Introduction
The classical interpretation of finite-time Lyapunov exponent (FTLE) fields
and the associated hyperbolic Lagrangian coherent structures (LCSs) provides
useful information about large-scale flow patterns and transport and mixing
phenomena in flow domains . There are an increasing number of studies that
apply various concepts of LCSs, based on the classic right Cauchy–Green
tensor, to describe and predict the time evolution of Lagrangian features in
geophysical systems . In some of these studies, geophysical
information (e.g., wind or oceanic velocity fields) have been used as the
input data, and Lagrangian results (e.g., the distribution of an oil spill in
the ocean or volcanic ash in the atmosphere) over a large area are compared
with the behavior of the geophysical system via satellite data or simulations
. A large-scale distribution of
particles is a common characteristic among these studies. In contrast, this
study is motivated by a series of field experiments regarding the
long-distance transport of airborne microorganisms where only a limited
number of localized and temporally consecutive measurements of the
atmospheric structure of microbial assemblages are available
. Therefore, there is a need to bridge the
powerful concept of FTLE and local field experiments.
(a) Separation of nearby particles during time interval T due to
the flow map ϕ. The two particles are released in the flow field at the
same time t0; isochronic particles. (b) ξ2 is the direction of
maximum growth at the initial point x, which evolves into the
direction r2 = Dϕt0t0+T(x)ξ2
at the evolved point ϕt0t0+T(x). The closer the initial
displacement vector δx is to the ξ2 direction, the more it will
be stretched to that maximum perturbation.
In this paper, we present two independent observations related to the
estimation of the local FTLE and the distance between destination (or source)
points of released (or collected) particles. These observations provide an
ansatz for bridging field experiment results with the concept of local
Lyapunov exponents and the direction of maximum expansion in ordinary
differential equation systems; however, a rigorous mathematical formalism for
non-autonomous dynamical systems is still needed . These observations may help
investigate long-distance transport phenomena as a possible cause of
variation in successively collected airborne samples such as the presence or
absence of a unique strain or species of microorganism. In addition, this
analysis is useful for planning geophysical sampling at a fixed location with
respect to forecast FTLE fields .
Because this study is motivated by aerial measurements in realistic
conditions, i.e., hundreds of collections of microorganisms from the
atmosphere with drones, it is necessary to consider the spatiotemporal
limitations of the available velocity field data. These limitations are
manifested in unresolved turbulence and impose uncertainties on the locations
of the source and destination points. For this reason, we use a Lagrangian
particle dispersion model to determine the probabilistic source (or
destination) regions and show how the concept of a local FTLE and
deterministic Lagrangian coherent structures (LCS) can explain the separation
between probabilistic source (or destination) regions, and may contribute to
understanding the geographic and genetic diversity observed in aerial samples
.
Results from this study can be applied to environmental applications such as
early warning systems for airborne pathogens, integrated pest management in
crops, and the collection of samplers from geophysical flows
.
This paper is outlined as follows. In Sect. we present
two observations associated with the estimation of the local FTLE and the
dispersion of destination (or source) points in flow fields. In
Sect. we show some numerical
examples and applications of presented observations in periodic and aperiodic
systems. In Sect. we consider the unresolved
turbulence and investigate the uncertainty of the backward and forward
trajectories and the resulting probabilistic source and destination regions.
Local finite-time Lyapunov exponent
In this section we present two independent observations related to the
estimation of the local FTLE and the distance between destination (or source)
points of successively released (or collected) particles in a time-varying
n-dimensional vector field,
dxdt=v(x,t),
where n = 2 for two-dimensional flows and n = 3 for
three-dimensional flows.
By local FTLE we mean the time-varying value of the FTLE field at an
arbitrary location x. Classically, the time-varying FTLE measures the
maximum separation rate between nearby particles when they are released in
the flow field at the same time (isochronic particles).
Figure a refers to this classical description. This
figure shows two isochrone particles that are close to each other at an
initial time t0. Under the effect of the flow field, the small
displacement vector between the two particles, δx, changes.
After an elapsed time T, the new vector between the two particles is
δxt0+T=ϕt0t0+T(x+δx)-ϕt0t0+T(x)=Dϕt0t0+T(x)+Oδxt02,
where ϕt0t0+T is the flow map for the vector field
(Eq. ) from time t0 to t0 + T,
Dϕt0t0+T = dϕt0t0+T(x)/dx
is the Jacobian of the flow map, and |⋅| is the Euclidean norm.
Two sequentially released (left panel)/collected (right panel)
particles at a fixed location shown by a bold cross. Particles 1 and 2 are
released/collected at t1 and t2 = t1 + δt,
respectively (the time interval between the two sequential samplings is
δt). The integration time between the destination/sources and the
release/sampling location is approximately T for both particles
(|T| ≫ δt). The displacement of the first particle during
δt is shown by
δ*.
Consider the right Cauchy–Green strain tensor, C(x, t0,
T) = Dϕt0t0+T(x)⊺Dϕt0t0+T(x).
For the sake of the following discussion, consider the situation of
incompressible two-dimensional flows, n = 2. The eigenvalues
λi and normalized eigenvectors ξi of C satisfy
Cξi=λiξi,ξi=1,i=1,2,0<λ1<1<λ2,ξ1⟂ξ2,
where the (x, t0, T) dependence of C, λi, and
ξi is understood. As illustrated in Fig. b,
the two eigenvectors, ξ1 and ξ2, are carried along by the flow
ϕt0t0+T to the two vectors r1 and r2, respectively, where
ri=Dϕt0t0+T(x)ξi,
whose lengths are scaled by a factor λi compared with the
normalized eigenvectors. The maximum possible separation between the released
particles after a time interval T, assuming a sufficiently small initial
distance |δx(t0)|, is
maxδxt0+T=λmaxCx,t0,Tδxt0
where λmax = λn.
The finite-time Lyapunov exponent (FTLE), with t0 and T fixed, is
considered a scalar field of the Lyapunov exponent as a function of initial
position, x,
σt0T(x)=1|T|lnλmax(C).
Similar to the calculation of maximum separation between two initially
neighboring points in a system of ordinary differential equations (ODEs) and
the corresponding maximum Lyapunov exponents, σt0T can be
used, via Eqs. ()
and (), to describe max|δx(t0 + T)|
as
maxδxt0+T=expσt0Tx,t0|T|δxt0.
In this study we are interested in particles that are released (or collected)
sequentially in time at a fixed location. Thus, the standard concept
of the FTLE, i.e., the separation rate of nearby isochronic points,
might not be applicable. Therefore, we present two independent observations
and show that we can (i) approximate the local FTLE by using the information
of local velocity and successive destination (or source) points, and
(ii) estimate the distance between the destination (or source) points by having
the local FTLE and velocity. These two observations require the assumption of
a time-dependent vector field, so that the initial displacement vector is not
along a common trajectory for sequential particles.
Referring to Fig. , left panel, assume
that two particles are sequentially released at t1 and
t2 = t1 + δt at the release location shown by the
×. The right panel corresponds to the analogous situation of
sequentially collected particles.
Observation I: the local FTLE value over the time
interval [t1, t2], given an appropriate
0 < δt ≪ |T|, can be approximated by
σ[t1,t2]T(x)=1|T|lnδx,T,t1,δtv‾x,t1,t2δt,
where δ(x,T, t1, δt) is the distance between
successive destination (or source) points corresponding to the elapsed
time T, and v‾(x, t1, t2) is the
(non-zero) average velocity at the release (or sampling) location during
[t1, t2].
Observation II: the distance between the
destination (or source) points of consecutively released (or collected)
particles can be estimated, given an appropriate δt, by the local
velocity and the true local FTLE at the release (or sampling) location as
δx,T,t1,δt=exp|T|σ(t1,t2)T(x)v‾x,t1,t2δt.
We suggest that observation I provides a recovered FTLE field, based on mild
assumptions that tend to hold in geophysical flows. Observation II is
important for sampling purposes, because it enables us to estimate the
distance between the source positions of consecutively collected particles,
if we have the local velocity and local FTLE data by separate means. We note
that observations I and II are independent, and the information on the
right-hand sides of Eqs. ()
and () is also assumed
known. For example, the local velocity could be obtained from an anemometer
or high-frequency radar in the ocean , or the local FTLE
could be obtained from a nowcast or forecast velocity field.
Forward FTLE field of a periodic double-gyre velocity field.
(a) The benchmark FTLE field at t0 = 1 calculated by
Eq. (). (b) The approximated FTLE field
calculated by Eq. (), and δt = 0.2
time units. A fourth-order Runge–Kutta integration scheme with constant
integration time step 0.01 and total integration time T = 15 time units
is implemented for both panels.
Two remarks are in order regarding these observations. (i) A proper choice of
δt, which depends on the spatiotemporal variability of the velocity
field, is critical for a good approximation of the local FTLE or the distance
between the destination (or source) points. If δt is chosen to be
too small, then variation of the velocity field would not be observed, the
two particles would essentially be following one another along a nearly
identical trajectory, and consequently the separation between the two
particles would lead to a null LE value, not maximal growth. However, if
δt is too large, then the initial particle separation at time
t2 is too large to justify the linearization assumption underlying FTLE
calculations; see, e.g., Eq. ().
Thus, a good selection of δt depends on the spatiotemporal
variability of the velocity field. (ii) A larger true local FTLE of the real
flow field yields a smaller error of estimations for the recovered local FTLE
and the distance between the destination (or source) points. This comes from
numerical evidence in the following sections.
The fundamental idea behind these observations is related to more general
methods of analysis of chaotic dynamical systems, often used in experimental
settings, namely, that the direction of maximum expansion dominates the
dynamics of typical displacement vector growth (see
Fig. b) . This
notion is generally accepted in settings assumed to be modeled by underlying
autonomous ODEs (whether known or unknown), but to our knowledge, there is no
similar theorem for non-autonomous ODEs. Our observations show that in a
time-dependent velocity field, with a proper choice of δt and
sufficiently large |T|, δ(x, T, t1, δt) is
often close to the maximum possible distance between the two particles. These
observations, at the present stage, are more of an ansatz, and may help
stimulate rigorous mathematical investigation related to separation of
non-isochronic nearby particles in a non-autonomous ODE setting.
In the following section we demonstrate some numerical verification and
applications of these observations.
Numerical examples and applications
Numerical examples of observations I and II, for periodic and aperiodic velocity fields
First, we study the well-known example of a periodic double-gyre. We consider
the same model and parameters introduced in Sect. 6, Example 1, of
. For observation I, we need to know
δ(x, T, t1, δt) and the local velocity. Therefore, we
use the double-gyre model to generate the velocity field and then exploit
that data to calculate the trajectories and the corresponding distance
between successively released particles after integration time T. We
repeat this procedure for all the grid points of the gyre domain [0, 2] × [0, 1].
Forward FTLE field of an aperiodic Rayleigh–Bénard convection
model. (a) The benchmark FTLE field at t0 = 1 calculated by
Eq. (). (b) The approximated FTLE field
calculated by Eq. (), and δt = 0.1
time units. A fourth-order Runge–Kutta integration scheme with constant
integration time step 0.01 and total integration time T = 75 units is
implemented for both panels.
Observations I and II at (x, y) = (0.3, 0.4) for a periodic
double-gyre velocity field. (a) Local benchmark and recovered
forward FTLE. (b) Benchmark and approximated final distance between
successively released particles corresponding to
T = 15.
Figure a shows the benchmark (true)
forward FTLE field corresponding to t0 = 1, calculated by
Eq. (), and Fig. b
shows the approximated forward FTLE field calculated by
Eq. (). A fourth-order Runge–Kutta integration
scheme with constant integration time step 0.01 and total integration time
T = 15 time units is implemented for both panels. For the recovered
(approximated) FTLE field, Fig. b, we
consider δt = 0.2 time units in
Eq. (). One can adjust parameters, e.g., T or
δt, to investigate their impact on the FTLE field.
To investigate observation I for an aperiodic system, we use a
two-dimensional model of the time-dependent Rayleigh–Bénard convection
model developed by and implemented by
to study unsteady flow separation on slip
boundaries. The streamfunction of this model is a function of position and a
stochastic time-dependent forcing term.
ψ(x,y,t)=Aksin{k[x-g(t)]}sin(2y)
Following , we generate the stochastic forcing,
g(t), based on a random Fourier spectrum with zero mean and unit impulse
covariance (see Fig. 5 in ).
Figure a shows the benchmark forward FTLE field
calculated by Eq. (), and Fig. b
shows the recovered forward FTLE field calculated by
Eq. () in the domain [0, 2] × [0,
π/2]. Two panels of this figure correspond to t0 = 1. Similar to
the previous example, a fourth-order Runge–Kutta integration scheme with
constant integration time step 0.01 is used. The total integration time is
T = 75 time units for both panels. For the recovered FTLE field,
Fig. b, we consider δt = 0.1 time
units in Eq. ().
By comparing the two panels of Figs.
and , respectively, one sees that the main features of the FTLE
field are recovered by Eq. (), and the benchmark
and approximated fields are highly correlated. However, in some areas (e.g.,
near (1.2, 0.5) in Fig. and (1.5, 0.8) in
Fig. ), we see discontinuities in the recovered
FTLE field. The reason might be that the selected δt is not a proper
choice in those regions. It is also important to note that we use a common
color scale for the two panels of Figs.
and , respectively. Therefore, minute differences
between the true and approximated fields are visually exaggerated because the
FTLE values are generally small in magnitude. Numerical comparison of the
results (see the next two numerical experiments) shows close approximation of
the recovered local FTLE to the true values.
Next, we investigate both observations I and II at an arbitrary point over
the time span [0, 10], which is one period of the double-gyre flow shown in Fig. . First, we
consider the point (x, y) = (0.3, 0.4) in the periodic double-gyre,
keeping all the parameters of FTLE computation the same as before (e.g.,
δt = 0.2). Figure a
shows the benchmark and approximated FTLE time series at that point. The
benchmark FTLE is calculated by Eq. () using the velocity
field information and the maximum eigenvalue of the Cauchy–Green strain
tensor. The approximated FTLE in this panel is calculated by
Eq. (). Information about δ and the local
velocity (i.e., v‾) are assumed to be known (in this
numerical experiment we obtain them from the velocity field).
Figure b shows the benchmark and
approximated distance between successively released particles after an
elapsed time T = 15. To calculate the benchmark time series, we use the
velocity field information to generate the trajectories and find the distance
between the successive particles. The approximated time series is generated
by Eq. (), with the provided
information about the local velocity and the local FTLE value.
Observations I and II at (x, y) = (1.3, 1.3) for an
aperiodic velocity field corresponding to a time-dependent
Rayleigh–Bénard convection model. (a) Local benchmark and
recovered forward FTLE. (b) Benchmark and approximated final
distance between successively released particles corresponding to
T = 75.
Next, we consider the point (x, y) = (1.3, 1.3) in the aperiodic
time-dependent Rayleigh–Bénard convection model, keeping all the
parameters of FTLE computation the same as before (e.g.,
δt = 0.1). Figure a
shows the benchmark and approximated FTLE time series at the selected point
and Fig. b shows the benchmark and
the approximated distance between successively released particles after an
elapsed time T = 75.
Figures
and show typical time series of the
recovered local FTLE and the distance between successively released (or
collected) particles. As one can observe, the two time series in panels a
and b are highly correlated, and the error of approximation is generally
small.
The error of approximation in observations I and II depends on many
parameters, for example, δt, T, and variation of the vector field
over the timescale δt. We leave the analysis of errors of
observations I and II for a future study.
Applications of the local FTLE observations I and II
Next, we consider the real-world wind data and focus on the backward FTLE
fields and the location of source points. This situation is important for
field studies for identifying potential source regions of plant pathogens and
their relative risk of transport to previously unexposed regions
. For this
purpose, we use observations I and II to compare the benchmark and the
recovered local backward FTLE time series and also the true and the estimated
distance of the source locations corresponding to the particles that were
collected at Virginia Tech's Kentland Farm, located at 37∘11′ N and
80∘35′ W. A large variety of microbial samples have been collected
at this location over the past 7 years (2006 to 2013) .
We refer to this point as (0, 0) in our plots.
(a) Trajectories of the collected particles during 24 h of
integration. (b) Sequential source points and the isochrone source
line. Sampling frequency is 1 h between 12:00 UTC 29 September and
12:00 UTC 30 September 2010, and the sampling location is at (0, 0)
(Virginia Tech Kentland Farm, 37∘11′ N and
80∘35′ W).
The flow maps are calculated by using numerical data corresponding to the
North America Mesoscale, NAM-218, provided by the National Oceanic and
Atmospheric Administration (NOAA) and National Centers for Environmental
Prediction's (NCEP) Operational Model Archive and Distribution
System (NOMADS) project.
The spatial resolution of this data set is about 12.1 km and the temporal
resolution is 3 h. All the trajectories are calculated by a fourth-order
Runge–Kutta integrator with a constant integration time step equal to
5 min. We use third-order splines for all necessary spatiotemporal
interpolations. We consider the time interval 12:00 UTC 29 September to
12:00 UTC 30 September 2010 for our numerical experiments and refer to it as
the interrogation window.
Figure shows the trajectories and the
initial positions of the indexed particles corresponding to the collected
particles at the sampling location during the interrogation window. The
frequency of sampling was 1 h and the backward time integration is 24 h for
all the particles. In addition, for simplicity and without losing generality
of the results, we perform the integration on a quasi-two-dimensional 850 mb
pressure surface . Indices of this figure
indicate the sampling times of the collected particles; for example,
index “12” that is located in the northwest of the figure refers to the
initial position of a particle that started at 12:00 UTC 28 September and
was collected 24 h later, i.e., 12:00 UTC 29 September, at the sampling
location. In terms of streaklines , this
line (see Fig. b) is composed of
contemporaneous points, e.g., 24 h, from the assembly of streaklines that
pass through the sampling location during the interrogation window. We define
this line as the isochrone source line since the integration time
from all points on it to the sampling location is equal, e.g., 24 h in this
example.
Following the assumptions of the local FTLE observations, i.e., a proper
δt with respect to the spatiotemporal variability of the velocity
field, we choose sampling periods from 0.1 to 1 h, and all the integration
is done in the same interrogation window.
Figure a shows the benchmark distance
between successive source points, i.e., δ(x, T, t1,
t2), during the interrogation window calculated from the available
velocity field data. We use the average velocity at the sampling location to
calculate δ* as |v‾(x, t1,
t2)δt|. Figure b shows the
recovered local FTLE time series, assuming that the true successive distances
are available.
(a) δ as the benchmark (true) distance between
successive source points corresponding to δt = 0.1, 0.5 and
1 h. The horizontal axis represents the averaged time corresponding to each
successive pair. (b) Approximated local FTLE for different
δts from 0.1 h (6 min) to 1 h. The interrogation window is
12:00 UTC 29 September to 12:00 UTC
30 September 2010.
Figures b
and demonstrate that we interpret a
local (backward) FTLE time series as differential stretching of line elements along an isochrone source line. To verify this result and to study
the effect of different δt's on the recovery of local FTLE time
series, we calculate the benchmark backward FTLE fields for the interrogation
window with integration time equal to 24 h.
Figure a shows an image of the true
time-varying FTLE field corresponding to 12:00 UTC 29 September 2010. To
give a sense of the changes of the FTLE field during the interrogation
window, we may describe the motion of the strong ridges of the field in
Fig. a toward a northwesterly direction, as
shown by the arrow. Figure b shows the
benchmark local FTLE value (black line) at the Kentland Farm during the
interrogation window. To generate this plot, we use Eq. ()
and calculate the backward FTLE field every 15 min; then, the time-varying
value of FTLE at (0, 0) is extracted. Also, to compare the results, the
recovered FTLE time series corresponding to δt = 0.1 h is
displayed in the same panel by the red line.
Figures b
and b indicate that (i) an optimal δt
for this example is between 0.1 and 0.5 h, and that (ii) the estimation
error is smaller for larger values of the true local FTLE. Therefore, we may
observe larger errors of estimation when (true) σ is close to zero,
e.g., between 00:00 and 04:00 UTC in Fig. b.
For δt = 0.1 h, we observe that the true and approximated local
FTLE time series are highly correlated, and their maxima (corresponding to
the local maxima of the FTLE field) are also at the same times (within
δt = ±0.1 h). Therefore, with a proper choice of
δts, the recovered local FTLE time series can accurately capture the
passage times of moving ridges of a FTLE field. Detecting these ridges is
important since they are candidates for hyperbolic LCSs in many geophysical
applications .
(a) The frozen image corresponding to 12:00 UCT
29 September 2010 of the backward FTLE field during the interrogation window.
Integration time is 24 h for FTLE calculations. The bold arrow shows the
general wind direction and the motion of the attracting LCS. (b) The
true (black) and recovered (red) local FTLE time series at the reference
point (0, 0). For the recovered time series (red), δt is equal to
0.1 h.
In addition, we investigate whether we can estimate the distances by using
observation II, provided there is necessary information about local velocity
and FTLE. Figure is a numerical example that
shows that the benchmark distance between the successive source points (black
line) is well approximated (red line) by observation II, i.e.,
Eq. (). Note that in this case we have
the data of the true local FTLE and the local velocity. In this figure we see
that at δt = 0.25 h, the estimated differential distance time
series is very close to the true answer, and it captures the correct times of
the local maxima.
This is an empirically important result, because one can schedule the
sampling of geophysical flows (e.g., with drones) based on the available
forecast FTLE fields and local velocity such that the successive collected
particles originate from the most possible diverse locations. In
Fig. it is evident that there are two
optimal time intervals, i.e., before and after 16:00 UTC, for maximal diversity
monitoring. To interpret this, consider
Fig. b and notice that the geographic
extent of the line segment from point 15 to point 16 is much larger than the
segment from point 13 to point 14.
Differential distance between the successive source points on the
isochrone source line corresponding to δt = 0.25 h. The black
line shows the benchmark and the red line shows the approximated time series
that is calculated by the local FTLE formula as
exp(|T|σ[t1,t2]T(x))|v‾(x,t1,t2)δt|.
The backward integration time for calculations of the flow maps is
T = 24 h and the interrogation window is 12:00 UTC 29 September to
12:00 UTC 30 September 2010.
A direct result of the local FTLE observations is the possibility of planning
for maximal geographic (and therefore also genetic) diversity monitoring such
that the collected particles come from the most separated source locations.
This means incorporating greater potential source areas, which could drive a
greater diversity of sample collection. Suppose that it is desired to
maximize the genetic diversity of microorganisms collected in a sample,
assuming that all the collected particles have approximately the same flight
time. Results of observation II indicate that to collect samples such that they originate from the most distant locations,
one should collect at times corresponding to the maxima of the local FTLE time series (note the high correlation between
the distance and the local FTLE time series in
Figs.
and b). To ensure that the particles are coming
from significantly separated locations, we may use the topology of the FTLE
field and collect the samples on either side of a strong attracting LCS
feature that corresponds to a local maximum of σ[t1,t2]T,
provided there is a short enough time between sampling periods. In this condition, a
high value of σ[t1,t2]T as the exponent in
Eq. () is the reason for having a large
δ. Figure schematically shows this strategy
when an attracting LCS feature passes over a fixed sampling location, causing
a dramatic change in the region of possible source points of collected
particles.
An attracting LCS feature (red) passes over the geographically fixed
sampling location (indicated by a bold ×). Black lines show
trajectories of hypothetical particles that are absorbed to a moving
attracting LCS. The bold arrow shows the general wind direction and the
motion of the attracting LCS at the specified interval. Collected samples on
either side of this attracting LCS feature come from two different regions.
As an example in realistic geophysical flow,
Fig. shows trajectories of three
hypothetical particles that are collected at (0, -100) km with respect to
the reference point. Backward integration time for specifying the
corresponding source points, i.e., A, B and C, and the trajectories is 40 h
for those three particles. The sampling times during the interrogation window
are 13:40 UTC for the red particle, 14:00 UTC for the blue particle and
14:10 UTC for the green particle. The green and blue particles are collected
on one side of an attracting LCS, but the red particle is collected on the
other side of the same LCS. As we observe, the source points corresponding to
blue and green particles, points B and C, are close. Meanwhile the source
point of the red particle, point A, is significantly far from the other two
particles. An interesting feature of this figure is that the separation of
the trajectories does not start from the sampling point, but as is shown, the
three trajectories remain close to each other for about 200 km and then
begin to diverge. This observation is directly related to the concept of the
FTLE, because σt0T is a function of the “final” separation
between nearby particles, and it does not specify the moment of divergence.
Three calculated trajectories of (hypothetical) collected samples.
The red and blue trajectories correspond to the samples on either side of a
LCS. The blue and green trajectories correspond to the samples on one side of
the same LCS. Sampling times are 13:40, 14:00 and 14:10 UTC during the
interrogation window (12:00 UTC 29 September to 12:00 UTC 30 September 2010)
for the red, blue and the green particles, respectively. Source points of the
collected particles are shown by A, B and C. The integration time for all
three particles is T = 40 h.
Referring to this example, observation II can help us to explain the seeming
association of sample diversity with high FTLE. There have been some reports
of significant characteristic variation of the collected particles, e.g.,
genetic types or aerial density of the microbial samples, during short
intervals when sampling coincides with a high-value local FTLE, or similarly,
passage of a strong LCS over the sampling location . In addition, a direct result of the local FTLE observations is
that when the local FTLE value is small during the sampling process, it is
expected that the collected particles originate from nearby source points,
assuming approximately the same flight times for them. This might be the
reason that the characteristics of the microbial samples remain
quasi-constant in consecutive collections, but differ as the time between
sample collections increases . This situation is similar
to sampling from a coherent set where the FTLE values are generally
small and the particles
have similar Lagrangian characteristics. Moreover, in cases where we observe
significant changes in collected samples while the local FTLE value is small,
we speculate that those changes are caused by local sources rather than
long-range transport phenomena . Thus, the local FTLE
concept helps us to include or exclude rare/unique microbes from specific
source regions. This sets the stage for additional work to be performed to
test hypotheses concerning the presence/absence of the unique microbes at the
potential source locations.
Unresolved turbulence and probabilistic regions
In this section we study the uncertainty in calculation of the source (or
destination) points due to unresolved turbulence and also the role of
high-value local FTLE and deterministic LCS in separation of the
probabilistic source (or destination) regions.
Precise calculation of the source (or destination) point of any collected (or
released) particle and the corresponding flow map require high-resolution
data of the velocity field. But geophysical data are always discrete and
spatially sparse. For example, spatial and temporal resolutions of
operational atmospheric data sets vary from the order of 10 to hundreds of kilometers and
3 h to longer intervals, respectively. Meanwhile, spatiotemporal scales of
atmospheric flows can be smaller than the resolution of the available data,
and we may lose important Lagrangian phenomena such as turbulent diffusion
and small-size eddies if we just consider available data
. Therefore, for realistic
calculation of the source (or destination) points, it is necessary to
consider the uncertainty of the trajectories. For this purpose, we consider a
Lagrangian particle dispersion model (LPDM) that provides the stochastic
component of the velocity with respect to the available deterministic
(background) data . In LPDM, the velocity vector at each point, v(x,
t), is assumed to be the sum of a deterministic term,
v‾(x, t), and a random variable, V(x,
v‾, t) that depends explicitly on the instantaneous
position of the particle x, its deterministic velocity
v‾ at that location and the time t; see
Eq. (). Later, we see how this dependency dictates
two different solutions for the calculations of the probabilistic source and
destination regions .
v(x,t)=v‾(x,t)+V(x,v‾,t)
The stochastic term of Eq. () is a Markov-chain
process as a function of the velocity deformation tensor and the Lagrangian
timescale of the flow field,
V(t+Δt)=RΔtVt+1-RΔt20.5N(0,1)κ/TL,
where V shows each component of the stochastic velocity term V,
and the correlation coefficient RΔt is a measure of the
association between stochastic velocities in successive time steps. Also,
N is a normal distribution with mean zero and unit standard
deviation. The correlation coefficient
RΔt=exp-Δt/TL
is a function of the integration time step, Δt, and the Lagrangian
timescale of the flow field, TL, which is on the order of
104 s. The term κ depends on the gradient of the instantaneous
deterministic velocity, v‾ = (u‾,
v‾), the grid size of the meteorological data, χ, and an
empirical constant, c:
κ=2-0.5(cχ)2∂v‾∂x+∂u‾∂y2+∂u‾∂x-∂v‾∂y20.5.
Because κ depends on the gradient of the background velocity, one can
easily use the set of
Eqs. ()–()
for forward integration. Using this set for simple backward
integration requires presumption about the position of a particle at specific
times, which leads to misleading results. Therefore, we have to consider two
distinct cases, (i) calculation of the probabilistic destination region of a
released particle, and (ii) calculation of the probabilistic source region of
a collected particle. In this study we discuss both cases, but like before,
emphasize the probabilistic source regions (corresponding to the backward
trajectories). We also revisit the problem of a local FTLE and successive
sampling in the presence of unresolved turbulence. Our numerical results show
that if successive sampling is performed on either side of a strong
attracting LCS (represented by the temporal peaks in the local backward FTLE
time series), the probabilistic source regions are significantly separated,
similar to the deterministic case.
To focus on the main concerns of this study and to avoid complexity, we
proceed with a two-dimensional velocity field similar to the previous
sections. However, this approach can be extended to
three-dimensional fields by adding an appropriate stochastic term in the extended
direction .
Probabilistic source and destination regions
(i) The probabilistic destination region is the probability distribution of
the final positions of virtually released particles after integration
time T when the initial position is known precisely, e.g., a Dirac delta
function. The case of forward integration and related calculations of a
probabilistic distribution is equivalent to solving the Fokker–Planck or
Kolmogorov forward equations ,
which describe the future of a probability distribution function of a known
initial condition that evolves under the dynamics of a system, e.g., a
diffusion process.
Because the time-varying vector fields are usually complicated, analytical
solutions for probabilistic destination regions are not available, and it is
necessary to use numerical solutions. For this aim, we discretize the domain
of our interest into sufficiently small boxes and use the Monte Carlo method
by releasing a sufficient number of independent particles from a box that
includes the release point.
Figure a shows this
procedure. By choosing an appropriate integration time step, we calculate the
trajectories. By completion of the integration process, we have a
distribution of particles in different boxes. If the total number of released
particles is sufficiently large and the boxes' dimensions are sufficiently
small, then the ratio of the virtual particles in each box to the total
number of released particles shows the probability distribution of the
destination region. By increasing the number of virtual particles and
decreasing the size of the boxes, the calculated distribution becomes
invariant.
(a) A solution for the probability distribution of a
forward case. Virtual particles are released from a box that includes the
release location. Distribution of the final positions after integration
time T would specify the probabilistic destination region. Calculation of
the probabilistic destination region is equivalent to the solution of a
Fokker–Planck equation for finding the future probability distribution of an
initially known distribution. Trajectories of the released particles from the
initial box are shown in green. (b) A solution for the probability
distribution of a source region. For a proper forward-time integration, the starting time is shifted to
t0 - T. Virtual particles are released from all the boxes
in the domain. Important particles are those that land in the target box,
which includes the sampling location. Trajectories of particles that land in
the target box are shown by green; other trajectories are shown by red. A
solution for the probabilistic source region is conceptually the same as the
solution of the backward Kolmogorov equation, where an initial probability
distribution is the desired solution such that in a future time the system
will have a specified probability
distribution.
(ii) The solution for a probabilistic source region is conceptually the same
as solving the Kolmogorov backward problem . In
mathematical terms, at time t0 - T (T > 0 is the integration
time), we investigate for a specific source distribution such that in a
future time, i.e., t0, the system will be in a given target set. A
probabilistic source region cannot be determined by simply performing
backward time integration, because κ in
Eq. () and consequently the
stochastic velocity term are determined by the instantaneous background
velocity that depends on the location and time. Naively applying the backward
time integration produces a series of “false” displacement vectors. The
cumulative effect of these false displacements yields a false probabilistic
source region. To solve this problem, we first discretize the domain of the
flow field into small boxes. Then, we shift the starting time to
t0 - T and consider the velocity field at this new time frame. By
this means, we convert this problem into a forward integration problem from
t0 - T to t0. At t0 - T we release a sufficient
number of independent particles from all boxes of the domain (this
step is the major difference between the current and previous case). By
forward integration from t0 - T to time t0, we find the
landing location of all released particles. The influential particles in this
procedure are those that land inside the sampling box, e.g., the particles
associated with the green trajectories in
Fig. b. In this
figure, those boxes that have contributed to the particles ending up in the
target box “j” are hatched. As we observe, there may be particles from
contributing boxes that do not land in the target box (shown by red
trajectories).
In Fig. b the boxes
are labeled by i = 1, 2, ⋯, nb, where nb
is the number of boxes and the sampling box is shown by index j. We denote
the number of particles that start from box i at time t0 - T and
are in box j at time t0 by ni→j. We calculate the relative
contribution of each source box as
γi=ni→j∑ini→j,
where ∑ini→j shows the total number of particles that land
in the sampling (target) box j and γi is the chance of a
collected particle coming from a specific box i. Therefore, the
distribution of γ over the domain approximates the probability
distribution of the source region. This procedure generates the correct
probabilistic source region, but its numerical efficiency is not high because
many, e.g., 106, independent particles are released from all boxes of
the domain, but only those particles that land in the sampling box are
counted. Thus, there are a huge number of calculated trajectories that are
left out. It is not the purpose of this study, but one can increase the
efficiency of this procedure by applying some optimization methods, for
example, sequential release of particles from large boxes that are inside a
circle centered at the sampling box and by identifying the regions with
maximum contributions. The radius of that circle can be determined by
statistical information about the mean velocity and the integration time.
After that, one may focus on those important regions by partitioning them
into smaller boxes and increasing the number of released particles to
determine fine structures of the probabilistic source region. For more
information regarding this problem, one can refer to the STILT
project .
Probabilistic source region and local FTLE observations
To investigate a realistic example of probabilistic source regions and the
applicability of the presented observations, we revisit the case study of
Sect. . Figure
shows one example of a probabilistic source region where the color intensity
determines the relative contribution of each source box. In this case the
sampling location is at (0, -100) km with respect to our reference point.
The sampling time is 14:10 UTC 29 September 2010 and the total elapsed time
for trajectory calculations is T = 40 h. This figure is the stochastic
equivalent of the source point of the particle whose trajectory is shown by
the green line in Fig. . For this
calculation, 105 particles are released from each 10 × 10 km
box. After trial and error experimentation, the search area for this specific
problem is considered to be the 900 km × 600 km rectangular grid shown in Fig. b.
Considering the size of the boxes, the total number of released particles and
calculated trajectories is 5.4 × 108 in each integration time
step.
(a) The probabilistic equivalent of the source point of the
(virtually) green particle in Fig. . The
sampling point S is located at (0, -100) km with respect to our reference
point and the sampling time is 14:10 UTC 29 September 2010.
(b) Details of the probabilistic source region that is composed of
5400 boxes, each 10 km × 10 km. Color intensity shows the relative
contribution of each source box. (c) γ, the relative
contribution of source boxes along the specified line
PQ.
An important point about probabilistic source and destination regions is that
although at each time step the stochastic velocity term has a Gaussian
distribution (recalling
Eq. ), the final distribution
of particles is not necessarily Gaussian. The reason for this fact is the
cumulative effects of the variability of the variance of normal distribution,
κ/TL, that is, a function of the gradient of
instantaneous velocity. In general, for small integration time, the
probability distribution of the source (or destination) region is close to a
Gaussian distribution, but as the integration time increases, the
corresponding distribution diverges from a normal one. For example, visual
inspection of Fig. b indicates that the final
distribution of the probable source points is not Gaussian. In
Fig. c the relative contribution of the source
boxes along the specified line PQ is shown. Standard statistical tests such
as the Kolmogorov–Smirnov test confirm that
the distribution is not Gaussian.
In Fig. we show that the source locations
of two collected particles on either side of an attracting LCS are much
further apart than the source points of two successive collected particles on
one side of the same LCS. We want to investigate whether this result is still
valid in the presence of unresolved turbulence. If that result holds, then in
practical applications such as sampling of the microbial structure of the
atmosphere, we can have reasonable confidence about the separation of the
probabilistic source regions based solely on a deterministic analysis, that
is, without performing bothersome probabilistic calculations. Therefore, we
study a case where we know its deterministic dynamics.
Figure shows the evolution of the
probabilistic source regions “A” and “B” (shown in
Fig. a) corresponding to the (virtually)
red and blue particles of Fig. ,
respectively. The total integration time for this example is 40 h. In each
panel of this figure we also show the attracting hyperbolic LCSs according to
and . For calculation of
each probabilistic region of this figure, 105 particles are released
from each small 10 km × 10 km box. By comparing
Figs.
and a, we observe that the probabilistic
source regions contain the deterministic source points, and they are
significantly separated from each other. Also, we see how the two
probabilistic regions contract and become closer to the attracting LCS as
they get closer to the sampling point. One noticeable feature in this figure
is the difference between the shapes of the two source regions, while the two
samplings are separated by only 20 min.
Sequence of the hyperbolic LCSs (blue) and two probabilistic source
regions corresponding to two successive samples. Probabilistic regions “A”
and “B” (a) correspond to the virtually red and blue particles in
Fig. . These six panels correspond to 40,
30, 20, 10, 5 and 0 h before collecting the corresponding samples at
13:40 and 14:00 UTC during the interrogation window (see the Supplement video).
Results of this example show that, similar to the consequences of observation
II, the probabilistic source regions corresponding to the collected particles
on either side of a deterministic attracting LCS are significantly separated
in backward time.
Conclusions
FTLE fields provide useful information about large-scale transport phenomena
and also Lagrangian structures of flow fields, particularly geophysical
flows. However, in field experiments the data are on a much more modest
scale. Therefore, it is necessary to bridge the gap between the concept of
large-scale FTLE fields and local experiments. To fill that gap, we propose a
methodology that is an ansatz that is closely related to the concept of local
Lyapunov exponents and the direction of maximum expansion in autonomous
ordinary differential equation systems (a rigorous mathematical formalism for
non-autonomous dynamical systems is still needed). Our observations
correspond to (i) estimation of the local FTLE, given the local velocity and
the distance between sequentially released (or collected) particles, and
(ii) estimation of the distances between the destination (or source) points
of sequentially released (or collected) particles assuming the availability
of the local velocity and local FTLE.
These observations were motivated in part by our previous work examining the
dynamics of assemblages of microorganisms in the lower atmosphere. We
numerically demonstrate the results of our observations for a periodic
velocity field (i.e., a double-gyre), an aperiodic system (i.e., a
Rayleigh–Bénard convection model) and real-world wind data. The
suggested notion is useful in practical cases where we have samples of
particles (e.g., microbes) collected at a fixed location, and we are
interested in formulating hypotheses about their origin, structure, and
potential transport phenomena driving their atmospheric movement. In
addition, we show that the concept of local FTLE and observation II can be
applied to scheduling of atmospheric sampling missions to collect high-diversity samples.
We also investigate the unresolved turbulence and the probabilistic
description of the source (or destination) points. We use the box
discretization method and discuss the important differences between
calculation methods of the probabilistic source and destination regions.
Furthermore, we show that because the stochastic velocity is a function of
instantaneous background velocity, the probabilistic source (or destination)
regions are not necessarily Gaussian. Finally, we study the probabilistic
source regions corresponding to successively collected particles on either side
of a strong hyperbolic attracting LCS – or equivalently, a local maximum of
the local FTLE time series – and demonstrate that one may trust the
estimated results of deterministic calculations of source (or destination)
points in realistic geophysical flows.
Results of this study can aid in optimizing the sampling schedules of passive
particles and understanding of the outcomes of local observations in
geophysical flows, based on large-scale transport features.