulations of turbulence-producing phenomena, such as the breaking of vertically propagating internal gravity waves (e.g., Lane and Knievel 2005) and KHI (e.g., Trier and Sharman 2016), are sensitive to model resolution.
Fig. 10

a Composite diagram of rainfall and convectively induced UTLS anticyclonic outflow (shown schematically by the bold curve with arrows) from a simulation described in Trier et al. (2012) using a model domain with 3.3-km horizontal grid spacing. b Vertical cross section of simulated meridional wind (5-m s−1 contour intervals) along transect SN in part (a) with time–space corrected observations of moderate-or-greater turbulence within 33.3 km of SN projected onto the cross section and described in the legend. c, d Vertical cross sections from a higher resolution simulation with 667-m horizontal grid spacing from the left inset region of part b of total cloud condensate (color shading; scale at right), vertical velocity (1-m s−1 contour interval; red lines ≥ 1 m s−1; brown lines  ≤ – 1 m s−1), where c has potential temperature (2-K contour intervals in black solid lines and d has winds parallel to the cross section (5-m s−1 contour intervals in black solid lines). The dashed vertically tilted lines in part c represent example phase lines of vertically propagating gravity waves, and the dashed horizontal lines in part d represent a range of estimates for the location of critical levels for vertically propagating waves of different frequencies. e Vertical cross section from the same higher resolution simulation as for parts c and d, but for the right inset region of part b of total cloud condensate (color shading; scale at right) and potential temperature (2-K contour intervals). From Trier (2016). Copyright Springer International Publishing

In this late-winter turbulence outbreak, there was widespread deep convection (Fig. 10a) and the turbulence occurred mostly above a strong upper tropospheric jet (Fig. 10b) that was oriented approximately south-to-north. In their simulations, Trier et al. (2012) found that the southerly jet was enhanced by  ~ 30% due to the convectively induced anticyclonic UTLS circulation indicated schematically by the bold arrows in Fig. 10a.

A vertical cross section (Fig. 10b) along the transect SN in the  = 3.3-km horizontal domain (Fig. 10a) indicates the most intense observed turbulence clustered in two regions including one from x = 640–760 km located directly above the jet maximum and another from x = 300–420 km located above the entrance region of the jet and relatively shallow moist convection. In the highest resolution horizontal domain with  = 667 m, the moist convection in the jet entrance region (Fig. 10a, b) excites vertically propagating internal gravity waves when it impinges on the region of larger static stability beginning near 8-km MSL (Fig. 10c). These waves break while propagating downshear when they reach a critical level, where U = c (Fig. 10d), similar to what is illustrated schematically in Fig. 5. Directly above the jet, overturning billows in the potential temperature field (Fig. 10e) implicate KHI as the turbulence mechanism. Though the KHI is horizontally remote from ongoing moist convection, note that it arises in response to strong vertical shear associated with convectively induced enhancement of the jet (Fig. 10a).

4.3 Turbulence Horizontally Displaced from Adjacent Deep Convection

In addition to occurring above cloud, CIT may also be displaced laterally from clouds associated with preexisting deep convection. In particular, the likelihood of moderate-or-greater turbulence is two times greater than the background occurrence within 75 km of deep convection (Fig. 4b). The ability of a current research model to simulate this phenomenon and its sensitivity to model resolution was analyzed by Barber et al. (2018). A specific example from Fovell et al. (2007) and Lane et al. (2012) occurred on 5 August 2005. In this case, EDR measurements documented moderate and severe turbulences at 11.3–11.9-km MSL encountered by two commercial aircraft approximately 20–30-km southeast of a small but intense thunderstorm complex (Fig. 7a).

CIT horizontally adjacent to, but outside of, deep convection has been less frequently studied and is not as well understood as CIT occurring directly above deep convection. However, some candidate mechanisms that could account for such turbulence have been hypothesized based on results from numerical simulations. For instance, Fovell et al. (2007) and Lane et al. (2012) proposed that vertically trapped gravity waves triggered by deep convection could propagate outward from the storm and result in localized regions of subcritical Richardson number within larger scale environments, where the background \(R_{i}\) is small.

Fovell et al. (2007) and Lane et al. (2012) presented calculations of the Scorer parameter \(l^{2}\) (Eq. 3) and, therefore, the susceptibility to wave trapping, for an idealized simulation based on the 5 August observed severe turbulence case at a downstream location a few tens of kilometers outside of the simulated cloud. These calculations are compared with those from a corresponding idealized simulation, where moist processes (e.g., deep convection) were excluded and both simulations have trapping layers centered in the upper troposphere near 11-km MSL and near the tropopause at around 13.5-km MSL (Fig. 11a). The full physics simulation that included moist processes also has a 0.5–1.0-km-deep layer of Ri  < 1 centered at 11.5-km MSL (Fig. 11b), which corresponds well with the altitude of the observed turbulence inferred from the EDR reports in Fig. 7a. The simulated Ri values at this altitude are smaller than the nearly subcritical values from the dry simulation of Ri~ 1 (Fig. 11b), and Fovell et al. (2007) and Lane et al. (2012) attribute these decreases to trapped internal gravity waves generated by upstream deep convection in the control (full physics) run. These Ri decreases in \({\text{control}} - {\text{dry}}\) (Fig. 11b) near and just above the bottom trapping layer near 11.5 km (Fig. 11a) are related to the combination of increases in vertical shear magnitude (Fig. 11c) and reductions in static stability (Fig. 11d) compared to the background environment signified by the dry simulation.
Fig. 11

Vertical profiles from a location southeast of the storm from control and dry simulations of the observed case in Fig. 7a. (a) Scorer parameter calculated using horizontal winds parallel to the wave propagation vector and a 30-m s−1 horizontal phase speed; (b) Gradient Richardson number and its contribution from (c) the squared magnitude of the vertical shear S, and (d) the squared Brunt-Väisälä frequency. From Lane et al. (2012). Copyright American Meteorological Society. Reprinted with permission

In the above example, short-wavelength (λ  ~ 10 km) trapped gravity waves are hypothesized to contribute to the onset of turbulence not through wave breaking but through shear instabilities resulting from local effects of the wave-induced perturbations on Ri. Figure 12 presents visual evidence of similarly trapped waves within an anvil extending downstream from a relatively isolated severe thunderstorm penetrating the tropopause in the central United States during 4 June 2015. In this case, there are turbulence PIREPs near wave fronts at the outer edge of the downstream anvil (Fig. 12). However, unlike in the 5 August 2005 case, only light intensities of turbulence were reported in the 4 June 2015 case.
Fig. 12

GOES 1-km visible satellite imagery from 0130 UTC 4 June 2015. The cross symbols indicate locations of turbulence PIREPs within 50 min of the time of the image

Zovko-Rajak and Lane (2014) analyzed high-resolution idealized simulations to diagnose the relationship between a severe thunderstorm similar to the 4 June 2015 case (Fig. 12), and turbulence within its upper level outflow downstream of the overshooting deep convection. Like in the 4 June 2015 case, Zovko-Rajak and Lane (2014) also found short-wavelength internal gravity waves that were trapped near the jet level. However, unlike in the 4 June 2015 case (Fig. 12), their waves did not emanate from the region of deep convection. Instead, their gravity waves and CIT developed locally 50–100-km downstream from the parent deep convection, and were attributed to KHI located both above and below the jet layer in which the waves were trapped. Nevertheless, the gravity waves may have played an important role in their simulated CIT by providing a mechanism that allowed communication and phase locking between the regions of instability in vertical shear layers both above and below the outflow jet. Their analysis underscores the importance of both vertical shear and static stability modifications to the background environment in storm outflows, and the potentially complex interplay between waves and instabilities in such outflows on the development of CIT.

4.4 Turbulence Near and Within MCS Anvils

Mesoscale convective systems (MCSs), which are ensembles of thunderstorms producing a contiguous precipitation area of 100 km or more in at least one horizontal direction, constitute a significant hazard to commercial aviation. A high-resolution LES of a quasi-2D MCS (Lane and Sharman 2014) illustrates elevated levels of turbulence (EDR  > 0.05) within deep convection near its leading edge (x = 350–400 km), wave breaking regions above deep convection (Sect. 4b), and within and near the cirrus anvil cloud that extends over 200 km from the deep convection region (Fig. 13).
Fig. 13

Spatial distribution of along-line maximum turbulence (ɛ1/3) obtained during the final 30 min of the simulation reported in Lane and Sharman (2014). Also shown is the along-line mean cloud outline (black contours defined by the 0.1 g kg−1 cloud mixing ratio isopleth). The approximate turbulence categories are based on comparisons to pilot reports as described by Sharman et al. (2014). From Lane and Sharman (2014) Fig. 4. Copyright American Geophysical Union. Reprinted with permission

As noted earlier, the heavily precipitating deep convection region in MCSs, where turbulence is typically most severe (Fig. 13), is easily avoided using on board and ground-based radar. The more extensive anvil regions, which themselves often contain significant turbulent vertical motions of order  ~ 1 m s−1 (e.g., Petre and Verlinde 2004; Muhlbauer et al. 2014), can also typically be avoided through visual identification.

Research dropsonde and research aircraft measurements have documented widespread areas of either reduced or subcritical gradient Ri in both midlatitude continental MCS (e.g., Collander et al. 2006; Gultepe and Starr 1995) and tropical cyclone (e.g., Molinari et al. 2014; Duran and Molinari 2016) anvils. However, because of their size (which can sometimes exceed several hundred km) and the lesser likelihood of severe turbulence than in deep convection, anvils are sometimes penetrated by commercial aviation. Thus, it is of interest to identify regions within anvils that are most susceptible to significant turbulence and to understand the variety of different mechanisms responsible for such turbulence.

The trailing anvil cloud and clear region beneath it (Fig. 13), respectively, comprise ascending and descending airstreams, which are driven by mesoscale buoyancy gradients within the MCS (Lafore and Moncrieff 1989; Weisman 1992). The region of enhanced EDR that extends downward beneath cloud base at the back edge of the anvil (Fig. 13) is associated with the descending mesoscale rear-inflow jet, which is a common feature in mature-to-dissipating MCSs (Augustine and Zipser 1987; Smull and Houze 1987). Collander et al. (2006) hypothesized that severe turbulence encounters within this region may be associated with KHI arising from strong vertical shear often found immediately beneath the rear-inflow jet. Furthermore, Ri can also be reduced by small static stability beneath the anvil cloud due to both subsidence and vertical gradients of diabatic cooling resulting from differences in the strength of sublimation or evaporation with depth beneath cloud base (e.g., Harris 1977; Knight et al. 2004).

Though they are most intense beneath the back edge of the anvil and in close proximity to the deep convection region, moderate-or-greater turbulence may occur with a shallow layer beneath cloud base along the entire horizontal extent of the anvil region in the Lane and Sharman (2014) simulation (Fig. 13). This is consistent with numerous studies of turbulence occurring underneath midlevel cirrus (e.g., Luce et al. 2010; Kudo 2013; Kudo et al. 2015), which has been attributed primarily to shallow regions of static instability developing from precipitation falling into approximately neutral subsaturated layers. In some cases (e.g., Luce et al. 2010), such turbulence may occur near or below mammatus clouds (Schultz et al. 2006) along the underside of the anvil.

At typical commercial aviation cruising altitudes (9–13-km MSL), EDR values consistent with light turbulence are found in the Lane and Sharman (2014) simulation extending nearly 100 km outside of the anvil outflow edge (Fig. 13). However, a shallow but horizontally extensive region of EDR values consistent with moderate turbulence is found near the top of the outer anvil (Fig. 13).

One common source of turbulence near the outer edge of the MCS anvil cloud shield is cirrus bands that extend radially outward from the storm center. Figure 14 presents an example from a high-resolution visible satellite image of this type of cloud organization, commonly referred to as “transverse bands” (Ellrod 1985; Lenz et al. 2009). Knox et al. (2010) note that transverse cirrus bands are also often present within tropical cyclone outflows and within upper level jet streams. The example from Knox et al. (2010) shows extensive banding with horizontal spacing between bands of less than 10 km and band lengths of about 100 km on the north side of deep convection within a small-to-medium sized MCS evident from the brighter white cloudiness (Fig. 14). These bands are nearly perpendicular (and hence transverse) to a strong 40–50-m s−1 precursor near-tropopause jet streak (not shown) that was likely further strengthened by anticyclonic outflow within the MCS.
Fig. 14

MODIS visible satellite image of transverse cirrus bands on the northern flank of a mesoscale convective system in the Midwestern United States at 1741 UTC 27 July 2008. Adapted from http://ge.ssec.wisc.edu/modis-today/images/terra/true_color/2008_07_27_209/t1.08209.USA3.143.250m.jpg

Lenz et al. (2009) found that transverse band events lasted an average of 9 h in approximately ½ of a sample of 136 large MCSs over the central U.S. during the 2006 warm season. In their climatology, they found a similar 5–10-km spacing between bands to that in Fig. 14 and noted at least one observation of light (moderate) commercial aviation turbulence deduced from in situ EDR measurements within or near the bands for 93% (44%) of their cases. They also found transverse bands associated with MCSs to be most common overnight and during the morning (Fig. 15a) and were often long-lasting (Fig. 15d), sometimes persisting several hours beyond MCS dissipation. They further noted that the bands lagged convection initiation within the MCS by an average of 7 h (Fig. 15b, c) and were typically restricted to one side of the MCS cloud shield (Fig. 16).
Fig. 15

Histograms showing the a temporal distribution of transverse bands, b initiation time of associated deep convection, c time lag between initiation and transverse band development, and d transverse band duration. Adapted from Lenz et al. (2009)

Fig. 16

Geographic distribution of MCS anvils (dark gray ovals) for May–August 2006 with transverse band locations indicated by light gray shading. Orientation of arrows indicate the directions of MCS propagation. Adapted from Lenz et al. (2009)

These aspects suggest that such bands are closely linked to earlier deep convection within the MCS and its associated upper tropospheric outflow. Fritsch and Maddox (1981) discuss how a wind speed maximum typically occurs north and northeast of the origin of convection in MCSs, which is summarized schematically in Fig. 17 (Trier and Sharman 2009). In this diagram, upper tropospheric outflow trajectories from the parent convection within the MCS are indicated by the purple curves, which become anticyclonic due to Coriolis effects acting over the t > 4-h timescale of the outflow. Fritsch and Maddox (1981) point out that the strongest winds are often found where the outflow adds to the background flow, which is in the north and northeast portion of the MCS for the case of the climatological background westerly jet in the midlatitude northern hemisphere (Fig. 17). While Lenz et al. (2009) found that transverse band formation was typically restricted to a single side of the MCS, there is not a clear bias in their cases towards the northern side (Fig. 16). However, the large variance and lack of a clear bias on the location of transverse bands in their observed northern hemispheric MCS cases could be associated with background flow differences related to synoptic variability among their different cases.
Fig. 17

Schematic diagram illustrating spatial relationships among the environmental winds, the strongest deep convection (green thunderstorm symbols), MCS anvil cloud (thin oval), streamlines of MCS upper level outflow (thick violet curves with arrowheads) and the location of strongest upper level outflow winds (gray shading). Adapted from Trier and Sharman (2009)

An example from 17 June 2005 of turbulence associated with transverse bands along the north side of a MCS is shown in Fig. 7b. Here, the observed bands in Fig. 7b appear less spectacular than in the Knox et al. (2010) example of Fig. 14, but this is largely an artifact of resolution differences (250 m versus 4-km pixel size) in the satellite images. Using a high-resolution horizontal nest with  = 600 m, Trier et al. (2010) successfully simulated these bands occurring in the northern portion of the MCS upper tropospheric outflow (Fig. 18a, b). The simulated bands (Fig. 18b) originated from a region of moist static instability (Durran and Klemp 1982) and were oriented approximately parallel to the vertical wind shear through the depth of the outer anvil (not shown). This organization is similar to cumulus cloud streets in the boundary layer (e.g., LeMone 1973), which arise from thermal-shear instability (e.g., Asai 1970). Thermal-shear instability directly leading to upper tropospheric cirrus banding has also been reported in simulations of observed CIT associated with oceanic cyclones (Kim et al. 2014) and convectively enhanced synoptic jet streams (Trier and Sharman 2016). These model results concerning the connection between transverse bands and thermal-shear instability are consistent with earlier speculation based on limited observations (Dixon et al. 2000).
Fig. 18

(Left) ARW–WRF model domains D1 (with 3-km horizontal grid spacing) and D2 (with 600-m horizontal grid spacing), 0800 UTC 17 June 2005 maximum radar reflectivity in a vertical column and 12.5-km horizontal winds for a control simulation and c simulation in which cloud-radiative feedbacks are disabled. (Right) Simulated brightness temperature and 12.1-km MSL smoothed moist static stability (2 × 10−5 s−1 with negative values dashed) at 0930 UTC 17 June 2005 in domain D2 for b control simulation and d simulation in which cloud-radiative feedbacks are disabled. Adapted from Trier et al. (2010)

Lenz et al. (2009) and Knox et al. (2010) indicate that in some cases, the cirrus bands occurring near the edges of MCS anvils are perpendicular to convectively generated gravity waves. Trier et al. (2010) found short-wavelength convectively generated gravity waves, which were trapped beneath the near-neutral layer in which the simulated transverse bands formed in their high-resolution simulation (Fig. 19). Here, local maxima in vertical velocity within the near-neutral layer are horizontally collocated with local minima in potential temperature within the gravity-wave layer below (Fig. 19). This suggests that convectively induced gravity waves could in some cases help release the moist static instability that evidently leads to transverse band formation. This hypothesis is consistent with observations of cirrus uncinus bands forming in near-neutral layers located above stable layers containing gravity-wave activity (Heymsfield 1975).
Fig. 19

a Simulated brightness temperature and 10-km MSL vertical velocity contoured in 1.5-m s−1 (− 1.5 m s−1) intervals with positive (negative) values in red (blue) starting at 0.75 (− 0.75) m s−1. b Vertical cross section averaged for 6 km across transect AB in part a of total cloud condensate (color scale) horizontal winds, potential temperature contoured in 2-K intervals and vertical velocity contoured in 1-m s−1 intervals with positive (negative) values in red (dashed blue) starting at 0.5 (− 0.5) m s−1. Adapted from Trier et al. (2010)

Trier et al. (2010) also found that both the moist static instability and the prominence of simulated cirrus banding were influenced by cloud-radiative feedbacks (cf. Fig. 18b, d) owing to differential radiative forcing related to longwave cooling at cloud top and warming at cloud base. However, this physical process does not explain the regionalization of the banding in the 17 June 2005 case and in the Lenz et al. (2009) climatology (Fig. 16), which is likely related to mesoscale asymmetries in the strength of the convectively induced upper tropospheric outflow (Fig. 17) that affect the vertical shear and static stability. Similarly, recall that convectively induced outflow can lead to CIT initiated by KHI (e.g., Trier et al. 2012; Zovko-Rajak and Lane 2014) in cases where it is not statically unstable (i.e., 0 < Ri < 1).

4.5 Remote Turbulence Associated with Convectively Enhanced Synoptic Jet Streams

Synoptic jet streams near the tropopause have long been recognized as locations predisposed to turbulence (e.g., Shapiro 1976; Kennedy and Shapiro 1975; Koch et al. 2005; Klostermeyer  and Rüster 1980) for a variety of different reasons (Sharman et al. 2012). The downstream (exit) regions of such jets are often generation sites of mesoscale inertia–gravity waves (e.g., Uccellini and Koch 1987; Plougonven et al. 2003; Zhang 2004). Like smaller scale internal gravity waves, mesoscale inertia–gravity waves can be associated with turbulence (e.g., Ellrod et al. 2015; Lane et al. 2004; Koch et al. 2005). Plougonven and Zhang (2016) note that the substantial vertical tilt of these longer waves makes them particularly effective in modifying the environmental vertical shear and static stability in a manner conducive to turbulence. Inertia–gravity waves near jet streams have a variety of different possible generation mechanisms (see, e.g., Plougonven and Zhang 2016 and references therein). Among these different generation mechanisms, organized deep convection has been examined in recent observational (e.g., Hoffmann and Alexander 2010) and modeling (e.g., Wei and Zhang 2014) studies.

In the remainder of this subsection, we provide an example of how convectively generated mesoscale inertia–gravity waves can lead to turbulence at considerable distances from their source location in the exit region of a near-tropopause synoptic jet stream. The turbulence in transverse cirrus bands within MCS anvils discussed in the previous subsection provided an example of CIT occurring due to processes within the upper level outflow emanating from deep convection up to several 100-km away. In the forthcoming example from Trier and Sharman (2016), the turbulence was related to, but was even farther removed (both spatially and temporally) from deep convection that had occurred earlier in the anticyclonically sheared exit region of a synoptic jet.

In this case over the North Atlantic Ocean on 15 Nov 2011, episodes of turbulence occurred in the vicinity of separate mesoscale clusters of upper tropospheric cirrus bands. This is a common situation that aviation forecasters recognize in their turbulence forecasting procedures (Melissa Thomas, private communication). The mesoscale regions of cirrus bands of 15 Nov 2011 (Fig. 20a) were well simulated by Trier and Sharman (2016) and diagnosed to result directly from thermal-shear instability. A supporting simulation where cloud-radiative feedbacks were disabled (Fig. 20b) was used to infer mesoscale reductions in static stability prior to the onset of cirrus banding in their higher resolution full physics control run (cf. Fig. 20a, b). Both the regions of cirrus banding in the control run and the regions of lowered static stability in the supporting simulation had a mesoscale horizontal wavelength of 400–500 km (Fig. 20b) and had a structure consistent with inertia–gravity waves (Trier and Sharman 2016, their Fig. 23a).
Fig. 20

a Simulated infrared satellite brightness temperature within a control simulation with 1-km horizontal grid spacing at 1530 UTC 15 November 2011 and b 11.25-km MSL horizontal winds, moist static stability (\(N_{m}^{2}\)), areas of negative Ertel’s potential vorticity (PV), and wind speed greater than 75 m s−1 (semitransparent gray shading) at 1400 UTC 15 November 2011. The annotations A, B, and C denote persistent negative PV features that were tracked using animations of model output having a frequency of 30 min. Corresponding regions of cirrus banding with turbulence that form in low moist static stability regions are indicated in part (a). Adapted from Trier and Sharman (2016)

The periodic mesoscale variations in upper tropospheric static stability in the control simulation are evident in a Hovmöller diagram (Fig. 21a). The three distinct episodes of cirrus banding A, B, and C formed about 4-h apart near −60° longitude (Fig. 20a) within eastward progressing regions of reduced static stability that originated farther west (Fig. 21a) over the eastern United States. The link to upstream deep convection is suggested by a comparison with a Hovmöller diagram from an otherwise identical dry simulation (Fig. 21b). Larger scale static stability variations occur in the dry simulation (Fig. 21b), including those associated with the passage of a synoptic trough, also evident in the full physics control simulation (Fig. 21a). However, the faster moving mesoscale stability minima (Fig. 21a), wherein the turbulent cirrus banding occurs, are absent in the simulation without upstream deep convection (Fig. 21b).
Fig. 21

Time-longitude Hovmöller diagrams of 11.5-km MSL dry static stability [K (hPa)−1] from a full physics control simulation, and b dry simulation reported by Trier and Sharman (2016). The sloping dashed line in a indicates the phase line of a leading high amplitude inertia–gravity wave from their higher resolution simulation, as shown in Fig. 20a. The annotations A, B, and C in part a indicate static stability minima in subsequent waves, which are associated with regions of cirrus banding shown in Fig. 20a. The sloping solid line in both parts a, b indicates the spatiotemporal location of a simulated upper level synoptic trough. Adapted from Trier and Sharman (2016)

The wavelike mesoscale regions of reduced static stability in Fig. 20b were located rearward and approximately 90° out of phase with regions of negative potential vorticity. Using surface rain rate as a proxy for diabatic heating, these regions of negative potential vorticity are more directly linked to much earlier deep convection occurring at distances greater than 1000-km upstream over the eastern United States.

The linkage of upper tropospheric negative potential vorticity with earlier deep convection may be understood by recognizing that in the absence of friction, the rate of change of Ertel’s potential vorticity, \(P = \left( {\zeta_{\text{a}} \cdot \nabla \theta } \right)/\rho ,\) following an air parcel is
$$\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{1}{\rho }\left( {\zeta_{\text{a}} \cdot \nabla {\text{d}}\theta /{\text{d}}t} \right).$$

The right side of the above equation indicates that diabatic heating (\({\text{d}}\theta /{\text{d}}t\)) along the 3D absolute vorticity (\(\zeta_{\text{a}}\)) vector is the physical process responsible for generating the negative potential vorticity. Hertenstein and Schubert (1991) demonstrated that vertical gradients of diabatic heating within large squall lines, such as those that occurred 12–18-h earlier over the eastern United States (Trier and Sharman 2016), can produce mesoscale regions of negative potential vorticity in the upper troposphere above where such heating is maximized. In the absence of subsequent diabatic sources, the negative potential vorticity is then advected downstream while undergoing substantial deformation in the exit region of the jet. Banded cirrus (Fig. 20a) with CIT develops in areas of reduced static stability that lags by ¼ wavelength the regions of negative potential vorticity (Fig. 20b).

4.6 Discussion

High-resolution numerical simulation research models have been used to investigate the onset of observed CIT in several cases influenced by diverse mechanisms. The turbulence in these cases is located either directly above or is horizontally displaced from tropopause-penetrating deep convection. In the latter, the CIT occurs at ranges from immediately adjacent to the parent deep convection to more than 1000-km away. A common aspect of each of these scenarios is that convectively generated gravity waves provide some link from the deep convection to subsequent CIT. However, both the horizontal scale of the gravity waves and the manner in which this linkage occurs varies substantially among different CIT types.

In the cases of turbulence occurring above deep convection, vertically propagating internal gravity waves of relatively short horizontal wavelengths (λH~ 5–10 km) play a direct role in initiating turbulence through wave breaking upon encountering critical levels. In other cases, where turbulence occurs adjacent to deep convection short-wavelength internal gravity waves can be vertically trapped and maintain coherence while propagating horizontally. In these cases, the waves play an important but less direct role in the initiation of CIT by either modifying the environment, so that shallow, highly localized regions of shearing or static instability occur, or by helping to initiate turbulence over larger regions that have been destabilized by convectively induced mesoscale circulations such as MCS outflows or other physical processes. In yet other cases, mesoscale inertia–gravity waves having horizontal wavelengths of up to λH ~ 500 km can themselves provide significant destabilization, thus facilitating turbulence at large horizontal distances (L ~ 1000 km) and with substantial temporal lags (t ~ 12–18 h) from the parent deep convection.

5 Forecasting CIT for Aviation

While Doppler weather radar can be used to detect in-cloud turbulence hazards (Williams and Meymaris 2016), it does not provide turbulence information in regions with low-signal-to-noise ratio (SNR) radar returns, contaminated data or poor radar coverage, and of course cannot be easily used as a forecasting tool. In addition, operational NWP models have limited utility in forecasting the location, strength, and type of convection due to latency issues, inadequate model resolution, and incomplete model physics (e.g., Lilly 1990; McNulty 1995; Bernardet et al. 2000; Weisman et al. 2008; Wakimoto and Murphey 2009). Furthermore, Lane and Knievel (2005) show a strong dependence of numerically simulated convective gravity-wave characteristics on model grid resolution. Finally, even if NWP models could properly forecast convection and convective gravity waves, explicit prediction of turbulence through direct numerical simulation (DNS) techniques or large eddy simulation (LES) techniques is computationally prohibitive, requiring semi-empirical (and less accurate) techniques to infer turbulence from the coarser resolution NWP model output (see, e.g., Sharman 2016; Knox et al. 2016 and references therein). These limitations make it extremely difficult to provide skillful CIT forecasts in an operational environment, although some of this uncertainty may be represented through probablitistic forecasts.

From theoretical studies and the examples from research simulations described in Sect. 4, it is obvious that the intensity of CIT is related to the size, depth, intensity, longevity, and other aspects of the deep convection. Also important are the characteristics of the near-storm environment including static stability, strength of the vertical shear, and the interaction of the convectively induced UTLS outflow with ambient winds. Although progress is being made in better understanding influences of thunderstorm–environment interactions on CIT, the case studies performed so far are insufficient to systematically or quantitatively characterize these relationships. Thus, empirical models have been sought to associate thunderstorm observations through satellite imagery, lightning characteristics, and radar parameters (reflectivity, spectral width) and NWP model derived relevant environmental parameters (e.g., CAPE, CIN, Ri), with observations of CIT.

One such empirical model, termed DCIT (Diagnosis of Convectively Induced Turbulence) is described in Williams (2014), where an artificial intelligence technique (viz., random forests) is used to fuse data from Doppler radar, geostationary satellites, total lightning data, and NWP model data to produce gridded deterministic and probabilistic CIT nowcasts at 6-km horizontal and 1000 ft vertical grid spacing updated every 15 min. The data fusion algorithm was trained using a retrospective dataset that includes in situ EDR reports and collocated predictor fields. The algorithm output was then evaluated on an independent test set using a common performance metric (receiver operating characteristic ROC curves, e.g., Gill 2016). Although the statistical evaluations show promising results for this technique, it is not yet available operationally.

Recent studies suggest that the lightning characteristics such as flash energy, length and extent densities may relate to turbulence intensity and location (Bruning and MacGorman 2013). Results based on comparisons of observational lightning to EDR data from Deierling and Williams (2016) suggest that the occurrence of moderate-or-greater turbulence in storms is related to storm total lightning flash rate, though the exact relationships depend on the storm type. The footprint of horizontal lightning flash extents is also observed to coincide with areas of moderate and greater turbulence.

Another approach is simply to infer CIT from CAT diagnostics (e.g., Sharman and Pearson 2017; Pearson and Sharman 2017) assuming that the NWP model captures large-scale perturbations due to convection. Similarly, cloud-top height (CTH) and other parameters can be used to produce a CIT likelihood (Kessinger et al. 2017). The CTH product maps the satellite cloud-top temperature (11 μm) to flight level using the global forecast system (GFS) model sounding at the cloud location. The convective diagnosis oceanic (CDO) product empirically determines the area of convective storms most hazardous for aviation by combining three satellite-based convection detection algorithms with global ground-based lightning strike data in a scaled and weighted data fusion scheme. Although not as sophisticated as the DCIT product, these products do provide global coverage, and also provide a means for uplinking the data to the cockpit on their electronic flight bag (EFB).

Empirical approaches such as these will only improve as higher resolution and cloud-resolving NWP models and other data sources (e.g., GOES-16 satellite) become available operationally. Convective gravity-wave parameterizations for use in larger scale NWP models are a continuing area of research (Beres 2004; Beres et al. 2004; Chun and Baik 1998, 2002; Kim et al. 2013; Stephan and Alexander 2015). Kim et al. (2017) have taken the next step of parameterizing CIT due to convective gravity-wave breaking. As our understanding of CIT increases through simulation case studies, these parameterizations will be become more sophisticated and likely more effective.

6 Summary and Future Research Needs

This review has attempted to demonstrate the utility of case studies based on actual CIT events, where high-resolution observations are used in conjunction with high-resolution numerical simulations. For the different mechanisms of CIT onset identified here, the high-resolution observations are provided either from digital flight recordings of an event or 1-min in situ EDR messages available from selected commercial aircraft, or to a lesser extant, PIREPs. Another approach would be to collect high quality atmospheric data from research aircraft involved in field campaigns, either through in situ measurements or remote sensing. Numerical simulation models must at a minimum be convection-resolving, which requires grid spacings of a few km or less (e.g., Bryan et al. 2003). This is usually sufficient to at least partly resolve convective generated gravity waves, and to provide an indication of conditions favorable to their breakdown into turbulence. However, much smaller grid spacings of about 100 m, would be required to start to resolve “turbulence” of relevance to aviation. Currently this can only be achieved using a grid nesting approach, but even then, the computational requirements can be severe.

The issue of CIT avoidance may be more pressing in the future, given that CIT frequencies could increase with increases in the frequency and intensity of thunderstorms implied by future climate projection models (e.g., Del Genio et al. 2007; Trapp et al. 2007; Diffenbaugh et al. 2013; Gensini et al. 2014; Romps et al. 2014; Prein et al. 2017). Climate change may affect clear-air turbulence as well (Williams 2017; Storer et al. 2017; Williams and Joshi 2013, 2016), further complicating the problem of turbulence forecasting.

Going forward, a multifaceted approach using dedicated research flights, expanded in situ EDR observations, and high-resolution modeling along with theoretical studies (to the extent possible) will continue to be required. Improvements in the representation of convective and turbulence processes within numerical simulation models will continue to be an active area of research. As computational resources increase, higher resolution grids can be accommodated over larger domains. High-resolution LES models that include the UTLS (Schalkwijk et al. 2015) have already been run operationally over limited horizontal domains.

Some fundamental research questions need to be addressed to promote enhanced understanding of CIT processes to ultimately be able to nowcast and possibly forecast CIT intensity, timing, and location. These questions include (from Sharman et al. 2016):
  • What is the spectrum of convective gravity waves and how does the energy associated with the waves relate to the region of wavelengths important to aircraft?

  • What is the relation of CIT to the convective cloud characteristics?

  • What is the global climatology of CIT?

  • How will climate change impact the intensity and frequency of CIT?

  • What is the best way to accommodate uncertainty in CIT forecasts?

  • What role do secondary gravity waves emitted from breaking convective gravity waves play in the turbulence process?

  • What are the predictability limits of CIT using NWP models?

  • What observations are needed to improve CIT?

Operationally, given the complexity of CIT, it does not appear that simple avoidance tactics as sought by the original FAA guidelines can be ever be developed. Knowledge of the thunderstorm structure and environmental parameters (e.g., the vertical distribution of potential temperature and wind throughout the UTLS) is required to provide a reasonable nowcast of CIT. Although such methods are beginning to be developed, a crucial aspect of tactical avoidance is to ensure timely transmission of these products to the cockpit. With onboard laptops or tablets the hazardous areas may be displayed for rapid evaluations, and hence facilitate better decisions for real-time avoidance and diversion strategies. Pilot training to promote a better understanding of CIT processes and the interpretation of uncertainty in forecasts is also required. This will undoubtedly reduce the number of turbulence encounters in the future. However, in the foreseeable future, improvements are likely to occur as only incremental steps towards fulfilling the operational goal of reducing the number of CIT encounters.

Notes

Acknowledgements

We thank Wiebke Deierling, Teddie Keller (each of NCAR), Gretchen Mullendore (University of North Dakota), and an anonymous reviewer for their comments of an earlier version of the manuscript, which helped improve the presentation. We also thank Julia Pearson (NCAR) for constructing Fig. 1 and Lara Ziady (NCAR) for Fig. 2. We acknowledge Earth Networks for providing ENTLN (lightning) data. This research is in response to requirements and funding by the Federal Aviation Administration (FAA). The views expressed are those of the authors and do not necessarily represent the official policy or position of the FAA.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Center for Atmospheric ResearchBoulderUSA

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