Large-Eddy and Flight Simulations of a Clear-Air Turbulence Event over Tokyo on 16 December 2014

R. Yoshimura aInstitute of Fluid Science, Tohoku University, Sendai, Japan

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K. Suzuki bInformation Infrastructure Department, Japan Meteorological Agency, Tokyo, Japan

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J. Ito cGraduate School of Science, Tohoku University, Sendai, Japan

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R. Kikuchi dOffice of Society Academia Collaboration for Innovation, Kyoto University, Kyoto, Japan

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A. Yakeno aInstitute of Fluid Science, Tohoku University, Sendai, Japan

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S. Obayashi aInstitute of Fluid Science, Tohoku University, Sendai, Japan

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Abstract

In this study, a clear-air turbulence event was reproduced using a high-resolution (250 m) large-eddy simulation in the Weather Research and Forecasting (WRF) Model, and the resulting wind field was used in a flight simulation to estimate the vertical acceleration changes experienced by an aircraft. Conditions were simulated for 16 December 2014 when many intense turbulence encounters (and one accident) associated with an extratropical cyclone were reported over the Tokyo area. Based on observations and the WRF simulation, the turbulence was attributed to shear-layer instability near the jet stream axis. Simulation results confirmed the existence of the instability, which led to horizontal vortices with an amplitude of vertical velocity from +20 to −12 m s−1. The maximum eddy dissipation rate was estimated to be over 0.7, which suggested that the model reproduced turbulence conditions likely to cause strong shaking in large-size aircraft. A flight simulator based on aircraft equations of motion estimated vertical acceleration changes of +1.57 to +0.08 G on a Boeing 777-class aircraft. Although the simulated amplitudes of the vertical acceleration changes were smaller than those reported in the accident (+1.8 to −0.88 G), the model successfully reproduced aircraft motion using a combination of atmospheric and flight simulations.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ryoichi Yoshimura, ryouichi.yoshimura.s2@dc.tohoku.ac.jp

Abstract

In this study, a clear-air turbulence event was reproduced using a high-resolution (250 m) large-eddy simulation in the Weather Research and Forecasting (WRF) Model, and the resulting wind field was used in a flight simulation to estimate the vertical acceleration changes experienced by an aircraft. Conditions were simulated for 16 December 2014 when many intense turbulence encounters (and one accident) associated with an extratropical cyclone were reported over the Tokyo area. Based on observations and the WRF simulation, the turbulence was attributed to shear-layer instability near the jet stream axis. Simulation results confirmed the existence of the instability, which led to horizontal vortices with an amplitude of vertical velocity from +20 to −12 m s−1. The maximum eddy dissipation rate was estimated to be over 0.7, which suggested that the model reproduced turbulence conditions likely to cause strong shaking in large-size aircraft. A flight simulator based on aircraft equations of motion estimated vertical acceleration changes of +1.57 to +0.08 G on a Boeing 777-class aircraft. Although the simulated amplitudes of the vertical acceleration changes were smaller than those reported in the accident (+1.8 to −0.88 G), the model successfully reproduced aircraft motion using a combination of atmospheric and flight simulations.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ryoichi Yoshimura, ryouichi.yoshimura.s2@dc.tohoku.ac.jp

1. Introduction

Atmospheric turbulence is a serious threat to aviation safety, and turbulence-related events have caused many aviation accidents. In particular, clear-air turbulence (CAT) is generated outside convective clouds, making it difficult for onboard radars to detect. Kelvin–Helmholtz (KH) instability can explain most causes of high-altitude (∼10 km) CAT (Ellrod and Knapp 1992). Turbulent eddies from KH instability decrease in size due to energy cascades, eventually reaching aircraft-size, and it is these small eddies that are mainly responsible for aircraft turbulence (Kim and Chun 2010; Joseph et al. 2004; Sekioka 1970). Breaking of gravity waves propagating from terrain or convection can also be a source of CAT (Sharman et al. 2012; Kim and Chun 2010; Trier et al. 2012; Lane et al. 2003; Lane and Sharman 2008). Other potential sources of turbulence include shear instability or shallow convective instabilities arising from modifications to environmental vertical shear and static stability in mesoscale outflows originating from horizontally distant convections (Trier and Sharman 2016; Trier et al. 2020).

Several organizations have developed technologies to prevent CAT-related accidents. For example, the Japan Aerospace Exploration Agency (JAXA) has developed and tested aircraft-mounted light detection and ranging (lidar) systems that can detect CAT ahead of aircraft and a framework to mitigate the impact of CAT using existing aircraft control surfaces (ailerons, elevators, etc.) in conjunction with the lidar observation information (Inokuchi et al. 2009, 2016; Machida et al. 2018; Kikuchi et al. 2020; Inokuchi and Akiyama 2019). The JAXA lidar system observes an infrared laser reflected from aerosol particles in front of an aircraft and estimates their speed, which is assumed to be the local wind speed, on the basis of the Doppler shift. Flight tests conducted in 2017 (Matayoshi et al. 2018) demonstrated the capability of the lidar system to observe CAT that is over 20 km away below 5000-ft and over 10 km away at 40 000-ft altitudes (1000 ft ≈ 305 m). The German Aerospace Center [Deutsches Zentrum für Luft- und Raumfahrt (DLR)] has also developed an aircraft-mounted lidar system with an improved algorithm to reproduce the forward wind field from lidar observations and validated their devices and attitude control systems using numerous experiments, including flight tests (Schmitt et al. 2007; Veerman et al. 2014; Fezans et al. 2019; Vrancken et al. 2016).

One challenge with the aforementioned developments is that not much actual measurement data of realistic CAT can be obtained. While flight testing is an important process to determine whether lidar devices can meet the performance specifics required under real flight conditions, the amount of data collected during flight tests can be insufficient because the number of flight tests is limited or because the testing aircraft rarely encounter strong turbulence. For example, there were few encounters with intense CAT during 33 h of flight tests conducted by DLR in 2013 (Veerman et al. 2014), and they succeeded in validating their system based on encounter data of less intense CAT. Validation of lidar systems in an environment with intense CAT, which is one of the major causes of aviation accidents, is generally difficult.

By establishing a method to simulate CAT and estimate its impact on aircraft, virtual flight testing under near realistic conditions becomes possible, facilitating more efficient development. Reproduction of detailed CAT structures using numerical methods enables virtual flight testing by inputting simulated CAT into aircraft equations of motion (Takase et al. 2017; Kato et al. 1982; McRuer et al. 1973) that have six (three in translation and three in rotation) degrees of freedom, and solving them. For a limited number of future flight tests, a preliminary investigation may be possible in numerical space to analyze the aircraft and attitude control system response when encountering CAT, and to validate the algorithm for wind field estimation from lidar observations. To obtain a CAT simulation close to reality, a numerical weather prediction model can be used to reproduce and forecast realistic weather.

The purpose of this study is to investigate the feasibility of virtual flight testing by performing a flight simulator in a wind field with CAT obtained from detailed simulations of turbulence using the large-eddy simulation (LES) option in the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008). Virtual flight testing requires a three-dimensional and high-resolution wind field; however, these types of data are not commonly available. Although there are several examples of numerical simulations of turbulence, including CAT (Trier et al. 2020; Trier and Sharman 2016; Trier et al. 2012; Sharman et al. 2012; Kim and Chun 2010), these efforts resolve turbulence using a combination of a planetary boundary layer and horizontal mixing parameterizations. Most LES applications for numerical weather prediction have focused on turbulence within the boundary layer (Wurps et al. 2020; Onishi et al. 2019; Shamsoddin and Porté-Agel 2017; Moeng et al. 2007), and there are almost no examples of LES at high altitudes. Misaka et al. (2008) succeeded in reconstructing a two-dimensional wind field using LES and variational data assimilation of flight data from past CAT encounters. However, their reconstruction was based on maximum likelihood estimation and not on physical equations, and it was not confirmed whether the wind field was meteorologically consistent. Moreover, there are no examples of flight simulations in a wind field simulated by a numerical weather prediction model.

To create a detailed turbulence dataset for input into a flight simulator, we used WRF to reproduce the conditions that led to an accident involving a Boeing 777-200 aircraft that occurred over Tokyo at 1035 UTC 16 December 2014 (1935 LST 16 December 2014), hereinafter referred to as the “target case” as shown in Fig. 1. The target case was chosen because the aircraft experienced significantly large acceleration changes (from +1.8 to −0.88 G; Japan Transport Safety Board 2016), there were many pilot reports (PIREPs) in the vicinity that detailed strong shaking from 730 to 1100 UTC, and it was relatively easy to acquire observation data given the dense observations around Tokyo. As shown in Fig. 1, area A includes the location of the target case and is the location of a particularly large number of PIREPs. We focus on area A in section 2.

Fig. 1.
Fig. 1.

Overview of the accident involving a Boeing 777-200 aircraft that occurred over Tokyo at 1035 UTC 16 Dec 2014 (1935 LST 16 Dec 2014; the target case). The flight path, time, and location of the target case are from the accident investigation report (Japan Transport Safety Board 2016).

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

2. Turbulence observations

On 16 December 2014, a developing extratropical cyclone moved northeastward along the southern coast of Japan. Most turbulence encounters were reported from 0730 to 1100 UTC in area A on Fig. 1 at the time when the cyclone was approaching Tokyo (Fig. 2; 1200 UTC 16 December 2014). Strong southwesterly winds dominated at an altitude of 300 hPa (Fig. 7a, described in more detail below).

Fig. 2.
Fig. 2.

Surface weather map at 1200 UTC 16 Dec 2014 and the path of the low pressure system over Japan (from the JMA). The star denotes the location of the target case.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

We refer to the Japan Airlines (JAL) PIREP to discuss the reports of turbulence encounters. In a PIREP, the intensity of turbulence is judged by the captain on the basis of the cabin condition, experience, and controllability. The JAL PIREP data contain six levels of intensity, as described in Table 1 (Japan Civil Aviation Bureau 2009). In this study, we used moderate-or-greater (MOG) turbulence reports to investigate the general spatiotemporal trends in the PIREP data.

Table 1

Turbulence intensity categories defined by JAL and the Federal Aviation Administration (FAA).

Table 1

Figure 3 shows that the location of the MOG PIREPs gradually moves northeastward over time, in the same direction as the cyclone. Most of the PIREPs were located between an altitude of 7 and 9 km (Fig. 3d), which approximately corresponds to the orange-colored points (FL251–FL300, or approximately 8–9 km altitude) in Figs. 3a–c. The target case (8.2 km altitude; 1035 UTC) was also located in this altitude range.

Fig. 3.
Fig. 3.

MOG-turbulence-level pilot reports in area A (a) from 0730 to 0859 UTC, (b) from 0900 to 1029 UTC, and (c) from 1030 to 1159 UTC, along with (d) a histogram of report altitudes; (c) includes the location of the target case and the radiosonde flight (Tateno, denoted by an X). The turbulence intensity and altitude [flight level (FL)] are represented by numbers (moderate = 4 and severe = 5) and colors, respectively. FL is altitude information at 100-ft (≈30.5 m) intervals, which is represented by a three-digit number.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

Figure 4 shows radiosonde observation data from 1200 UTC 16 December 2014 at Tateno (Fig. 3c), the closest radiosonde location to the targeted case. A strong vertical wind shear is apparent at 7–9 km, just under the jet stream axis located at an altitude of 10 km. The range of 7–9 km corresponds to the altitude at which a large number of MOG-level PIREPs were reported. Therefore, we suggest that the vertical wind shear below the jet stream axis contributed to turbulence generation at this altitude.

Fig. 4.
Fig. 4.

Radiosonde data from Tateno (refer to Fig. 3c) at 1200 UTC 16 Dec 2014.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

Figure 5 shows the Japan Meteorological Agency (JMA) radar echo-top height at 0900 (Fig. 5a), 1030 (Fig. 5b), and 1200 (Fig. 5c) UTC 16 December 2014, corresponding to the PIREPs shown in Fig. 3. The precipitation area moves northeastward along with the cyclone, and the echo-top height is below 8 km in most areas of area A. Figure 5b shows that the altitude of the target case (8.2 km) is approximately the same as the observed echo-top height. In the accident report by the Japan Transport Safety Board (2016), the crew described the conditions at the time of the encounter as “It was still dark and thin clouds were sometimes seen outside the windows of the cockpit and an airborne radar did not display the clouds expected to cause the big shake” (p. 3), suggesting that the aircraft was flying outside the precipitation cloud at the time.

Fig. 5.
Fig. 5.

Observed echo-top height at (a) 0900, (b) 1030, and (c) 1200 UTC 16 Dec 2014. In (b), the flight path and the turbulence location of the target case are superimposed on the observed echo-top height. The outline of area A is shown as a dotted rectangle.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

To summarize, cyclone-related MOG-level turbulence events were frequently reported at 7–9 km in the vicinity of Tokyo (Fig. 1; area A), and the target case occurred at the same time. Radar and radiosonde observation data suggest that the 7–9-km altitude was between the cloud top (Fig. 5) and the jet stream axis (Fig. 4). The captain’s testimony suggests that the flight was outside the precipitation clouds. Therefore, some local turbulence existed above (not inside) precipitation clouds in area A, and the target case occurred when the aircraft was flying around the cloud top.

3. Model configuration

We conducted numerical experiments with horizontal grid spacings of 3, 1, and 0.25 km using the WRF-ARW, version 3.8, model. Figure 6 shows the computational domain and terrain height. JMA Mesoscale Model (MSM) analysis (5-km grid spacing; JMA 2013) defined the initial/lateral boundary conditions above the ground. NCEP 1° × 1° reanalysis (NOAA/NCEP 2000) was used for the soil conditions. The model incorporates 100 vertical layers from the surface to 100 hPa; vertical grid spacing increases with altitude and becomes constant (dz ∼ 180 m) above approximately 2 km. The start time of the numerical experiments was 1800 UTC 15 December 2014, approximately 16.5 h prior to the onset of the turbulence events. We turned on the two-way nesting option (feedback = 1).

Fig. 6.
Fig. 6.

Model domains 1 (1920 × 1920 km), 2 (649 × 649 km), and 3 (195 × 150 km). Domain 3 resolves part of area A, which includes the location of the target case (red cross).

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

The WRF single-moment 6-class graupel scheme (Hong and Lim 2006) was used for microphysics parameterization. This scheme predicts the mixing ratios of water vapor, cloud water, rainwater, ice, snow, and graupel. A cumulus parameterization scheme was not used in the present study. The MYJ turbulent kinetic energy (TKE; Janjić 1994, 2001), horizontal Smagorinsky, and Monin–Obukhov schemes governed subgrid vertical mixing, subgrid horizontal mixing, and surface-layer physics, respectively. The MYJ TKE scheme solves the one-dimensional equation for subgrid TKE and diagnostic equations for the vertical subgrid-scale fluxes not only in the planetary boundary layer but also in the free atmosphere.

For domain 3, an LES was performed using a Smagorinsky first-order closure scheme (Moeng et al. 2007; Mirocha et al. 2010). Subgrid-scale mixing in the horizontal and vertical directions were predicted by subgrid-scale stress, which is the product of wind shear and eddy viscosity. The eddy viscosity depends on the local grid size and the magnitude of the wind shear. For horizontal grid interval, dx = 250 m can resolve the first vortices that appear from the KH instability, the largest eddies that affect the motion of aircraft, with several points in this case.

4. Results

a. Synoptic-scale simulation (domain 1)

Figure 7 compares simulated 300-hPa wind and height and radar echo-top height with JMA MSM data and radar observations, respectively, for domain 1 at 1000 UTC (Fig. 7a,b) and 1030 UTC (Fig. 7c,d) 16 December 2014. The WRF simulation estimates the MSM analysis well. The observed echo-top height was reproduced well in domain 1 for its maximum height (8–10 km) around 139°–141°E, 35°–37°N.

Fig. 7.
Fig. 7.

Comparisons of the (a) JMA MSM analysis and (b) WRF domain-1 simulated 300-hPa wind and height at 1000 UTC 16 Dec 2014 and (c) observed and (d) WRF domain-1 simulated radar echo-top height at 1030 UTC 16 Dec 2014. The location of the target case is denoted by a red cross. Radar reflectivity tends to be weaker in the distant sea.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

Figure 8 shows the wind field at z = 8 km at 1000 UTC in domain 1, representing an altitude close to that of the target case. Small-scale up/downdrafts are apparent over an area of high total cloud condensate and form wave trains. Up/downdrafts possibly due to vertically propagating mountain waves from the mountainous region along the 138.5°E line (near the west edge in Fig. 8) are also apparent; however, these waves have a different spatial scale than the wave trains. Therefore, over the region where many MOG-level PIREPs were reported, regular fluctuations in vertical wind speed, not attributed to terrain, are widely distributed over the clouds.

Fig. 8.
Fig. 8.

WRF domain-1 simulated horizontal wind barbs (long barbs 10 m s−1, short barbs 5 m s−1) and |vertical wind component| > 1 m s−1 (red/blue shading) at an altitude of 8.0 km (the approximate altitude of the target case), and 5-km total cloud condensate (gray shading) at 1000 UTC 16 Dec 2014. The transect AB refers to the location of the vertical cross sections presented in Fig. 9, below.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

b. Turbulence generation mechanism (domain 1)

To understand the generation of wave trains, Fig. 9 shows the cross section A–B along one of the wave trains observed in Fig. 8. The cross section confirms a band with Richardson numbers (Ri) < 0.25, (green shading) at 6–8 km altitude, and the fluctuation of vertical winds (red/blue shading) centered around the band extends to 12–13 km through gravity waves. Figure 9a highlights A′ as the height region where most of the PIREPs were reported, and it is apparent that A′ corresponds to vertical wind fluctuations along the Ri < 0.25 band. Ri is a dimensionless number defined by
Ri=N2VWS2,
where N2 is the Brunt–Väisälä frequency and VWS represents the vertical wind shear. For two-dimensional stratified shear flow, Ri < 0.25 is a necessary condition of shear-layer instability (Nappo 2002; Drazin and Reid 1981).
Fig. 9.
Fig. 9.

WRF domain-1 simulated Richardson number (Ri; green shading) and the |vertical wind component| > 1 m s−1 (red/blue shading) along the A–B cross section shown in Fig. 8 with (a) isentropes at 4-K intervals and (b) isotachs at 6 m s−1 intervals (black contours) at 1000 UTC 16 Dec 2014.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

According to Scorer (1969), the range of wavelengths that gives a positive growth rate to a sinusoidal wave on a vertical shear layer of Δz width can be approximated using Ri, as shown in
λ<λmax=πΔzRi

In Fig. 9a, the average wavelength of waves at x = 25–70 km, where the direction of the wave train is parallel to the cross section, is λ = 14.7 km. The average width of the Ri < 0.25 region (Δz) is 1.26 km. The λmax = 15.9 km obtained based on these parameters suggests that the wave trains were generated by shear-layer instability. Because the Ri < 0.25 region is located near the upper end of the region where the potential temperature gradient becomes small (Fig. 9a), and in the large vertical shear region (Fig. 9b), we suggest that both the reduction of atmospheric stability and the reinforcement of vertical wind shear would contribute to the reduction of the Ri.

Figure 10 shows shear vectors from 6 to 8 km, calculated by subtracting 6-km wind vectors from 8-km wind vectors. Because the 6-km wind vector is approximately 35 m s−1 northwesterly and the 8-km vector is approximately 63 m s−1 west-southwesterly over Tokyo (35.75°N, 139.75°E), the shear vector is westerly. Over Tokyo, the shear vector and wave train are almost parallel. Figure 11 shows simulated radar reflectivity and potential temperature along the vertical cross section A–B (Fig. 8). We can see that the wave train along the Ri < 0.25 band is located above the high radar reflectivity region. This means that the Ri < 0.25 region, the possible source of the turbulence, is placed over the precipitation clouds, consistent with the captain’s testimony in section 2.

Fig. 10.
Fig. 10.

Shear vectors, defined as the 6–8-km horizontal wind difference and the |vertical wind component| > 1 m s−1 (red/blue shading) at an altitude of 8.0 km, at 1000 UTC 16 Dec 2014 in domain 1.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

Fig. 11.
Fig. 11.

WRF domain-1 simulated radar reflectivity, areas with Ri < 0.25 (green shading), and isentropes (K; black contours) along the A–B cross section shown in Fig. 8.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

c. LES results (domain 3)

Figure 12 shows the horizontal distribution of vertical wind speeds at 8.2 km, the altitude of the target case, and 2-km radar reflectivity at 1021 UTC 16 December 2014 for the domain-3 simulation. Similar to Fig. 8, wave trains are apparent over regions of high total cloud condensate. Figure 13 shows the simulated Richardson number, potential temperature, and vertical winds along the vertical cross section C–D (Fig. 12). The magnitude of the vertical wind fluctuation peaks at an altitude of approximately 6–8 km, where the Ri < 0.25. It is apparent that the strong vertical wind fluctuation reaches an altitude of 8.2 km, the flight level of the target case. In terms of magnitude, the domain-1 simulation estimates a small fluctuation of ±8 m s−1 or less. In contrast, the domain-3 simulation reproduces fine-scale fluctuations possibly formed by breaking of the wave trains and estimates an amplitude from +20 to −12 m s−1 (refer to section 5). Furthermore, domains 1 and 3 estimated different values for the spatial frequency of the waves. As mentioned in section 4b, domain 1 estimated a wavelength of approximately 15 km for the first instability mode from shear-layer instability. However, in domain 3 the wavelength of waves thought to be the first instability mode is approximately 10 km, 30–40 times the horizontal grid size. Overturning isentropes are found at an altitude of approximately 6–8 km, indicating wave breaking and the likely onset of turbulence.

Fig. 12.
Fig. 12.

|Vertical wind component| > 1 m s−1 (red/blue shading) at 8.2-km altitude and 5-km total cloud condensate at 1021 UTC 16 Dec 2014 simulated in domain 3. The red cross indicates the location of the target case.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

Fig. 13.
Fig. 13.

Vertical cross section along C–D (Fig. 12) showing domain-3 simulated Ri (green shading), isentropes (K; black contours), and the |vertical wind component| > 1 m s−1 (red/blue shading).

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

We investigate the intensity of turbulence simulated in domain 3 by computing the energy dissipation rate (EDR) ε to the power of 1/3; ε1/3 is known as an objective index of turbulence intensity (International Civil Aviation Organization 2016). According to Schumann (1991), ε can be approximated using subgrid TKE, as shown in
ε0.84e3/2Δ,
where Δ is the length scale based on grid spacing Δ = (ΔxΔyΔz)1/3 = 224 m, and we use resolved TKE as e to make a rough estimation of EDR. The resolved TKE is calculated for domain 3 based on locally averaged wind components using a Gaussian filter [Eqs. (4)(6)] subtracted from local wind components. We use a Gaussian filter to obtain the local average that follows the wind speed variation well and removes high-frequency fluctuations. Here, σ is the radius of localization, which is set to 500 m to evaluate TKE due to near aircraft-size wind variation:
e(lon,lat)=i=13[ui(lon,lat)]22,
ui(lon,lat)=ui(lon,lat)x,yF(x,y)ui(lonx,laty), and
F(x,y)=1(2πσ)2exp(x2+y22σ2).

Figure 14 shows the horizontal distribution of ε1/3 and the ±1 m s−1 isoline of the vertical wind speed at 1021 UTC 16 December 2014 in the same area, as shown in Fig. 12. EDR is large along with the wind variation due to the shear-layer instability, which peaks in the region of large wind fluctuations (see Figs. 12 and 14). In most areas, ε1/3 is less than 0.7 m2/3 s−1, which is the threshold value between moderate and severe turbulence categories described in the International Civil Aviation Organization (ICAO) Annex 3 (International Civil Aviation Organization 2016). However, some values exceed 0.7, and the maximum value is 0.86 in the white dashed circle. For computation with σ = 1000 m, which is sufficiently larger than aircraft size, the maximum EDR increases to 1.49, because the local wind fluctuation becomes larger.

Fig. 14.
Fig. 14.

EDR (ε) and vertical velocity = ±1 m s−1 (gray contours, with solid line being +1 m s−1 and dashed line being −1 m s−1) at 1021 UTC 16 Dec 2014 in domain 3. Maximum EDR is found in the thick white dashed circle.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

5. Simulated aircraft response

a. Methods

We use the aircraft equations of motion (Kato et al. 1982; Takase et al. 2017; McRuer et al. 1973) to estimate the vertical acceleration experienced by an aircraft flying in an airspace with the turbulence reproduced in section 4c. Aircraft motion can be approximated by solving the six-degrees-of-freedom equations of motion about an aircraft’s center of gravity, as shown in Eqs. (7)(12). Equations (7)(9) solve the motion by translational forces (Fx, Fy, and Fz), and Eqs. (10)(12) solve the rotational motion by moments (Mx, My, and Mz). The equations are defined on a coordinate system whose origin is located at the aircraft’s center of gravity. The variables m and I represent the aircraft mass and moment of inertia, respectively. These forces and moments are caused by gravity or the interaction between the aircraft and air. Figure 15a defines the coordinate system and directions of the forces and moments:
mdudt=Fxmgsinθ,
mdυdt=Fy,
mdwdt=Fz,
IxdpdtIxzdrdt=Mx,
Iydqdt=My,and
IzdrdtIxzdpdt=Mz.
Fig. 15.
Fig. 15.

(a) Forces and moments defined on the aircraft coordinate plane, and (b) state variables for aircraft motion.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

In this study, we only partially solved these equations to simulate vertical acceleration changes. Equations (7)(12) can be separated into two independent groups of equations by linearization of the external force terms using the methods introduced in the appendix: equations for longitudinal motions that include vertical acceleration information and equations for lateral motions. The former group contains the following three equations:

  1. Eq. (7), the temporal change in the horizontal speed u of the aircraft due to thrust, drag, and the x-axis component of the gravity force,

  2. Eq. (9), the temporal change in the vertical speed w of the aircraft due to the lifting force, and

  3. Eq. (11), the temporal change of the rotational angular velocity about the y axis due to the aerodynamically generated angular moment about the y axis.

We do not solve Eq. (7) because changes in the angle of attack (the degree between the x axis and the aircraft direction of travel α; refer to Fig. 15b) caused by vertical wind changes generally have a much greater impact on vertical acceleration variation relative to horizontal wind changes. This means that the horizontal speed of an aircraft is set to a constant value of U0. The equations of motion used in this study are given by Eqs. (13) and (14):
ddt[αq]=[ZαU01+ZqU0MαMq][αq]+1U0[ZuZαU0ZqXMuMαU0MqX][ugwg] and
Mu=(Mu+MαZuU0),Mα=(Mα+MαZαU0),Mq=[Mq+Mα(U0+Zq)U0].
The left-hand side in Eq. (13) indicates the temporal changes, the first term on the right-hand side is the aerodynamic force depending on the attitude of the aircraft (α and q), and the second term is the additional force due to turbulence (ug and wg). Figure 15b shows the variables in the equations. The angle of attack α is approximated from the vertical speed of aircraft w by using α = atan(w/U0) ≈ w/U0, under the assumption of a very small α. The q is the angular velocity along the y axis (q = dθ/dt). Both α and θ have the same unit (rad); however, the former is the angle based on the aircraft’s direction of travel, whereas the latter is the angle based on the Earth coordinate system. In other words, α defines the aircraft’s direction of travel, and θ defines the angle of the aircraft from Earth’s coordinate system. The coefficients (such as Zα and Mq) for expressing aerodynamic forces and moments in combination with these state variables (α and q = dθ/dt) or turbulence (ug and wg) are unique to each aircraft. The definitions of the coefficients, estimation, and validity of the estimation method are given in the appendix.

b. Computational conditions

We set the conditions for the flight simulation to be close to those of the target case. We estimated the aerodynamics of the Boeing 777-200 (coefficients Zα, Mq, etc.) and substituted them into Eqs. (13) and (14). The aircraft flies at U0 = 258 m s−1 along the path from I to J that crosses the region of peak EDR (the white dash circle in Fig. 14), on the 8.2-km cross section at 1021 UTC 16 December 2014. Figure 16 shows the initial location of the assumed aircraft and its flight direction. We solve Eqs. (13) and (14) for 70 s (18.3 km), from the initial point I in the atmospheric field with the time fixed at 1021 UTC 16 December 2014, using the WRF simulation results as turbulence (ug, wg). In addition, the aircraft does not have any control rules; we do not assume the pitching control embedded in the Boeing 777 aircraft (C* law; Spitzer 2001) in this study.

Fig. 16.
Fig. 16.

Computational settings for the flight simulation. (left) The location of the cross section I–J and domain-3 simulated |vertical wind component| > 1 m s−1 (red/blue shading) at 8.2 km at 1021 UTC 16 Dec 2014. (right) the vertical cross section along I–J showing the initial location of the aircraft, the |vertical wind component| > 1 m s−1 (red/blue shading), and the horizontal wind (solid lines).

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

c. Results

Figure 17a shows the simulated aircraft response. While flying a distance of 18.3 km, the aircraft experiences up to |ΔG| = 1.49 G due to a large wind change at 12–14 km. The maximum and minimum accelerations are 1.57 and 0.08 G, respectively, and the maximum deviation from 1 G is 0.92 G, which is classified as moderate turbulence according to Aircraft Meteorological Data Relay (AMDAR) turbulence categories (WMO 2003). This suggests that WRF-LES data and flight simulations can reproduce a situation that causes moderate shaking in the AMDAR scale on a Boeing 777-class aircraft without pitching control.

Fig. 17.
Fig. 17.

(a) Simulated flight path and the time series of vertical acceleration. The vertical wind field at 1021 UTC 16 Dec 2014 is shown by red/blue shading, the flight path is a black solid line, and the history of vertical acceleration is a brown solid line. Vertical acceleration is nondimensionalized with the acceleration of gravity and has the unit of G. (b) Simulated vertical acceleration and the vertical wind components experienced by the aircraft.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

Figure 17b shows the histories of the observed vertical/horizontal winds and the vertical acceleration experienced by the aircraft. The aircraft experiences a local minimum acceleration of 0.08 G due to vertical wind changes from +20 to −12 m s−1 around t = 50 s. The contribution of each term in the equations of motion to the acceleration changes is also shown. Acceleration due to horizontal wind changes Gug = (Zu/U0)ug (red dashed line) is very small relative to acceleration due to vertical wind changes Gwg = (Zα/U0Zq∂/∂X)/U0wg (blue dashed line) and aircraft attitude (α, q) Gattitude = Zα/U0α + (1 + Zq/U0)q (green dashed line). The Gwg almost follows the vertical wind changes (black solid line); however, it is apparent that Gattitude is generated in a direction to cancel Gwg. This is because the aircraft has characteristics to stabilize its attitude changes, such as α and q, in this case. In particular, Zα is a negative value (see the appendix), which causes a strong restoring force against α changes (w change, as αw/U0). The Galtitude increase lags against that of Gwg because Gattitude depends on α and q, which vary more slowly than wg owing to the large aircraft mass m and moment of inertia I. Therefore, there is a slight lag between the gust and the Gattitude.

In this study, we investigated the aircraft response to the turbulent eddies larger than that of an aircraft. By using the above method, we expect that it will become possible to check how the attitude control system introduced in section 1 behaves in response to a realistic wind field with large spatial scales.

Since severe turbulence (from +1.8 to −0.88 G; Japan Transport Safety Board 2016) was reported in the target case, the aircraft should have experienced more intense shaking than was simulated in this study. To further investigate the feasibility of virtual flight testing, we would need to simulate turbulence with a grid resolution on the order of 10 m using WRF-LES.

6. Conclusions

In this study, we investigated the feasibility of virtual flight testing by performing a flight simulation in an airspace with turbulence reproduced using the WRF-LES model. In actual flight testing, it is not always possible to encounter intense CAT, which makes the validation of lidar and surrounding devices using a large amount of observation data difficult. If virtual flight testing in a numerically generated turbulence field is enabled, it becomes possible to conduct many flight tests under near realistic conditions, which can provide preliminary test data for future flight testing that investigates the validity of algorithms for aircraft control and forward wind estimation. However, there are almost no examples of LES for CAT at high altitudes and flight simulations in wind fields generated by numerical weather prediction.

The target case was an accident that occurred at an altitude of 8.2 km near Tokyo on 16 December 2014. Observation data suggested that local CAT was generated in the airspace between the jet axis and precipitation clouds. First, we performed a WRF simulation to reproduce the turbulence at the time of the accident using a three-domain model configuration with the minimum horizontal grid spacing of 250 m. The synoptic-scale simulation reproduced the observed conditions well and estimated relatively small-scale (up to 15 km) waves of vertical wind speed fluctuations near the location of the accident. Furthermore, these waves were found to be instability waves because the Ri was reduced by both the vertical wind shear below the jet axis and the reduction of stability around the top of the precipitation clouds. In the LES domain, we confirmed stronger wind variations relative to the synoptic-scale simulation and found overturning isentropes in the resolved waves. The maximum EDR was over 0.7, which was classified as severe by ICAO. This indicates that the model could reproduce conditions that can cause significant aircraft shaking.

We estimated the vertical acceleration experienced by a Boeing 777-class aircraft without control by solving the aircraft equations of motion in the wind field generated by the WRF-LES. The 70-s flight simulation, which traversed the area of the maximum EDR, estimated 0.08–1.57-G acceleration changes at the point where a 32 m s−1 wind change was observed. This change was equivalent to moderate conditions on the AMDAR scale. Based on the promising results from this study, we expect further progress in terms of the feasibility of virtual flight testing based on WRF-LES and flight simulation.

In future studies, we will perform LES on a finer grid that can resolve eddies smaller than aircraft to simulate flights more accurately, for example, a turbulence that shakes a B747 at its eigenfrequency (1.29 Hz; Table A5, below) as the pitching motion shakes the aircraft with a larger amplitude than other turbulences with different frequencies. The corresponding wavelength to the eigenmotion is 200 m [=258 (m s−1)/1.29 (Hz)], which is smaller than the grid spacing (250 m) for WRF domain 3, suggesting that 10–20-m grid spacings may be required.

Acknowledgments.

Numerical simulations were performed on the Supercomputer system “AFI-NITY” at the Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku University. This work was partially supported by JSPS KAKENHI Grant 20J21567 and Program for Promoting Technological Development of Transportation of Ministry of Land, Infrastructure, Transport and Tourism of Japan. We thank Japan Airlines for permission to use the JAL PIREP. Insightful comments and suggestions for the flight simulation configurations from Mr. H. Takami of Tohoku University are greatly acknowledged.

Data availability statement.

Because of confidentiality agreements, JAL PIREP data cannot be made openly available. WRF netCDF data that support the findings of this study are available from the corresponding author, Ryoichi Yoshimura, upon reasonable request.

APPENDIX

Estimating the Aerodynamic Characteristics of an Aircraft

This section describes the method used to estimate the aircraft-dependent constant coefficients included on the right-hand side of Eq. (13). We decompose the forces/moments A′ to the steady part A¯ and the unsteady part A and assume that a linear relationship can precisely approximate these unsteady forces/moments (Z and M) within very small changes to the state variables (u, α, and q). Therefore, Eq. (13) only solves the additional motion of the aircraft from its steady flight condition. For the example of moment M, the relationship between the additional moments and state variable disturbances is approximated linearly by Eq. (A1). These coefficients (Mu, Mα, and Mq) provide a first-order approximation for the moments/forces by multiplying with the state variable disturbances (u, α, and q):
M(u,α,q)=MM¯Muu+Mαα+Mqq=Muu+Mαα+Mqq.
In the equations of motion, the total two force/moment (Z and M) depends on u, α, and q, which means there are six types of aerodynamic coefficients:
Zu, Zα,Zq, Mu,Mα,andMq.

a. Measurement of aircraft geometry

The aircraft geometry required for coefficient estimation is measured on the basis of airport planning data from Boeing (B777-200: Boeing 2011a; B747-100: Boeing 2011b). Figure A1 shows an example of the shape of a B747-100. The outer mold line, which is gray in color, is the aerodynamic shape of a B747 (Boeing 2011b), and the shape enclosed by thick black solid lines is the simplified B747 shape composed of elements that mainly contribute to the overall aerodynamic performance. We use the latter shape to define aircraft geometry for coefficient estimation. In this study, we neglect the aerodynamics of the vertical stabilizer and engines because their contribution to the vertical acceleration changes due to turbulence is small.

Fig. A1.
Fig. A1.

Aircraft geometry.

Citation: Journal of Applied Meteorology and Climatology 61, 5; 10.1175/JAMC-D-21-0071.1

The shape of the fuselage (body) is approximated by a combination of cones and cylinders, as shown in Fig. A1. The bottom shape is a circle whose area is as large as the cross-sectional area (Boeing 2011b). To estimate Mα, we estimate the moment in the pitching direction (about the y axis) using Munk’s airship theory (Munk 1924). We assume tip/rear body shapes consisting of semiellipsoids (shown in Fig. A1 in blue).

The two-dimensional shape of the main wing is defined as a trapezoid whose base is located on the x axis, and its shape and the outer mold line are consistent on the outboard part of the wing. We identified the location of the mean aerodynamic chord (MAC), where the moment of the lifting force generated by the main wing becomes zero, based on point A, which is the intersection of two lines: the line defined by the measured wing tip chord length ct and the measured wing root chord length cr, and the 50% chord length line that connects the midpoints of the leading (front) and trailing (back) edges on the main wings. We define x% MAC as the x% location from the leading edge to the trailing edge of the MAC. The location of the aerodynamic center, on which the average lifting force of the main wing acts, and the center of gravity are set as 25% MAC and 35% MAC, respectively. We define the sweep angle Λ as the angle between the y axis and the 25% chord length line. The wing area S is 2 times as large as the area of the trapezoid, and its aspect ratio (AR) is 1/S multiplied by the square of the span length b, 2 times as long as the length between the x axis and the wingtip.

The two-dimensional shape of the horizontal stabilizer is defined as a trapezoid whose base is located at the intersection of the 50% chord line and the outer mold line. Shape parameters such as MAC, S, and AR are measured in the same way as the main wing. The horizontal distance between the center of gravity and the aerodynamic center of the horizontal stabilizer is called the moment arm lt.

b. Estimation of each coefficient

The following method follows Kato et al. (1982).

1) Coefficients for the force in the z direction
The Zu is the change in the z direction force with respect to airspeed u. Because the left-hand side of the equation of motion is divided by the aircraft mass m or the moment of inertia, they are included in these coefficients:
Zu=Zu=ρU0S2m(Czu2CL).
We consider a nondimensionalized form of Zu; Czu is a derivative of Zu nondimensionalized by the wing area S, the steady flight speed U0, and the density of air ρ. For moments, they are also nondimensionalized by MAC length c¯. In Eq. (A3), ρU0S/m has a dimension of Zu. The Czu is estimated as
Czu=Czu=M0CLM0,
where M0 is U0 divided by the sound speed. However, this coefficient can be neglected in this study. The CL is the nondimensionalized form of the lifting force of the entire aircraft during steady flight and is estimated by nondimensionalizing the equilibrium of forces in the z direction; the lifting force is equal to the aircraft weight. In Eq. (A5), ρU02S transforms a force into a nondimensional value:
CL=mg0.5ρU02S.
The Zα is the change in the z-direction force with respect to the angle of attack α:
Zα=Zα=ρU02S2mCzα.
The nondimensional coefficient Czα is given by α differentiation of the equilibrium of nondimensionalized force in the z direction:
Czα=CLα.
The CLα is the α derivative of CL. Assuming that only the main wings and horizontal stabilizers generate the lifting force of the entire aircraft, CL is estimated as a linear function of α; CLα is estimated as follows:
CLα=CLα=aw[1+atawStS(1δα)].
The aw and at are α derivatives of the nondimensionalized lifting force generated on the main wing and the horizontal stabilizer. Using the following equation, they can be empirically estimated by substituting sweep angles Λ, aspect ratios AR for each wing, and a0 (5.7 is set in this study), which is the α derivative of the nondimensionalized lifting force of a two-dimensional airfoil:
ai=a0cosΛi1+a0cosΛiπARi.
The ∂δ/∂α is the change in downwash δ, which is the change in the direction of airflow at the location of the horizontal stabilizer brought by the aerodynamic action of the main wing, with respect to α, and is approximated by the following equation:
δα=2awπAR.
The Zq is the change in the z-direction force with respect to the angular velocity around the y-axis q. The q derivative of lifting force changes due to the pitching angular velocity q gives Zq and its nondimensional form Czq as follows:
Zq=Zq=ρU0Sc¯4mCzq,Czq=2Vh*at.
The Vh* is called the nondimensional tail volume, defined as Vh*=ltSt/(c¯S), where lt is the moment arm and St is the area of the horizontal stabilizer.
2) Coefficients for the moment about the y axis
The Mu is the change of the moment about the y axis with respect to the airspeed u:
Mu=Mu=ρU0Sc¯2IyyCmu,Cmu=M0CMM0.
We assume that this coefficient can be neglected. The Mu contributes to long-period aircraft motion. For the gust response, which is a short-period motion, the contribution of this coefficient is considered to be small.
The Mα is the change in the moment about the y axis with respect to the angle of attack α. The Cmα is estimated by the α derivative of the sum of the moments around the center of gravity. We consider that the fuselage, main wing, and horizontal stabilizer generate moments:
Mα=Mα=ρU02Sc¯2IyyCmα,Cmα=CMα=aw[(hhnw)Vh*ataw(1δα)]+Mfus.
Here, h and hnw are the locations of the center of gravity and the aerodynamic center (% MAC) of the main wing. The term Mfus represents the pitching instability induced by the fuselage. We estimate this term using Munk’s airship theory (Munk 1924).
The Mα˙ is the change in the moment about the y axis with respect to the temporal derivative of the angle of attack α˙=dα/dt. The Cmα˙ is approximated using the α˙ derivative of the increment of the downwash δ at the location of the horizontal stabilizer due to α changes as follows:
Mα˙=Mα˙=ρU0Sc¯24IyyCmα˙,Cmα˙=2Vh*(ltc¯)atδα.
The Mq is the change in the moment about the y axis with respect to the angular velocity around the y-axis q. Similar to the Czq estimation, Cmq is obtained by the q derivative of the moment of the lifting force acting on the horizontal stabilizer generated by the angular velocity q:
Mq=Mq=ρU0Sc¯24IyyCmq,Cmq=2Vh*(ltc¯)at.

c. Obtained coefficients and validity of the method

Table A1 lists the estimated coefficients of forces/moments substituted into the equations of motion. We estimated coefficients for the B777-200 that encountered turbulence in the target case by using the method introduced in appendix section b above. The geometry of the B777-200 was taken from airport planning data (Boeing 2011a). The aircraft was assumed to be in a steady flight condition at 8200 m (ρ = 0.514 kg m−3) and U0 =258 m s−1 (Mach number M0 = 0.84). The lifting coefficient was set at CL = 0.324 in reference to Eq. (A5). The aircraft mass was m = 2.47 × 105 kg, and the moment of inertia around the y axis was estimated to be Iyy = 3.15 × 107 kg m s2 as based on the moment of inertia of the B747-100 multiplied by the mass and length ratios of the B777 and B747.

Table A1

Estimated aerodynamics of Boeing 777-200 aircraft.

Table A1

To demonstrate the validity of the method presented in section b, we investigate the estimation error between the B747-100 coefficients from Heffley and Jewell (1972) and coefficients estimated using this method. The reference coefficients of the B747-100 and its flight conditions are listed in Tables A2 and A3. The air density ρ was set from the International Standard Atmosphere.

Table A2

Reference coefficients of Boeing 747-100 aircraft.

Table A2

Table A4 lists the coefficients estimated on the basis of the geometry defined using airport planning data and the method in appendix section b under the flight conditions listed in Table A3.

Table A3

Reference flight conditions.

Table A3
Table A4

Estimated aerodynamics of Boeing 747-100 aircraft.

Table A4

We compared both sets of coefficients, focusing on one of the eigenmotions of aircraft characterized by these coefficients, which is short-period, vertical motion. Because we simulated the aircraft response to gusts in this study, it was appropriate to evaluate the method and coefficients for this short-period motion. This motion is an oscillation with strong decay, where the angle of attack and the pitching angle interact with each other, and its frequency and damping rate are unique to each aircraft. Equation (13) is the equation of motion without the equation for the horizontal speed u, and the eigenmode of the equation without external force due to turbulence is only the short-period mode. The Laplace transform of the equation yields the following quadratic equation:
s2+(ZαU0MqMα˙)s+(Mα+ZαU0Mq)=s2+2ωξs+ω2=0,
and we can compute the characteristic frequency ω and damping rate ξ as follows:
frequency:ω=Mα+ZαU0Mq and
damping rate:ξ=(ZαU0MqMα˙)/(2ω).

Table A5 compares the frequencies and damping rates obtained using our estimation and Heffley and Jewell (1972) data. The estimation errors for the frequency and damping rate are both approximately 0.05 s−1, which means that the short-period motion by the coefficients obtained using our method approximates the motion well in comparison with the reference coefficients provided by Heffley and Jewell (1972). Therefore, the method presented in appendix section b is reasonable.

Table A5

Comparison of eigenmodes from the reference and estimated aerodynamics of B747-100 aircraft.

Table A5

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  • Vrancken, P., and Coauthors, 2016: Flight tests of the DELICAT airborne lidar system for remote clear air turbulence detection. EPJ Web Conf., 119, 14003, https://doi.org/10.1051/epjconf/201611914003.

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  • Fig. 1.

    Overview of the accident involving a Boeing 777-200 aircraft that occurred over Tokyo at 1035 UTC 16 Dec 2014 (1935 LST 16 Dec 2014; the target case). The flight path, time, and location of the target case are from the accident investigation report (Japan Transport Safety Board 2016).

  • Fig. 2.

    Surface weather map at 1200 UTC 16 Dec 2014 and the path of the low pressure system over Japan (from the JMA). The star denotes the location of the target case.

  • Fig. 3.

    MOG-turbulence-level pilot reports in area A (a) from 0730 to 0859 UTC, (b) from 0900 to 1029 UTC, and (c) from 1030 to 1159 UTC, along with (d) a histogram of report altitudes; (c) includes the location of the target case and the radiosonde flight (Tateno, denoted by an X). The turbulence intensity and altitude [flight level (FL)] are represented by numbers (moderate = 4 and severe = 5) and colors, respectively. FL is altitude information at 100-ft (≈30.5 m) intervals, which is represented by a three-digit number.

  • Fig. 4.

    Radiosonde data from Tateno (refer to Fig. 3c) at 1200 UTC 16 Dec 2014.

  • Fig. 5.

    Observed echo-top height at (a) 0900, (b) 1030, and (c) 1200 UTC 16 Dec 2014. In (b), the flight path and the turbulence location of the target case are superimposed on the observed echo-top height. The outline of area A is shown as a dotted rectangle.

  • Fig. 6.

    Model domains 1 (1920 × 1920 km), 2 (649 × 649 km), and 3 (195 × 150 km). Domain 3 resolves part of area A, which includes the location of the target case (red cross).

  • Fig. 7.

    Comparisons of the (a) JMA MSM analysis and (b) WRF domain-1 simulated 300-hPa wind and height at 1000 UTC 16 Dec 2014 and (c) observed and (d) WRF domain-1 simulated radar echo-top height at 1030 UTC 16 Dec 2014. The location of the target case is denoted by a red cross. Radar reflectivity tends to be weaker in the distant sea.

  • Fig. 8.

    WRF domain-1 simulated horizontal wind barbs (long barbs 10 m s−1, short barbs 5 m s−1) and |vertical wind component| > 1 m s−1 (red/blue shading) at an altitude of 8.0 km (the approximate altitude of the target case), and 5-km total cloud condensate (gray shading) at 1000 UTC 16 Dec 2014. The transect AB refers to the location of the vertical cross sections presented in Fig. 9, below.

  • Fig. 9.

    WRF domain-1 simulated Richardson number (Ri; green shading) and the |vertical wind component| > 1 m s−1 (red/blue shading) along the A–B cross section shown in Fig. 8 with (a) isentropes at 4-K intervals and (b) isotachs at 6 m s−1 intervals (black contours) at 1000 UTC 16 Dec 2014.

  • Fig. 10.

    Shear vectors, defined as the 6–8-km horizontal wind difference and the |vertical wind component| > 1 m s−1 (red/blue shading) at an altitude of 8.0 km, at 1000 UTC 16 Dec 2014 in domain 1.

  • Fig. 11.

    WRF domain-1 simulated radar reflectivity, areas with Ri < 0.25 (green shading), and isentropes (K; black contours) along the A–B cross section shown in Fig. 8.

  • Fig. 12.

    |Vertical wind component| > 1 m s−1 (red/blue shading) at 8.2-km altitude and 5-km total cloud condensate at 1021 UTC 16 Dec 2014 simulated in domain 3. The red cross indicates the location of the target case.

  • Fig. 13.

    Vertical cross section along C–D (Fig. 12) showing domain-3 simulated Ri (green shading), isentropes (K; black contours), and the |vertical wind component| > 1 m s−1 (red/blue shading).

  • Fig. 14.

    EDR (ε) and vertical velocity = ±1 m s−1 (gray contours, with solid line being +1 m s−1 and dashed line being −1 m s−1) at 1021 UTC 16 Dec 2014 in domain 3. Maximum EDR is found in the thick white dashed circle.

  • Fig. 15.

    (a) Forces and moments defined on the aircraft coordinate plane, and (b) state variables for aircraft motion.

  • Fig. 16.

    Computational settings for the flight simulation. (left) The location of the cross section I–J and domain-3 simulated |vertical wind component| > 1 m s−1 (red/blue shading) at 8.2 km at 1021 UTC 16 Dec 2014. (right) the vertical cross section along I–J showing the initial location of the aircraft, the |vertical wind component| > 1 m s−1 (red/blue shading), and the horizontal wind (solid lines).

  • Fig. 17.

    (a) Simulated flight path and the time series of vertical acceleration. The vertical wind field at 1021 UTC 16 Dec 2014 is shown by red/blue shading, the flight path is a black solid line, and the history of vertical acceleration is a brown solid line. Vertical acceleration is nondimensionalized with the acceleration of gravity and has the unit of G. (b) Simulated vertical acceleration and the vertical wind components experienced by the aircraft.