Blake Rutherford NWRA Collaborators Michael Montgomery Tim Dunkerton Mark Boothe Pouch boundaries in unsteady flows The pouch model assumes a steady flow in a waverelative frame and the streamlines of the cats eye in that frame act as boundaries to protect the pouch ID: 802590
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Slide1
Lagrangian flow boundaries in developing tropical cyclones
Blake Rutherford, NWRA
Collaborators: Michael
Montgomery, Tim Dunkerton, Mark
Boothe
Slide2Pouch boundaries in unsteady flows
The pouch model assumes a steady flow in a wave-relative frame, and the streamlines of the cat’s 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.
Slide3Stable and Unstable manifolds
In 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.
Slide4Lagrangian
boundaries,
Vorticity
and moisture
transport
The
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.
Slide5Permeability of the pouch
Entrainment 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 Gaston
The lobe dynamics of Nate
Slide6Importance 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.
Slide7Importance of the Lagrangian boundaries
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.
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.
Slide8Accumulation of Unstable manifold near center
In 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.
Slide9Representing the inner pouch
The 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.
Slide10Characteristics of the
Lagrangian
OW field
The
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 cat’s 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)
Slide12Radial profiles of the
Lagrangian
OW field
As 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.
Slide13Radial profiles of the
Lagrangian
OW field
As 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.
Slide14Nadine merger sequence prior to a shear sheath
For 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.
Slide15Dynamic 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.
Slide16Conclusions:
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.