The interaction of a tropical disturbance with its environment is thought to play an important role in whether a disturbance will develop or not. Most developing disturbances are somewhat protected from the intrusion of environmental dry air at mid-levels. For African easterly wave (AEW) disturbances, the protective boundary is approximated by closed streamlines in the wave-relative frame, and their interior is called the wave pouch. The dynamic and thermodynamic processes of spin-up occur inside the pouch.

In this study, we define the kinematic boundaries for a non-AEW disturbance in the Bay of Campeche that originated along a sharp frontal boundary in a confluent region of low pressure. We examine these boundaries during the genesis of Hurricane Nate (2011) to show how a pouch boundary on isobaric levels in the Lagrangian frame may allow for some transport into the pouch along the frontal boundary while still protecting the innermost development region. This result illustrates a generic property of weakly unsteady flows, including the time-dependent critical layer of AEWs, that lateral exchange of air occurs along a segment of the boundary formed by the instantaneous, closed translating streamlines.

Transport in the Lagrangian frame is simplest when measured with respect to the stable and unstable manifolds of a hyperbolic trajectory, which are topologically invariant. In this framework, an exact analysis of vorticity transport identifies the primary source as the advection of vorticity through the entrainment and expulsion of bounded material regions called lobes. We also show how these Lagrangian boundaries impact the concentration of moisture, influence convection, and contribute to the pouch vertical structure.

The question of development versus non-development of tropical disturbances is a complex problem that has
seen significant interest yet has an inherently high amount of
unpredictability. There are many known factors that influence development,
such as sea surface temperatures, available moisture and vorticity, vertical
wind shear, and the timing and distribution of convection; see, e.g.,

As seen by

It is at the pouch boundary that the interaction of the proto-vortex with its
environment occurs, and transport of any air across the boundary alters the
physics within the pouch. When mixed into a vortex, dry air may quench
convection, reducing the total latent heat release and subsequent convergence,
thus reducing the rate of or preventing spin-up of the vortex; see

Permeability of the pouch boundary allows environmental air to enter a
disturbance, which may prevent development if enough dry air reaches the
circulation center, as was shown for Gaston (2010) by

A pouch whose boundary is open to transport on one side may also favor
development, as

In AEW flows, the pouch boundary enclosing a region of recirculation can be
seen by assuming a steady flow in the co-moving frame of the parent wave with

The vector

In this study, we consider the kinematic boundaries and their impermeability with respect to advection on constant pressure levels and three-dimensional non-advective fluxes for non-AEW disturbances that form along a frontal boundary. We show that the boundary limits the advection of environmental dry air to that contained within a single closed material region plus non-advective fluxes – those fluxes not proportional to horizontal velocities – along the boundary. We extend the wave critical layer theory and its associated translating critical points and manifolds for two reasons. First, there is no distinguished frame of reference provided by the AEW, and second, the time dependence of the flow causes trajectory paths to cross Eulerian streamlines in any translating or rotating frame.

A surface low formed along a frontal boundary in the Bay of Campeche on
6 September and the National Hurricane Center classified this disturbance as
a tropical storm on 7 September. The pre-Nate pouch was first identified by
the Montgomery Research Group on 6 September as P25L

The pre-Nate pouch was called P25L as the 25th pouch of the 2011 Atlantic season. This name is used to show the location of the storm for the remainder of this paper.

. One or more vorticity filaments extending across the Gulf of Mexico from the predecessor, Tropical Storm Lee, connected to Nate's region of formation. These filaments were associated with a strong horizontal gradient of water vapor orthogonal to the frontal boundary. An obvious question arises as to whether and to what extent the “anti-fuel” behind the frontal zone would affect Nate adversely. During the development of Nate, the frontal zone itself was deformed into a graceful S curve by the combined action of Lee and Nate at opposite ends of the frontal zone.Over the next few days, dry air to the north of the frontal boundary, with
ECMWF relative humidity values less than 20 % throughout the
mid-troposphere, was in close proximity to Nate, yet Nate was still able to
intensify. Since satellite visible imagery indicated that dry air remained
approximately 1

Sea surface temperature data are from the NCDC Optimum Interpolation Sea Surface Temperature (OISST) .25 resolution data set

greater than 29In this study, we examine the 2-D Lagrangian flow structures on isobaric surfaces from ECMWF model analyses to describe the transport of dry air and evolution of vorticity at mid-levels. The Lagrangian manifolds defined in the upcoming section indicate what flow features, including the remnants of Tropical Storm Lee, contributed to the circulation of Nate and measure the impact of dry air that Nate interacted with after genesis. The Lagrangian boundaries are also shown in relation to regions of convection to show that convection is typically located interior to Lagrangian boundaries.

The outline of the remainder of this paper is as follows. Section 2 provides an introduction to the mathematical methods for the location of Lagrangian boundaries. In Sect. 3, we show numerical details of the computations and data sets. In Sect. 4, we describe the genesis of Nate from the perspective of the evolution of Lagrangian manifolds. In Sect. 5, we provide a detailed description of the interaction of Nate with its environment, and show how the Lagrangian flow boundaries offer both protection from the outer environment and help to concentrate vorticity into the vortex core. Conclusions and a discussion of future work are provided in Sect. 6.

In generic time-dependent flows in a distinguished frame of reference, flow
boundaries are a set of distinguished material curves called the stable and
unstable manifolds of a hyperbolic trajectory; see, e.g.,

Due to the difference in the direction of time in the stable and unstable
cases, the manifolds may cross at points other than hyperbolic stagnation
points, forming material regions, called lobes, that are enclosed by multiple
manifold segments. The time evolution of lobes describes the transport of
material “across” the Eulerian boundary and is called lobe dynamics; see

Boundaries of physically important regions in time-dependent flows may be
formed by connected stable and unstable manifold segments that form an
enclosure called a separatrix

In idealized theoretical models of
nonlinear critical-layer flows, the cat-eye boundary is a separatrix of
total (wave plus mean) stream function inside of which absolute vorticity is
advected passively to leading order; see, e.g.,

In the case of Rossby-wave critical-layer flows, a cat-eye region of
recirculation is expected

A schematic of the time evolution of manifolds of hyperbolic trajectories

The change in system circulation due to the advection of each lobe may be
computed using Stokes' theorem along the lobe boundary.

Since the unstable manifold is attracting under a forward time integration,
tracer-like quantities such as equivalent potential temperature

In their mature stage, tropical storms exhibit an additional inner pouch
boundary as a Rankine-like vortex core with solid-body rotation isolated
from the pouch exterior by a ring of strong differential rotation (azimuthal
shearing); see

Diagram of lobe transport for a
time-dependent cat-eye flow where the contents of the lobes are transported
across the separatrix boundary formed by the unstable (blue and cyan) and
stable (red and magenta) manifolds of a pair of hyperbolic trajectories. The
lobes labeled

While the Lagrangian manifolds characterize the pouch as semi-permeable to
transport, whether these intrusions disrupt the internal processes is
determined by the existence of an additional boundary between the core and
the cat-eye boundary called a shear sheath. In most developing
disturbances, the region of high vorticity in the center behaves as a
finite-time Kolmogorov–Arnold–Moser (KAM) torus (see

For this study, we use operational ECMWF model analyses constructed at the
start of each assimilation cycle to initialize forecast models with
velocities and thermodynamic variables given on a regular
0.25

To compare the ability of the ECMWF analyses to correctly represent the Lagrangian topology with 6 h data, we also use Weather Research and Forecasting (WRF) model simulations at 10 km horizontal grid spacing with 10 min output intervals. The WRF simulations were initialized with 25 km ECMWF analyses. The ECMWF analyses were used to control the boundary conditions for the entire simulation.

Trajectory computations use isobaric velocities with bi-cubic interpolation in space and time. Manifold computations and Lagrangian scalar fields come from sets of particle trajectories, which are computed using a fourth-order Runge–Kutta method with an intermediate time step of 15 min for the ECMWF trajectories and, at the model, an output time step of 10 min for the WRF simulations to accurately represent the curvature of particle paths.

Manifolds in time-dependent flows require the location of a hyperbolic
trajectory and its local stable and unstable manifold segments. The initial
segment used to construct Lagrangian manifolds is typically a line segment
that straddles a hyperbolic trajectory. Hyperbolic trajectories are difficult
to locate prior to manifold computation since they require first identifying
a distinguished frame of reference and then locating quasi-steady saddle
points in that frame. Alternatively, material surfaces called Lagrangian
coherent structures (LCSs) may still be found which strongly influence nearby
trajectories. If LCSs can be located, manifold segments are initialized along
attracting LCSs found in a forward direction in time for the unstable
manifold segments and in a backward direction in time for the stable manifold
segments. Ridges of the finite-time Lyapunov exponent (FTLE) field have been
used for initial segment location for the polar vortex by

The unstable (yellow, blue) manifolds are overlaid on the ECMWF potential vorticity
(PV) field (K m

Manifolds are advected using the algorithms of

Tropical Storm Lee left the Gulf of Mexico on 3 September, making landfall in
Louisiana and traveling northeast. From 4 to 6 September, southerly flow in
the Gulf of Mexico guided moisture and remnant vorticity from Lee into the
Bay of Campeche where a frontal boundary between moist air to the south and
very dry air to the northeast was established. The potential vorticity (PV) field
showing the pre-Nate region and TS Lee is shown at 00:00 and 12:00 UTC on 5 September in Fig.

National Hurricane Center
Tropical Storm Nate Discussion Number 1.

We consider now the period of formation by analyzing the Lagrangian manifolds
from 00:00 UTC on 6 September to 00:00 UTC on 9 September to see the sources of
vorticity and the establishment of a pouch boundary as a barrier to very dry
air. The manifolds are overlaid on the

The stable (red, magenta) and unstable (blue, cyan) manifolds are
overlaid on the ECMWF PV field (K m

The unstable manifolds comprising the pouch boundary are attracting regions
that can actually be observed before 6 September. On 5 September, there are
two attracting lines in the confluent region from the Lee flow: one from the
southern side that is already elongated at 00:00 UTC and one from the
northern side that emerges as a short segment at 12:00 UTC. These lines are
shown as blue and yellow curves, respectively, in Fig.

The stable (red, magenta) and unstable (blue, cyan) manifolds are
overlaid on the latitude tracer field (degrees)

At 12:00 UTC on 5 September, the area of

The non-advective isobaric absolute vorticity vector differs from the advective vector and therefore can cross material contours.

. The circulation ofThe more complicated structure of the Lagrangian manifolds and their
additional intersections allowed the formation of lobes. The lobe

In addition to

The stable (magenta, red) and unstable (blue, cyan) manifolds from
the WRF simulation using non-uniform SSTs are overlaid on the vertical
vorticity (s

The Lagrangian boundaries are closely related to the transport of tracers as

Advected tracers form steep gradients purely from advective transport and can
be seen by plotting the initial value of the advected quantity at the final
location of the particle. Behavior similar to that of the other physical
tracers can be visualized by the latitude tracer field (conserving initial
latitude along trajectories), which shows the initial latitude of particles
(Fig.

We examine now whether the accumulation of moisture and confluent flow along
the unstable manifold impacts the location of convection. The 700 hPa stable
and unstable manifolds are overlaid on GOES shortwave infrared
3.9

The stable (magenta, red) and unstable (blue, cyan) manifolds from
the WRF simulation at the

The WRF model simulation is used to compare how temporal resolution, spatial
resolution, divergence, and varying versus constant SSTs affect the manifold
topology. Lobe transport is shown for the complimentary WRF simulation in
Fig.

Due to the existence of the frontal boundary and large moisture and
temperature gradient on the north side of the pouch, one may question whether
manifolds computed with isobaric velocities represent realistic particle
motions. The manifolds using isentropic velocities at the 315 K level,
representative of the

Using the full flow field on constant

The role of varying SSTs is investigated by an additional WRF simulation
using temporally constant SSTs at the initial time of 00:00 UTC on 6 September
so that the upwelling that occurred from Nate has no feedback into the
simulation. The manifolds at

The WRF simulations collectively demonstrate that varying SSTs and resolution
have little impact on manifold structure or on the contribution of the
manifolds to the circulation since the small filaments emanating from the
lobes contain very little circulation. The primary Lagrangian structures,

The circulation (m

We now consider the radial profiles of important kinematic quantities
including the strain rates, where radius is taken with respect to the
best-track storm location. Both OW and the sum of the squares of strain rates
are translation invariant quantities, so they do not depend on the choice of
translating Eulerian reference frame, while the strain rates depend
individually on the choice of coordinate system. We orient the coordinate
system along the direction given by the tangent to particle motion by the
transformed velocities

The time evolution of radial profiles for OW, shear strain, relative
humidity, and vorticity from the ECMWF data is summarized in
Fig.

The radial (degrees) profiles of OW (s

Backward trajectories provide additional details about the impermeability of
the inner core. Trajectories were seeded uniformly at 00:00 UTC on 9 September
within the inner core boundary defined as a circle 1

The 850 and 925 hPa trajectories are shown in Fig.

An idealized study by

The values of

We now examine the vertical structure of the pouch by identifying the
manifold structure on other levels. The manifolds are shown from 18:00 UTC on 6 September to 06:00 UTC on 8 September at 850 hPa (left column) and 500 hPa
(right column) in Fig.

At 500 hPa, the structure is very similar to the structure at 700 hPa, where the manifolds form a complete pouch boundary and allow only a small intrusion of dry air from the north that is contained within a lobe. A very similar structure (not shown) can be observed at 400 and 600 hPa, though it does not extend above 400 hPa.

At 850 hPa, the manifold structure differs from those found from 700 to 500 hPa in that the stable manifolds do not have additional intersections with the unstable manifolds other than at the locations of hyperbolic trajectories. As the manifolds evolve, the northern unstable manifold is entrained inward, leaving a large region of dry air that can enter the vortex. Lobe transport does not apply here and entrainment of dry air is not limited to the contents of the lobes but rather to the total flux through the open pathway.

At both 500 and 850 hPa, unstable manifolds are entrained into a limit cycle
of circular flow with no further entrainment, and like at 700 hPa, the
change of hyperbolic to shear stability of the manifold forms a shear sheath
that provides some protection to the inner core, defined kinematically as the
region with strong recirculation seen, e.g., by large positive OW values,
from the intruding dry air (Fig.

Stable (red, magenta) and unstable (blue, cyan) manifolds are
overlaid on the ECMWF

In this paper, we have explored how the rearrangement of Lagrangian flow boundaries may limit the dry air in the vicinity of a tropical disturbance from interacting with the disturbance. Hurricane Nate developed despite a region of very dry air in close proximity to the storm. While the storm-relative frame showed closed streamlines, the stable and unstable manifolds defined invariant regions called lobes that can transport intruding dry air into the pouch toward the storm center, but failed to penetrate the core of the proto-vortex. A shear sheath served to protect the center by maintaining a strongly deforming radial shear which, in turn, allowed vorticity concentration of the core to continue, undiluted with dry air and weaker vorticity.

We offered a dynamic view of the pouch for Nate that describes the entire storm evolution at fixed vertical (e.g., isobaric) levels based on the evolution of Lagrangian flow boundaries. In this view, we found that the Lagrangian pouch structure showed the sources of air that were advected into the pouch. We also showed that the advective fluxes of vorticity into the pouch can be measured by lobe transport and account for over half of the vorticity that Nate acquired. The transport that we see in this case is consistent with the radial profiles which showed the accumulation of the manifolds, an increasing tracer gradient, and a shear sheath that marks an additional transport boundary to the inner core.

The Lagrangian boundaries lead us to a material description of the transition
from a large-scale pouch boundary that blocks environmental dry air during
genesis to a much smaller vortex core that is present after genesis:

The frontal boundary rolls up and combines two air masses, one from each side of the frontal boundary.

During the roll-up, hyperbolic trajectories become detectable along the boundary, indicating the existence of stable and unstable manifolds.

The folding of manifolds near the hyperbolic trajectories implies the existence of additional intersection points of the manifolds that are not the hyperbolic trajectories, but these intersection points travel cyclonically toward the hyperbolic trajectory. The manifold segments between adjacent intersection points mark the lobe boundaries.

Within the pouch, wrapping of the unstable manifold reaches a limit cycle surrounding the inner core as convergence concentrates vorticity from the moist region into a vortex core.

In a competing process, additional entrainment of lobes allows the import of dry air towards, but not necessarily into, the core.

As vorticity is concentrated into the core, the unstable manifold lengthens, and the elongated manifolds
and lobes form a shear sheath barrier to transport of additional dry air into the core

In this case,
entrainment of manifolds forms a transport barrier at the edge of the inner core where the entrainment of
lobes is due to time dependence of velocities. The end result is similar to the effect of divergence in
steady flow studied in

There are two configurations of manifolds that describe the transport of dry air toward the storm center along the manifolds. At 700 and 500 hPa, lobe transport and a rearrangement of a separatrix allowed a region of dry air to enter the pouch. However, the dry air region was contained and did not penetrate the inner core boundary due to the presence of the shear sheath. At 850 hPa, the manifolds showed that there was a direct pathway of transport into the pouch that still reached a limit cycle before reaching the circulation center, and the pathway was wider than suggested by translating Eulerian streamlines. These two mechanisms for dry air transport compete with the aggregation of cyclonic vorticity. Lobe transport is a slower process which limits the amount of dry air entering the vortex, while an open pathway allows continual entrainment of dry air.

Based on companion numerical integrations of this case, the advection of manifolds is somewhat sensitive to SST, pressure level, integration time, and numerical model (details). While the fine-scale details of the manifolds may differ considerably, the differences between manifolds computed in different ways are confined to filaments that have little circulation and are homogenized. The conclusion, that dry air from the north of Nate entered in and corresponded to the transport of one lobe, while moist air that entered Nate was confined to another lobe, is robust.

Further study on the rate of entrainment versus the rate of vorticity aggregation in an idealized setting and in modeling studies will help clarify the role of dry air intrusions that are partially ingested into developing storms that are slowing but still allowing development. The techniques used here should be useful for those studies.

All of the manifolds computed in this study are available as a data set in the Supplement.

The authors declare that they have no conflict of interest.

The authors would like to thank Dave Ahijevych, Michael Riemer, and an anonymous referee for their comments on earlier versions of the manuscript. The authors would like to acknowledge NSF grants AGS-1313948, AGS-1439283, AGS-0733380 (now expired), and NASA grant NNG11PK021. The authors would also like to thank Gerard Kilroy and Roger Smith and the German Weather Service for providing us with the ECMWF global model data for this basic research investigation. Edited by: Heini Wernli Reviewed by: Dave Ahijevych, Michael Riemer, and one anonymous referee