ENVIRONMENTAL FORCING OF HURRICANE MOTION AND
INTENSIFICATION:
ARE THE BETA GYRES NORMAL MODES?
Principal Investigator:
H. E. Willoughby
Objective: To determine whether the streamfunction dipoles induced by the beta effect on
simulated hurricane-like vortices are normal modes or continuous-spectrum solutions.
Rationale: A substantial body of work (e.g. Willoughby 1992, hereafter W92) interprets the
linear asymmetry and motion of a shallow water barotropic vortex on a beta plane as the result of
excitation of a normal mode with nearly zero frequency. In these simulations, the dipole
asymmetry and vortex motion grow linearly with time, as one would expect for an oscillator
forced at its resonant frequency. If the beta effect is turned off, the unforced solution persists for
a long time. On the other hand, similar asymmetries in beta-plane calculations by Montgomery
and co-workers (Nicklas and Mongomery 1997, Montgomery et al. 1998, hereafter MMN)
appear to be continuous spectrum solutions. The gyre amplitude and vortex motion approach a
finite limit after a few days. The gyres rotate as a result of advection by the mean vortex flow.
They are subject to axisymmetrization and vortex filamentation through differential tangential
advection. Why are the results of these superficially similar calculations so different? What are
the true dynamics of the beta gyres and of the beta drift?
Method: The model of W92 can be modified to reproduce Montgomery's results. W92's
calculation is set in translating coordinates that track the axis of vortex rotation closely. The time
marching process uses an extrapolated vortex motion inherited from previous calculations and
updated periodically. Every five simulated minutes, the algorithm relocates the center,
interpolates the vortex position from the extrapolated track to the new center, and corrects the
vortex translation for use in future calculations. By contrast, MMN's calculation integrates the
equations in coordinates fixed to the earth. It periodically locates the center and repositions the
vortex to the new location. W92 accomplishes nearly all of the vortex translation during the time
marching; whereas MMN keeps the vortex at a fixed position and relocates it separately from the
time marching. Modification of W92's algorithm so that the extrapolated vortex center position
is replaced by the most recent past position makes the model behave similarly to MMN.
Accomplishment: Figure 1 shows that the different treatment of vortex motion leads to large
differences in the vortex path. The vortex in translating cylindrical coordinates accelerates
toward the northwest according to the normal mode paradigm. The vortex in repositioned
coordinates initially accelerates toward the northwest then slows and turns westward. It
eventually tracks just slightly to the south of due west as it continues to decelerate. Examination
of the potential vorticity fields (not shown) associated with these solutions offers clues to the
differences between them. In both cases nearly all of the asymmetric PV lies near the edge of the
vortex where both the axisymmetric swirling flow's advection and its radial gradient that causes
axisymmetrization are weak. Thus, relatively slow vortex-Rossby wave propagation can
maintain the gyres' position. Little energy is lost back to the symmetric vortex far from the
center. In the translating coordinate formulation, the wavenumber one PV near the center is very
nearly zero in a large region around the vortex center (Fig. 2). There, the balance in the PV
equation is between radial advection of the axisymmetric PV and the vortex motion---termed
"self advection." In repositioned coordinates there is a substantial wavenumber one PV gradient
around the vortex center so that the gyres are more subject to tangential advection and
axisymmetrization.
These differences are consistent with the wavenumber one energy balance. In translating
coordinates about half of the energy derived from beta effect forcing supports growth of the
gyres' amplitude. A quarter is lost to axisymmetrization and another quarter to radiation out of
the domain. In repositioned coordinates, the energy grows initially through beta forcing. Then
symmetrization reduces it dramatically. As the axis between the gyres and the motion rotate
south of west, the beta forcing, which supplies energy to the gyres in a cyclonic vortex when
outflow correlates with larger value of the Coriolis parameter, actually consumes some of the
energy as the outflow rotates to smaller values of f.
The repositioned coordinate formulation yields such different results because the self advection
balance in the vortex is not possible when all of the effect of the motion takes place during
interpolation to a new vortex position. As the time marching proceeds, unbalanced radial
advection creates an artificial asymmetry that is subject to tangential advection and
axisymmetrization. Although much of this asymmetry is eliminated during interpolation to a
new position, absence of the key motion term from the PV equation makes the difference
between a singular set of equation and an nonsingular one. The translating coordinate
formulation derives from the conventional fixed coordinate form of the Navier Stokes Equations
with few, easily justified approximations. I argue that this more accurate and consistent
formulation captures the true linear dynamics of a vortex on a beta plane.
Key References:
Montgomery, M. T., J. D. Moller, and C. T. Nicklas, 1998: Linear and Nonlinear Vortex Motion
in an Asymmetric Balance Shallow Water Model. J. Atmos. Sci., 55, (in press).
Nicklas, C. T., and M. T. Montgomery, 1997: Hurricane motion on a beta plane in an
axisymmetric balance model. Colorado State University, Department of Atmospheric Science,
Paper No. 610, 80 pp.
Willoughby, H. E. 1996: Linear motion of a shallow-water barotropic vortex as an initial-value
problem. J. Atmos. Sci., 49, 2015-2031.
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