## 1. Introduction

The eyewall of tropical cyclones (TCs) has been the subject of much recent research (Schubert *et al.*, 1999; Montgomery *et al.*, 2006), which has led to a significant increase in our understanding of the physical mechanisms sustaining and the dynamics inherent in hurricanes, as well as to improved methods for predicting intensity. The dynamics of an eyewall is very complex in both theoretical models and observational data, due in large part to the 3D nature of the storms, and the time dependence and size of the data sets. Although a great deal of research has been conducted on dimensionally reduced models, in particular axisymmetric (Rotunno and Emanuel, 1987) and planar (Schubert *et al.*, 1999) models, real data and large-scale 3D simulations have shown the importance of non-axisymmetric structures and vertical variations. Representing key dynamical structures of a fully 3D TC as low-dimensional datasets should provide significant gains in computational efficiency as well as a better understanding of essential properties of hurricane dynamics. The objective of this paper is to extract a low-dimensional representation of a Lagrangian eye–eyewall interface from 3D simulation data to better understand the dynamics in a simplified framework.

In several recent studies of the structure of the eye and eyewall (see, for example, Montgomery *et al.*, 2006), it has been noted that intensification is coincident with the vanishing of higher wavenumber asymmetries, leaving low wavenumber asymmetries. Mixing of angular momentum inward into a ring-like structure strengthens an axisymmetric vortex (Kossin and Eastin, 2001), while the collapsing of angular momentum into a monotonic profile is present in steady or weakening storms (Schubert *et al.*, 1999). According to the asymmetric balance (AB) theory of Shapiro and Montgomery (1993), the decay of higher wavenumber asymmetries is caused by shear. A quantitative study of the process of asymmetrization based on AB theory has been performed by Smith and Montgomery (1995) and showed that time-scales for particle separation are related to asymmetry decay in dependence of the azimuthal wavenumber involved. In Reasor *et* *al.* (1999) it is found that low-wavenumber asymmetries are dominant in model simulations and in the recorded data of Hurricane Olivia (1994), though the degree of asymmetry was sensitive to the location of vortex centres. Similar structures in 3D vortices have been reported in Nolan and Montgomery (2002).

Although the aforementioned studies highlight the importance of the role of asymmetries, there is no agreement on how to define the structure of the inner core. A tropical cyclone eye–eyewall interface is accepted as the region between the fast-moving circulation and updraughts of the eyewall, and the region of slower solid body rotation contained within the eye. Common measures of vorticity and shear, as well as thermodynamic properties, are often used to describe storm structure (Zhang *et al.*, 2002). Thermodynamically, the eye and eyewall have very different characteristics. The eye contains air with high potential temperature, low moisture and low central pressure (Braun, 2002; Smith *et al.*, 2005). The eyewall has a much higher moisture content and is seen in observations as a cylindrical wall of cumulonimbus convection with strong radial pressure gradient (Willoughby, 1998), extending from the sea surface to the upper troposphere containing the radius of maximal winds (Emanuel, 2003; Smith *et al.*, 2008). Maximal ascent velocities are generally found in the upper troposphere (Jorgenson *et al.*, 1985), although the vertical velocity field is difficult to determine from observations (Holton, 2004). The eyewall is commonly measured through radar reflectivity measurements as lines of intense reflectivity at altitudes below 3 km (Marks *et al.*, 2008). The basic problem with any of these thermodynamic fields is that they are not continuously advected through the storm. In fact, higher wavenumber asymmetries have been shown to cause irreversible diffusion of many of these fields (Hendricks and Schubert, 2009).

An objective definition of an eye–eyewall boundary taken only from the wind fields of an idealized simulation designed to study asymmetric three-dimensional aspects in intensification was proposed by Rutherford and Dangelmayr (2010), which labelled the boundary as a two-dimensional boundary separating the eye and eyewall. The definition was based on particle transport rather than thermodynamic quantities, which has the advantage of being frame independent for time-varying flows, and requiring no prior knowledge of storm location. The eyewall from a simulation of Hurricane Isabel was found by Sapsis and Haller (2009) by computing direct Lyapunov exponents from trajectories governed by inertial particle dynamics.

Flow boundaries may be computed by utilizing and extending recent techniques for locating and extracting Lagrangian coherent structures from 2D time-dependent velocity fields (Haller and Poje, 1997; Haller and Yuan, 2000; Haller, 2005; see Koh and Legras (2002) and Joseph and Legras (2001) for an application to geophysical flows) to the 3D case and discriminating between particle transport due to hyperbolicity and shear. Finite-time Lyapunov exponents (FTLEs), which are a common measure of particle separation, do not differentiate between hyperbolicity and shear. A circular band of shear in the eyewall region produces a broad region of high values in the FTLE field of the mature hurricane simulation used in this study. The high-resolution simulation used in this study has stronger shear and greater symmetry in the mature phase than the storm studied by Sapsis and Haller (2009). In the framework used in this paper, a Lagrangian eye–eyewall interface (LEEI) was identified with a Lagrangian hyperbolic structure that provides a barrier to transport between the eye and eyewall regions, and varies continuously with the initial time (time at which trajectories are seeded). The structure was computed as a maximal ridge surface of a specific Lagrangian field that measures hyperbolic separation orthogonal to trajectories. A benefit of the approach developed in Rutherford and Dangelmayr (2010) is fast convergence of the newly introduced Lagrangian fields, which resolved the LEEI in a 1 h integration time, in contrast to the 6 h integration time required to detect a maximal surface from FTLEs. A ridge extraction algorithm also locates discrete points on a hyperbolic structure exactly.

The resulting LEEI is an asymmetric cylindrical surface with the degree of asymmetry quantified by the amplitudes of the azimuthal modes. The degree of azimuthal asymmetry within a 3D storm varies during intensification, but studies (Montgomery *et al.*, 2006) have noted that while many of the early asymmetries vanish after maturation into a single vortex, the low wavenumber asymmetries, specifically wavenumbers 1 and 2, remain. True axisymmetry occurs when all of the asymmetries of the primary vortex vanish during the mature phase. The remaining asymmetries can be due to convective bursts or material transport between the eye and eyewall. A major objective of this study is to examine the role of transport in the evolution of asymmetries during a fully formed TC with periods of intensification.

In this paper, we extend the ridge extraction algorithm from Rutherford and Dangelmayr (2010) to a moving frame, and provide a good approximation to the true maximal hyperbolic surfaces that is continuous, closed and computed in an automated manner, by using a representation of the surface in terms of Fourier descriptors. Combining the Fourier descriptors with a fit of radial basis functions to capture the dependence on the vertical coordinate yields a smooth representation of the LEEI. A similar procedure is applied to compute a maximal shearing surface.

The combined amplitudes of the azimuthal and vertical modes are then analysed, which results in a dynamical model that captures the temporal evolution of the LEEI, and allows quantification the asymmetries due to particle transport. By using a shape-based coordinate system, the vortex centre is unambiguously defined as the centroid of the LEEI.

Energy considerations show that the LEEI structure is well approximated by the first three Fourier modes and 15 radial basis functions vertically (yielding 90% of the energy). Thus our approach yields a low-dimensional representation of a dynamically evolving LEEI.

The outline of this paper is as follows. In section 2 we introduce the Lagrangian fields which form the basis for our study. In section 3 a fully automated (over *z*-levels and across initial time) ridge extraction algorithm is developed that generates closed maximal ridge curves on *z*-slices, which are smoothed by truncating an expansion in terms of Fourier descriptors. The Fourier descriptors provide information about the degree of axisymmetry and decay of azimuthal wavenumbers during a mature hurricane. Matching the variation of the ridge curves with *z*-levels to vertical basis functions then leads, in section 4, to a 3D continuous and low-dimensional representation of the LEEI. Concluding remarks and an outlook on future work are given in section 5.

### 1.1. Model

The model from which the velocity data are calculated is the fifth generation Penn State/NCAR mesoscale (MM5) 3D hurricane model (Grell *et al.*, 1995; Dudhia, 1993), which was used in Rutherford and Dangelmayr (2010). The initialization used in this study was a non-hydrostatic axisymmetric vortex initialization from experiment 12 of Nguyen *et al.* (2008), which studied the role of asymmetric convective structures during intensification. The model physics used is a bulk-aerodynamic boundary-layer scheme, with a simple moisture scheme. Output wind fields are given on a three-layer mesh, with data on the innermost mesh given on an equidistant *xy* grid with d*x* = d*y* = 1.67 km, and vertical information computed on pressure levels, which are converted to *z*-levels. The temporal output used for trajectory computations is d*t* = 2 min.