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 etal. (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.
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 dx = dy = 1.67 km, and vertical information computed on pressure levels, which are converted to z-levels. The temporal output used for trajectory computations is dt = 2 min.
2. Lagrangian field definitions
The use of Lagrangian scalar fields for identifying stable and unstable manifolds in unsteady flows (Haller and Poje, 1997; Haller, 2000; Haller and Yuan, 2000; Haller, 2003) has proven effective for locating flow boundaries in complex geophysical datasets (e.g. D'Ovidio et al., 2004; Green et al., 2006; Salman et al., 2008), and has been applied to atmospheric flows by Koh and Legras (2002), Joseph and Legras (2001) and Tang et al. (2009), and to hurricanes by Rutherford et al. (2010a, 2010b), Rutherford and Dangelmayr (2010) and Sapsis and Haller (2009). Maximal surfaces of scalar fields represent Lagrangian coherent structures (LCSs), which are related to the stable and unstable manifolds of a hyperbolic fixed point for many aperiodic flows (Shadden et al., 2005), partition the flow and dictate trajectory transport.
The fields computed in this paper are the same as those computed by Rutherford and Dangelmayr (2010), which measure the separation of trajectories along defined subspaces. We briefly review the definition of these fields; for details we refer to Rutherford and Dangelmayr (2010).
If ξ is a perturbation of a trajectory x(t) for a 3D time-dependent velocity field u, , then ξ grows according to the variational equation
Defining a continuously varying orthogonal coordinate system by the tangent unit vector t = u/|u| and two unit vectors n and b in the normal plane of a trajectory, the transformation ξ = Tη, where T = [t,n,b], orients the growth of the perturbation along the Lagrangian velocity. If the time-dependent terms are small compared to velocity gradients, the transformed system has the form
where A = [a1,T∗a2,T∗a3] (an asterisk denotes the transposed vector or matrix), and
We use the particular choice of n and b as in Rutherford and Dangelmayr (2010), with n oriented radially outward from u, and we require that n and b vary continuously along trajectories, but are otherwise not fixed by t. Examples of trajectories along with the continuously rotating coordinate frame are shown in Figure 1. The neglected terms in the approximation of (2) contain the explicit time derivatives of t, n and b. We justify the approximation since the change in t, n and b due to time variation (Figure 2(a)) is small compared to the total variation (Figure 2(b)). This approximation holds except in localized regions of strong convection, which tend to reside outside of the LEEI.
We see that the growth of the normal component in the direction outward from the Lagrangian velocity is given by Ψ22. This component captures only the hyperbolic separation of nearby trajectories originating in the same x–y plane, and the maximal ridge surface of this field is defined as the hyperbolic LEEI. The LEEI separates two distinct regions of trajectory behaviour. Inside the LEEI, trajectories experience little mixing, which can be seen as a small change in radius from the storm centre in trajectories (Figure 3). Outside the LEEI, trajectories experience far more change in distance from the storm centre, and more mixing which can be seen by filamentation. Thus the LEEI is the structure which separates the air in the eye with little mixing and the air outside the eye with strong mixing.
Shear is captured by a rotation of the normal subspace onto the Lagrangian velocity. We view the angle of rotation ϕ2, defined by
as a measure of shear. Maximal shear corresponds to values of ϕ2 near .
3. Ridge extraction
As in Rutherford and Dangelmayr (2010), the LEEI is identified with the maximal surface of Ψ22 that encloses the z-axis, and the maximal shear surface is identified with the maximal ridge surface of ϕ2. In this paper, the ridge extraction algorithm of Rutherford and Dangelmayr (2010) is extended in an automated manner across varying initial time, and at each z-level the ridge is smoothed by representing it in terms of a reduced set of Fourier descriptors. Since the data are rather noisy, the smoothing procedure combined with gradient climbing (see Rutherford and Dangelmayr, 2010), yields a ‘best-fit surface’.
At given z-levels, a ridge curve is computed by evolving an initial guess of an ordered set of points towards the ridge through gradient climbing. A change to our previous calculation is that the points are uniformly azimuthally distributed, and the gradient climbing is restricted to the radial direction. Let R = (Rx,Ry) be the initial guess. The Fourier descriptors of R are defined by FR = ℱ(Rx + iRy), where ℱ denotes the discrete Fourier transform (computed through the fast Fourier transform algorithm). The curve is smoothed by zeroing higher-order Fourier descriptors and mapping back to Cartesian coordinates via inverse Fourier transform. This yields a new ordered set of points R′, which is again evolved through radial gradient climbing. The algorithm for ridge extraction at a given z-level can be summarized as follows:
1.Choose an ordered set of azimuthally uniformly distributed points as an initial guess.
2.Evolve points through gradient climbing in the radial direction.
3.Apply a fast Fourier transform to the evolved points.
4.Set higher-order Fourier descriptors to zero and apply inverse fast Fourier transform.
5.Continue 2–4 until convergence is reached.
In our computation, we have used 70 points and kept 10 Fourier descriptors for Fourier inversion. Examples of the resulting ‘best-fit ridge curves’ are shown in Figure 4. The best-fit ridges on horizontal levels have average logΨ22 values of 2.6 near the sea surface, decreasing to 1 at upper levels. Near the sea surface, all of the ridge points have positive Lagrangian values, whereas at upper levels 73% of ridge points have positive logΨ22 values. The Lagrangian fields have values ranging from −4 to 4, with mean values slightly less than zero. From this, we conclude that a strong majority of approximated ridge points lie on the actual ridge. Since very few points have negative values, the points that are not directly on ridges are located close to the ridges and not in valleys, which would show negative logΨ22 values.
The advantage of this method is that the ridge points for varying z-levels have identical azimuthal distributions and so are stacked to form a ridge surface that can be represented in a matrix. This leads to easy visualization (see Figures 5 and 6) and the possibility to build a continuous dynamical model of the surface. In addition, at a given z-level, the method provides a centre given by the zero frequency Fourier descriptor, and higher-order Fourier descriptors provide information about the degree of asymmetry.
We note that the algorithm used here is a priori designed to find a closed ridge curve on each z-level. In our previous computation, without smoothing and gradient climbing restricted to the radial direction, some of the LEEI sections show gaps which allow transport between the eye and eyewall. The signature of those gaps can be found in the higher-wavenumber Fourier descriptors.
3.1. Continuation across z-levels and initial time
The ridge extraction on a fixed z-level described above is extended across varying initial times as follows:
1.For fixed initial time, the ridge curves are advanced from bottom to top by using the converged ridge from the previous computation as initial guess for the computation at the next z-level.
2.The bottom ridge curve computed at a given initial time is used as initial guess for the computation of the bottom ridge curve at the next initial time.
3.2. Areas and volume
The size of the eye is measured by the areas of the LEEI sections and by the total volume. The area, A, of the eye at a z-slice is computed from the closed ridge curve using Green's theorem, according to
where R(s) is arclength parametrization of the curve R = (Rx(s),Ry(s)). Given the areas at all z-levels, the volume of the eye, V, is approximated by
where N is the number of z-levels and An is the eye area at level z = nΔz. In our calculation, N = 40 and Δz = 250 m. Areas and volume as a function of initial time varying between 50 and 80 h are depicted in Figure 7. The volume of the eye contracts just before times of 60, 70, and 72 h, which are times when intensity increases (Figure 6(b)).
The relative contributions of the individual Fourier descriptors to the total energy provide information about the degree and structure of the asymmetries contained in the LEEI. In Figure 8 we show the contributions of the wavenumber 1 and 2 components as functions of initial time and z-level, which indicates that these two modes contain almost 90% of the energy. In particular, from t =70 to 80 h, a very high percentage of the energy is contained within the first two wavenumbers. A comparison to maximum tangential winds (Figure 9), and minimum pressure on z =1 km (Figure 10), shows that the higher contribution by the lower wavenumbers, and thus a vanishing of higher wavenumber asymmetries, is coincident with intensification. Shorter time intervals of dominant contributions by wavenumbers 1 and 2 occur at 54, 60 and 64 h, and the same times as sharp increases in intensity.
A comparison of volume and wavenumber suggests that the lower wavenumbers also show a stronger contribution as the size of the eye decreases. Thus axisymmetry is a more valid assumption of smaller eyes. The eye area on z-levels (Figure 7(a)) shows an inverse relationship to maximal winds on z-levels (Figure 9(a)). During periods of intensification, including when the storm is nearing peak intensity at t = 80 h, higher winds are present not just above the sea surface, but also up to 10 km, which suggests that even during periods of low asymmetry the intensification mechanisms are inherently 3D. At the same time, there is a notable contraction in the size of the eyewall, and the shape becomes more cylindrical as there is less variation of area between z-levels.
While the low-wavenumber asymmetries contribute to the primary shape and size of the eyewall, higher wavenumber asymmetries are associated with the tangle of LCSs that corresponds to enhanced mixing between the eye and eyewall. The declining contribution of higher-wavenumber asymmetries is consistent with a more resilient vortex as the model attains higher intensity and lower volume of the eye.
4. Dimensionally reduced model
The ridge extraction of section 3 produces a representation of the LEEI that is continuous in the azimuthal variable φ, and discrete in the vertical coordinate z. Setting c = x + iy, this representation can be written as
where ℱR,mn is the Fourier descriptor at level zn, (0 ≤ n ≤ N) for wavenumber m, and M = 35, N = 40 in our calculations. A continuous representation across z is achieved by matching (9) to vertical basis functions Bk(z), 0 ≤ k ≤ N, as
The condition that (10) coincides with (9) at z = zn yields a linear equation for the coefficient vectors cm = (cm0,,cmN):
which has a unique solution for each m and t if the basis functions are linearly independent.
We consider two choices for the vertical basis functions. The first choice is Fourier basis functions:
and the second choice is radial basis functions (RBFs):
where ϕ is any RBF and the dk are uniformly distributed centres in the vertical range. After experimenting with different RBFs, we found that the choice
with β = 20 works well, although the results are not very sensitive to variations in β.
The approximations of the first three Fourier descriptors at a fixed initial time as functions of z-level in terms of five and ten basis functions are shown in Figures 11 and 12 respectively, along with the relative errors in these approximations. Using 5 and 10 RBFs captures 90% and 95% of the information in the vertical variation. The weights |cmk|2 + |c−mk|2 for 10 RBFs are shown in Figures 13 and 14 as functions of t for m = 1 and m = 2 respectively, which indicate that most of the information is contained in the RBFs centred near the middle of the vertical domain. A noticeable spike in the RBF weights at mid levels near 67 h and lesser spikes, more visible in the second Fourier descriptor at 53, 59, and 62 h, are consistent with decreases in tangential winds. Although the tendency exists for the isolated spikes, there is little correlation between the RBF weights and maximal tangential wind time series. With 10 RBFs and three Fourier descriptors, our approach yields a 30-dimensional model for the dynamics of the LEEI.
The dynamics of a 3D Lagrangian eye–eyewall interface were studied by applying Lagrangian methods for its construction, and using methods of data reduction in the extraction and analysis of this structure. The LEEI was defined as a ridge of a Lagrangian field that acts as a barrier for particle transport and varies with initial time. The construction of a reduced-dimensional model using Fourier descriptors and appropriate vertical modes showed that the structure can be viewed as lower dimensional, both in the azimuthal and the vertical directions. Moreover, the amount of information contained within the lower-dimensional dataset characterizes the degree of axisymmetry in the model during a mature state, and the ‘shape-based coordinate’ eliminated the problem of a non-stationary vortex centre. The temporal evolution of the LEEI in both azimuth and height was shown to coincide with changes in intensity, and shows that the contraction of the LEEI, and thus the contraction of material within the eye, contributes to higher intensity. The combined importance of the vertical eyewall structure and radial asymmetries shows that intensification is a 3D process, but much of the information can be contained in a reduced-dimensional model. Further work in this direction will include a similar analysis for an intensifying storm, where the complexity increases due to the non-stationary evolution and interaction of mesovortices.
The authors would like to thank the editor and anonymous referees for their helpful comments, which improved the quality of this paper. The authors would also like to thank Michael Montgomery and John Persing for their helpful discussions. This work was supported by the National Science Foundation under NSF Cooperative Agreement ATM-0715426.