NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-379-2017Insights into the three-dimensional Lagrangian geometry of the Antarctic polar vortexCurbeloJezabelGarcía-GarridoVíctor JoséMechosoCarlos RobertoManchoAna Mariaa.m.mancho@icmat.esWigginsStephenNiangCoumbaInstituto 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, 28049 Madrid, SpainDepartment of Atmospheric and Oceanic Sciences, University of California at Los Angeles, Los Angeles, California, USASchool of Mathematics, University of Bristol, Bristol BS8 1TW, UKLaboratoire de Physique de l'Atmosphere et de l'Ocean Simeon Fongang, Ecole Superieure Polytechnique, Universite Cheikh Anta Diop, 5085, Dakar-Fann, SenegalAna Maria Mancho (a.m.mancho@icmat.es)25July201724337939210February201716February20178June201716June2017This 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/24/379/2017/npg-24-379-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/379/2017/npg-24-379-2017.pdf
In this paper we study the three-dimensional (3-D) Lagrangian structures in the
stratospheric polar vortex (SPV) above Antarctica. We analyse and visualize
these structures using Lagrangian descriptor function M. The procedure for
calculation with reanalysis data is explained. Benchmarks are computed and
analysed that allow us to compare 2-D and 3-D aspects of Lagrangian
transport. Dynamical systems concepts appropriate to 3-D, such as normally
hyperbolic invariant curves, are discussed and applied. In order to
illustrate our approach we select an interval of time in which the SPV is
relatively undisturbed (August 1979) and an interval of rapid SPV changes
(October 1979). Our results provide new insights into the Lagrangian
structure of the vertical extension of the stratospheric polar vortex and its
evolution. Our results also show complex Lagrangian patterns indicative of
strong mixing processes in the upper troposphere and lower stratosphere.
Finally, during the transition to summer in the late spring, we illustrate
the vertical structure of two counterrotating vortices, one the polar and the
other an emerging one, and the invariant separatrix that divides them.
Introduction
Over the past several decades the mathematical theory of dynamical systems
has provided a fruitful framework to describe the transport and mixing
processes that take place in fluids and to understand the underlying flow
structures associated with these phenomena. The seminal paper by
on chaotic advection sparked interest in this perspective,
which is inspired by the work of Poincaré. For the understanding of
particle dynamics, Poincaré sought a geometrical approach that was based on
geometrical structures and their role in organizing all trajectories into
regions corresponding to qualitatively different dynamical fates. These
structures have been referred to as Lagrangian coherent structures (LCS) in
the fluid mechanics community .
Many studies of LCS in the atmosphere and in the ocean have been performed in
a two-dimensional (2-D) scenario. This is because in an appropriate range of
space scales and timescales a Lagrangian property of the particles is
approximately unchanged in time. Hence, the flow can be assumed to occur on
surfaces on which that property is constant. For instance, stratospheric
flows on the timescale of stratospheric sudden warmings (∼ 10 days)
are, to a first approximation, adiabatic and frictionless, and thus fluid
particles and their trajectories are constrained to remain on surfaces of
constant specific potential temperature (isentropic surfaces).
and have examined transport processes across
the Antarctic stratospheric polar vortex (SPV) on isentropic surfaces, which
are quasi-horizontal in the atmosphere. Also, for oceanic flows it is often
assumed that fluid parcels remain on surfaces of constant density
(isopycnals), which are quasi-horizontal. , and
followed the isopycnal approach for oceanic applications in
the Mediterranean Sea, for the Gulf of Mexico and
for other ocean areas.
Geophysical flows, however, are not 2-D. The study of transport processes in
3-D flows brings into the discussion issues about the three-dimensional (3-D)
visualization of Lagrangian structures (see e.g. ). In
idealized 3-D time-dependent flows Poincaré sections have been used to
recognize significant Lagrangian structures .
Invariant manifolds acting as transport barriers in 3-D flows may have the
structure of convoluted 2-D surfaces embedded in a volume . In
oceanic contexts these surfaces have been identified by stitching together
2-D Lagrangian structures in different layers or connecting
ridges computed from finite-size Lyapunov exponent fields . In
the field of atmospheric sciences, these structures have similarly been
obtained by connecting ridges computed from finite-time Lyapunov exponents
(FTLE) . More recently, also in atmospheric
contexts, 3-D Lagrangian information has been extracted by means of 2-D
slices of the full 3-D FTLE field computed from 3-D trajectories
.
The methodology used in this paper for visualizing 3-D Lagrangian structures
in the stratosphere is based on the Lagrangian descriptor (LD) known as the
M function . So far, in the stratosphere
context, the M function has been used to gain insight into key dynamical
and transport processes in 2-D settings
. More recently,
and have applied the M function to the
visualization of structures in idealized 3-D flows. have
applied the M function to visualize coherent structures in full 3-D direct
numerical simulations of the compressible magnetohydrodynamic equations. The
M function has the advantage of highlighting simultaneously invariant
manifolds by means of singular features and also tori-like coherent
structures (see ). Here we apply
M to produce a full 3-D description from 3-D flows above Antarctica during
a period in the spring of 1979 in which the stratosphere was both rather
stable (August) and subjected to rapid changes (October) . In
this region, the later period selected for analysis comprises an interval
when the winter circulation, which is characterized by a strong circumpolar
westerly (cyclonic) flow known as the SPV, breaks down as the final warming to summer conditions
develops. Although final warmings in the Southern Hemisphere are broadly
similar each year , they can be punctuated by periods of rapid
changes. During the final warming of 1979, the transition from winter to
summer circulation accelerated during mid-October, a period when perturbing
waves were very active and the vertical energy flux from the troposphere
intensified . Our aim is to describe and visualize these
phenomena from a full 3-D perspective using reanalysis data. To our
knowledge, this is the first time that the potential of M to achieve this
goal is explored.
The article is organized as follows. Section 2 describes the dynamical
systems approach to the analysis of 3-D Lagrangian structures. Section 3
describes the dataset used in this study, the calculation of M in 3-D, the
data post-processing needed to this end, the computational procedures and
other issues involved in this task. Section 4 discusses some benchmark
calculations and their interpretation. Section 5 provides the results and
findings of M on the 3-D Lagrangian structure of the polar vortex on August
and in mid-October 1979. Finally, Sect. 6 includes a discussion and presents
our conclusions.
The dynamical systems approach to the analysis of 3-D Lagrangian structures
The theory of dynamical systems provides an ideal framework for studying
nonlinear transport and mixing processes in the atmosphere. The geometrical
structures that vertebrate the Lagrangian skeleton act as material barriers
that fluid particles cannot cross. A key element in the dynamical description
is the presence of hyperbolic regions defined by rapid fluid contracting and
expanding rates along directions that are respectively associated with the
stable and unstable manifolds. In 2-D flows, these manifolds are curves,
while in 3-D settings, other possibilities arise. We discuss some
particularities for the system under study next.
If we assume that air parcels are passively advected by the flow, the
dynamical system that governs the atmospheric flow is given by
x˙=vxt,t,xt0=x0,
where
x(t;x0) represents the trajectory of an air parcel that
at time t0 is at position x0, and v is the velocity
field. For the geophysical context we are focusing on, the velocity
components will be supplied by the ERA-Interim Reanalysis dataset produced at
the European Centre for Medium-Range Weather Forecasts, as explained in
detail in the next section. This dataset confirms that the magnitude of
the vertical velocity component in the stratosphere is very
small, so that vertical displacements of fluid parcels compared to the
horizontal displacements are also small for the timescales of interest in
this study (∼ days). These considerations motivate the discussion of a
system with the particular structure given in Eq. () as it will
support the interpretation of the findings described in Sect. 5 in this
region of the atmosphere.
For two-dimensional flows hyperbolic trajectories and their stable and
unstable manifolds are the key kinematical features responsible for the
geometrical template governing transport. However, in three dimensions there
are new types of three-dimensional structures that may form a geometrical
template that governs transport. The “weak three-dimensionality” of many
geophysical flows, like the one considered in this paper, gives rise to a
(normally) hyperbolic invariant curve (i.e. not a single trajectory) that has
two-dimensional stable and unstable manifolds embedded in the 3-D space. In
this case the stable and unstable manifolds of this invariant curve are
codimension one in the flow and therefore provide barriers to transport.
Moreover, since in this case the stable and unstable manifolds are both
codimension one, they can intersect to form lobes, resulting in a
three-dimensional version of lobe dynamics. We now describe the special form
of the flow giving rise to this structure. The form of the flow follows from
and was described in the context of fluid mechanics in
.
Here we examine a simple model three-dimensional velocity field that captures
the form of the velocity field given by the dataset that we study and,
therefore, allows us to describe this less familiar notion of a normally
hyperbolic invariant curve in a simple setting. The velocity field has the
following form:
dxdt=∂H(x,y,z,t)∂y=vx(x,y,z,t)dydt=-∂H(x,y,z,t)∂x=vy(x,y,z,t)dzdt=0,
where H(x,y,x,t)=A(z)sin(πy)sin(πx)/π and A(z)=1+sin(πz/2).
The system is defined in the domain (x,y,z)∈[0,1]×[-1,1]×[-1,1]. We refer to x and y as the horizontal coordinates and to z
as the vertical coordinate. This velocity field is certainly “weakly”
three-dimensional as there is no motion in the vertical direction, yet the
horizontal motion does depend on the height. This model contains the essence
of the geometrical structures governing transport in the dataset that we
analyse.
More specifically, note that for each z the system () has a
hyperbolic fixed point at (x=0,y=0), for which the linearized system is
dxdt=A(z)πx,dydt=-A(z)πy.
The curve (0,0,z) is clearly an invariant curve (in particular, it is a
curve of fixed points). The linearized stability described by Eqs. (3) and
() quantifies linearized stability normal to the
invariant curve (0,0,z) and, hence, is the origin of the phrase
normal hyperbolicity. For each z, the fixed point has a
one-dimensional stable manifold and a one-dimensional unstable manifold.
Therefore as a function of z the curve (0,0,z) has two-dimensional
stable and two-dimensional unstable manifolds in three dimensions. Hence,
these two-dimensional invariant surfaces provide barriers to transport in the
three-dimensional flow . The geometrical
representation of the manifolds is shown in Fig. a.
The 3-D Lagrangian structure of Eq. () visualized in
Fig. b is achieved by means of the M function, which is
represented on slices intersecting the manifolds. Other slice choices are
possible, but here we just show two simple possibilities on perpendicular
planes which help to capture the essential features of the full 3-D motion.
The M function is defined as follows:
Mx0,t0,τ=∫t0-τt0+τ‖v(x(t;x0),t)‖dt,
where v(x,t) is the velocity field and ‖⋅‖
denotes the Euclidean norm. At a given time t0, M corresponds to the
length of the trajectory traced by a fluid parcel starting at x0=x(t0) as it evolves forwards and backwards in time for a time
interval τ. For sufficiently large τ values the sharp changes that
occur in narrow gaps in the scalar field provided by M, which we will refer
to as singular features, highlight the stable and unstable manifolds and, at
their crossings, hyperbolic trajectories as confirmed by
Fig. b. Recently, have established a rigorous
mathematical foundation for specific LDs for a class of examples in
continuous time-dynamical systems.
(a) Stable (blue) and unstable (red) manifolds of the
normally hyperbolic invariant curve in Eq. ();
(b) representation of the Lagrangian structures at specific slices
by means of the M function.
Figure b shows that the M function also has the capability
of revealing vortices present in the fluid. In particular for this example
two counterrotating vortices which are vertically extended are visible. The
yellowish colours highlight the parts of the vortices with the highest
speeds. Typically vortex- or jet-like structures (for periodic domains) are
related to 2-tori in three-dimensional flows. This is discussed in
and . This notion is related to fluid
regions trapping fluid parcels in their interior and isolating them from the
surrounding fluid, as for instance is the case for the circulating strong jet
forming the SPV. There exist formal results linking contour lines of the time
average of M with tori-like invariant sets. In this manner singular lines
in Fig. b highlight invariant manifolds and contour lines of
converged averages of M highlight invariant tori (see ).
Dataset and computation of the M functionERA-Interim Reanalysis dataset
We use the ERA-Interim Reanalysis dataset produced by European Centre for
Medium-Range Weather Forecasts (ECMWF; ). The usefulness
of this dataset for the Lagrangian study of atmospheric flows from the
dynamical systems perspective is established by the results of several
previous studies. For instance, applied Lagrangian
descriptors to study the structure of the SPV during the southern spring of
2005, to support the interpretation of several features found in the
trajectories of superpressure balloons released from Antarctica by the
VORCORE project . ERA-Interim covers the period from 1979 to
the present day , and can be downloaded from
http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=sfc/.
From the ERA-Interim dataset we extract the horizontal wind velocity components u and v, vertical velocity ω (dP/dt), temperature, specific humidity,
potential vorticity, surface pressure, and the geopotential field. In the
version of the dataset that we selected for the present study, these physical
variables are available four times daily (00:00, 06:00, 12:00, 18:00 UTC)
with a horizontal resolution of 0.75∘× 0.75∘ in
longitude and latitude. The velocity fields, temperature and specific humidity extracted correspond to the
60 hybrid-sigma levels of the model component of the reanalysis system (from
the Earth's surface to the 0.1 hPa level). We also take from the dataset
potential vorticity at 15 levels of potential temperature (265, 275, 285,
300, 315, 330, 350, 370, 395, 430, 475, 530, 600, 700, and 850; K) and
geopotential at field pressure levels (1, 2, 3, 5, 7, 10, 20, 30, 50, 70, 100
to 250 by 25, 300 to 750 by 50, 775 to 1000 by 25; hPa).
Computation of the M function
The procedure to obtain the M function in (lat, lon, height) coordinates
from the data described in the previous section consists of the following
steps.
Step 1. The data are downloaded from ERA-Interim in .grib format
and on a monthly basis.
Step 2. The data files are converted from .grib to .nc format with the Climate Data Operator (CDO) software
(available at https://code.zmaw.de/projects/cdo). This is done with the
copy command, setting as arguments -t ecmwf to indicate
that the data are from ERA-Interim and -f nc to specify that the
output complies with NetCDF.
Step 3. The vertical velocity ω, the temperature and the specific humidity are concatenated using the CDO command merge to provide
the input to the function vertwind to compute the vertical velocity
w in metres per second (m s-1).
Step 4. Then, the 3-D velocity (u,v,w) and 2-D surface pressure data are also concatenated
to produce the input required in the next step. using the merge CDO
command.
Step 5. The data are converted from sigma levels to the height levels
specified for the analysis, and required data are produced by interpolation,
with all velocity components expressed in metres per second (m s-1). This is done using the
CDO command ml2hlx applied to the NetCDF files. The resulting 3-D
velocity field has a horizontal resolution of
0.75∘× 0.75∘ in longitude and latitude, with 80
height levels ranging from 0 to 47600 m at 600 m intervals. Each wind data
variable is stored in separate files by day using the selvar and
selday commands. This
procedure avoids the handling of very large data files when computing
particle trajectories, which requires interpolation in time and space.
Step 6. To avoid issues at the pole in the calculation of trajectories from
equations expressed in spherical coordinates, the velocities are written in
Cartesian coordinates from the velocity data available in spherical
coordinates (see ); the velocity components in Cartesian
coordinates are given byvx=wcosλcosϕ-usinλ-vcosλsinϕ,vy=wsinλcosϕ+ucosλ-vsinλsinϕ,vz=wsinϕ+vcosϕ,where u, v, and w are the zonal, meridional, and vertical
velocity components, respectively, λ is longitude, and ϕ is
latitude.
Step 7. The trajectories are calculated in a Cartesian coordinate system,
which are obtained by solvingdxdt=vx(λ,ϕ,h,t)dydt=vy(λ,ϕ,h,t)dzdt=vz(λ,ϕ,h,t).
Step 8. The M function is obtained by approximating the integral in
Eq. () by the sum of the lengths (in the Euclidean space) of the
segments linking the position of the integrated particle trajectory at two
successive time steps. As τ increases, a richer Lagrangian history is
incorporated into M, and a more complex and detailed dynamical description
is obtained.
M calculated on 15 August 1979 at 00:00:00 UTC:
(a) using a 2-D approach where particle trajectories are constrained
to the 850 K potential temperature level and τ=5 days;
(b)M computation for τ=5 days using a full 3-D computation
of trajectories, and it is represented at a constant height level of h=31.3 km (which corresponds approximately to the 850 K isentropic surface);
(c) the same as panel (a) with τ=10 days;
(d) the same as panel (b) with τ=10 days;
(e) the same as panel (a) with τ=20 days;
(f) the same as panel (b) with τ=20 days.
Benchmarks
To benchmark the procedure described in the previous section, we compare the
Lagrangian outputs obtained from the full 3-D scenario at constant heights
(spherical shells) with those obtained from the 2-D scenario for potential
temperature surfaces that are approximately at the same height. Figure compares, for
different τ values, the evaluation of the M function for a 2-D
integration in the 850 K isentropic surface with that obtained on a
spherical shell at 31.3 km height, which approximately corresponds to the
850 K surface. For τ=5, 10 days, the figure highlights well-defined
structures that are very similar in both cases. These consist
of a large circulating coherent vortex in yellow, representing high values of
M that are related to particles exhibiting large displacements and blueish
zones corresponding to calmer regions. Lobes eroding the outer part of the
vortex are clearly identified by coloured filaments. Also, crossings of
contours of M highlighting hyperbolic trajectories are noticed along
longitudes 15∘ W and 165∘ E. These results confirm that in the upper stratosphere the potential temperature surface and the spherical shell are almost identical.
M calculated on 15 August 1979 at 00:00:00 UTC:
(a) using a 2-D approach where particle trajectories are constrained
to the 330 K potential temperature level and τ=5 days;
(b) the same with τ=10 days; (c)M calculated for
τ=5 days using a 3-D computation of trajectories and represented at a
constant height level of h=10 km (which corresponds approximately to the
330K isentropic surface); (d) the same with τ=10 days.
Increasing the τ values to 20 or 30 days adds more Lagrangian detail to
the figures, which thus contain long-term transport issues reflected in very
thin and long filamentous structures. However, the figures at constant
potential temperature and constant height are still very similar and contain
the gross structure already observed at smaller τ. Differences are
restricted to those filamentous structures which are difficult to follow and
to compare. Given that long-term transport issues are difficult to interpret
and that we do not require them to describe the phenomena of our interest
which occur in time intervals varying from hours to 10 days, we fix in this
range the selected τ values for this article.
Comparison of geopotential at constant pressure 10 mb, potential
vorticity at 850 K and M function at z=31.3 km for several days of
October 1979.
Comparisons of transport between both approaches are rather different, as
Fig. confirms. The first row in this figure shows the 2-D
calculations for a 2-D integration in the 330 K isentropic surface for a
period of τ=5, 10 days, and the second row a 3-D output at the
corresponding height (10 km) for the same integration intervals. The panels
on the top row of Fig. (at a constant potential
temperature) are quite similar to those in the bottom row (at constant
height), although differences are more evident than those in
Fig. for the upper stratosphere. At this level constant
potential temperature surfaces are not so close to a sphere, but flow
structures in the troposphere are deep, i.e. they have a vertical scale of
about 10 km, which is larger than the vertical separations between points on
the 10 km high sphere and 330 K surfaces. Results discussed in the next
section will show further evidence of the 2-D–3-D motion transitions.
Figure contrasts the more classical description of events
in the middle–upper stratosphere with the one provided by our Lagrangian
descriptor. The left-hand column of Fig. shows the
geopotential height at 10 mb. This field is particularly useful because it
is equivalent to the velocity streamfunction at the corresponding pressure
level in the extratropics (see ). This field provides a purely
Eulerian description of the flow; however, in the time-dependent case this
point of view is limited as it does not address issues regarding the fate of
particle trajectories. The middle column shows the picture provided by the
potential vorticity at the 850 K equivalent temperature (isentropic)
surface. This field is conserved along particle trajectories; thus, the image
is Lagrangian in the sense that we observe the field of a purely advected
quantity. The right-hand column shows the M function at the surface z=31.3 km obtained from particle trajectories in full 3-D calculations.
Similarly to the second column, the displayed information is Lagrangian, but
here we obtain more fundamental information in this regard. This figure
provides feedback for characterizing the time evolution of any purely
advected scalar field, while the previous one displays just the realization
of one particular initial datum. More specifically, the third column
highlights the position and evolution of two hyperbolic points in the outer
part of the vortex, as well as the vortex itself. As discussed by
, hyperbolic points are responsible for filamentation processes.
Whether or not these filaments are eventually observed depends on the
distribution of the scalar field. For instance, if the scalar field is
completely uniform in the whole domain, then its time evolution will show
nothing about the features highlighted by M. How the features of the M
field are visible in a scalar field depends on how the initial distribution
of the advected field is with respect to the features of M.
Figure illustrates these facts in a very simple example.
Figure a highlights the Lagrangian skeleton as obtained for
the stationary cat eyes, i.e. x˙=y, y˙=0.5sinx. The
hyperbolic fixed point at the origin and its stable and unstable manifolds
are clearly visible. Also, elliptic fixed points at positions (-π,0) and
(π,0) are visible. Figure b shows a set of initial
scalar fields and c shows the evolution of the three patches: the one in cyan
shows filamentation, the other staying coherent (in blue) and the third done
showing a tongue formation (in red) similar to the one observed for the
potential vorticity in Fig. . Colours are chosen to
highlight the analogous features visible in Fig. . Clearly
Fig. b and c show time-dependent patterns; however, the
Lagrangian skeleton shown in Fig. a is stationary.
(a) Lagrangian skeleton for the stationary cat eyes;
(b) three initial patches of a purely advective field;
(c) evolution of the patches at time t=5.6.
Three-dimensional Lagrangian structures over Antarctica
The strong and cyclonic SPV characteristic of the winter circulation above
Antarctica has been typically represented in the literature by cross sections
such as those in Figs. and . In this
section we will improve this representation with figures that are more
revealing of the full 3-D description of the circulation.
Figure shows for a day in late winter 1979 (15 August) the
representation of M obtained for τ=5 for the vertical slice passing
through longitudes 90∘ W and 90∘ E. An outstanding feature
in this representation is the bright yellow colour highlighting a coherent
structure. This feature captures the SPV as a tubular structure, similar to
those described in Fig. b, with walls around 60∘ S
and an approximately vertical axis coinciding with that of the Earth and
extending from the uppermost level of data down to between 15 and 20 km in
height, i.e. the transition between the troposphere and stratosphere
(tropopause). In this case however the average of M does not converge as
the flow is aperiodic (see ), and thus contours of M do
not strictly represent invariant sets. Despite this, the setting is analogous
to that described for the vortices in the example of Sect. 2. Additionally,
the greenish colours that extend equatorward between 15 and 20 km, both in
the west and the east, capture the upper tropospheric subtropical westerly
jets. Specifically in the west, the greenish colours extend downwards up to
the Equator (90∘ W), suggesting that the structures involve the
entire atmospheric layer. We can also clearly see evidence of the very
different dynamical characters of the troposphere and the stratosphere.
Whilst the former region is practically dominated by the SPV, the latter
shows much finer detail reflected by an intricate line pattern. This tangled
pattern is the manifestation of crossings of stable and unstable manifolds,
which are associated with strong and fast mixing processes at the lowest
atmospheric levels. This image is consistent with and complementary to the
projection at 10 km presented in Fig. c and d and also to
the one described next.
The description of the vortex on 15 August 1979 is supplemented by
Fig. , which shows the M function computed for τ=10
along a vertical slice through latitude 60∘ S, where most of the SPV
walls are. The intricate structures in the troposphere are also apparent in
this figure. In addition, a wavy structure is clearly visible at the boundary
between troposphere and stratosphere. In terms of a Fourier decomposition of
M at constant height in Fig. , we can see the classical
pattern of longer wavelengths dominating the field as height increases. At
tropopause level, a wavenumber 4 component is clearly visible, while in the
upper part of the vortex, a wavenumber 2 is evident . The
vertical propagation of these features across the stratosphere from 8 to
21 August of 1979 is clearly visible in the attached Movie S1 in the
Supplement.
15 August 1979 00:00:00 UTC. M for τ=5 days displayed
along the vertical slice passing through longitudes 90∘ W and
90∘ E. The black lines on the stratosphere correspond to heights 10,
20, 30 and 40 km and black lines on the Earth's surface correspond to
latitudes 15, 30, 45, 60, and 75∘ S.
We next focus on the description of the 3-D Lagrangian structures for the
period 6–18 October 1979. Figure shows the rapid changes in
the SPV that took place in October 1979 as the lower polar stratosphere
warmed up strongly during the spring season . The figure shows
the M function computed for τ=5 along a vertical slice passing through
latitude 60∘S for several October dates. Figure a, b, c
and d confirm that Lagrangian structures in the stratosphere become more
complex in the warming period. On 18 October no yellow coherent features are
visible in the upper stratosphere.
15 August 1979 00:00:00 UTC. M for τ=5 days displayed
along the vertical slice passing through latitude
60∘ S.
Vertical slices showing M for τ=5 at constant latitude
60∘ S on 4 selected days in October. The colour scale is the same in all figures.
6 October 1979 00:00:00 UTC. (a) Slices of M at constant
heights z=10, 21.2, 31.3 and 40 km with τ=5; (b) slice
of M at constant longitude λ=90∘ W and λ=90∘ E. (b) The black lines on the stratosphere correspond
to heights 10, 20, 30 and 40 km and black lines on the Earth's surface
correspond to latitudes 15, 30, 45, 60, and 75∘ S.
(c) Slices of M at constant heights z=21, 30, and 39 km; the
red line indicates the position of the hyperbolic point marked with a white
arrow in panel (a) for different heights, i.e it is the normally
hyperbolic invariant curve.
To help in the interpretation of Fig. ,
Fig. displays horizontal sections of M on 6 October
at different heights z=10, 21.2, 31.3 and 40 km as well as a vertical
section along meridians 90∘ E and 90∘ W.
Figures and show important differences
with the winter conditions 2 months earlier, visible in Figs.
and . The 60∘ S section in Fig. a
intersects the cyclonic vortex four times, mostly in the Western Hemisphere.
Another deep, anticyclonic vortex appears in the Eastern Hemisphere above
25 km. These cyclonic and anticyclonic vortices are also evident in
Fig. a at z=31.3 km and z=40 km, and in
Fig. b. In particular in the projection of
Fig. a at z=31.3 km, along the longitude 0∘,
between latitudes 30 and 45∘ S, crossing lines marked with a white
arrow show the presence of a hyperbolic point, its unstable manifold being
the separatrix between the two vortices. The vertical extension of this
hyperbolic trajectory is depicted with the red line in
Fig. c. This figure shows the analogue to the normally
hyperbolic invariant curve explained in Sect. 2 with the additional feature
that here the flow is time-dependent, while, for simplicity, the example in
Sect. 2 was stationary. Figure b sketches the corresponding
typical configuration for two counterrotating vortex tubes illustrating this
description.
Evaluation of the M function with a black particle on it.
(a) The black particle is placed exactly over an invariant manifold
on a 2-D slice obtained at a height of 31.3 km on 6 October 1979,
00:00:00 UTC; (b) the same black particle on the same day and time
placed on a 2-D slice obtained at longitude 90∘ E; (c) the
same black particle 6 h later on a 2-D slice of M obtained at the
corresponding height of the particle at that time; (d) the same
black particle 6 h later on a 2-D slice of M obtained at the corresponding
longitude of the particle at that time.
The vertically extended unstable manifold of the normally hyperbolic
invariant curve that separates the two vortex tubes is captured by M and is
visible in Figs. a and b as a narrow dark
blue line in an analogous way to that presented in Fig. . The
first of these two figures shows it near the edge of the polar vortex, around
45∘ E marked with a white arrow, and for the latter a white arrow
also points to the described feature. This manifold structure separates the
two counterrotating vortex tubes just described. It acts as a vertical
barrier, which is several kilometres deep, as is the case for the unstable manifolds associated with a normally
hyperbolic invariant curve formed from hyperbolic trajectories at each h
level. Further information on the singular features is given in
Fig. . The black dots in Fig. a and b represent,
for 6 October 1979, 00:00:00 UTC, the horizontal and vertical positions of a
particle located on the feature indicated by the dark blue line at a height
of 31.3 km, longitude 90∘ E and latitude 77.74∘ S. The
black dots in Fig. c and d show the corresponding locations of
the same particle 6 h later. The invariant character of the singular
structure is confirmed as the particle remains on it during its evolution,
and its unstable character is confirmed by the fact that the particle moves
away from the hyperbolic point.
have suggested that the fact that the preferred
geographical location (ridge south of Australia) for the development of this
anticyclonic vortex in this particular event indicates that the stratospheric
circulation is governed to a significant extent from below (see also
). This anticyclonic vortex will strengthen and
eventually dominate at high levels. In terms of Fourier components, a
quasi-stationary wave 1 amplifies on this date, in conjunction with the
displacement of the cyclonic vortex from the polar position.
Conclusions
In the present paper we discuss the visualization of three-dimensional
Lagrangian structures in atmospheric flows. Specifically, we have explained
mathematical aspects about the Lagrangian geometrical structures to be
expected in the atmospheric setting in 3-D and have introduced the concept of
normally hyperbolic invariant curves in a specific example which recovers
features of those observed in the stratosphere. The algorithm used to
represent the 3-D Lagrangian structures is based on the methodology of
Lagrangian descriptors (LDs). We have explored the application of the full
power of the M function computed with 3-D trajectories, which hitherto had
been used in 2-D settings. The consistency of our development has been
verified by comparing the 3-D scenario results at a constant height with
those obtained from the 2-D scenario in potential temperature surfaces at
equivalent heights.
To demonstrate the methodology we have applied it to a numerical dataset
describing the flow above Antarctica during the southern mid–late winter and
spring. The dataset was obtained from ERA-Interim Reanalysis data provided by
the ECMWF. Our findings show the vertical extension and structure of the
stratospheric polar vortex and its evolution. We also characterize, from the
Lagrangian point of view, the boundary between the troposphere and the
stratosphere. Very complex Lagrangian patterns are identified in the
troposphere, which support the presence of strong mixing processes. The
“final stratospheric warming” is characterized by the breakdown of the
westerly SPV during the transition from winter to summer circulation. Our
results confirm that the onset of this process is characterized by an initial
decay of the vortex in the upper stratosphere where the circulation weakens,
albeit remaining strong at lower heights. We have also captured the
anticyclonic circulation that develops during October preferentially above
the southern part of Australia. We illustrate the vertical structure of these
two counterrotating vortices, and the invariant separatrix that divides them.
The particular feature found is several kilometres deep and we demonstrated
that fluid parcels remain in this feature during intervals of the order of
days. Such features highlight the complexities in the transport of chemical
tracers in the stratosphere.
The datasets used in this work are described in Sect. 3,
where links are 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-24-379-2017-supplement.
The authors declare that they have no conflict of
interest.
This article is part of the special issue “Current perspectives
in modelling, monitoring, and predicting geophysical fluid dynamics”. It is
not part of a conference.
Acknowledgements
Jezabel Curbelo, Víctor José García-Garrido and Ana Maria Mancho
are supported by MINECO grant MTM2014-56392-R. Coumba Niang acknowledges
Fundacion Mujeres por Africa and ICMAT Severo Ochoa project SEV-2011-0087 for
financial support. Ana Maria Mancho and Coumba Niang are supported by CSIC
grant COOPB20265. The research of Stephen Wiggins is supported by ONR grant
no. N00014-01-1-0769. Carlos Roberto Mechoso was supported by U.S. NSF grant
AGS-1245069. We also acknowledge support from ONR grant
no. N00014-16-1-2492. Thanks are owed to CESGA and ICMAT for computing facilities. Edited by: Emilio Hernández-García
Reviewed by: two anonymous referees
References
Aref, H.: Stirring by chaotic advection, J. Fluid Mech., 143, 1–21, 1984.
Bettencourt, J. H., López, C., Hernández-García, E., Montes, I., Sudre,
J., Dewitte, B., Paulmier, A., and Garçon, V.: Boundaries of the
Peruvian oxygen minimum zone shaped by coherent mesoscale dynamics, Nat.
Geosci., 8, 937–940, 2014.Bowman, K. P.: Large-scale isentropic mixing properties of the Antarctic
polar vortex from analyzed winds, J. Geophys. Res., 98,
23013–23027, 10.1029/93JD02599, 1993.
Branicki, M. and Kirwan Jr., A. D.: Stirring: The Eckart paradigm
revisited, Int. J. Eng. Sci., 48, 1027–1042, 2010.
Branicki, M. and Wiggins, S.: An adaptive method for computing invariant
manifolds in non-autonomous, three-dimensional dynamical systems, Physica D,
238, 1625–1657, 2009.
Branicki, M., Mancho, A. M., and Wiggins, S.: A Lagrangian description of
transport associated with a Front-Eddy interaction: application to data from
the North-Western Mediterranean Sea, Physica D, 240, 282–304, 2011.
Cartwright, J. H. E., Feingold, M., and Piro, O.: Chaotic adection in
three-dimensional unsteady incompressible laminar flow, J. Fluid Mech., 316,
259–284, 1996.Dee, D. P., Uppala, S. M., Simmons, A. J., Berrisford, P., Poli, P.,
Kobayashi, S., Andrae, U., Balmaseda, M. A., Balsamo, G., Bauer, P.,
Bechtold, P., Beljaars, A. C. M., van de Berg, L., Bidlot, J., Bormann, N.,
Delsol, C., Dragani, R., Fuentes, M., Geer, A. J., Haimberger, L., Healy, S.
B., Hersbach, H., Hólm, E. V.,Isaksen, L., Kållberg, P., Köhler,
M., Matricardi, M., McNally, A. P., Monge-Sanz, B. M., Morcrette, J.-J.,
Park, B.-K., Peubey, C., de Rosnay, P., Tavolato, C., Thépaut, J.-N., and
Vitart, F.: The ERA-Interim reanalysis: configuration and
performance of the data assimilation system, Q. J. Roy.
Meteor. Soc., 137, 553–597, 10.1002/qj.828, 2011.
de la Cámara, A., Mancho, A. M., Ide, K., Serrano, E., and Mechoso, C.:
Routes of transport across the Antarctic polar vortex in the southern
spring., J. Atmos. Sci., 69, 753–767, 2012.
de la Cámara, A., Mechoso, R., Mancho, A. M., Serrano, E., and Ide., K.:
Isentropic transport within the Antarctic polar night vortex: Rossby wave
breaking evidence and Lagrangian structures, J. Atmos. Sci., 70,
2982–3001, 2013.
d'Ovidio, F., Isern-Fontanet, J., López, C., Hernández-García, E., and
García-Ladona, E.: Comparison between Eulerian diagnostics and
finite-size Lyapunov exponents computed from altimetry in the Algerian
basin, Deep-Sea Res. Pt. I, 56, 15–31, 2009.
du Toit, P. C. and Marsden, J. E.: Horseshoes in hurricanes, J. Fix. Point
Theory A., 7, 351–384, 2010.García-Garrido, V. J., Mancho, A. M., Wiggins, S., and Mendoza, C.: A
dynamical systems approach to the surface search for debris associated with
the disappearance of flight MH370, Nonlin. Processes Geophys., 22, 701–712,
10.5194/npg-22-701-2015, 2015.García-Garrido, V. J., Ramos, A., Mancho, A. M., Coca, J., and Wiggins, S.: A
dynamical systems perspective for a real-time response to a marine oil
spill, Mar. Pollut. Bull., 112, 201–210,
10.1016/j.marpolbul.2016.08.018, 2016.García-Garrido, V. J., Curbelo, J., Mechoso, C. R., Mancho, A. M., and
Wiggins, S.: A simple kinematic model for the Lagrangian description of
relevant nonlinear processes in the stratospheric polar vortex, Nonlin.
Processes Geophys., 24, 265–278, 10.5194/npg-24-265-2017, 2017.
Guha, A., Mechoso, C. R., Konor, C. S., and Heikes, R. P.: Modeling Rossby Wave
Breaking in the Southern Spring Stratosphere, J. Atmos. Sci., 73, 393–406,
2016.
Haller, G. and Yuan, G.: Lagrangian coherent structures and mixing in
two-dimensional turbulence, Physica D, 147, 352–370, 2000.
Joseph, B. and Legras, B.: Relation between Kinematic Boundaries, Stirring, and
Barriers for the Antarctic Polar Vortex, J. Atmos. Sci., 59, 1198–1212,
2002.Lekien, F. and Ross, S. D.: The computation of finite-time Lyapunov exponents
on unstructured meshes and for non-Euclidean manifolds, Chaos, 20, 017505,
10.1063/1.3278516, 2010.Lopesino, C., Balibrea-Iniesta, F., García-Garrido, V. J., Wiggins, S., and
Mancho, A. M.: A theoretical framework for lagrangian descriptors, Int. J.
Bifurcat. Chaos, 27, 1730001, 10.1142/S0218127417300014, 2017.Madrid, J. A. J. and Mancho, A. M.: Distinguished trajectories in time
dependent vector fields, Chaos, 19, 013111, 10.1063/1.3056050, 2009.
Mancho, A. M., Hernández-García, E., Small, D., Wiggins, S., and
Fernández, V.: Lagrangian transport through an ocean front in the
North-Western Mediterranean Sea, J. Phys. Oceanogr., 38, 1222–1237,
2006.
Mancho, A. M., Wiggins, S., Curbelo, J., and Mendoza, C.: Lagrangian
descriptors: A Method for Revealing Phase Space Structures of General Time
Dependent Dynamical Systems, Commun. Nonlinear Sci., 18, 3530–3557, 2013.Manney, G. L. and Lawrence, Z. D.: The major stratospheric final warming in
2016: dispersal of vortex air and termination of Arctic chemical ozone loss,
Atmos. Chem. Phys., 16, 15371–15396, 10.5194/acp-16-15371-2016, 2016.
Manney, G. L., Farrara, J. D., and Mechoso, C. R.: The behavior of wave 2 in
the southern hemisphere stratosphere during late winter and early spring., J.
Atmos. Sci., 48, 976–998, 1991.Mechoso, C. R. and Hartmann, D. L.: An Observational Study of Traveling
Planetary Waves in the Southern Hemisphere, J. Atmos.
Sci., 39, 1921–1935,
10.1175/1520-0469(1982)039<1921:AOSOTP>2.0.CO;2, 1982.
Mechoso, C. R., O'Neill, A., Pope, V. D., and Farrara, J. D.: A study of the
stratospheric final warming of 1982 in the Southern Hemisphere, Q. J.
Roy. Meteor. Soc., 114, 1365–1384, 1988.Mendoza, C. and Mancho, A. M.: The hidden geometry of ocean flows, Phys. Rev.
Lett., 105, 038501, 10.1103/PhysRevLett.105.038501, 2010.Mendoza, C., Mancho, A. M., and Wiggins, S.: Lagrangian descriptors and the
assessment of the predictive capacity of oceanic data sets, Nonlin. Processes
Geophys., 21, 677–689, 10.5194/npg-21-677-2014, 2014.
Mezić, I. and Wiggins, S.: On the integrability and perturbation of
three-dimensional fluid flows with symmetry, J. Nonlinear Sci., 4,
157–194, 1994.
Mezic, I. and Wiggins, S.: A method for visualization of invariant sets of
dynamical systems based on the ergodic partition, Chaos, 9, 213–218, 1999.Moharana, N. R., Speetjens, M. F. M., Trieling, R. R., and Clercx, H. J. H.:
Three-dimensional Lagrangian transport phenomena in unsteady laminar flows
driven by a rotating sphere, Phys. Fluids, 25, 093602, 10.1063/1.4819901, 2013.
Pouransari, Z., Speetjens, M. F. M., and Clercx, H. J. H.: Formation of
coherent structures by fluid inertia in three-dimensional laminar flows, J.
Fluid Mech., 654, 5–34, 2010.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P.:
Numerical Recipes in C: The Art of Scientific Computing, Cambridge University
Press, New York, NY, USA, 1992.Quintanar, A. I. and Mechoso, C. R.: Quasi-stationary waves in the Southern
Hemisphere. Part I: Observational data, J. Climate, 4, 2659–2672, 1995.
Rabier, F., Bouchard, A., Brun, E., Doerenbecher, A., Guedj, S., Guidard, V.,
Karbou, F., Peuch, V., El Amraoui, L., Puech, D., Genthon, C., Picard, G.,
Town, M., Hertzog, A., Vial, F., Cocquerez, P., Cohn, S. A., Hock, T., Fox,
J., Cole, H., Parsons, D., Powers, J., Romberg, K., VanAndel, J., Deshler,
T., Mercer, J., Haase, J. S., Avallone, L., Kalnajs, L., Mechoso, C. R.,
Tangborn, A., Pellegrini, A., Frenot, Y., Thépaut, J., McNally, A.,
Balsamo, G., and Steinle, P.: The Concordiasi Project in Antarctica, B. Am.
Meteorol. Soc., 91, 69–86, 10.1175/2009BAMS2764.1, 2010.
Rempel, E. L., Chian, A. C.-L., Brandenburg, A., Munuz, P. R., and Shadden,
S. C.: Coherent structures and the saturation of a nonlinear dynamo, J. Fluid
Mech., 729, 309–329, 2013.
Rutherford, B. and Dangelmayr, G.: A three-dimensional Lagrangain hurricane
eyewall computation, Q. J. Roy. Meteor. Soc.,
136, 1931–1944, 2010.Rutherford, B., Dangelmayr, G., and Montgomery, M. T.: Lagrangian coherent
structures in tropical cyclone intensification, Atmos. Chem. Phys., 12,
5483–5507, 10.5194/acp-12-5483-2012, 2012.Rypina, I. I., Pratt, L. J., Wang, P., Özgökmen, T. M., and Mezic, I.:
Resonanace phenomena in a time-dependent, three-dimensional model of an
idealized eddy, Chaos, 25, 087401, 10.1063/1.4916086, 2015.
Shadden, S. C., Lekien, F., and Marsden, J. E.: Definition and properties of
Lagrangian Coherent Structures from finite-time Lyapunov exponents in
two-dimensional aperiodic flows, Physica D, 212, 271–304, 2005.
Simmons, A., Uppala, S., Dee, D., and Kobayashi, S.: ERA-Interim: New
ECMWF reanalysis products from 1989 onwards, ECMWF Newsletter, 110, 25–35,
2007.
Smith, M. L. and McDonald, A. J.: A quantitative measure of polar vortex
strength using the function M, J. Gephys. Res.-Atmos., 119, 5966–5985, 2014.
Wiggins, S.: Global bifurcations and chaos: analytical methods, vol. 73,
Springer Verlag, New York, USA, 1988.Wiggins, S.: Coherent structures and chaotic advection in three dimensions, J.
Fluid Mech., 654, 1–4, 10.1017/S0022112010002569, 2010.
Yamazaki, K. and Mechoso, C. R.: Observations of the Final Warming in the
Stratosphere of the Southern Hemisphere during 1979, J. Atmos. Sci., 42,
1198–1205, 1985.