Advances in Meteorology

Volume 2015, Article ID 816727, 13 pages

http://dx.doi.org/10.1155/2015/816727

## Lagrangian Coherent Structure Analysis of Terminal Winds: Three-Dimensionality, Intramodel Variations, and Flight Analyses

^{1}School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA^{2}Hong Kong Observatory, Hong Kong

Received 2 November 2014; Revised 20 February 2015; Accepted 20 February 2015

Academic Editor: Luis Gimeno

Copyright © 2015 Brent Knutson et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present a study of three-dimensional Lagrangian coherent structures (LCS) near the Hong Kong International Airport and relate to previous developments of two-dimensional (2D) LCS analyses. The LCS are contrasted among three independent models and against 2D coherent Doppler light detection and ranging (LIDAR) data. Addition of the velocity information perpendicular to the LIDAR scanning cone helps solidify flow structures inferred from previous studies; contrast among models reveals the intramodel variability; and comparison with flight data evaluates the performance among models in terms of Lagrangian analyses. We find that, while the three models and the LIDAR do recover similar features of the windshear experienced by a landing aircraft (along the landing trajectory), their Lagrangian signatures over the entire domain are quite different—a portion of each numerical model captures certain features resembling those LCS extracted from independent 2D LIDAR analyses based on observations.

#### 1. Introduction

A Lagrangian framework for terminal wind hazard detection near Hong Kong International Airport (HKIA) has been recently developed and locations of such disturbances have been compared against flight data [1–3]. The approach outlined in this series of work focuses on obtaining signatures of convergence and divergence of fluid parcel trajectories based on two-dimensional (2D), near-ground velocity data retrieved from light detection and ranging (LIDAR) equipment [4]. The disturbances extracted from the Lagrangian methods are found to be in close proximity of real jolts experienced by landing aircraft. Validating with lengthy flight data over several months, it is found that this approach outperforms traditional Eulerian measures, such as velocity fluctuation measurements [5] as they provide better Receiver Operating Characteristic (ROC) graphs [6, 7], and matches closely with an operational algorithm based on a scanning pattern that follows the actual aircraft landing trajectories [8].

One limitation of the aforementioned methodology is the lack of three-dimensional (3D) data from the 2D LIDAR output. Indeed this is a common limitation shared by all other methods based on 2D LIDAR data, and the Lagrangian framework outperforms traditional methods in part due to its capability to better infer the signatures transversal to the 2D plane-position-indicator (PPI) scanning cone. Variational wind retrieval algorithms in three dimensions are also available [9–11], but they are more time-consuming and relevant to operational forecasts at HKIA, and PPI scans are only available at a few elevation angles. Henceforth, it is beneficial to verify that sophisticated results based on 2D data are useful in operational applications.

In this study, we aim to explore to what extent the transversal signatures inferred from 2D scans represent true 3D structures—Do we find correspondence of 3D vertical structures at the locations where 2D convergence and divergence are the strongest, at least near the center of the LIDAR scanning cone? Does this interpretation successfully extrapolate to data at the peripheral of the LIDAR scanning cone, where the vertical elevation could be over 100 meters above mean sea level, and an argument of strong two-dimensionality near ground may not apply? What extra information does 3D data reveal that is absent from 2D analyses?

Towards this end, we have generated three independent numerical simulations of the regional atmospheric flows near HKIA, for a case of strong windshear on December 27th, 2009. This case corresponds to an airstream associated with a ridge of high pressure along the southeastern coast of China meeting a cold front from inland, resulting in aircraft diverting to Shenzhen because they could not land at HKIA [12]. Two of the simulations are based on numerical weather prediction models—the Regional Atmospheric Modelling System (RAMS) [13] and the Weather Research and Forecasting model (WRF) [14]; the third simulation uses the FLOWSTAR package, which is analytically based and depends more on the terrain data than the physics [15]. In terms of the initialization of the simulations, the two weather forecast models are driven by global forecast system (GFS) data [16] and use nested grids to achieve high resolution over HKIA, whereas the latter uses upstream observational data (independent from LIDAR) as the constant boundary conditions for computation of a steady state solution.

To realize our goal on validation of the 2D analyses, we make the following comparisons. Firstly, we contrast LCS among different models to obtain a full picture of the 3D flow structures. This comparison reveals the variability among models. Secondly, within each model, we compare LCS obtained from full 3D data to those derived from 2D wind fields interpolated on the LIDAR scanning cone. Two schemes are considered for the 2D data generation. One is the horizontal wind speed interpolated directly from 3D model. This wind field closely mimics the resolved model flow, so the comparison directly reflects 2D signatures of a 3D field. The other uses line-of-sight (LOS) velocity from the models and goes through the 2D wind retrieval scheme [4]. The retrieved wind is then used to generate 2D LCS. This effectively mimics the procedure of LIDAR measurements—LCS generation. Consequently 2D information loss and modeling assumed in the wind retrieval scheme are tested with 3D data. Thirdly, we contrast the 3D and 2D LCS from models with 2D LCS obtained from the actual LIDAR measurements to check for any correspondence. Lastly, we compare the LCS from these analyses with data collected from a landing aircraft. This brings all models and schemes to the ultimate test for possible operational implementation.

The rest of the paper is organized as follows. In Section 2 we briefly review the wind retrieval, extrapolation, and LCS generation algorithms. In Section 3 we introduce the three numerical model data sets. In Section 4 we discuss the various comparisons among 3D, 2D, and measurement data. In Section 5, we draw conclusions and discuss further studies underway.

#### 2. Wind Retrieval and LCS Generation Algorithms

We briefly summarize the algorithms used to generate 2D wind retrieval from LIDAR scans and to extract LCS based on the retrieved data. The 2D wind retrieval algorithm for LIDAR is modified from a two-step variational method for RADAR [11]. The cost function to be minimized is given bywhere and are the components of the retrieved wind field, subscript is the background field, generated from LIDAR radial velocity in the way described in [11], is the retrieved radial velocity, superscript is the observed values, and are the horizontal grid point, and is the time index (three consecutive scans are used in each analysis). The weights are (after the first step retrieval), , , and . They are chosen empirically in this paper to ensure that the constraints have proper orders of magnitude. The model-emulated LOS velocity is subject to this retrieval algorithm for emulated 2D wind and subsequent LCS analyses. For more discussions of this algorithm, the readers are referred to [2, 4].

The retrieved wind is extrapolated beyond the LIDAR resolved range as a global linear flow that best fits the 2D wind retrieval [1]. This best reveals the nonlinearity inside the LIDAR observational domain while it avoids addition of extra nonlinearity from data outside when they are completely unknown. In terms of the current study, this extrapolation is applied to the true and emulated LIDAR data (so as to measure the performance of the 2D LCS retrieval algorithm). The numerical models have data coverage outside of the LIDAR range and those data are used in the 3D computations as they provide the true LCS pertinent to the corresponding velocity fields.

For a Cartesian grid in a rectangular region, where the coordinate axes have been chosen such that the domain center corresponds to the origin, the closest linear incompressible flow which minimizes error in the Euclidean norm is where is the spatial average of a function over the grid .

The global flow is constructed on the whole plane by lettingwhere is a filter function that takes value 1 inside a subset of and value 0 in the exterior of . In between we have a buffer zone of width where smoothly transition between 1 and 0. This allows smooth trajectories to leave the LIDAR domain.

We use a common measure, the finite-time Lyapunov exponents (FTLE), to reveal the LCS field. Given a velocity field (2D or 3D), the Lagrangian position of a fluid parcel at time which started at location and time can be computed numerically, and the FTLE is defined as

An auxiliary grid approach is used to approximate the FTLE field with better accuracy [17]. Also, trajectories near ground are allowed to slide along the surface with horizontal velocity at 10 m. This helps remove structures due to the no-slip boundary conditions and better reveals the structures more relevant to airflow disturbances that affect the airport.

The FTLE field [3] and its associated gradients [6] have been used to compare LCS signatures of airflow hazard and jolts experienced from flight data. We will contrast the LCS analyzed here with flight data for the corresponding event in a similar fashion in Section 4.

#### 3. Numerical Model Data for the Windshear Event

The numerical model data based on RAMS and FLOWSTAR have been reported previously [12, 18], but they focus on comparisons between the headwind profile from measurements onboard a flight and in numerical predictions. Both models capture the general trend of the measured headwind profile. In this study, these two model data sets, along with a third one generated using WRF, are used for LCS analyses.

The RAMS model is nested within the 20 km resolution Operational Regional Spectral Model (ORSM) of the Hong Kong Observatory (HKO). RAMS was run with nested grids having spatial resolutions of 4 km, 800 m, and 200 m with two-way nesting. The smallest nest is sufficient to resolve the mountains on Lantau Island, immediately south of HKIA. The Mellor–Yamada turbulence parameterization scheme was used in the first grid and the Deardorff scheme in the other two grids. The model run started at 1200 UTC, 26 December 2009, and was carried out for 12 h.

The FLOWSTAR model is based on topographic data and meteorological inputs and generates steady-state solutions of velocity. The boundary conditions for the FLOWSTAR runs were defined using the observed 10 m wind upstream of Lantau Island and by the radiosonde ascent for midnight UTC 27 December 2009 at King’s park for the wind speed and direction, vertical temperature structure, and hence buoyancy frequency. Three cases were considered for the wind speed and direction and boundary layer height. In this study we use the data set that best matches the measured headwind profile. This corresponds to an upstream wind speed of 7.3 at m height, wind direction of 140°, boundary layer height m, buoyancy frequency profile for heights , for heights , temperature step of 7.19°C at , and surface roughness of 0.5 m. The readers are referred to [12, 18] for more thorough discussions of these two models.

Finally, the WRF model is initiated from the GFS data and runs with nested grids having spatial resolutions of 51.2 km, 12.8 km, 3.2 km, 800 m, and 200 m with two-way nesting. The model is centered at 22.313 N and 113.92 E. In an attempt to match the high horizontal resolution, each nest also had 85 vertical levels. In the 3 coarsest grids the YSU (Yonsei University scheme) boundary layer parameterization was used, whereas in the 2 finest resolution nests the boundary layer parameterization was turned off in order to have full 3D diffusion. Full diffusion (diff_opt = 2) was selected, paired with the 1.5 order TKE prediction for turbulence parameterization. Each nest used the Dudhia shortwave radiation and RRTM longwave radiation schemes. Five-layer thermal diffusion was used as the land surface option, whereas the MM5 similarity scheme was selected for the surface layer and the Kessler scheme was used for microphysics. No cumulus parameterization was used. The simulation is initiated at 1800 UTC, 26 December 2009, and ran for 6 h.

As a first comparison, we contrast some Eulerian velocity data among models and measurements in Figure 1, for a case of missed approach. The approach is from left to right, as indicated by the black arrow above Figure 1(a). Figures 1(a)–1(c) show comparison of the headwind for the three numerical models along the landing corridor. The aircraft trajectory is also shown in each of the panel as the thick black solid curve. It is seen that all three models capture a region of strong headwind near the runway threshold. Flying through this patch leads to significant windshear. The two regional models appear to capture the same trend of this patch from bottom left to top right. This patch is more columnar in the FLOWSTAR data. The sharp transition in the FLOWSTAR model at 400 m indicates the height of the inversion layer. To better infer the flow topology, we use the velocity vectors projected on the vertical plane of the landing corridor (headwind and vertical velocity). It seems that RAMS has the weakest vertical velocity in this plane; FLOWSTAR shows more wave undulation as the aircraft approaches the runway threshold, yet WRF shows a hint of flow reversal at −1 NM from the runway threshold between 300 and 500 m in altitude. This feature possibly leads to strong LCS. In Figure 1(d), the headwind profile is shown from different data sets. In this panel, the thick black solid curve is the actual measurement of headwind onboard the landing aircraft. The magenta curve shows the headwind profile extracted from LIDAR conical scans. This data closely follows the velocity ramp from onboard measurement at about −1 nautical mile (NM) from the runway threshold. The blue dashed curve (RAMS), red dash-dotted curve (FLOWSTAR), and green dotted curve (WRF) differ from the onboard measurement further, but they do capture the general trend of the windshear (with weaker velocity gradients). A cross correlation study shows that, at the respective maxima, the LIDAR data is 71% correlated with onboard head wind measurements, followed by WRF (63%), FLOWSTAR (62%), and RAMS (56%). Lagrangian signatures of these resolved flow data are discussed in the following section.