of betweenness in the two cases are very much correlated, and the Pearson correlation
coefficient is larger than 0.9. Hence, despite the MPPs represent a small portion of the
paths in the KIJM() subsets (between 3 − 10% for = 0.1, depending on the value of M),
which is itself a very small fraction of the full set of paths in the network, they seem to be representative of the main spatio-temporal structures describing the global dynamics.
Indeed, center and right panels of Fig. 8.1show that most of the relevant paths remain
spatially close to the MPP. This observation is confirmed by calculations of the spatial
dispersion between paths in KIJM(), whose average turns out to be of the order of the
size of the boxes defining the nodes.
8.3
Comparing fastest and most probable paths
In the study of temporal networks, the concept of fastest path has been put forward as a natural extension of the shortest path of static networks. In our work we define and analyze a different type of relevant path which is the Most Probable Path (MPP). It is important to address the differences between most probable and fastest paths, here we illustrate them with data from the temporal network of surface flow in the Mediterranean Sea.
Figure 8.5:We show the fastest -MPP (blue) and the absolute-MPP (red), between an origin node I (black star) and a destination node J (black triangle). The considered full set of paths ranges The fastest -MPP reaches the destination node in 4 steps
of τ = 10 days while the absolute-MPP needs 8 steps i.e. 40 days more. The
probability associated to the fastest -MPP is 5.9 × 10−7 and the probability of the
CHAPTER 8. ADVECTION CORRIDORS IN THE MEDITERRANEAN SURFACE FLOW 0 0.5 1 1.5 2 2.5 3 x 107 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 path ranking P IJ fast , P IJ MPP
Figure 8.6: Ranking plot in which PIJf astcorrespond to the probability of fastest -
MPPs and PMPP
IJ correspond to the probability of absolute-MPPs. The range of
probability values can be read from the vertical axis (logarithmic scale). The total number of optimal paths can also be read-off from the horizontal axis.
The MPP refers to the path transporting the maximum fraction of water (or of probabil- ity) between two nodes, and the fastest path to the pathway linking the two nodes in the shortest time. This second concept can not be implemented when the number of time- steps is fixed. However we can reclaim the concept of fastest path within a multistep approach, i.e. by looking at a time window specified by a range of values for the num-
ber of time steps M. We can then define the set M[Mmin,Mmax]
IJ of (Mmax−Mmin+ 1)-MPPs
for the pair I, J for M ∈ [Mmin, Mmax], and the fastest -MPP as the MPP in M[MIJ min,Mmax]
corresponding to the smaller M. On the other side we can also define an absolute-MPP, i.e. the MPP in M[Mmin,Mmax]
IJ having the highest probability. By comparing the set of
absolute-MPPs with the set of fastest -MPPs we can address the question: is the fastest path necessarily the most probable?
In Fig.8.5we show that the fastest -MPP among two nodes of the network is different to
the absolute-MPP and that its probability, in several cases, can be orders of magnitude
smaller. We considered for this example paths ranging from M= 3 to M = 9 steps of 10
days (i.e. a time scale of 1 − 3 months) with starting date January 1st 2011. The results show the importance to distinguish between the connections realized in the shortest time and the connections that carries most of the transported mass (the most probable).
To display in a more systematic way the differences between fastest and absolute MPPs
across the network we study the rank plot of the whole set of paths during ten years (2002 − 2011) ranging from 3 to 9 steps of 10 days starting at January 1st of each year
8.4. CONCLUSIONS
sets sorted in decreasing order. We see a gap in probabilities between the two sets of about one order of magnitude in most of the range displayed. The fastest -MPPs have probabilities significantly smaller than absolute-MPPs.
Finally, we also evaluate how these differences are reflected in the betweenness mea- sures. We define the relative difference among the betweenness computed using the set of fastest -MPPs and absolute-MPPs for the node K as:
∆K= 2 Babs K − B f ast K Babs K + B f ast K , (8.1)
where BabsK is the betweenness computed using absolute-MPPs and BKf astthe between-
ness computed using fastest -MPPs. We consider once more paths ranging from 3 to 9 steps of 10 days with starting date January 1st 2011 and we compute the spatial-average
value for the module of the relative difference finding h|∆K|iK = 0.32. This means that,
on average, the difference between the two measures is about 30%.
We stress that all the comparisons above are among paths that are already MPPs linking a pair of nodes. Considering still fastest paths (for example the one by which the very first particle from one node reaches the other) will lead to much stronger differences. In summary, the results show the importance to distinguish between the connection realized in the shortest time and the connection that carries most of the transported mass. This gains even more relevance when considering possible applications such as, rescue operations, pollutant-spreading or biological connectivity.
We considered paths in a temporal flow network describing surface flow in the Mediter- ranean Sea. We quantifed the relative importance of the most probable path between two nodes with respect to the whole set of paths, and to a subset of highly probable paths which incorporate most of the connection probability. Despite MPPs represent only a small fraction of the whole set of paths, we found that they suffice to highlight the main transport pathways across our network. We provide an alternative definition of betweenness centrality. High betweenness areas are organized in one dimensional-like narrow pathways that connect different regions of the ocean. Finally we showed the difference among fastest paths and MPPs stressing the different physical interpretation of such paths in temporal networks.
8.4
Conclusions
To conclude, we considered paths in a temporal flow network describing surface flow in the Mediterranean Sea. We quantified the relative importance of the most probable path between two nodes with respect to the whole set of paths, and to a subset of highly probable paths which incorporate most of the connection probability. Despite MPPs represent only a small fraction of the whole set of paths, we found that they suffice to highlight the main transport pathways across our network. We provide an alternative definition of betweenness centrality. High betweenness areas are organized in one dimensional-like narrow pathways that connect different regions of the ocean. Finally we showed the difference among fastest paths and MPPs stressing the different physical interpretation of such paths in temporal networks.
CHAPTER
9
Transport pathways in an
atmospheric blocking event
9.1
Introduction
Eastern Europe and Western Russia experienced a strong, unpredicted, heat wave dur- ing the summer of 2010. Extreme temperatures resulted in over 50000 deaths and inflicting large economic losses to Russia. The heat wave was due to a strong atmo- spheric blocking that persisted over the Euro-Russian region from late June to early
August (Matsueda, 2011). During July the daily temperatures were near or above
record levels and the event covered Western Russia, Belarus, Ukraine, and the Baltic na- tions. Physically, the origins of this heat wave were in an atmospheric blocking episode that produced anomalously stable anticyclonic conditions, redirecting the trajectories of migrating cyclones. Atmospheric blockings can remain in place for several days (sometimes even weeks) and are of large scale (typically larger than 2000 km). In par- ticular, the Russian blocking of summer 2010 was morphologically of the type known as Omega blocking that consists in a combination of low-high-low pressure fields with
geopotential lines resembling the Greek letterΩ (see Fig.9.1). Omega blockings bring
warmer and drier conditions to the areas that they impact and colder, wetter conditions
in the upstream and downstream (Black et al.,2004). Despite these type of events have
been well-investigated over the years, a complete understanding and prediction is still missing.
Here we present a characterization of this flow pattern based on the study of fluid transport as a Lagrangian flow network. The most probable paths linking nodes of this atmospheric network reveal the dominant pathways traced by atmospheric fluid particles. We study the concrete period extended from the July the 20th to July 30th (Ser-Giacomi et al.,2015c).
CHAPTER 9. TRANSPORT PATHWAYS IN AN ATMOSPHERIC BLOCKING EVENT
Figure 9.1:Geopotential height at 500hPa (contours, in m) and temperature (color code, in degrees C) on July 24th, 12:00 UTC.
9.2
Atmospheric setup
Atmospheric data were provided by the National Centers for Environmental Prediction (NCEP) Climate Forecast System Reanalysis (CFSR) through the Global Forecast System
(GFS) (Saha and coauthors,2010). This reanalysis was initially completed over the 31
year period from 1979 to 2009 and extended to March 2011. Data can be obtained with
a temporal resolution of 1 hour and a spatial horizontal resolution of 0.5◦ × 0.5◦
. The
spatial coverage contains a range of longitudes of 0◦E to 359.5◦E and latitudes of 90◦S
to 90◦N. The variables needed as input to the Lagrangian dispersion model described
in the next section include dew point temperature, geopotential height, land cover, planetary boundary layer height, pressure and pressure reduced to mean sea level, relative humidity, temperature, zonal and meridional component of the wind, vertical velocity and water equivalent to accumulated snow depth. All these fields are provided by CFSR data on 26 pressure levels.
As mentioned, the idea is to obtain the effective velocity field felt by any fluid particle. Then the Lagrangian dispersion model (see next subsection) will integrate it to provide as output the three-dimensional positions of the particle at every time step. The numer- ical model used to integrate particle velocities and obtain trajectories is the Lagrangian
particle dispersion model FLEXPART version 8.2 (Stohl et al.,2005,2011). FLEXPART
simulates the long-range and mesoscale transport, diffusion, dry and wet deposition,
and radioactive decay of tracers released from point, line, area or volume sources. It most commonly uses meteorological input fields from the numerical weather prediction model of the European Centre for Medium-Range Weather Forecasts (ECMWF) as well