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Modeling Rossby Wave Breaking in the Southern Spring Stratosphere
ANIRBAN GUHA* AND CARLOS R. MECHOSO
Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
CELAL S. KONOR AND ROSS P. HEIKES
Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
(Manuscript received 17 March 2015, in final form 29 September 2015)
ABSTRACT
Rossby wave breaking (RWB) plays a central role in the evolution of stratospheric flows. The generation
and evolution of RWB is examined in the simple dynamical framework of a one-layer shallow-water system
on a sphere. The initial condition represents a realistic, zonally symmetric velocity profile corresponding to
the springtime southern stratosphere. Single zonal wavenumber Rossby waves, which are either stationary or
traveling zonally with realistic speeds, are superimposed on the initial velocity profile. Particular attention is
placed on the Lagrangian structures associated with RWB. The Lagrangian analysis is based on the calculation of trajectories and the application of a diagnostic tool known as the ‘‘M’’ function. Hyperbolic trajectories (HTs), produced by the transverse intersections of stable and unstable invariant manifolds, may yield
chaotic saddles in M. Previous studies associated HTs with ‘‘cat’s eyes’’ generated by planetary wave breaking
at the critical levels. HTs, and hence RWB, are found both outside and inside the stratospheric polar vortex
(SPV). Significant findings are as follows: (i) stationary forcing produces HTs only outside of the SPV and
(ii) eastward-traveling wave forcing can produce HTs both outside and inside of the SPV. In either case, HTs
appear at or near the critical latitudes. RWB was found to occur inside the SPV even when the forcing was
located completely outside. In all cases, the westerly jet remained impermeable throughout the simulations.
The results suggest that the HT inside the SPV observed by de la Cámara et al. during the southern spring 2005
was due to RWB of an eastward-traveling wave of wavenumber 1.
1. Introduction
The southern stratosphere during the winter and
spring seasons is characterized by a primarily westerly
flow that defines the polar night vortex [or stratospheric
polar vortex (SPV)], which gradually weakens with time
(Mechoso et al. 1988). This westerly flow is perturbed by
planetary-scale disturbances representing Rossby waves.
Stratospheric flow features observed during these seasons
are broadly consistent with the ideas of Charney and
Drazin (1961) and Matsuno (1970), according to which
only waves with very long wavelengths (typically zonal
* Current affiliation: Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India.
Corresponding author address: Dr. Anirban Guha, Department
of Mechanical Engineering, Indian Institute of Technology Kanpur, SL 109, Kanpur 208016 UP, India.
E-mail: anirbanguha.ubc@gmail.com
DOI: 10.1175/JAS-D-15-0088.1
Ó 2016 American Meteorological Society
wavenumbers 1–3) can propagate upward from the troposphere, provided the mean flow is westerly and not too
strong. Among the planetary-scale disturbances in the
southern stratosphere, the quasi-stationary wave of zonal
wavenumber 1 (S1 ) is the most apparent (Quintanar and
Mechoso 1995). Another prominent wave feature, especially in October, is an eastward-propagating Rossby
wave of zonal wavenumber 2 (T2 ) (Manney et al. 1991;
Hio and Yoden 2004). The planetary-scale waves induce
quasi-horizontal tracer advection, which is a trademark of
the winter and spring seasons in the stratosphere.
Rossby wave propagation on isentropic surfaces is
associated with reversible deformation of potential
vorticity (PV) contours. Rossby wave breaking (RWB),
which occurs when the wave amplitude becomes sufficiently large, is associated with irreversible deformation.
RWB in the ‘‘surfzone,’’ where material filaments are
pulled out of the SPV edge and mixed with the exterior
flow (McIntyre and Palmer 1983, 1984, 1985), sharpens
the PV gradient surrounding the SPV, making the SPV
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edge a barrier to horizontal transport of air parcels
(Juckes and McIntyre 1987).
The discovery of the Antarctic ozone hole, and the
great interest in understanding its generation and
evolution, motivated many researchers in the last few
decades to study tracer advection in the stratosphere.
Particular emphasis has been given on the permeability
of the SPV. In the southern spring stratosphere, the
transport barrier mentioned in the previous paragraph
is robust in the presence of waves with slower phase
speeds (Bowman 1996). The strong SPV retains its
material identity, and the air in its interior remains
chemically isolated from the rest of the stratosphere
(Juckes and McIntyre 1987; Haynes 2005). Therefore,
dynamical processes have important implications with
respect to the formation and dissipation of the ‘‘Antarctic ozone hole’’ (Shepherd 2007; de la Cámara
et al. 2012).
Significant advancements have been made in the understanding of isentropic transport in stratospheric flows
using Lagrangian methods along with the application of
concepts from dynamical systems theory (Pierrehumbert
1991a,b; Bowman 1993; Ngan and Shepherd 1997, 1999;
Mizuta and Yoden 2001; Koh and Legras 2002; Rypina
et al. 2007; Beron-Vera et al. 2010). Equivalent studies
have been performed in oceanography as well; see
Mendoza and Mancho (2010) and Olascoaga et al. (2013).
Recently, de la Cámara et al. (2012, 2013) examined the
trajectories of superpressure balloons released during the
springs of 2005 and 2010 from McMurdo, Antarctica, by
the Vorcore and Concordiasi components of the Stratéole project [see Hertzog et al. (2007) and Rabier et al.
(2010), respectively]. The balloons drifted in the lower
stratosphere for several months, providing approximations to quasi-isentropic fluid trajectories. For analysis of
balloon behavior, de la Cámara et al. (2012, 2013)
applied a global Lagrangian descriptor, ‘‘M’’ (Jiménez
Madrid and Mancho 2009; Mancho et al. 2013), to European Centre for Medium-Range Weather Forecasts
(ECMWF) interim reanalysis (ERA-Interim) velocity
data. Previous work by Mendoza and Mancho (2010)
have shown that M displays stable and unstable manifolds
from all possible hyperbolic trajectories (HTs, which are
the traces of material hyperbolic points) in the neighborhood of a region, without the need to identifying these
trajectories a priori. Using the descriptor M, de la Cámara
et al. (2013) identified HTs in the flow field. These HTs
were interpreted as evidence of Kelvin’s ‘‘cat’s eye’’
patterns (Stewartson 1977; Warn and Warn 1978). Cat’seyes form around critical layers, which develop in shear
flows at locations where the wave phase speed matches
the background velocity. These concepts are further discussed in section 3 below.
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Several numerical studies on RWB and associated
stratospheric transport have been performed using
shallow-water equations on a sphere using realistic initial and boundary conditions (Juckes 1989; Polvani et al.
1995; Ngan and Shepherd 1999; Rong and Waugh 2004).
However, these studies concentrated on the processes
and mechanisms at work outside the SPV (specifically,
the surfzone dynamics). By contrast, only a few studies
have addressed RWB inside the SPV, for example, see
Mizuta and Yoden (2001) and Nakamura and Plumb
(1994). These particular studies used either highly idealized velocity fields (Nakamura and Plumb 1994) or
implemented a dynamically consistent model (Mizuta
and Yoden 2001). Our objective in this paper is to systematically investigate RWB both outside and inside of
the SPV. Our motivation for studying RWB inside the
SPV is primarily derived from the recent work of de la
Cámara et al. (2013), who found evidence of HT inside
the SPV during the southern spring of 2005. De la
Cámara et al. (2012) also discovered that ‘‘lobes’’ associated with HTs (see Koh and Plumb 2000) outside and
inside of the SPV can intersect, thereby establishing
routes of transport across the SPV edge and challenging
its permeability. The authors conjectured that the HT
observed inside the SPV is also due to RWB, but they
did not determine the properties of the breaking wave
(i.e., its wavenumber and frequency). In addition, de la
Cámara et al. (2013) mentioned that associated transports may bring ozone-rich air from the vortex periphery
to its inside. We undertake this investigation of RWB
during southern winter/spring conditions in the simple
dynamical framework provided by a single-layer
shallow-water numerical model with a free surface on a
sphere. Tessellation of the domain is done using geodesic (icosahedral) grids, which avoid the ‘‘pole problem’’ of longitude–latitude grids. The model’s initial
conditions correspond to a realistic, zonally symmetric
balanced flow. The bottom topography provides a perturbation with a single zonal wavenumber, for which
different amplitude, phase speed, and meridional
structures are considered. Depending on the background flow and phase speed, critical latitudes can exist
both outside and inside the SPV. For analysis, we apply
the Lagrangian descriptor M to the model’s results and
search for HTs in the flow field. Finally, we associate the
HTs to cat’s-eyes and hence to RWB.
The outline of the paper is as follows. In section 2 we
describe the shallow-water model and present the prescribed initial conditions and bottom topography. In
section 3, we discuss the Lagrangian analysis performed
through offline integration of particle trajectories. We
also provide a brief description of the Lagrangian descriptor M and its implementation in our context. In
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section 4 we present the results of several simulations.
The cases examined can be broadly divided into two
categories: (i) stationary forcing and (ii) eastwardtraveling forcing. We perform a Lagrangian analysis
on the velocity data obtained from each test case. In
particular, we emphasize the contour plots of j=Mj, from
which HTs clearly emerge at the critical latitudes, signifying RWB. Finally, conclusions are drawn in section
5, and implications of our study with respect to observed
stratospheric features are discussed.
2. Shallow-water model
a. Model equations
An important technical difficulty in setting up a numerical study of the flow inside the polar vortex is the
need to resolve motions near or over the pole. The use of
spherical coordinates for discretization of the model’s
equations leads to the standard pole problem (Heikes
and Randall 1995a). This occurs because in a longitude–
latitude-based coordinate system with constant resolution in latitude, the distance between adjacent grid
points along a latitude circle continuously decreases
toward the poles. Thus, a very small time step must be
used in order to avoid linear computational instability.
For the present study we selected the shallow-water
system on the surface of a sphere described by Heikes
and Randall (1995a). This model is free from the pole
problem and provides quasi-homogeneous and quasiisotropic resolution over the entire sphere (Heikes and
Randall 1995a,b; Heikes et al. 2013). In the following we
give a brief model description that highlights recent
model updates.
The model’s prognostic variables are vorticity, (velocity) divergence, and fluid depth. The absolute vorticity za is predicted by
›za
5 2= (za y) ,
›t
(1)
^ = 3 y is relative vorticity, f is
where za [ z 1 f , z [ k
^ is the
the Coriolis parameter, t is time, y is velocity, and k
vertical unit vector. The (velocity) divergence d [ = y
is predicted by
›d
5 = (za =c) 1 J(za , x) 2 =2 [g(h 1 hb ) 1 K],
›t
(2)
^ (=a 3 =b) is
where c is the streamfunction; J(a, b) [ k
the Jacobian operator; x is the velocity potential;
=2 [ = = is the Laplacian operator; g is acceleration
due to gravity; h is fluid depth; hb is the height of the
bottom topography; and K is kinetic energy, defined as
follows:
FIG. 1. The icosahedral grid of the shallow-water model. For illustration purposes, we show G2 resolution yielding 162 grid cells
and approximately 1908-km grid distance. Dots and ‘‘P’’ indicate
cell centers and pentagons, respectively. Cell centers, walls, and
corners are marked with thin arrows. Normal velocities for the
walls of a hexagon are marked with thick arrows.
1
K [ [= (c=c) 2 c=2 c 1 = (x=x) 2 x=2 x] 1 J(c, x) .
2
(3)
The fluid depth is predicted by
›h
5 2= (hy) .
›t
(4)
The streamfunction and velocity potential are obtained
from the relative vorticity and divergence through
solving the following elliptic equations:
=2 c 5 z,
=2 x 5 d.
(5)
Finally, the velocity is obtained from the streamfunction
and velocity potential through
^ 3 =c 1 =x .
y[k
(6)
b. Discretization of equations on an icosahedral grid
The model equations are discretized on the hexagon–
pentagon icosahedral grid sketched in Fig. 1. The grid
generation, optimization, and performance of finitedifference operators are discussed by Heikes et al.
(2013). In the integrations performed for this study, we
use the tweaked G6 grid with 40 962 grid cells, yielding
approximately 120-km grid distance. Regardless of resolution, 12 of the cells are pentagons. Note that the
icosahedral grid system used here is not twisted. The
geographic poles are placed in diametrically opposite
pentagons— for example, the one marked with ‘‘P’’ in
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Fig. 1. Following the Z-grid convention introduced by
Randall (1994), the vorticity, divergence, and fluid
height are predicted at the cell centers (marked with
dots in Fig. 1). An important advantage of the Z grid
over the A and C grids is the exclusion of computational
modes. The model uses the multigrid solver described in
Heikes et al. (2013) to obtain the streamfunction and
velocity potential at the cell centers. The only velocity y
defined in the discrete system is the one normal to the
cell walls pointing from one cell center to the next one as
shown for a selected hexagon in Fig. 1. To obtain this
velocity, the streamfunction is interpolated to the cell
corners from cell centers using a quadratic interpolation
technique requiring information from six surrounding
cell centers. The absolute vorticity flux za y and mass flux
hy are computed at the cell walls using a third-order
upstream finite-difference scheme.
c. Initial conditions
The initial wind profile is zonally symmetric and is
prescribed as follows:
f 1 C2
u(f) 5 C1 cos(f) sech
C3
C 2f
.
2 C4 cos(f) sech 5
C6
(7)
Here C1 5 118 m s 21 , C2 5 1:09 rad, C3 5 0:185, C4 5
20 m s21, C5 5 0:5 rad, C6 5 0:7, and f is latitude (rad).
This velocity profile yields a maximum jet speed of
umax 5 55:4 m s21 (at approximately 608S), and a zerowind line at 288S; see Fig. 2a. The velocity profile represented by Eq. (7) is similar to the one used by Ngan
and Shepherd (1999); however, the constants are suitably chosen so as to mimic the zonal-mean zonal wind
distribution at the 475-K isentropic surface on 17 September 2005. The choice is representative of the zonal
flow in the lower stratosphere during the Vorcore
campaign.
The initial height field is obtained by using the gradient wind balance equation,
h(f) 5 h0 2
ðf
Ru(f0 )
tan(f0 )
f1
u(f0 ) df0 , (8)
g
R
where h0 5 5985 m is the free-surface height at the South
Pole and R is Earth’s radius. Following Polvani et al.
(1995) the average shallow-water height is chosen to be
8 km. Physically this height represents the vertical scale
of the Rossby waves that would be produced when the
flow is perturbed (hence this height does not necessarily
correspond to the actual height of the 475-K isentropic
surface). Figure 2b shows the variation of h with latitude.
FIG. 2. (a) Initial zonal wind profile. The gray dashed line corresponds to the zonal-mean zonal wind distribution on 17 Sep 2005
at the 475-K isentropic surface. The thick solid black line, given by
Eq. (7), provides a simple analytical representation of the real
profile. Thin solid black lines are Rossby wave phase speeds
[computed from Eq. (10)] for different simulation test cases; see
Table 1. The intersection of thin and thick solid lines occurs at
critical latitudes, marked by open circles. The dashed–dotted gray
line below represents the latitudinal variation of the topography
given by Eq. (9). (b) The solid line represents the latitudinal variation of the initial shallow-water height, which is obtained from Eq.
(8). The dashed line is the mean shallow-water height.
d. Topography
The height of the bottom topography is prescribed by
#
" #
f 2 f0 2
cos(f)
exp 2
hb 5 2B
cos(f0 )
Df
t
cos(kl 2 vt) ,
3 1 2 exp 2
tS
"
(9)
where B 5 392 m (it corresponds to a 400-m-high
mountain), f0 5 2p/3 (unless mentioned otherwise),
Df 5 p/18, tS 5 10 days, l is longitude (rad), k is a
nondimensional zonal wavenumber, and v is frequency
(day21). The value of B for the given shallow-water
height is chosen such that the forcing amplitude is relatively weak. Stronger forcing simply accelerates the
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TABLE 1. Simulation test cases. The values of fcrit are predicted from linear theory.
Case
Label
k
f0 (8)
T (days)
finside
(8)
crit
foutside
(8)
crit
Stationary wave 1
Stationary wave 2
Stationary wave 3
Traveling wave 1
Traveling wave 2
Traveling wave 3
Traveling wave 2 (fast)
Traveling wave 2 (fast, different f0 )
S1
S2
S3
T1 (7:72)
T2 (7:72)
T3 (7:72)
T2 (3:86)
T^2 (3:86)
1
2
3
1
2
3
2
2
260
260
260
260
260
260
260
245
‘
‘
‘
7.72
7.72
7.72
3.86
3.86
—
—
—
275
283
287
275
275
228
228
228
250
243
239
250
250
evolution of the events. Our preference is to stay close to
linearity and delay the appearance of chaotic flow.
The phase speed of Rossby waves induced by this
topography is given as follows:
c(f) 5
vR cos(f)
.
k
(10)
Our choices of v and k are given in Table 1. We select
k 5 1, 2, and 3 because these are the zonal wavenumbers
typically observed in the winter/springtime stratosphere.
The point(s) of intersection of the u(f) profile given by
Eq. (7) and the c(f) profiles given by Eq. (10) for the
selected values of v and k, respectively, correspond to
the critical latitudes; see Fig. 2a. There is only one critical latitude when the wave is stationary, and two critical
latitudes when the wave is eastward traveling at a speed
of 0 , c , umax . Therefore, we expect RWB outside of
the SPV in the stationary forcing cases, and both outside
and inside of the SPV when the wave is traveling. The
simulation cases listed in Table 1 are integrated for
60 days, except for one (S1 ), which is integrated for
70 days. In all cases the model’s time step is 1 min and
the output is saved every 1 h.
3. Lagrangian analysis
The trajectory of the tracer parcels are calculated as
follows:
dx(t)
5 y(x, t) ,
dt
(11)
where the velocity field is provided by the output of the
shallow-water model, which is available hourly. For
these calculations, the velocity data are interpolated to
parcel locations using a bicubic spline interpolation
scheme. Parcels march forward in time using a fourthorder Runge–Kutta scheme, with a time step of 1 h. The
results obtained are largely insensitive to halving the
time step.
The Lagrangian descriptor M is defined as follows
(Jiménez Madrid and Mancho 2009):
Mt (x0 , t0 ) 5
ð t0 1t
t0 2t
jy[x(t), t]j dt .
(12)
Here x0 5 x(t0 ) is the location of a parcel at t 5 t0 . Thus,
the values of M correspond to the length of the trajectories in the phase space (which, in this case, is same as
the physical space) passing through x0 at t 5 t0 over the
time interval [t0 2 t, t0 1 t]. Therefore, calculation of M
implies integration of backward trajectories from t0 2 t
to t0 and forward trajectories from t0 to t0 1 t. The
function M, as defined in Eq. (12), provides a global
dynamical picture of any arbitrary time-dependent flow.
As noted in Mancho et al. (2013), convergence of the
structure of M toward the stable and unstable manifolds
requires sufficiently large values of t. We have chosen
t 5 15 days for stationary forcing cases and t 5 20 days
when forced via traveling waves. It is important to note
here that M depends on t. For instance, for low t the
appearance of M is rather simple, almost without
structure and resembling that of Eulerian currents. For
larger t, the structure of M becomes more and more
refined and more details of the invariant manifolds
are observed. This is similar to what is observed in
Lyapunov exponents or in the direct calculation of the
manifolds: the longer the integration period, the more
detailed and complex is the structure of the manifold
(Mendoza and Mancho 2010, 2012; Mancho et al. 2013;
Lopesino et al. 2015).
The chaotic saddles in M portray the transverse intersections of stable and unstable invariant manifolds,
which provide the evidence of HTs (see Fig. 2 of de la
Cámara et al. 2013). Proving the existence of chaos in
the current setting is beyond the scope of this article.
However. we have noted similarities between the expected evolution of trajectories within the domains
bounded by transverse intersections (in our numerical
experiments) and those of the chaotic invariant set obtained from the ‘‘Smale horseshoe map’’ [see Mendoza
and Mancho (2010), Mancho et al. (2013), and Mendoza
and Mancho (2012) for details on a rigorous proof
for the existence of chaos on autonomous and nonautonomous maps]. These analogies suggest the presence
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FIG. 3. PV contours for (a) S1 on day 50, (b) S2 on the day 25, and (c) S3 on day 30.
of chaotic advection, which occurs in temporally aperiodic systems when well-defined heteroclinic structures
associated with aperiodic HTs exist in the flow field.
Ngan and Shepherd (1999) proposed a necessary condition for the existence of chaotic advection in dynamical flows—there should be an ‘‘organizing structure’’
(which includes hyperbolic points) in the flow field and
that should be robust. These structures are also known as
Lagrangian coherent structures (LCS) in the literature—
for example, see Beron-Vera et al. (2010) and references
therein.
De la Cámara et al. (2013) suggested that chaotic
saddles in M, and hence HTs, are representative of cat’seye LCSs associated with planetary wave breaking at the
critical levels. The flow structures referred to as cat’seyes form around ‘‘critical layers,’’ which develop in
shear flows at locations where the wave phase speed
matches the background velocity. In a reference frame
traveling with the Rossby wave phase speed, the flow
near a critical layer exhibits Kelvin’s cat’s-eye patterns
(Stewartson 1977; Warn and Warn 1978). Hyperbolic
points are at the locations where the cats’ eyelids meet.
Perturbation of the cat’s-eyes results in irreversible deformation of material contours, signifying RWB. This
can lead to chaotic advection of tracers, which can be
understood using Hamiltonian chaos theory (Pierrehumbert
1991a,b; Wiggins 1992; del-Castillo-Negrete and Morrison
1993; Samelson and Wiggins 2006; Rypina et al. 2007).
4. Results and discussion
a. Critical layer outside the SPV
We start by considering stationary forcing cases with
k 5 1, 2, and 3. These cases are referred to as S1 , S2 , and
S3 , respectively; see Table 1. According to the predictions of linear theory, stationary forcing results in the
formation of a critical layer at the zero-wind line. For the
background zonal flow prescribed in Eq. (7), the zerowind line is located in the subtropics at 288S (see Fig. 2a)
and is therefore far outside the SPV (note that the SPV
extends up to ’608S). Figure 3 shows contour plots of
PV (equal to za /h) for each of the three cases considered.
Each panel in Fig. 3 displays anticyclonic features
around the critical layer (the surfzone), signifying RWB.
Initially, the number of anticyclones (or cat’s-eyes) is
equal to the k corresponding to the forcing. As the
simulation progresses, secondary cat’s-eyes may appear,
for example, see Fig. 3a. Another outstanding feature is
the deformation of the SPV, which is most prominent for
the S3 case.
The time evolution of the tracer field corresponding to
each k is shown in Fig. 4. Although significant latitudinal
transport is observed in the surfzone, there is virtually
no such transport across the SPV boundary. This observation is in agreement with a large body of literature
that examined the impermeability of the SPV boundary
under stationary (especially k 5 1) forcing, for example,
Juckes (1989), Polvani et al. (1995), and Ngan and
Shepherd (1999). RWB and the appearance of a complex surfzone occur earlier in time as k is increased (note
that the forcing amplitude B has been held constant in
all cases). Merging of secondary and primary cat’s-eyes
is also observed—see day 31 and day 40 plots for the S2
case in Fig. 4b.
We next apply the Lagrangian descriptor M discussed
in section 3 to visualize the invariant manifolds and locate the HTs. In the following, we have opted to plot
j=Mj instead of M as in the previous papers. The term
=M approximately aligns with the manifolds, and its
magnitude provides direct information on the regions
where M has strong horizontal variations. In this context, HTs are captured by the intersection of contour
lines where j=Mj has local maxima. Figure 5 shows
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GUHA ET AL.
399
FIG. 4. Tracer evolution for (a) S1 , (b) S2 , and (c) S3 . Color contours represent the initial latitudinal position of tracers. The day corresponding to each plot appears to its left.
contours of j=Mj for S1 , S2 , and S3 . The HTs in these
cases are labeled ‘‘H’’ in the different panels of the figure. As expected, these HTs are close to the zero-wind
lines, and their locations are practically independent of
time for a few days. Noting that our flow is fully nonlinear, deviations from stationary conditions in space
predicted by linear theory can be expected. The latitude
f at which HTs appear correspond to c ’ u, and the
number of HTs corresponds to the zonal wavenumber of
the perturbation k. Hence, the frequency v can be calculated from Eq. (10).
Secondary cat’s-eyes may form during the later stages
of integration, producing additional or secondary HTs.
These are labeled ‘‘H’’ in gray in Figs. 5a and 5b.
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FIG. 5. The j=Mj contours for (a) S1 with t0 5 day 50, (b) S2 with t0 5 day 25, and (c) S3 with t0 5 day 30. The black ‘‘H’’ denotes HTs from
the primary cat’s-eyes, while the gray one denotes HTs from the secondary cat’s-eyes.
Primary HTs are related to stronger gradients in M;
hence, they are defined by darker shades.
b. Critical layers both outside and inside the SPV
To examine these cases, the model is forced with
eastward-traveling Rossby waves by setting v . 0 in Eq.
(9). Similar to the stationary case, we take k 5 1, 2, and 3
but choose v corresponding to a period of 7.72 days. The
justification for this choice is that waves with zonal
wavenumbers 1–3 and a period of 7–8 days have been
observed in the Southern Hemisphere during winter/
springtime (Hio and Yoden 2004). We refer to these
cases as T1 (7:72), T2 (7:72), and T3 (7:72); see Table 1.
According to Fig. 2a, the c profile for each of these cases
intersects the u velocity profile at two locations. Hence,
there are two critical layers, one outside and one inside
the SPV. The outer critical layer is located in the midlatitudes, as shown in Fig. 6. The signature of the inner
critical layer is the deformation of PV contours, which is
prominent in Figs. 6a and 6b but not in Fig. 6c. We find
that the critical layer(s) for traveling disturbances appear(s) to be weaker/thinner than that of stationary
disturbances. The narrow width of the cat’s-eyes for
traveling waves is a direct consequence of the nature of
the streamfunction created by superimposing a wave
on a background shear. The width of a cat’s-eye in the
steady case is proportional to the amplitude of the wave
and the value of the shear. In the unsteady case, the
width of the chaotic layer will depend on the size of the
unperturbed cat’s-eye, as well as on the amplitude and
nature of the transient parts of the flow. These findings
are in agreement with the observations of Ngan and
Shepherd (1999).
From the tracer plots shown in Fig. 7, it is difficult to
discern whether the clearly seen flow structures represent cat’s-eyes. For proper visualization, one should
FIG. 6. PV contours for (a) T1 (7:72) on day 30, (b) T2 (7:72) on day 30, and (c) T3 (7:72) on day 25.
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FIG. 7. Tracer evolution for (a) T1 (7:72), (b) T2 (7:72), and (c) T3 (7:72). Plot properties are similar to that of Fig. 4.
work in a comoving reference frame in which the wave
appears to be stationary. Structures inside the SPV are
not visible in the T3 case as shown in both Figs. 6c and 7c.
One possible reason for this absence is that the critical
latitude in this case is inside the SPV finside
crit , which is at
about 878S. Therefore, the effect of topographic forcing
is considerably
(which is centered near 608S) at finside
crit
weak, and it does not lead to RWB. Another possibility
is that the model’s horizontal resolution is insufficient at
these high latitudes.
Similar to the stationary forcing case, we apply j=Mj
to visualize invariant manifolds, especially HTs. The
presence of HTs will indicate whether the observed
structures are indeed LCSs, and therefore signify RWB.
In Fig. 8a, HTs clearly emerge both outside and inside
the SPV. In other words, the ‘‘8-day wave’’ of zonal
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JOURNAL OF THE ATMOSPHERIC SCIENCES
VOLUME 73
FIG. 8. The j=Mj contours for (a) T1 (7:72) with t0 5 day 30, (b) T2 (7:72) with t0 5 day 30, and (c) T3 (7:72) with t0 5 day 25.
wavenumber 1 breaks both inside and outside of
the SPV.
Unlike the T1 case, HTs are not visible inside in the
other two cases; see Figs. 8b and 8c. It is important to
note here that structures are observed in the PV field for
the T2 case. The critical latitude in both T2 and T3 are
close to the South Pole (see Table 1). For T2 case we
believe those structures to be LCSs but were unable to
obtain HTs. We expect that a higher-resolution simulation and a larger value of t might reveal HTs in this
case. However it is unlikely for the T3 case to have HTs
inside the SPV.
To visualize HTs for a higher wavenumber, we performed another simulation with k 5 2 and a period of
3.86 days. This case is referred to as T2 (3:86) in Table 1,
to occur at 758S. Physand linear theory predicts finside
crit
ically, T2 (3:86) corresponds to the ‘‘4-day wave’’ seen in
the southern stratosphere during wintertime (Mizuta
and Yoden 2001). Figure 9a reveals four HTs, two outside and two inside the SPV.
We also carried out a model integration with the
mountain (nearly) centered at 458S. This case is referred
to as T^2 (3:86) in Table 1. Although the mountain in this
case is (almost) completely outside the SPV, the forcing
is strong enough to deform its periphery and trigger
Rossby waves inside the vortex. Evidence of HTs in the
T^2 (3:86) case is given in Fig. 9b. We find that there are
four primary HTs, of which two are outside and two are
FIG. 9. The j=Mj contours on t0 5 day 25 for the cases (a) T2 (3:86) and (b) T^2 (3:86). The black ‘‘H’’ denotes HTs
from the primary cat’s eyes, while the gray one denotes HTs from the secondary cat’s-eyes.
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GUHA ET AL.
FIG. 10. HTs outside and inside the SPV for the case T1 (7:72).
Successive markers are 1 day apart.
inside the SPV. Although Figs. 9a and 9b have been
plotted for the same t0 , the outcome in each case is
substantially different from the other. There are also
two secondary HTs (labeled with gray ‘‘H’’) outside for
the T^2 (3:86) case, as indicated by the grayscale shading
in the figure. Since the peak amplitude of the forcing is at
, secondary cat’s-eyes are easily
458S and close to foutside
crit
formed. On the other hand, the inner critical layer is far
from the location predicted by linear theory. This is
probably due to the nonlinearity in the background flow,
which makes the velocity field less zonal with time, especially at high latitudes.
Figures 8a and 9a resemble Fig. 10 of Mizuta and
Yoden (2001). Note that the procedure for visualizing
HTs described by these authors requires a corotating
frame of reference in which the wave appears to be
stationary. Although this is a standard technique for
visualizing the breaking of a single wave in a Cartesian
system [e.g., see Samelson and Wiggins (2006) and references therein], implementing it in a realistic scenario
is challenging. It is unclear at which phase speed the
frame should rotate when multiple waves are present.
Also, unlike the b-plane approximation, phase speed in
the spherical coordinate system is a function of latitude
according to Eq. (10). The function M, by contrast, is
frame independent and does not require a priori
knowledge of the underlying dynamical system.
HTs can be directly used to obtain the phase speed
and frequency of the breaking Rossby waves. As an
example, we have shown the HTs of T1 (7:72) in Fig. 10.
The figure shows that during the period investigated, the
outer HT is located very close to 508S, while the inner
HT is located near 758S. These two latitudes are foutside
crit
and finside
crit , respectively, predicted by the linear critical
layer theory (see Table 1), which gives an excellent
prediction in this case. Two successive markers along
each HT are 1 day apart; hence, phase speed can be
easily evaluated. It is important to note here that
T1 (7:72) is an ideal case in which the phase speed can
403
also be well approximated from the background zonal
flow (c 5 u at the critical layer, where u is the zonalmean zonal wind). In realistic flows, neither the background flow nor the Rossby wave phase speed is strictly
zonal. Even in our idealized simulations, we found a large
departure from the linear theory predictions for RWB
inside the SPV; see case T^2 (3:86). Therefore, HTs captured using M provide an accurate, frame-independent
way for evaluating the phase speed of a breaking Rossby
wave. Once the phase speed is known, the frequency can
be obtained from Eq. (10).1 We confirmed that the wave
period obtained from each HT is approximately 7.72 days.
Although forcing via traveling waves can eventually
produce chaotic regions both outside and inside of the
SPV, in our simulations these regions are always separated by the westerly jet. The jet acts like an effective
barrier to lateral transport and remains impermeable for
the entire length of our simulations. Similar behavior
was also observed by Rypina et al. (2007), who studied
the permeability of a Bickley jet. The fact that the jet is a
strong transport barrier was explained by Kolmogorov–
Arnold–Moser (KAM) theory—the jet is composed of
an invariant KAM tori (Rypina et al. 2007; Beron-Vera
et al. 2010). We are currently performing several analyses in order to gain insight into these issues, which may
require a more complex framework for research.
5. Summary and conclusions
Recently de la Cámara et al. (2013), on the basis of
results obtained with reanalysis products for the southern winter/spring of 2005, found HTs both outside and
inside the SPV (see Fig. 2 in their paper). On the basis of
linear critical layer theory, they argued that such flow
structures are indicative of RWB. The present paper
exploits this argument in an idealized framework, in
which a circumpolar vortex with a realistic velocity
profile was forced by perturbations with a single zonal
wavenumber. Our methodology was based on a spherical single-layer shallow-water model with free surface
in a geodesic (icosahedral) grid system. Numerical instabilities at the poles (known as the ‘‘pole problem’’) do
not occur when such a grid system (see Fig. 1) is used.
The initial velocity profile, given by Eq. (7), approximated the zonal flow in the lower stratosphere on a
given day in the southern spring of 2005. The
1
Secondary cat’s-eyes, if present, can produce a higher number
of HTs. In a real scenario, k is unknown,and is found by counting
HTs. This may lead to an erroneous determination of v, which can
be avoided by performing two checks: (i) the value of c is close to u
and (ii) the HTs are related to the intersection of darker j=Mj
contours.
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JOURNAL OF THE ATMOSPHERIC SCIENCES
perturbations were provided by a bottom topography
consisting of a single wavenumber in the zonal direction
and a (nearly) Gaussian profile in the meridional direction [see Eq. (9)]. The perturbations at the model’s
lower boundary had the longest planetary waves
(k 5 1–3), which could be either stationary or eastward
traveling. With this system we performed a systematic
series of numerical experiments (see Table 1).
First we focused on cases in which the forcing was
stationary. In all of these cases, critical layers develop in
the tropics at around 288S—that is, far outside the SPV
edge that is approximately at 608S. Three outstanding
features were observed: (i) RWB around the critical
layer leads to the formation of ‘‘cat’s eyes’’, which are
LCSs signifying chaotic advection (see Fig. 4). (ii) Initially the number of HTs corresponded to the zonal
wavenumber with which the system was forced. At later
times, however, secondary cat’s-eyes appeared in the
flow, which led to the formation of additional HTs. The
j=Mj plots in Fig. 5 clearly depicted all these features.
(iii) Forcing could distort the SPV, and the distortion
bore the signature of the forcing wavenumber. This was
particularly true whenever k . 2. For example, a triangularshaped SPV was observed in the S3 case; see Fig. 3c.
Next, we turned to eastward-traveling cases. Unlike
the stationary forcing cases, two critical layers were
formed, one outside the SPV (in the midlatitudes) and
the other inside of it; see Fig. 2a. Based on observation
data, we chose the forcing frequency to correspond to a
period of approximately 8 days. These cases were given
by T1(7.72), T2(7.72), and T3(7.72) in Table 1.
The significant findings in this regard are as follows: (i)
Typical cat’s-eye structures were not detected in either
PV plots (Fig. 6) or tracer plots (Fig. 7). This is because
appropriate visualization of such structures requires the
observer to be in a comoving reference frame where the
breaking wave appears to be stationary. Nevertheless,
structures were seen outside, and in some cases, inside
the SPV as well. Lagrangian analysis was performed to
search for HTs in order to confirm that these structures
were indeed cat’s-eye LCSs. (ii) Plots of j=Mj in Fig. 8
showed HTs outside the SPV in all cases but inside the
SPV only for T1 (7:72). Thus, we concluded that the
8-day wave with k 5 1 breaks both outside and inside of
the SPV. (iii) In the T3 (7:72) case, RWB did not occur
inside the SPV because the location of the critical latitude was very close to the South Pole, where wave amplitude is very small and model resolution may not
suffice. Relatively coarse resolution and smaller length
of integration t probably contributed to the difficulties
in not observing HTs for the T2 (7:72) case.
We made additional numerical experiments with
traveling waves by either changing the wave period or
VOLUME 73
the location of the peak of the bottom topography. Our
findings are as follows: (iv) By forcing with a (nearly)
4-day wave of k 5 2 [case T2 (3:86) in Table 1], we were
able to observe HTs both outside and inside of the SPV;
see Fig. 9a. (v) By making f0 5 458S in Eq. (9), the location of the mountain was made outside of the SPV.
Without changing the mountain amplitude, we were
able to generate HTs both outside and inside of the SPV
as shown in Fig. 9b. Thus, a traveling perturbation localized outside the SPV could still produce RWB inside
of it.
In both stationary and traveling forcing cases, the
Lagrangian descriptor M was found to be a powerful
diagnostic tool for RWB. In fact, wave properties like
frequency, wavenumber, and phase speed could be
evaluated from the HTs; see Fig. 10.
In all simulations, we found the westerly jet to be an
impermeable transport barrier. Even though RWB and
chaotic transport might occur on both sides of the jet,
strong KAM stability makes it impermeable. This result
is in agreement with the findings of Rypina et al. (2007).
Transport across the jet, as observed by de la Cámara
et al. (2013), might be the result of more complex events,
which need to be investigated in the future.
Finally, we conclude that the HT inside the SPV observed during early spring of 2005 by de la Cámara et al.
(2013) was due to the breaking of an eastward-traveling
Rossby wave of the zonal wavenumber 1. A more detailed justification for this assertion will be given in
future work.
Acknowledgments. We are grateful to Alvaro de la
Cámara and Ana Mancho for the useful discussions,
and their generous help regarding the ‘‘M’’ function.
We also thank the anonymous referees for their helpful suggestions and constructive comments. Anirban
Guha and Carlos R. Mechoso were supported by the
U.S. NSF Grant AGS-1245069. Anirban Guha is also
grateful to the initiation grant provided by IIT Kanpur
(IITK/ME/2014338) for partial support of this research work. Celal S. Konor and Ross P. Heikes were
partially funded by the U.S. DOE under Cooperative
Agreement DE-FC02-06ER64302 to Colorado State
University and the DOE under Grant DE-SC07050.
Also, Celal S. Konor was partially funded by the NSF
Grant AGS-1062468.
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