NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-265-2017A simple kinematic model for the Lagrangian description of relevant nonlinear processes in the stratospheric polar vortexGarcía-GarridoVíctor JoséCurbeloJezabelMechosoCarlos RobertoManchoAna Maríaa.m.mancho@icmat.esWigginsStephenInstituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, C/ Nicolás Cabrera 15, Campus de Cantoblanco UAM, 28049 Madrid, SpainDepartamento de Matemáticas, Facultad de Ciencias, Universidad Autonóma de Madrid, Madrid, SpainDepartment of Atmospheric and Oceanic Sciences, University of California at Los Angeles, Los Angeles, CA, USASchool of Mathematics, University of Bristol, Bristol, UKAna María Mancho (a.m.mancho@icmat.es)9June201724226527813December201614December20167April201725April2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/265/2017/npg-24-265-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/265/2017/npg-24-265-2017.pdf
In this work, we study the Lagrangian footprint of the planetary
waves present in the Southern Hemisphere stratosphere during the exceptional
sudden Stratospheric warming event that took place during September 2002. Our
focus is on constructing a simple kinematic model that retains the
fundamental mechanisms responsible for complex fluid parcel evolution, during
the polar vortex breakdown and its previous stages. The construction of the
kinematic model is guided by the Fourier decomposition of the geopotential
field. The study of Lagrangian transport phenomena in the ERA-Interim
reanalysis data highlights hyperbolic trajectories, and these trajectories are
Lagrangian objects that are the kinematic mechanism for the observed
filamentation phenomena. Our analysis shows that the breaking and splitting
of the polar vortex is justified in our model by the sudden growth of a
planetary wave and the decay of the axisymmetric flow.
Introduction
The availability of high-resolution and high-quality reanalysis data sets
provides us with a powerful tool for obtaining a detailed view of the
space–time evolution of the stratospheric polar night vortex (SPV), which has
implications for the geophysical fluid dynamics of the entire Earth. The
complexity of such a detailed view, however, makes it difficult to extract
the physical mechanisms underlying notable transport features in the observed
behaviour. The goal of this work is to gain new insights into the fundamental
mechanisms responsible for complex fluid parcel evolution, since these lie at
the heart of our understanding of the dynamics and chemistry of the
stratosphere. To this end, we extract, directly from the data, a simple model
with stripped-down dynamics in order to directly probe, in a controlled and
systematic manner, the physical mechanisms responsible for the key observed
transport features of the SPV. Models of this kind, termed “kinematic
models”, have provided a simple approach for studying Lagrangian transport
and exchange associated with flow structures such as meandering jets and
travelling waves . Other works have used
analytical kinematic models to illustrate phenomena in planetary atmospheres
(e.g. ). In the present paper, we focus on SPV
transport processes associated with filamentation and vortex breaking, of
which the dynamical structure is not fully understood.
The importance of an increased understanding of the SPV was dramatically
demonstrated by the intense research effort that followed the discovery of
the “Antarctic ozone hole” phenomenon in the 1970s
. In the following decades, during which
monitoring of ozone in atmospheric columns above Antarctica showed little
interannual variability, in situ measurements corroborated by satellite data
revealed that ozone was systematically decreasing in the Antarctic lower
stratosphere during the southern spring season. Whilst this was immediately
associated with the simultaneous increase in atmospheric pollution by
anthropogenic activities, several key questions arose :
Why does this occur over Antarctica and not over the Arctic (since pollution sources are
stronger in the Northern than in the Southern Hemisphere)?
Why does this occur in the
spring season?
Will ozone depletion extend worldwide?
The research
demonstrated that, indeed, increased atmospheric pollution was to be blamed
for the ozone depletion and identified the participating substances and
special mechanisms. The research also demonstrated that the unique
atmospheric conditions above Antarctica were responsible for the geographic
preference for ozone destruction. In particular, it was shown that the strong
circumpolar and westerly SPV characteristic of the southern winter and spring
stratosphere contributes to the isolation of the cold polar region, setting up a
favourable environment for the special chemistry to act. The new knowledge led
to the formulation of international agreements that resulted in a negative
answer to question (3) above. The analysis of transport of fluid parcels
outside the region isolated by the SPV also showed strong stirring and
mixing of the flow. In this “surf zone” , air parcels can
travel long distances away from the SPV in an environment where contours of
long-lived tracers, such as potential vorticity, can stretch and form complex
patterns. In this region, Rossby wave breaking (RWB) is associated with
irreversible deformation that pulls material filaments of the outer edge of
the SPV and enhances mixing with the exterior flow
. Such a process makes the SPV edge a
barrier to horizontal transport of air parcels while
continuously eroding and regenerating the SPV edge by filamentation
. and examined, in an idealized
setting, the way in which Rossby waves break and eject SPV material outward.
The latter conceived a similar setting in which Rossby waves also break inward.
Dynamical systems theory provides valuable insights into the transport
processes described in the previous paragraph. Tools of the theory include
the geometrical structures, referred to as hyperbolic trajectories (HTs),
their stable and unstable manifolds, and their intersection in homoclinic and
heteroclinic trajectories that provide the theoretical and computational
basis for describing the filamentation process. A challenge in the
application of these concepts to realistic geophysical flows is that while
the structures mentioned are defined for infinite-time autonomous or periodic
systems, geophysical flows are typically defined as finite-time data sets and
are not periodic. addressed this challenge for realistic ocean
flows by identifying special hyperbolic trajectories in the finite data set,
called distinguished hyperbolic trajectories (DHTs), and by computing stable
and unstable manifolds as curves advected by the velocity field. A pioneering
effort for identifying HTs for the stratosphere was due to .
, and suggested that HTs are responsible for the
cat-eye structures associated with planetary wave breaking at the critical
levels, i.e. where the wave phase speed matches the background velocity
. HTs are at the locations where the cat-eye
structures meet. Perturbation of the cat eyes results in irreversible
deformation of material contours, signifying Rossby wave breaking.
and identified HTs both within and outside the
SPV, thus suggesting that Rossby wave breaking can occur in either of those
regions. The former authors worked with reanalysis data, while
used a dynamical model based on the shallow-water equations in which the
perturbing waves are produced in a controlled manner. Therefore, HTs are
essential features for tracer mixing both outside and inside the vortex, and
for occasional air crossings of the vortex edge.
We focus on the SPV behaviour during the major stratospheric sudden warming
that occurred in the southern stratosphere during September 2002. In this
unusual event, the SPV broke down in the middle stratosphere
. We begin by identifying key Lagrangian
features of the flow in reanalysis data fields. Next, we build a kinematic
model of the event that emulates the behaviour of planetary waves observed in
the data. We show that our model produces strikingly similar transport
features to those found in the reanalysis data, confirming the key role
played by the HTs during vortex filamentation and breakdown.
The structure of the paper is as follows. Section 2 describes the data and
methods we used. Section 3 describes the planetary waves in the reanalysis
data in the year 2002 in the stratosphere at selected pressure levels
(10 hPa). We relate these to filamentation phenomena and the polar
vortex breakdown that occurred in that year. Section 4 reproduces the
findings obtained with our analytical kinematical model, confirming the role
played by the HTs in the 2002 vortex filamentation and breakdown. Section 5
discusses the consistency of the kinematic model as representative of
atmospheric flows that conserve potential vorticity. Finally, in Sect. 6, we
present the conclusions.
Data and methodsERA-Interim reanalysis data
To achieve a realistic representation of the atmospheric transport processes,
it is crucial to use a reliable and high-quality data set. We use in this work
the ERA-Interim reanalysis data set produced by a weather forecast
assimilation system developed by the European Centre for Medium-Range Weather
Forecasts (ECMWF; ). obtained encouraging
results on the suitability of the ERA-Interim data set for Lagrangian studies
of stratospheric motions in their comparison of parcel trajectories on the
475 K isentropic surface (around 20 km) using this data set and the
trajectories of super-pressure balloons released from Antarctica by the
VORCORE project during the spring of 2005 .
The ERA-Interim data set that we selected for this study is available four
times daily (00:00, 06:00, 12:00 and 18:00 UTC), with a horizontal resolution of
1∘×1∘ in longitude and latitude and 60σ levels
in the vertical from 1000 to 0.1 hPa. The data cover the period from 1979
to the present day and they can be downloaded from
http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=sfc/. In
particular, we will use the data for the geopotential height on surfaces of
constant pressure and wind fields on isentropic surfaces for the period August–September 2002.
The geopotential height Z on constant pressure surfaces p is defined as
the normalization to g0=9.80665 ms-2 (standard gravity at mean
sea level) of the gravitational potential energy per unit mass at an
elevation s (over the Earth's surface) and has the form
Z(λ,ϕ,p,t)=1g0∫0s(p,t)gλ,ϕ,zdz,
where g is the acceleration due to gravity, λ is longitude, ϕ
is latitude and z is the geometric height (). In the
quasi-geostrophic approximation, the geopotential height is proportional to
the streamfunction of the geostrophic flow .
For the analysis of planetary waves, we apply a zonal Fourier decomposition
to the geopotential height field on the 10 hPa pressure level (approximately
850 K potential temperature). The zonal wave decomposition yields
Z=Z0ϕ,p,t+∑k=1∞Zkλ,ϕ,p,t.
The mean flow is defined as
Z0ϕ,p,t=12π∫02πZλ,ϕ,p,tdλ,
and the different modes Zk with wave number k≥1 have the sinusoidal description:
Zkλ,ϕ,p,t=Bkϕ,tcoskλ+φkϕ,p,t,
where λ∈[0,2π) is longitude, ϕ∈[-π/2,π/2] is
latitude, Bk is the amplitude of the wave and φk its
phase. During the warming event that occurred in the southern stratosphere during
September 2002, the flow was dominated by the contributions of the mean flow
and the two longest planetary waves (Z1 and Z2;
)
Lagrangian descriptors
Dynamical systems theory provides a qualitative description of the evolution
of particle trajectories by means of geometrical objects that partition the
phase space (the atmosphere in our case) into regions in which the system
shows distinct dynamical behaviours. These geometrical structures act as
material barriers to fluid parcels and are closely related to flow regions
known as hyperbolic, where rapid contraction and expansion takes place.
Several Lagrangian techniques have been developed in order to detect such
structures in geophysical fluids. This is challenging because, while
classical dynamical systems theory is defined for infinite-time autonomous or
periodic systems, in geophysical contexts, the velocity fields are generally
time dependent, aperiodic in time and defined over a finite discrete
space–time domain. Among others, techniques developed are finite-size
Lyapunov exponents (FSLEs) and finite-time Lyapunov exponents
(FTLEs) (). Other techniques include
DHTs and the direct
calculation of manifolds as material surfaces ,
the geodesic theory of Lagrangian coherent structures (LCS) and the variational theory of
LCS , etc. Our choice in this work will be the use of the
Lagrangian descriptor (LD) function M introduced by and .
The function M has been applied in a variety of geophysical contexts. For
example, in the ocean, it has been used to analyse the structure of the
Kuroshio current , to discuss the performance of different
oceanic data sets , to analyse and develop search and rescue
strategies at sea and to efficiently manage in real time the
environmental impact of marine oil spills . In the field of
atmospheric sciences, M has been used to study transport processes across
the southern SPV and RWB and to
investigate the Northern Hemisphere major stratospheric final warming in 2016
.
The dynamical system that governs the atmospheric flow is given by
x˙=vxt,t,xt0=x0,
where xt;x0
represents the trajectory of a parcel that, at time t0, is at position
x0, and v is the wind velocity field. Since our interest is
in the timescale of stratospheric sudden warmings (∼10 days), we can
assume to a good approximation that the fluid parcels evolve adiabatically.
Therefore, trajectories are constrained to surfaces of constant specific
potential temperature (isentropic surfaces). We will concentrate on the
850 K surface, which is in the middle stratosphere and approximately
corresponds to the 10 hPa level. In Sect. 3, we expand on the reasons for
this choice.
To compute fluid parcels trajectories, it is necessary to integrate
Eq. (). As the velocity field is provided on a discrete
spatiotemporal grid, the first issue to deal with is that of interpolation.
We apply bicubic interpolation in space and third-order Lagrangian
polynomials in time (see for details). Moreover, for the
time evolution, we have used an adaptive Cash–Karp Runge–Kutta method. It is important to
remark that, as done in for the computation of particle
trajectories, we use Cartesian coordinates in order to avoid the singularity
problem arising at the poles from the description of the Earth's system in
spherical coordinates. For our Lagrangian diagnostic, we use the M function
defined as follows:
M(x0,t0,τ)=∫t0-τt0+τ‖v(x(t;x0),t)‖dt,
where ‖⋅‖ stands for the modulus of the velocity vector. At a given
time t0, the function M(x0,t0,τ) measures the arc length
traced by the trajectory starting at x0=x(t0) as it evolves
forward and backward in time for a time interval τ. Sharp changes of
M values (what we call singular features of M) occur for sufficiently
large τ, for very close initial conditions and highlight stable and
unstable manifolds.
have performed systematic numerical computations of
invariant manifolds and found that they are aligned with singular features of
M. They also provide examples in geophysical flows where manifolds are
defined in a constructive way. Invariant manifolds are mathematical objects
classically defined for infinite time intervals. The unstable (stable)
manifold of a hyperbolic fixed point or periodic trajectory is formed by the
set of trajectories that in negative (positive) infinity time approach these special
trajectories. In geophysical contexts, this definition is not realizable,
because only finite-time aperiodic data sets are possible. Nevertheless,
manifolds can still be defined constructively with the following procedure.
At the beginning time, these curves are approximated by segments with short
length, aligned with the stable and unstable subspaces of the DHT identified
with algorithms described in and . This starting step aims to
build a finite-time version of the asymptotic property of manifolds. Next,
segments are advected forward and backward in time by the velocity field.
Due to the large expansion and contraction rates in the neighbourhood of the
DHT, the curves grow rapidly in forward and backward time and specific
issues are addressed by the procedure described in and . The
procedure provides curves, manifolds, that by construction are barriers to
transport in geophysical flows. In this way, since manifolds are aligned with
singular features of M, the latter belong to invariant curves of system
Eq. (), and therefore their crossing points are indeed trajectories
of system Eq. (). The capability of LDs, in general, and M in
particular, to reveal invariant manifolds was analysed in detail in
. and have discussed, in discrete
and continuous-time dynamical systems, respectively, a theoretical framework
for some particular versions of LDs in specific examples.
The consistency between the output field of Eq. () and FTLE ridges has
been discussed in some references (see ).
The integral expression in Eq. () can be split in two terms: one for
forward time and the other for backward time integration. Explicit calculations
discussed in for a linear saddle, show that singular features
of the first term are aligned with the stable manifolds while those for the
backward time integration are aligned with the unstable manifolds. This is
similar to what is obtained with FTLEs that highlight stable and unstable
manifolds, respectively, for forward and backward time integration
intervals. The fact that we choose to add both fields is advantageous for
highlighting hyperbolic trajectories at the crossing points of the singular
features.
As an example relevant to the case that motivates the present study, we show
in Fig. the evaluation of M over the Southern Hemisphere
using τ=15 on the 850 K isentropic level for 5 August 2002. The
representation shows a stereographic projection (see ) in
which the SPV is clearly visible by the bright yellow colour as well as the
filamentation phenomena ejecting material both from the outer and inner parts
of the jet. These filaments are related to the presence of hyperbolic
trajectories highlighted in the figure. The fact that these saddle points of
the LD field are hyperbolic trajectories of system Eq. () is
numerically supported. To this end, show that, for similar
ERA-Interim fields, these points belong to the intersection of stable and
unstable manifolds highlighted by the singular features of the field (see their
Fig. 2). In what follows, all figures showing M were computed
with τ=15. This choice of τ is made based on the fact that
diabatic heating/cooling processes in the extratropical stratosphere
generally have longer timescales than those of horizontal advection. Hence,
air parcels move on two-dimensional isentropic surfaces to a good
approximation (they stay within 850 K for 30 days) .
Moreover, diabatic heating rates in the Antarctic mid-stratosphere are on the
order of 0.5 K day-1, although uncertainties in this magnitude
remain large . During the time interval of our calculations
of isentropic trajectories (τ=15 days, i.e. time period of 30 days),
the material surface would experience an increase of potential temperature of
around 15 K. Nevertheless, calculations of M using wind fields at 850 K
and 700 K (not shown) produce qualitatively similar results. This suggests
that horizontal motions of the parcels will be affected by similar geometric
structures at those isentropic levels and that the isentropic approach is
justified in our problem.
Stereographic projection of Lagrangian descriptors evaluated using
τ=15 on the 850 K isentropic level for 5 August 2002 at
00:00:00 UTC. The SPV is clearly visible, as well as three hyperbolic
trajectories (HTs) outside the vortex (marked with white arrows), two
northeast and one southwest of it. Filamentation phenomena, which occur in
the neighbourhood of HTs, are visible both inside and outside the vortex, where
the outer filamentous structures play the role of eroding the jet barrier.
Notice also the presence of two eddy-coherent structures over the South
Atlantic and south of Australia.
Data analysis
As we indicated in the previous section, in order to characterize the
planetary waves that propagate in the stratosphere, we carry out a Fourier
decomposition of the geopotential height. In Fig. , we show
the axisymmetric mean flow together with waves 1 and 2 in the geopotential
field for 22 September 2002 on the 10 hPa pressure surface. The time
evolution of these waves is described in the movies S1–S4 in the Supplement. Movies S1–S3
show components 0, 1 and 2 separately for the time period of interest, while
S4 shows the superposition of these three waves. It is important to reiterate
that, since the geopotential provides a good approximation of the streamfunction
of the large-scale flow in the extratropical regions, its analysis
will provide us with guidance on the building of the simple kinematic model
presented in the next section.
On the 10 hPa pressure level, the winter SPV in the Southern Hemisphere can
be broadly defined as a circumpolar westerly jet. Figure a and b
illustrate the evolution of the circulation during August–September 2002.
We can clearly see the gradual deceleration of the SPV and the abrupt change
in direction from westerly to easterly velocities at high latitudes that
occurred on 22 September. This was a unique major sudden stratospheric warming (SSW) in the southern
stratosphere. Planetary waves in the southern stratosphere were very active
during the period where the 2002 SSW developed. Figure c presents
a time series of the ratio between the amplitudes of waves 1 and 2. Increased
wave 1 amplitude results in a displacement of the SPV vortex from a
circumpolar configuration, while increased wave 2 amplitude results in a stretching the
SPV in one direction and contraction (or “pinching”) in the orthogonal
direction. According to Fig. c, the amplitude of wave 1 was
generally larger than that of wave 2 during the entire period, confirming the
major role of this wave. Finally, Fig. d displays the variations
in time of the ridges of wave 1 and wave 2. Note that wave 1 is
quasi-stationary, while wave 2 propagates eastward, as is typical in the
southern stratosphere during early spring .
Stereographic projection of the geopotential height field and its
Fourier decomposition for the 10 hPa pressure level on 22 September 2002
at 00:00:00 UTC: (a) geopotential height; (b) mean flow;
(c) Fourier component Z1; (d) Fourier
component Z2. Observe how the amplitude of the planetary wave
with wave number 1 can be at least 3 times larger than that of wave number
2.
On the 10 hPa pressure level: (a) time evolution of the
geopotential height corresponding to the mean flow. (b) Time
evolution of the mean flow velocity. Notice the change in wind direction from
westerly to easterly that takes place from 22 to 24 September 2002.
(c) Time series of the ratio of the amplitudes of waves 1 and 2.
(d) Hovmöller (time–latitude) showing the position of the ridges
of waves 1 and 2 at latitude 60∘ S.
The contribution of these different waves to the evolution of the SPV and
their transport implications are clearly observed in movie S5. A regime giving
rise to the stretching of material lines and the appearance of hyperbolic
regions and the associated filamentation processes is observed. These
filamentous structures and HTs are clearly highlighted by the application of
LDs to the wind fields, as shown in Figs. and .
Filamentation phenomena occur both inside and outside the vortex, where the
outer filamentous structures play the role of eroding the jet material
barrier. Also, the presence of HTs in the flow (see captions of
Figs. and ) indicates regions subjected to
intense deformation and mixing (see ). We emphasize that HTs
appear both inside and outside the SPV. Finally, the breakup of the SPV on
24 September 2002 depicted in Fig. b (see also movie
S5) occurs when manifolds associated with an HT that forms within the SPV
connect the interior and the exterior of the jet, allowing for the
interchange of parcels through the barrier. The pinching of the SPV takes
place off the pole because Z1 has large amplitudes in the days
preceding the breakup. As we approach 24 September, Z2
becomes the same order as Z0, and the jet elongates and
flattens. At this point, the mean flow reversal is crucial for completing the
pinching process and the appearance of a HT in the interior of the vortex as
this splits apart.
The kinematic model
Kinematic models have a long history in the geophysical fluid dynamics
community. They allow for a detailed parametric study of the influence of
identified flow structures on transport and exchange of fluid parcels. All
early studies utilizing the dynamical systems approach for understanding
Lagrangian transport and exchange associated with flow structures, such as
meandering jets and travelling waves, have employed kinematic models (see
).
Continuing in this spirit, we propose a kinematic model that allows us to
identify, in a controlled fashion, the characteristics of the distinct
propagating waves that are responsible for the different Lagrangian features
observed in the SPV. Our kinematic model is inspired by the Fourier component
decomposition of the geopotential extracted from the ERA-Interim data as
discussed in the previous section. The analysis of data from August and
September 2002 shows a mean axisymmetric flow, disturbed mainly by waves with
planetary wave numbers 1 and 2 whose amplitudes and phase speeds vary in a
time-dependent fashion. Therefore, we propose a kinematic model in the form of
a streamfunction given by
Ψ=ε0Ψ0+ε1Ψ1+ε2Ψ2,
where ε0,ε1,ε1 are the perturbation
parameters, which we will refer to as amplitudes, and Ψi are the
Fourier components, along the azimuthal direction with wave numbers i=0,1,2
respectively, which we describe next.
We will work in a plane (x,y) that is the orthographic projection of the
Southern Hemisphere onto the equatorial plane (). For
simplicity, and in order to highlight the periodicity along the azimuthal
direction, the components of the streamfunction are given in terms of polar
coordinates satisfying x=rcos(λ) and y=rsin(λ), where the
azimuthal direction λ is related to the geographical longitude and
r is related to the geographical latitude.
The particular forms of Ψ0, Ψ1 and Ψ2 are inspired by the
Fourier decomposition of the geopotential field shown in
Fig. for the 10 hPa pressure level on
22 September 2002. Starting with the mean zonal velocity,
we will assume a jet with the following expression:
vλ=r(r-a)e-r.
Therefore, vλ=0 only at r=0 and r=a, and the velocity decreases
exponentially away from the pole. Changing the values of a will allow us to
consider variations in the position of the jet maxima. Integration with
respect to r gives
Ψ0=e-r(ar+a-r(r+2)-2).
The other streamfunction components are
Ψ1=-r2e-r2sin(λ)
and
Ψ2=(r/d)2e-r2/dsin(2λ+ω2t+π/4),
where d and w2 are also tunable constants, and the phase π/4 was
added so that the relative positions of waves 1 and 2 at t=0 resemble
those in Fig. . Positive values of ω2 correspond
to clockwise rotation. Note that Eq. () can represent a wave that
propagates in the azimuthal direction λ if w2 is different than
zero. Figure shows the streamfunctions Eqs. (),
() and () in the horizontal plane for the particular
set of parameters indicated in the corresponding caption. In the panels of
Fig. and the following, the centre represents the South Pole and the
circular dashed line indicates the Equator. The similarity between
Figs. and for the selected set of parameters is
evident, taking into consideration that they correspond to stereographic and
orthographic projections, respectively.
Stereographic projection of the M function calculated using τ=15 on the 850 K isentropic level for (a) 22 September 2002
at 00:00:00 UTC and (b) 24 September 2002 at 00:00:00 UTC.
Filamentation phenomena and hyperbolic trajectories (marked with white
arrows) are nicely captured in these simulations both in the interior and the
exterior of the SPV. Observe how the vortex breakdown on 24 September
occurs when, in the interior of the vortex, a HT allows the transport and
mixing of parcels across the barrier.
Representation of the three components of the streamfunction.
Panel (a) indicates ε0Ψ0 for a=2 and ε0<0;
(b)Ψ1; and (c)Ψ2 for d=1, w2=0.
Some illustrative parameter choices for the kinematic model.
(a) A representation of the mean flow azimuthal velocity (dotted
line), the azimuthal velocity of wave 2 for the stationary case along
λ=0 (dashed line), the total azimuthal velocity along λ=0
(solid line), the phase velocity for ω2=0.1 (green line) and the
phase velocity for ω2=4π (red line); (b) representation of
the M function for a kinematic model considering a mean flow (a=2) plus a
stationary wave 2 (d=η2=1); (c) the same as (b) for a
rotating wave 2 with ω2=0.1; (d) the same as (b)
for a rotating wave 2 with ω2=4π.
Lagrangian patterns obtained at t=0 for τ=15 and different parameter
settings in the kinematic model. (a) Fourier components Ψ0 and
Ψ2 with the latter adjusted to perturb the vortex in its outer part;
(b) Fourier components Ψ0 and Ψ1 with the latter adjusted to
perturb the vortex in its outer part; (c) the model keeps Ψ0,
Ψ1 and Ψ2; (d) the model keeps Ψ0, Ψ1 and
Ψ2 with parameters adjusted differently to (c).
Lagrangian patterns obtained at t=0 for τ=15 and different parameter
settings in the kinematic model. In panel (a), the model keeps Ψ0 and
Ψ2 adjusted to perturb the vortex in its interior part; in panel
(b), the model keeps Ψ0 and Ψ1 adjusted to perturb the
vortex in its interior part.
The velocity of fluid parcels in the Cartesian coordinates (x,y) is given by Hamilton's equations:
dxdt=-∂Ψ∂y,dydt=∂Ψ∂x.
We take the amplitudes to be time dependent in order to emulate changes in
magnitudes. Let us start with ε0 constant and
ε1=η1(1+sin(μt+π)),ε2=η2(1+sin(μt)).
Here, η1 and η2 are constants. The time dependence of
ε1 and ε2 allows us to analyse each wave either
separately or together and their transient effect on the observed Lagrangian
structures and therefore their transport implications. The time dependence in
Eq. () is such that one amplitude decreases when the other
increases, roughly allowing conservation of the total energy when both waves
are present. In the simulations presented below, μ=2π/10.
We begin by considering the case of a mean flow with a=2 and just wave 2
rotating at different speeds. Furthermore, d=1 and η2=1. Let us start
with ω2=0, i.e. the stationary case. For this case, the dotted line
in Fig. a shows the azimuthal velocity of the mean flow for
ε0=-2.5, the dashed line is the azimuthal velocity of wave 2 at
λ=0, where the radial velocity cancels, the solid line is the total
azimuthal velocity and the blue line is the wave phase speed. According to
Fig. a, there are two points where the total velocity cancels, one
being the origin. We can also easily see that there are additional fixed
points at the r coordinate where the dotted and dashed curves intersect,
but placed along the lines λ=π/2,3π/4. This gives a total of
five points in the hemisphere. Figure b shows the M function for
τ=15 evaluated on this stationary field at t=0. The minima of M
highlighting the five fixed points are evident. Moreover, we can see
two hyperbolic points in the outer part of the vortex.
Next, we consider the case with the same parameters except for ω2.
Figure c shows how this picture changes when ω2=0.1, i.e.
for the slow rotation rate of wave 2. The total azimuthal velocity of the wave,
in this case, is given by the dashed line in Fig. a plus the phase
velocity represented by the green line in the figure. If this total azimuthal
velocity of the wave is added up to the mean flow, two points are found in
which the total azimuthal velocity cancels. Additionally, for a slow rotating
wave, similarly to the previous case, the total azimuthal velocity of the
wave can still be equal to the zonal mean velocity at some points in the
domain. Therefore, Fig. c is similar to Fig. b except
for a rotation. However, for a fast rotation of wave 2 (ω2=4π; red
line), the total azimuthal velocity of the wave will be larger than the zonal
mean velocity at all points in the domain. In this case, the pattern of M
(Fig. d) is very different from the pattern in Fig. b
showing no HTs.
Evolution of the Lagrangian template for the case in which the mean
flow decreases and wave 2 increases. The sequence reproduces many of the
Lagrangian features observed in the splitting event that occurred at the end
of September 2002 (see movie S5). (a)t=3; (b)t=4;
(c)t=5; (d)t=6.
Evolution of a vorticity patch. (a) Initial vorticity
distribution at time t=0; (b) evolution of the vorticity at time
t=2.
Evolution of the scaled lower boundary h. (a) The
function h/D at time t=0; (b) evolution of h/D at time t=2.
Figure displays the function M obtained from the kinematic model
for the same mean flow of Fig. a and different parameters for waves 1 and 2. All the representations are for t=0 and τ=15. Figure a is for the same case as Fig. b, except that the
amplitude of wave 2 changes in time (η1=0,η2=1). Again, two HTs are
visible in the external jet boundary along which filamentation occurs. Figure b corresponds to just wave number 1, changing amplitude in time
(η1=1,η2=0). We can see one HT at the outer boundary of the jet
where material of the vortex is being ejected. In these figures, transport
processes producing filamentation-ejecting material have close connections
to those present in
Figs. and
a), which have been linked to Rossby wave breaking at midlatitudes .
In Fig. c, the mean flow is perturbed by the not-rotating wave 2 of
Fig. a and wave 1 of Fig. b (η1=1,η2=1). In
Fig. d, the parameters are the same as Fig. c, except
that wave 2 rotates (ω2=2π/15). The jet shape and filamentary
structures greatly resemble those present in the reanalysis data, as shown in
Figs. and a.
Figure present a jet which in the interior is eroded by waves 2
and 1, respectively. To achieve such a configuration, free parameters are
specifically tuned, including a zonal mean flow with negative velocities near
the pole. In Fig. a, the mean flow obtained with parameters
ε0=2.6 and a=0.75 is perturbed by just a travelling wave 2
(η1=0,η2=1,ω2=-4π/25) with d=2. Two filaments projecting
material from the interior of the vortex are observed, and they are related
to the presence of interior HTs. In Fig. b, the mean flow is
obtained with the parameters ε0=2.5 and a=0.5. This mean flow
is perturbed by just wave 1 with amplitude that varies in time (η1=1,η2=0). A protruding material filament from the interior of the vortex is
observed, which is related to the presence of an interior HT. The interior
filaments in these figures recover features that are identified as interior
Rossby wave breaking phenomena in and , and are also
visible from the reanalysis data, as shown in Figs. and
a.
Figure b shows the pinching of the SPV in the observations on
24 September 2002, which is before its breakup. In the kinematic model,
this structure can be obtained with a strong Ψ2 and a substantial
contribution from Ψ1 to have a displacement from the pole. Movies S1,
S2, S3 and S4 in the Supplement illustrate such structures. In order to
reproduce the splitting, we do not need to consider the displacement, and thus
we neglect mode 1 in what follows. Figure shows a sequence
of M patterns obtained with the amplitude of mean flow is given by
ε0=η0(1+sin(μt+π)),
where η0=-2.5 and μ=2π/10, and a stationary wave 2 (ω2=0)
with amplitude given by Eq. (). Note that, in this way, the mean flow
weakens as wave 2 strengthens, and vice versa. The parameters fit a
streamfunction which at t=0 coincides with that used in Fig. a.
The development of an hyperbolic point at the pole in the observations
(Fig. b) can be clearly seen in Fig. a. The
two vortices have completely split at t=6.
Kinematic models and conservation of potential vorticity
In this section, we discuss the connection between the kinematic model
introduced in the previous section and a fundamental dynamical principle of
geophysical fluids. Geophysical flows that are adiabatic and frictionless
conserve the potential vorticity Q along trajectories. Conservation of Q
is expressed as follows:
dQdt=0.
Here, d /dt stands for the material derivative. A natural question
here is to discuss whether the proposed kinematic model conserves Q. Let us
assume that our setting is described by the quasi-geostrophic motion of simple
vortices in a shallow-water system (see ) in which Q
is given by
Q=f0+∇2Ψ-γ2Ψ+f0hD.
Here, f0 is a constant related to the rotation rate, D is the mean depth
of the shallow-water system, D-h is the total depth, h is the bottom
boundary of the fluid layer, which is small when compared to D, and
γ=f0/g0D, where g0 is the gravity constant. Ψ is the
geostrophic streamfunction for the horizontal velocity field, in our case
given by Eq. (), with parameters corresponding to those of
Fig. d, i.e. ε0=-2.5, η1=1, η2=1, a=2,
d=1 and ω2=2π/15.
We assume that at the initial time, t=0, the vorticity Q consists of a
circular patch with constant vorticity Q0 inside and vorticity Q1
outside. At a later time, t=2, the vorticity distribution that preserves
Eq. () is obtained by advecting the circular contour at t=0,
according to motion Eq. (), with algorithms described in
. Figure summarizes the evolution of the vorticity.
In order to preserve Eq. () from time t=0 to time t=2, and
assuming the barotropic approach in which γ=0, h is solved from
Eq. () as
hD=Qf0-∇2Ψf0-1.
Figure shows the evolution of the function h/D between t=0 and
t=2. In particular, the figure shows results for Q0=2, Q1=1.8 and
f0=20. We note that this calculation could have been repeated for any
initial distribution of Q defined as a piecewise constant function. The
lower boundary h is thus a time-dependent function adjusted to preserve the
conservation of the potential vorticity. Without this forcing, kinematic
models would not preserve potential vorticity.
Conclusions
In this work, we propose a simple kinematic model for studying transport
phenomena in the Antarctic polar vortex. We are interested in gaining
insights into the processes which carry material outward from the vortex
structure and inward to the vortex structure.
The construction of the kinematic model is realized by analysing geopotential
height data produced by the ECMWF. In particular, our focus is on the
stratospheric sudden warming event that took place in 2002, producing the
pinching and then breaking of the stratospheric polar vortex. We identify the
prevalent Fourier components during this period, which consist of a mean
axisymmetric flow and waves with wave numbers 1 and 2. The kinematic model
is based on analytical expressions which recover the spatial structures of
these representative Fourier components. The model can be controlled so that
waves with wave numbers 1 and 2 can be switched on and off independently.
We are also able to adjust the relative position of the waves with respect to
the mean axisymmetric flow.
The study of Lagrangian transport phenomena in the ERA-Interim reanalysis
data by means of Lagrangian descriptors highlights hyperbolic trajectories.
These trajectories are Lagrangian objects “seeding” the observed
filamentation phenomena. The Lagrangian study of the kinematic model sheds
light on the role played by waves in this regard. The model is adjusted to a
stationary case which considers a mean flow and a stationary wave 2 that
perturbs the mean flow in its outer part, producing hyperbolic trajectories.
For the stationary case, hyperbolic trajectories are easily identified. This
framework is modified by transforming it to a time-dependent problem by
making the wave phase speed different from zero or by introducing time-dependent
amplitudes. This allows to relate the time-dependent structures
with those easily identified in the stationary case. The setting is repeated
with wave 1 and both wave 1 and wave 2 together. The joint presence of
these waves produces complex Lagrangian patterns remarkably similar to those
observed from the analysis of the complex reanalysis data and confirms the
findings discussed by . Further adjustment of some model
parameters is able to produce erosion by means of filaments just in the
interior part of the flow. Finally, we point out that our analysis shows that
the breaking and splitting of the polar vortex is justified in our model by
the sudden growth of wave 2 and the decay of the axisymmetric flow.
The data used in this work are described in Sect. 2.1,
where links are also provided to the official websites from which they have been downloaded.
The Supplement related to this article is available online at https://doi.org/10.5194/npg-24-265-2017-supplement.
The authors declare that they have no
conflict of interest.
Acknowledgements
V. J. García-Garrido, J. Curbelo and A. M. Mancho are supported by MINECO
grant MTM2014-56392-R. The research of C. R. Mechoso is supported by the U.S.
NSF grant AGS-1245069. The research of S. Wiggins is supported by ONR grant
no. N00014- 01-1-0769. We also acknowledge support from ONR grant no.
N00014-16-1-2492.Edited by: C. López
Reviewed by: two anonymous referees
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