PowerPoint Presentation

Lagrangian flow boundaries in developing tropical cyclonesBlake Rutherford, NWRACollaborators: Michael Montgomery, Tim Dunkerton, Mark BoothePouch boundaries in unsteady flowsThe pouch model assumes a steady flow in a wave-relative frame, and the streamlines of the cats eye in that frame act as boundaries to protect the pouch. In the steady flow, these boundaries are the stable and unstable manifolds of a saddle point. Within the boundary, 3D processes amplify vorticity.Closed streamlines in any Eulerian frame are not closed to transport.

Stable and Unstable manifoldsIn a time-dependent flow, the stable and unstable manifolds of a hyperbolic trajectory separate flow regions.Since the manifolds move with the flow, and may intersect, there is additional freedom for entrainment. These manifolds show an accurate description of time-dependent transport at the pouch scale.

Lagrangian boundaries, Vorticity and moisture transportThe Lagrangian boundaries are in close proximity to tracer gradients, e.g. PV, equivalent potential temperature, or ozone.These boundaries give a better description of the dynamics, thermodynamics, and the coupling between them than steady flow approximations.Intrusions into a pouch can be defined by lobes. The advective change in circulation in the pouch is determined by the circulations of the exchanged lobesLikewise, nonadvective fluxes can be measured within this framework.

The stable and unstable manifolds are shown with a theta_e tracer field for Nadine at 12 UTC 8 Sept. Permeability of the pouchEntrainment into the pouch can occur in 2 ways:An open pathway to entrainment due to convergent flow or when two hyperbolic points are not connected by a streamline.Exchange of material through lobe dynamics caused by small oscillations.

The open pathway of GastonThe lobe dynamics of NateImportance of the Lagrangian boundaries in describing the permeability of the pouch.The Lagrangian boundaries offer a description of the permeability of the pouch when the Eulerian description is ambiguous. So far, there are 3 configurations where the use of these boundaries has given a more complete description of transport.Examples: Gaston (2010), An apparently closed Eulerian boundary was open in the Lagrangian flow allowing dry air with low vorticity to be entrained laterally. Nate (2011), Dry air from nearby had limited depth of entrainment and accumulated outside of center, and was described by lobe dynamics. Sandy (2012) and Edouard (2014), Lagrangian boundaries to the north gave protection but open boundaries to the south allowed air with high vorticity to be entrained, supplementing what was otherwise a weak wave.

The stable and unstable manifolds of Gaston at 700 mb indicate an open pouch though streamlines were closed.Importance of the Lagrangian boundariesThe Lagrangian boundaries offer a description of the permeability of the pouch when the Eulerian description is ambiguous. So far, there are 3 configurations where the use of these boundaries has given a more complete description of transport.Gaston (2010), An apparently closed Eulerian boundary was open in the Lagrangian flow allowing dry air with low vorticity to be entrained laterally.Nate (2011), Dry air from nearby had limited depth of entrainment and accumulated outside of center, and was described by lobe dynamics.Sandy (2012), Lagrangian boundaries to the north gave protection but open boundaries to the south allowed air with high vorticity to be entrained, supplementing what was otherwise a weak wave.The moist pathway during the genesis of Hurricane Sandy.

Accumulation of Unstable manifold near centerIn developing systems, the unstable manifold tends to accumulate near the center, becoming elongated, and making lobe dynamics impractical after sufficient lengthening. Inside of the accumulation region, a vortex core can be seen that is in nearly solid-body rotation (high OW values with very little deformation).

In addition, remnant manifolds from rotating convection accumulate outside the region of solid body rotation.

The accumulation of 2D and 3D manifolds creates a second inner pouch, the shear sheath, that protects the core.

Nate (2011): The unstable manifold (yellow) forms a limit cycle around the core.Manifolds from idealized MM5 simulation.Representing the inner pouchThe inner pouch is more easily seen by a Lagrangian scalar field. At a time t, a scalar is defined as the accumulated eigenvalue of the velocity gradient tensor along particle trajectories.

Since rotation dominated regions have negative eigenvalues and strain dominated regions have positive eigenvalues, we differentiate these two regions in a single field by

The integration is along particle trajectories, and gives a time-smoothing of velocities while retaining characteristics of the Lagrangian flow. An integration time of 72 hours is sufficient to resolve the inner and outer pouch boundaries.

Characteristics of the Lagrangian OW fieldThe Lagrangian OW field retains the solid-body rotation and deformational characterization of the flow from OW.The features that can be seen are:Vortex cores, seen as maximal circles.Shear sheaths, seen as minimal discs outside vortex cores.Outer pouch boundary, seen as minimal curves in a cats eye configuration.

The Lagrangian OW field as an indicator of tropical storm strength disturbances.The Lagrangian OW field shows both a maxima near center and a shear sheath around the periphery for tropical storms in ECMWF model analysis data at 700 hPa.

Nadine (2012)Lisa (2010)Oscar (2012)Radial profiles of the Lagrangian OW fieldAs a coordinate system, we use contours of a non-divergent streamfunction.The contour values are monotonic from the pouch boundary to the center. The shape of these contours is a good representation of the Lagrangian flow. Taking the radial profile in this coordinate, we see that outside center, the Lagrangian OW field has a disc with negative Lagrangian OW values.The radial profile of the flux normal to the contours shows a radial transport minimum at the location of the most negative Lagrangian OW value.

The shear sheath acts to dynamically protect the core.

Contours of the streamfunction (top) and radial profiles plotted against mean radius (bottom) for Nadine before genesis.Radial profiles of the Lagrangian OW fieldAs a coordinate system, we use contours of a non-divergent streamfunction.The contour values are monotonic from the pouch boundary to the center. The shape of these contours is a good representation of the Lagrangian flow. Taking the radial profile in this coordinate, we see that outside center, the Lagrangian OW field has a disc with negative Lagrangian OW values.The radial profile of the flux normal to the contours shows a radial transport minimum at the location of the most negative Lagrangian OW value.The shear sheath acts to dynamically protect the core. The shear sheath is present at the time of TS formation in almost every case.

Nadine Lagrangian OW field at time of TS formation.Radial profiles at time of TS formation.Nadine merger sequence prior to a shear sheathFor Nadine (2012), there were 3 pouches in a row before Nadine developed. A first merger occurred between Nadine and the trailing P25L that combined the circulations of the two pouches. After the merger, a shear sheath developed, and Nadine could not acquire the vorticity from P23L as it was wrapped around the periphery of the pouch.A shear sheath blocks the entrainment of dry air but may also limit what can enter the pouch.

Dynamic protection for a long lifecycle

The shear sheath appeared for Nadine near the time of genesis and remained in place during the extremely long lifecycle.

Conclusions:The Lagrangian boundaries describe the difference between a storm and its environment.

The concept of a layer-wise 2D boundary extends from the pouch scale down to the scale of the inner core.

The shear sheath for Nadine was present for its entire lifecycle.

The Lagrangian OW field and radial profiles are included as part of the MRG pouch products.