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Barriers to the Transport of Diffusive Scalars in Compressible Flows

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Our recent work identifies material surfaces in incompressible flows that extremize the transport of an arbitrary, weakly diffusive scalar field relative to neighboring surfaces. Such barriers and enhancers of transport can be located directly from the deterministic component of the velocity field without diffusive or stochastic simulations. Here we extend these results to compressible flows and to diffusive concentration fields affected by sources or sinks, as well as by spontaneous decay. We construct diffusive transport extremizers with and without constraining them on a specific initial concentration distribution. For two-dimensional flows, we obtain explicit differential equations and a diagnostic scalar field that identify the most observable extremizers with pointwise uniform transport density. We illustrate our results by uncovering diffusion barriers and enhancers in analytic, numerical, and observational velocity fields.
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Barriers to the Transport of Diffusive Scalars in
Compressible Flows
George Hallera,
, Daniel Karraschband Florian Kogelbauera
February 27, 2019
(a) Institute for Mechanical Systems, ETH Zürich, Leonhardstrasse 21, 8092 Zürich, Switzerland
(b) Zentrum Mathematik M3, Technische Universität München, Boltzmannstrasse 3, 85748
Garching, Germany
Abstract
Our recent work identifies material surfaces in incompressible flows that extremize the trans-
port of an arbitrary, weakly diffusive scalar field relative to neighboring surfaces. Such barriers
and enhancers of transport can be located directly from the deterministic component of the
velocity field without diffusive or stochastic simulations. Here we extend these results to com-
pressible flows and to diffusive concentration fields affected by sources or sinks, as well as by
spontaneous decay. We construct diffusive transport extremizers with and without constrain-
ing them on a specific initial concentration distribution. For two-dimensional flows, we obtain
explicit differential equations and a diagnostic scalar field that identify the most observable
extremizers with pointwise uniform transport density. We illustrate our results by uncovering
diffusion barriers and enhancers in analytic, numerical, and observational velocity fields.
1 Introduction
Transport barriers can informally be defined as observed inhibitors of the spread of substances in a
flow. They are well documented in geophysics [33], classical fluid dynamics [23], plasma fusion [6],
reactive flows [27] and molecular dynamics [32], yet no general theory for them has been available
until recently. In [13], we have put forward such a theory for incompressible flows and weakly
diffusing substances by defining and solving an extremum problem for material surfaces that block
the diffusive transport of passive scalars more than neighboring surfaces do.
The a priori restriction of this optimization problem to material surfaces (codimension-one in-
variant manifolds of the flow in the extended phase space of positions and time) is justified by the
complete lack of advective transport across such invariant surfaces. Indeed, for small enough dif-
fusivities, pointwise, finite-time, advective transport through any non-material (i.e., non-invariant)
surface is always larger than diffusive transport. As a consequence, one should seek universal,
diffusion-independent transport barriers among material surfaces.
Such barriers turn out to be computable and depend on the structure of the diffusion tensor
but not on the actual value of the diffusivities [13]. Thus, diffusion barriers remain well defined
in the limit of non-diffusive, purely advective transport. In that limit, they form surfaces that
will prevail as transport inhibitors or enhancers under the presence of the slightest diffusion or
uncertainty modeled by Brownian motion. This gives a transport-barrier definition in the advective
limit, independent of any preferred coherence principle and based solely on the physically well-
defined and quantitative notion of diffusive transport through a surface. This limiting property
of diffusion barriers eliminates the current ambiguity in locating coherent structures in finite-time,
advective transport where different coherence principles give different results on the same flow [9].
Corresponding author. Email: georgehaller@ethz.ch
1
arXiv:1902.09786v1 [physics.flu-dyn] 26 Feb 2019
Unlike set-based approaches to coherent advective transport, the approach in [13] does not require
diffusion barriers to be closed, and hence also finds open bottlenecks to transport such as fronts and
jets. This feature of the method is also important for closed diffusion barriers to remain detectable
even if they do not lie entirely in the domain where velocity data is available.
While valid in arbitrary dimension, the results in [13] rely explicitly on the assumption that the
flow carrying the concentration field of interest is incompressible. Fluid flows arising in applications
are indeed practically incompressible, but air flows are relatively easy to compress. This precludes
the application of [13] to atmospheric transport problems, such as the identification of temperature
barriers surrounding the polar vortices (cf. [2, 4, 17, 29]). Notable compressibility also arises in two-
dimensional velocity fields representing horizontal slices of planetary atmospheres, obtained from
observations [10] or from numerical models [1]. The dramatic accumulation of oil and flotsam on the
ocean surface [5], as well as the characteristically non-conservative surface patterns formed by algae
[34], also necessitate the use of two-dimensional numerical models with significantly compressible,
two-dimensional velocity fields.
These examples of compressible velocity fields nevertheless invariably conserve mass. For in-
stance, oil remains buoyant and hence confined to the ocean surface, thus there is no significant loss
in the total oil mass in the absence of other processes eroding it. Accordingly, a velocity field model
for surface oil transport should be mass conserving. Inspired by such examples, we consider here
diffusive transport in the presence of a carrier flow that may be compressible but conserves mass.
At the same time, we also allow for variations of the transported concentration field due to effects
beyond diffusion. These effects include contribution from distributed sources and sinks, as well as
spontaneous concentration decay in time governed by a potentially time-dependent decay rate.
A number of prior approaches exist to weakly diffusive transport (see, e.g., [33] for a survey), but
only a handful of these target structures in the compressible advection-diffusion equation. Among
these, [30, 31] recast the advection-diffusion equation in Lagrangian coordinates and suggest a quasi-
reduction to a one-dimensional diffusion PDE along the most contracting direction. While this
approach yields formal asymptotic scaling laws for stretching and folding statistics along chaotic
trajectories, such trajectories become undefined for finite-time data sets that we seek to analyze
here. The residual velocity field of [24] offers an attractive visualization tool for regions of enhanced
or suppressed transport, but requires already performed diffusive simulations as input, rather than
providing predictions for them from velocity data. The popular effective diffusivity approach of [22] is
based on the assumption of incompressibility (conservation of area), and hence becomes inapplicable
to compressible flows. We finally mention recent work in [15] which provides an advection–diffusion
interpretation for the compressible dynamic isoperimetry methodology developed in [8]. This lat-
ter, set-based approach targets metastable or almost-invariant sets in a purely advective transport
context.
Our analysis here considers a mass-based (rather than volume-based) concentration field c(x, t).
In the absence of diffusion, spontaneous concentration-decay and concentration sources, the trans-
port of c(x, t)in and out of an evolving material volume V(t)would be pointwise zero due to the
conservation of the mass of V(t)by the flow. Source terms and spontaneous concentration decay
add a deterministic evolution to the concentration along particle trajectories, and hence the initial
concentration remains deterministically reproducible, i.e., a conserved quantity along all particle
motions in the absence of diffusion.
The presence of diffusion, however, erodes this conservation law, as if trajectories were stochastic
and hence the value of the initial concentration along them could not be immediately reproduced
just from the knowledge of the present concentration, the initial time and initial location along a
fluid particle trajectory. Here, we will seek transport barriers as material surfaces across which this
diffusive erosion of initial concentrations is stationary when compared to nearby material surfaces.
For incompressible flows, this barrier concept will turn out to simplify exactly to the concept of
most impermeable material barriers to diffusion, as developed in [13]. In the present work, we will
collectively refer to stationary surfaces (minimizers, maximizers and saddle-type surfaces) of diffusive
transport as transport barriers without performing a second-order calculation to identify their types.
2
Beyond adding the effects of compressibility, sources, sinks and spontaneous decay, our present
analysis performs the transport extremization both with and without conditioning it on a known
initial concentration field. In this context, unconstrained barriers are material surfaces that prevail
as stationary surfaces of transport even under concentration gradients initially normal to them.
Constrained barriers, in contrast, are stationary surfaces of transport under a fixed initial diffusion-
gradient configuration. We derive mathematical criteria for both types of barriers and illustrate
these criteria first on explicitly known Navier–Stokes flows, then on observational and numerical
ocean data.
Our analysis and examples show that several well documented features in a diffusing scalar
field, such as jets and fronts, are technically not minimizers of diffusive transport when constrained
on a given initial scalar field. This is at odds with the usual terminology by which surfaces of
large concentration gradients are generally referred to as transport barriers, even though the actual
transport through them appears large precisely because of those large gradients. This paradox has
already been pointed out in the Eulerian frame but has remained unresolved [22]. Here we recover
the same effect in rigorous terms in the Lagrangian frame, and find that barriers are transport
maximizers with respect to all localized perturbations
2 Set-up
We consider the compressible advection-diffusion equation for a mass-unit-based concentration field
c(x, t)in the general form
t(ρc) + ∇· (ρcv) = ν·(ρDc)k(t)ρc +f(x, t)ρ, (2.1)
c(x, t0) = c0(x), ρ(x, t0) = ρ0(x).
Here denotes the gradient operation with respect to the spatial variable xURnon a compact
domain Uwith n1;v(x, t)is an n-dimensional smooth, mass-conserving velocity field generating
the advective transport of c(x, t)whose initial distribution is c0(x);D(x, t) = DT(x, t)Rn×n
is a dimensionless, positive definite diffusion-structure tensor describing possible inhomogeneity,
anisotropy and temporal variation in the diffusive transport of c;ρ(x, t)>0is the mass-density of
the carrier medium; ν0is a small diffusivity parameter rendering the full diffusion tensor νDsmall
in norm; k(x, t)is a space- and time-dependent coefficient governing spontaneous concentration decay
in the absence of diffusion; and f(x, t)describes the spatiotemporal sink- and source-distribution
for the concentration. We assume that the initial concentration c(x, t0) = c0(x)is of class C2, and
D(x, t),k(t)and f(x, t)are at least Hölder-continuous, which certainly holds if they are continuously
differentiable.
Without the spontaneous decay and source terms, eq. (2.1) was apparently first obtained by
Landau and Lifschitz [16] as a compressible, non-Fickian advection-diffusion equation for ρc (see
also Thiffeault [31]). We note, however, that with the modified velocity field
w=v+ν
ρDρ, (2.2)
eq. (2.1) can also be recast as
t(ρc) + ∇· (ρcw) = ν·(D(ρc)) k(t)ρc +f(x, t)ρ, (2.3)
an advection-diffusion equation with classic Fickian diffusion for the scalar field ρc under the modified
velocity field w.
Given a carrier velocity field v(x, t)of general divergence ∇ · v(x, t), the density ρfeatured in
eqs. (2.1)-(2.3) must satisfy the equation of continuity
tρ+∇· (ρv)=0.(2.4)
3
Combining the continuity equation (2.4) with (2.1) gives an alternative form of the compressible
advection-diffusion equation as
Dc
Dt =1
ρν·(ρDc)kc +f. (2.5)
The flow map induced by the velocity field vis Ft
t0:x07→ x(t;t0,x0), mapping initial positions
x0Uto their later positions at time t. We assume that all trajectories stay in the domain
Uof known velocities, i.e., Ft
t0(U)Uholds for all times tof interest. We will be studying
diffusive transport through material surfaces which are time-dependent families of codimension-one
differentiable manifolds satisfying
M(t) = Ft
t0(M0)U,
with M(t0) = M0denoting the initial position of the material surface. Note that (M(t), t)is an n-
dimensional invariant manifold in the extended phase space of the non-autonomous ODE ˙
x=v(x, t).
We denote by 0Ft
t0the gradient of Ft
t0with respect to initial positions x0, satisfying
det 0Ft
t0(x0) = exp Zt
t0
·vFs
t0(x0), sds. (2.6)
The equation of continuity (2.4) together with (2.6) then yields the relation
ρFt
t0(x0), t=ρ0(x0) exp Zt
t0
·vFs
t0(x0), sds=ρ0(x0)
det 0Ft
t0(x0).
The smallness of the diffusivity parameter νis not just a convenient mathematical assumption:
most diffusive processes in nature have very small νvalues (i.e., large Péclet numbers) associated
with them (see, e.g., Weiss and Provenzale [33]). The smallness of ν, however, does not automatically
allow for simple perturbation approaches, because (2.1) is a singularly perturbed PDE for such ν
values.
3 The compressible diffusion barrier problem
To formulate the compressible diffusion barrier problem outlined in the Introduction in precise terms,
we first observe that for ν= 0, eq. (2.5) is solved by
c(x, t) = eRt
t0k(s)dsc0(Ft0
t(x)) + Zt
t0
eRt
sk(σ)f(Fs
t(x), s)ds.
Therefore, the function
µ(x, t):=eRt
t0k(s)dsc(x, t)Zt
t0
eRs
t0k(σ)f(Fs
t(x), s)ds, (3.1)
returning the initial concentration at time t0along characteristics of (2.5), is conserved along tra-
jectories, given that µ(x(t), t) = c0(Ft0
t(x)) c0(x0).
For nonzero νvalues, µis no longer conserved along trajectories of v(x, t). In that case,
D
Dt µ(x(t), t)measures the irreversibility in the evolution of c(x, t)along trajectories. Specifically, we
have
D
Dt µ(x, t) = D
Dt eRt
t0k(s)dsc(Ft
t0(x0), t)Zt
t0
eRs
t0k(σ)f(Fs
t0(x0), s)ds
=k(t)eRt
t0k(s)dscFt
t0(x0), t+eRt
t0k(s)ds D
Dt cFt
t0(x0), teRt
t0k(s)dsfFt
t0(x0), t
=ν1
ρ(x, t)·ρ(x, t)D(x, t)µ(x, t) + Zt
t0
eRs
t0k(σ)f(Fs
t(x), s)ds.(3.2)
4
This latter PDE for the evolution of µ(x, t)can also be rewritten in Lagrangian coordinates applying
the formulas of Tang and Boozer [30] and Thiffeault [31] that transform the classic advection-diffusion
equation to Lagrangian coordinates. When applied to ˆµ(x0, t) := µ(Ft
t0(x0), t), those formulas
directly give
tˆµ(x0, t) = ν1
ρ0(x0)0·Tt
t0(x0)0[ˆµ(x0, t) + b(x0, t)],(3.3)
with the notation
Tt
t0(x0):=ρ0(x0)0Ft
t0(x0)1D(Ft
t0(x0), t)0Ft
t0(x0)T,(3.4)
b(x0, t):=Zt
t0
eRs
t0k(σ)f(Fs
t0(x0), s)ds.
The definition of the transport tensor Tt
t0(x0)in (3.4) is similar to that in [13] but the present
definition also contains the initial density ρ0(x0)as a factor and no longer assumes the flow map to
be volume-preserving.
Remark 1.It will be crucial in our present derivation that no spatially dependent terms beyond the
initial density remain in front of the divergence operation in eq. (3.2). That is only the case if the
flow map of the characteristics of eq. (2.1) is linear in x0for ν= 0. This, in turn, only holds if the
right-hand side of (2.1) is a linear function of c(x, t), as we have assumed. Consequently, (2.1) is the
broadest class of PDEs to which our present approach is applicable.
The following result is critical to our analysis, establishing a leading-order formula for the total
transport of the scalar field µ(x, t)field through an evolving material surface.
Theorem 1. The total transport of µthrough an arbitrary, evolving material surface M(t) =
Ft
t0(M0)over the time interval [t0, t1]is given by
Σt1
t0(M0) = νZt0
t0ZM0Tt
t0(0c0(x0) + 0b(x0,)) t, n0dA0dt +o(ν),(3.5)
with o(ν)denoting a quantity that, for ν0,tends to zero pointwise at any x0∈ M0even after
division by ν.
Proof. See section §A.
Next, we will seek diffusion barriers as codimension-one stationary surfaces of the leading-order
term in the expression of Σt1
t0(M0)in two different settings. First, we consider the initial tracer
concentration unknown or uncertain, and assume the most diffusion-prone initial concentration
distribution for calong each material surface. Next, we consider an arbitrary but fixed initial
concentration and seek material surfaces that render diffusive transport stationary under this initial
concentration.
4 Unconstrained diffusion barriers
To compare the intrinsic ability of different material surfaces to withstand diffusion, we now subject
each material surface to the same, locally customized initial concentration gradient setting that
makes the surface a priori maximally conducive to diffusive transport. Specifically, we initialize the
initial concentration along the initial position M0of any material surface in such a way that
0c0(x0) = K0
ναn0(x0), ν > 0,x0∈ M0,(4.1)
for some constant α(0,1). In other words, we prescribe uniformly high concentration gradients
along M0that are perfectly aligned with the normals of M0and grow as ναas ν0. We will
5
refer to (4.1) as the uniformity assumption. This assumption focuses our analysis on the intrinsic
ability of a material surface to block diffusion, rather than on its position relative to features present
in a specific initial concentration field.
Remark 2.The uniformity assumption in [13] is a specific case of (4.1) with α= 0.If sinks and
sources are absent (f(x, t)0), as is the case in [13], we can also select α= 0 in (4.1) and still
obtain the upcoming results of this section.
By the compactness of Uand the time interval [t0, t1],we can also select a constant bound
M0>0so that
(t1t0)Zt1
t0
01
ρ0(x0)0·Ts
t0(x0)0b(x0, s)dsM0
for all x0U, given that Ts
t0(x0)0b(x0, s)is assumed C1for all svalues. We can then rewrite
Σt1
t0(M0)in (3.5) as
Σt1
t0(M0) = ν(t1t0)KZM0¯
Tt1
t0n0,n0dA0+o(ν) + OνM0ρ0να
K0.
This leads to the normalized total transport of µ(x, t)in the form
˜
Σt1
t0(M0) := Σt1
t0(M0)
νK (t1t0)A0(M0)=Tt1
t0(M0) + o(να), α (0,1],(4.2)
where the transport functional,
Tt1
t0(M0) := RM0n0,¯
Tt1
t0n0dA0
RM0dA0
,(4.3)
measures the leading-order diffusive transport of c(x, t)through the material surface M(t)over the
period [t0, t1]. This functional formally coincides with the transport functional obtained in [13] for
incompressible flows with k(t)f(x, t)0in the PDE (2.1). The only, minor difference here is
in the definition (3.4) of ¯
Tt1
t0, which now features the initial density field ρ0(x0). Importantly, the
present theory returns the results of the incompressible theory when applied to incompressible flows
with uniform density. We stress that Tt1
t0(M0)can be computed for any initial surface M0directly
from the trajectories of vwithout solving the PDE (2.1).
We propose that what makes a barrier most observable is a near-uniform concentration jump
along it. By continuity with respect to all quantities involved over finite time intervals, however,
surfaces delineating near-uniform concentration jumps form continuous families and hence cannot
be uniquely identified. The theoretical centerpiece of such a family is still well defined by uniform
barriers which are characterized by constant pointwise transport density at leading order. By formula
(4.3), these surfaces satisfy
n0,¯
Tt1
t0n0=T0=const. (4.4)
Because of the formal coincidence between the transport functional Tt1
t0(M0)defined in (4.3) and
that arising in the incompressible case, the general necessary criterion of [13] for uniform barriers
remains valid here and can be stated with the help of the tensor family
ET0(x0) := ¯
Tt1
t0− T0I(4.5)
as follows.
Theorem 2. Under assumption (4.1), a uniform minimizer M0of the transport functional Tt1
t0is
necessarily a non-negatively traced null-surface of the tensor field ET, i.e,
hn0(x0),ET0(x0)n0(x0)i= 0,trace ET0(x0)0,(4.6)
6
holds at every point x0∈ M0with unit normal n0(x0)to M0. Similarly, a uniform maximizer M0
of Tt1
t0is necessarily a non-positively traced null surface of the tensor field ET, i.e,
hn0(x0),ET0(x0)n0(x0)i= 0,trace ET0(x0)0,(4.7)
holds at every point x0∈ M0.
Finally, by direct analogy with the incompressible case treated in [13], a diagnostic field measuring
the strength of unconstrained diffusion barriers is given by the Diffusion Barriers Strength (DBS)
field, defined as
DBSt1
t0(x0) = trace ¯
Tt1
t0(x0).(4.8)
This follows because, as shown in [13], the leading-order change in the non-dimensionalized transport
functional Tt1
t0under a small, surface-area-preserving perturbation localized in an O()neighborhood
of a point x0∈ M0is given by DBSt1
t0(x0). Of all uniform extremizers, therefore, those along the
ridges of DBSt1
t0(x0)will prevail as the strongest inhibitors or enhancers of diffusive transport.
4.1 Relationship between diffusion barriers and classic invariant mani-
folds
Assume that the diffusion structure tensor is DI(homogeneous and isotropic diffusion), ρ0(x0) =
ρ0=const. (homogeneous initial density) and the conservation law (5.4) holds for t1(t0,)over
the material surface evolving from M0under the flow map. This implies
n0,1
t1t0Zt1
t0
ρ00Ft
t0(x0)10Ft
t0(x0)Tdtn0=T0,
or, equivalently,
1
t1t0Zt1
t00Ft
t0(x0)Tn0(x0)
2dt =T0
ρ0
=const., t1(t0,).
At the same time, the normal component of an initially normal unit perturbation to M0, rep-
resented by n0(x0), is given by the orthogonal projection of the advected normal 0Ft
t0(x0)n0(x0)
onto the unit normal
nFt
t0(x0)=0Ft
t0(x0)Tn0(x0)
0Ft
t0(x0)Tn0(x0)
,
i.e., by the the normal repulsion rate σt
t0(x0), computed as
σt
t0(x0) = *0Ft
t0(x0)n0(x0),0Ft
t0(x0)Tn0(x0)
0Ft
t0(x0)Tn0(x0)+
=1
0Ft
t0(x0)Tn0(x0)
.
Therefore, we have
1
t1t0Zt1
t0
1
σt
t0(x0)2dt =T0
ρ0
=const.,
or, equivalently,
σt
t0(x0)1
L2(t0,t1)=T0
ρ0
=const.
This shows that the temporally L2-normed reciprocal of the normal stretching rate along the manifold
M0should be spatially constant along diffusion barriers. This is certainly satisfied for t1→ ∞ along
stable manifolds of fixed points and periodic orbits, as well as along quasiperiodic invariant tori.
7
4.2 Unconstrained diffusion barriers in two-dimensional flows
In arbitrary dimensions, the first equation defining null surfaces in (4.6) and (4.7) are partial differ-
ential equations with a priori unknown solvability properties. For two-dimensional flows, however,
the null-surfaces become curves satisfying ordinary differential equations that turn out to be explic-
itly solvable (cf. [13]). These differential equations can be expressed in terms of the invariants of the
time-averaged, diffusion-structure-weighted Cauchy–Green strain tensor
¯
CD(x0):=1
t1t0Zt1
t0
det DFt
t0(x0), tTt
t0(x0)1dt
as follows.
Theorem 3. For two-dimensional flows (n= 2), let ξi(x0)R2denote the unit eigenvectors
corresponding to the eigenvalues 0< λ1(x0)λ2(x0)of the positive definite tensor ¯
CD(x0). A
uniform extremizer M0of the transport functional Tt1
t0is then necessarily a trajectory of the direction
field family
x0
0=η±
T0(x0):=sλ2(x0)− T0
λ2(x0)λ1(x0)ξ1(x0)±sT0λ1(x0)
λ2(x0)λ1(x0)ξ2(x0),x0UT0,(4.9)
defined on the spatial domain UT0={x0U:λ1(x0)≤ T0λ2(x0)}.
The domain of definition UT0of the direction field family η±
T0(x0)is precisely the spatial domain
where the tensor ET0(x0)defined in (4.5) is indefinite (Lorentzian) and hence indeed have well-
defined null-surfaces. Formula (4.9) enables a detailed computation of diffusion barriers in two-
dimensions based on a numerical approximation of the flow map Ft
t0(x0)obtained on a grid of
initial conditions (see [13] for details).
As shown in [13] for two-dimensional flows, we have ¯
Tt1
t0= det ¯
CD¯
C1
D.This implies that the
DBS field defined in (4.8) in the present, two-dimensional case, can be evaluated as
DBSt1
t0(x0) = trace ¯
Tt1
t0(x0) = det ¯
CD(x0)trace ¯
C1
D(x0) = trace ¯
CD(x0).(4.10)
The results in Theorems 1 and 2 are directly applicable to incompressible flows as well, in which
case they agree with the results in [13]. The derivation in [13], however, does not apply to the
present, compressible case, while our argument leading to the normalized total transport ˜
Σt1
t0(M0)
in (4.2) is general enough to cover the incompressible case as well.
5 Constrained diffusion extremizers
The above formulation is independent of the initial concentration c0(x)and assumes large enough
initial gradients along the initial surface M0that continue to dominate contributions from con-
centration decay and source terms (cf. the uniformity assumption (4.1)). If, however, we wish to
extremize diffusive transport with respect to a specific initial concentration field c0(x0), as opposed
to unspecified or uncertain initial concentrations, then we can no longer prescribe (4.1) along an ar-
bitrary material surface M0. Indeed, some material surfaces will experience 0c0vectors favorable
to cross-surface diffusion while others will not.
Considering 0c0as a given quantity and using formula (3.5), we rewrite the normed, total
transport of µ(x, t)as
˜
Σt1
t0(M0):= 1
RM0dA0Zt1
t0Z∂V (t0)Ts
t0(0c(x0) + 0b(x0, s)) ,n0dA0ds +o(να)
=RM0¯qt1
t0,n0dA0
RM0dA0
+o(να),(5.1)
8
with the help of the transport vector field
¯qt1
t0(x0) := Zt1
t0Tt
t0(x0) (0c0(x0) + 0b(x0, t))dt. (5.2)
The sign of the net total transport ˜
Σt1
t0(M0)through M0is now not necessarily positive, which
necessitates extremizing the normed transport (the time-integral of the geometric flux in the ter-
minology of MacKay [20]). To this end, we seek to extremize the area-normalized, leading-order,
normed diffusive transport
˜
E(M0) := RM0¯qs
t0,n0dA0
RM0dA0
with respect to the initial surface M0, for which a necessary condition is the vanishing Gâteaux
derivative
δ˜
E(M0)=0.(5.3)
For such stationary surfaces of ˜
E, we have the following result.
Theorem 4. Along any solution M0of the variational problem (5.3), there exists a constant CR
such that
¯qt1
t0(x0),n0(sx0)− T0pdet G(sx0) = C, x0∈ M0,
with T0:= ˜
E(M0)denoting the value of the normed transport functional on M0.
Proof. See section §B.
As argued in [13], the most observable stationary surfaces of diffusive transport are those with
nearly uniformly high gradients along them, associated with a nearly uniform pointwise transport
density. The theoretical centerpieces of such regions are provided by surfaces with perfectly uniform
transport density, which, by the statement of Theorem (4) with C= 0, satisfy the implicit equation
¯qt1
t0(x0(s)),n0(sx0(s))=T0(5.4)
for any selected pointwise transport density T00.
Remark 3.By analogy with the DBS field defined in (4.8), a direct measure of the local strength of
a uniform constrained barrier is
DBSt1
t0(x0) = ¯qt1
t0(x0),(5.5)
a predictive diagnostic field applicable in any dimension. This diagnostic emerges as the normed,
leading-order change in the functional E(M0)under small, localized perturbations to a stationary
surface M0. Ridges of DBSt1
t0(x0)are expected to highlight the strongest diffusive transport barriers
over the time interval [t0, t1].
Remark 4.When k(t)0and f(x, t)0holds (no sources or sinks), then we have ˆµ(x0, t)
c(Ft
t0(x0), t)and ˜
Σt1
t0(M0)in (5.1) can exactly—i.e., without the o(ν)error—be represented as
˜
Σt1
t0(M0) = ν1
RM0dA0Zt1
t0ZM0Tt
t0(x0)0c(Ft
t0(x0), t),n0(x0)dA0dt, (5.6)
as ones verifies from eq. (A.3) of section §A. Furthermore, noting that in this case, ˆµ(x0, t)
c(Ft
t0(x0), t),we obtain from the chain rule that
Tt
t0(x0)0c(Ft
t0(x0), t) = ρ0(x0)0Ft
t0(x0)1D(Ft
t0(x0), t)0Ft
t0(x0)T0c(Ft
t0(x0), t)
=ρ0(x0)0Ft
t0(x0)1D(Ft
t0(x0), t)c(Ft
t0(x0), t).(5.7)
9
Therefore, formulas (5.6)-(5.7) give an exact expression for ˜
Σt1
t0(M0)with a redefined form of ¯qt1
t0as
˜
Σt1
t0(M0):=RM0¯qt1
t0,n0dA0
RM0dA0
,
¯qt1
t0(x0):=Zt1
t0
ρ0(x0)0Ft
t0(x0)1D(Ft
t0(x0), t)c(Ft
t0(x0), t)dt. (5.8)
Using the form (5.8) of ¯qt1
t0(x0)in all our results below increases the accuracy of transport extremizer
detection. At the same time, formula (5.8) requires explicit knowledge of the current concentration
c(x, t),which generally necessitates the numerical solution of the advection-diffusion equation (2.1).
A notable case in which (5.8) is useful is when c(x, t) = ω(x, t)is the scalar vorticity associated
with a two-dimensional velocity field v(x, t).In that case, once v(x, t)is known, ω(x, t)is readily
obtained as the plane-normal component of ω(x, t)without the need to solve the vorticity-transport
equation. In this latter case, no assumption is needed on the smallness of the viscosity ν.
5.1 Perfect constrained barriers and enhancers to diffusion
In any dimension, a distinguished subset of uniform constrained barriers, the perfect barriers, inhibit
transport completely pointwise at leading order, i.e., are characterized by T0= 0. By eq. (5.4), the
time-t0positions of perfect constrained barriers satisfy
¯qt1
t0,n0= 0.(5.9)
Therefore, the vector ¯qt1
t0(x0)is necessarily tangent to a perfect barrier at every point x0. In
other words, time-t0positions of material surfaces acting as perfect constrained material barriers to
diffusive transport are necessarily codimension-one invariant manifolds of the autonomous dynamical
system
x0
0=¯qt1
t0(x0) = ¯
Tt1
t0(x0)0c0(x0) + ¯
Bt1
t0(x0),(5.10)
with
¯
Bt1
t0(x0):=Zt1
t0
Tt
t00b(x0, t)dt.
In the absence of sources and sinks (b(x0, t)0), eq. (5.10) simplifies to
x0
0=¯
Tt1
t0(x0)0c0(x0),(5.11)
which leads to the following result.
Proposition 1. Consider the time t0position of a perfect constrained diffusion barrier along which
0c0is not identically zero. Then the barrier can contain no homoclinic, periodic, quasiperiodic or
almost periodic orbit.
Proof. We use the function V(x0) = c0(x0)to note that
d
dsV(x0(s)) = 0c0·x0
0=0c0,¯
Tt1
t00c0
along trajectories of (5.11). Since ¯
Tt1
t0is positive definite, V(x0(s)) strictly increases at points where
0c0does not vanish. This excludes the existence of any recurrent motion that contains at least
one point where 0c0does not vanish.
A consequence of Proposition 1 in two-dimensions: no closed perfect constrained barriers can
exist apart from closed ridges and trenches of the initial concentration field. In three dimensions,
Proposition 1 implies that no two-dimensional, quasiperiodic invariant tori can arise as perfect
10
constrained diffusion barriers, apart from toroidal ridges or trenches of the initial concentration
field. Finally, for perfect constrained barriers, the diffusion barrier strength field defined in (5.5)
simplifies to
DBSt0
t1(x0) = ¯
Tt1
t0(x0)0c0(x0).(5.12)
In contrast to perfect barriers to diffusion, perfect enhancers to diffusive transport can be defined
as material surfaces that pointwise maximize diffusive transport. By eq. (5.4), the time-t0positions
of such perfect constrained enhancers must have unit normals n0satisfying
¯qt1
t0(x0),n0(x0)=¯qt1
t0(x0),(5.13)
Note that the norm of ¯qt1
t0is not necessarily constant along such surfaces, and hence perfect transport
enhancers are generally not solutions of the constrained variational problem (5.3). Instead, perfect
enhancers to transport are simply surfaces that are pointwise normal to ¯qt1
t0, thus experiencing the
locally strongest transport possible at each of their points.
In two dimensions, perfect transport enhancers are curves satisfying the ODE
x0
0=Ω¯qt1
t0(x0) = ¯
Tt1
t0(x0)0c0(x0) + ¯
Bt1
t0(x0),(5.14)
along which ¯qt1
t0has constant norm. Here we have used the notation
:= 0 1
1 0 (5.15)
for planar 90-degree rotations. In the absence of sources and sinks (b(x0, t)0), eq. (5.14) simplifies
to
x0
0=¯
Tt1
t0(x0)0c0(x0),(5.16)
which leads to the following result.
Proposition 2. In a two-dimensional flow, consider the time t0position of a closed, perfect con-
strained diffusion enhancer along which 0c0is not identically zero. Then this closed enhancer
cannot be fully contained in a domain where the symmetric tensor ¯
Tt1
t0(x0)¯
Tt1
t0(x0)is defi-
nite.
Proof. As in Proposition 1, we use the function V(x0) = c0(x0)to obtain
d
dsV(x0(s)) = 0c0·x0
0=0c0,¯
Tt1
t00c0
=0c0,¯
Tt1
t0¯
Tt1
t00c0
along trajectories of (5.16). Under the assumptions of the proposition, V(x0(s)) strictly increases
or decreases on domains where ¯
Tt1
t0(x0)¯
Tt1
t0(x0)is definite, which excludes the existence of
any closed trajectory for eq. (5.16).
5.2 Constrained diffusion barriers in two-dimensional flows
The expression (5.4) is a PDE in three and more dimensions. In two dimensions, however, it is
equivalent to two ODEs, as we spell out in the following result.
Theorem 5. In two-dimensional flows, time-t0positions of constrained material diffusion barriers
with uniform, pointwise transport density T0satisfy the following necessary conditions:
(i) Constrained uniform transport maximizers M0are necessarily solutions of the differential equa-
tion family
x0
0=q¯qt1
t0(x0)
2− T 2
0
¯qt1
t0(x0)
2¯qt1
t0(x0)±T0
¯qt1
t0(x0)
2Ω¯qt1
t0(x0),T0>0.(5.17)
11
(ii) If such a uniform transport maximizer M0is a closed orbit of (5.17) or a homoclinic or hete-
roclinic orbit connecting a zero of the ¯qt1
t0(x0)vector field to itself, then the symmetric matrix
L=ZM0
sign ¯qt1
t0(x0(s)) ,Ωx0
0(s)2
x0x0¯qt1
t0(x0(s)) ,Ωx0
0(s)ds (5.18)
must be negative semidefinite.
(iii) If a uniform transport maximizer M0satisfies
¯qt1
t0(x0(si))=T0, i = 1,2,¯qt1
t0(x0(s1)) k¯qt1
t0(x0(s2)) ,
¯qt1
t0(x0(si))=T0at its endpoints, M0
¯qt1
t0(x0(si)) kx0
0(si)kx0
0(sj), i, j = 1,2, i 6=j,
then
hLΩx0
0,Ωx0
0i ≤ 0(5.19)
must hold along M0.
(iv) Constrained uniform transport minimizers must necessarily be perfect barriers, i.e., satisfy the
differential equation
x0
0=¯qt1
t0(x0).
Proof. See section §C.
Remark 5.The argument in the proof of (i) of Theorem 5 is not applicable to perfect diffusion
barriers, as for such material lines, the leading-order term in the second variation of E(M)vanishes.
In the absence of sources or sinks (i.e., for b(x0, t)0in (5.2)), eq. (5.17) simplifies to
x0
0=1
¯
Tt1
t00c0
2A±(x0;T0)0c0(x0),(5.20)
with the tensor A±R2defined as
A±(x0;T0) = T0±q¯
Tt1
t00c0
2− T 2
0I¯
Tt1
t0.(5.21)
In this case, the following result is helpful in the numerical identification of closed diffusion barriers
as limit cycles of (5.20).
Proposition 3. Eq. (5.20) will have no closed (periodic or homoclinic) orbits contained entirely
in spatial domains where the symmetric part of the matrix A±R2is definite and 0c0is not
identically zero.
Proof. For the Lyapunov function
V(x0) = c0(x0),
we obtain that
d
dsV(x0(s)) = 0c0·x0
0
=1
¯
Tt1
t00c0
20c0,A±0c0.
Therefore, V(x0(s)) is strictly monotonically increasing or decreasing at least at one point of any
orbit of (5.20) that lies entirely in a domain in which A±(x0;T0)is definite and 0c0is not identically
vanishing. This implies the statement of the proposition.
12
Note that At1
t0is certainly definite for T0= 0 at any point where 0c0is nonzero, and hence
(5.20) has no closed orbits for T0= 0, except possibly for ones along which the initial concentration
gradient vanishes (curves of critical points, which is non-generic, yet abundant in areas of constant
initial concentration). This statement remains valid for small enough T0on compact domains by
the continuous dependence of the eigenvalues of A±(x0;T0)on the parameter T0.
6 Particle transport barriers in stochastic velocity fields
We showed in [13] how our results on barriers to diffusive scalar transport carry over to probabilistic
transport barriers to fluid particle motion with uncertainties, modeled by diffusive Itô processes.
Our derivation, however, specifically exploited the incompressibility of the deterministic part of the
velocity field. Here we show how similar results continue to hold for compressible Itô processes of
the form
dx(t) = v0(x(t), t)dt +νB(x(t), t)dW(t).(6.1)
Here x(t)Rnis the random position vector of a particle at time t;v0(x, t)denotes the deterministic,
generally compressible drift component in the velocity of the particle motion; and W(t)in an m-
dimensional Wiener process with diffusion matrix νB(x, t)Rn×m. Here the dimensionless,
nonsingular diffusion structure matrix Bis O(1) with respect to the small parameter ν > 0.
We let p(x, t;x0, t0)denote the probability density function (PDF) for the current particle posi-
tion x(t)with initial condition x0(t0) = x0. This PDF satisfies the classic Fokker–Planck equation
(see, e.g., Risken [26])
pt+·(pv0) = ν1
2··BB>p,(6.2)
or, alternatively,
pt+·(p˜v0) = ν·1
2BB>p,˜
v0=v0ν
2·BB>.(6.3)
This latter equation is of the advection–diffusion-form (2.3) if we select
c:=p
ρ,D:=1
2BB>,w:=˜
v0=v0ν·D, k(t) = f(x, t)0.(6.4)
Consequently, the Fokker–Planck equation (6.3) is equivalent to the advection–diffusion equation
(2.1) with the velocity field
v:=wν
ρDρ=v0ν·Dν
ρDρ. (6.5)
Since the equation of continuity,
tρ+∇· (ρv)=0,(6.6)
must hold for the velocity field vfor our formulation to apply, substitution of the definition of v
from (6.5) into (6.6) gives
tρ+∇· ρv0ν
ρDρν·D= 0,
or, equivalently,
tρ+∇· (ρv0) = ν∇· (·(Dρ)) ,(6.7)
which is the same PDE (6.3) that the probability-density psatisfies.
With the above choice of vand ρin (6.5) and (6.7), all results in the earlier sections on material
diffusion extremizers in compressible flows carry over to material diffusion barriers of the density-
weighted probability-density function c=p/ρ with respect to the velocity field vin (6.5) if we
re-define the transport tensor Tt
t0(x0)as
Tt
t0(x0):=1
2ρ0(x0)0Ft
t0(x0)1B(Ft
t0(x0), t)B>(Ft
t0(x0), t)0Ft
t0(x0)T.(6.8)
13
Furthermore, let us denote the initial probability-density function by p0(x) := p(x, t0;x0, t0)and
assume the initial carrier fluid density ρ0(x)as given. Then, with the help of the vector field
(cf. (5.2))
¯qt1
t0(x0) = Zt1
t0Tt
t0(x0)0
p0(x0)
ρ0(x0)dt, (6.9)
we collect the related results in the following theorem.
Theorem 6. (i) Unconstrained, uniform barriers of transport for the mass-based PDF, p/ρ, of
particle positions satisfy Theorems 2–5 with the transport tensor field Tt
t0(x0)defined as in (6.8).
(ii) Constrained, uniform barriers of transport for the mass-based PDF, p/ρ, satisfy Theorems
4–3 with the transport vector field ¯qt1
t0(x0)defined as in (6.9).
Proof. Indeed, for the case of an unspecified initial density c0(x), we obtain the normalized total
transport of µ(x, t)in the form (cf. (4.2))
˜
Σt1
t0(M0) = Σt1
t0(M0)
νK (t1t0)A0(M0)=RM0n0,¯
Tt1
t0n0dA0
RM0dA0
+o(να), ν (0,1] (6.10)
with the transport tensor (6.8), where Ft
t0(x0)is the flow map associated with the velocity field
v0(x, t)and ρ0(x0)is the initial density field of the carrier fluid, serving as initial condition for the
density evolution equation (6.7). The formulas (6.10)-(6.8) follow because the flow map of the full
velocity field vdefined in (6.5) is at least O(ν)C0-close to the flow map Ft
t0(x0)of v0(x, t)over
the finite time interval [t0, t1].As a consequence, only the leading order term, Ft
t0(x0), of the flow
map generated by (6.5) appears in the transport tensor (6.8). Higher-order corrections to the the
full flow map can be subsumed into the o(να)term in (6.10). With this observation, statements (i)
and (ii) can be deduced in the same fashion as Theorems 2–3 and Theorems 4–5.
Based on Theorem 6, the arguments leading to the diffusion barrier strength indicator in eqs. (4.8)-
(5.5) continue to apply, with the DBS field simplified to
DBSt1
t0(x0) =
¯
Tt1
t0(x0)0
p0(x)
ρ0(x)
.
7 Examples
7.1 Two-dimensional channel flow
An unsteady solution of the 2D, unforced Navier-Stokes equations is given by the decaying channel
flow
v(x, t) = eνt acos y
0,(7.1)
whose vorticity field
ω(x, t) = aeνt sin y
satisfies the advection–diffusion equation
tω+ω·v=νω, (7.2)
i.e., the two-dimensional vorticity-transport equation with viscosity ν. The simplest member of a
more general Navier-Stokes solution family (see, e.g., Majda and Bertozzi [21]), the velocity field (7.1)
describes a decaying horizontal shear-jet between two no-slip boundaries at y=±π
2(see Fig. 7.1).
14
y=
π
/ 2
y=
π
/ 2
v(x,t)
x
y
Figure 7.1: Unsteady, horizontal jet with a jet core at y= 0.
The jet core is given by the horizontal line y= 0. The constant aR+governs the strength of
shear within the jet. If we define the variable xto be spatially periodic, the flow becomes a model
of a perfectly circular vortical flow in an annulus with no-slip walls.
All horizontal lines are invariant material lines in (7.1). Out of these invariant lines, the jet core
at y= 0 is the most often noted barrier to the diffusion of vorticity, keeping positive and negative
vorticity values apart for all times. Indeed, for large values of a, the norm of the vorticity gradient
ω(x, t) = eνt 0
acos y
maintains its global maximum along the jet core for all times. The upper and lower channel bound-
aries at y=±π/2technically also block the diffusion of vorticity into the wall, but vorticity tapers
off to zero anyway as one approaches these boundaries in the vertical direction.
As the initial distribution of ω(x, t)is constrained by the velocity field, our theory of constrained
diffusion barriers is applicable to barriers to the transport of vorticity. To see the predictions of this
theory, we first note that the flow map Ft
t0(x0)in this example is
Ft
t0(x0) = x0a
ν(eνt eν t0) cos y0
y0,
which gives
0ωFt
t0(x0), t=aeνt 0
cos y0,
Tt
t0(x0) = 1 + a2
ν2(eνt eν t0)2sin2y0a
ν(eνt eν t0) sin y0
a
ν(eνt eν t0) sin y01.
Therefore,
¯
qt1
t0(x0) = 1
t1t0Zt1
t0
Tt
t0(x0)0ωFt
t0(x0), tdt
=1
2ν(t1t0)Asin 2y0
Bcos y0,
where
A=a2
νsin 2y01
2e2νt1+1
2e2νt0eν(t1+t0), B =aeνt0eν t1.
Consequently, the ODE family describing the time t0position of uniform constrained barriers is
given by
x0
0=1
2ν(t1t0)
q¯qt1
t0(x0)
2− T 2
0
¯qt1
t0(x0)
2Asin 2y0
Bcos y0+T0
¯qt1
t0(x0)
2Bcos y0
Asin 2y0
(7.3)
15
for some value of the transport density T0R. For the choice
T0=¯qt1
t0(x0)y0=0 =B
2ν(t1t0),(7.4)
the ODE (7.3) becomes
x0
0|y0=0 =B
2ν(t1t0)B
0kΩ¯qt1
t0(x0)|y0=0.(7.5)
Therefore, y0= 0 is an invariant line for equation (7.3) for the parameter value T0selected as in (7.4).
Consequently, the jet core at y0= 0 is a uniform, constrained barrier to vorticity diffusion along
which the pointwise diffusive transport of vorticity is equal to (7.3). As noted earlier, a barrier (as
a stationary surface of the transport functional) is not necessary a minimizer of transport. Indeed,
any other horizontal material curve admits a strictly lower transport density than the jet core.
In contrast, choosing the constant
T0= 0
in (7.3) gives the ODE
x0
0=1
2νa (t1t0)¯qt1
t0(x0)Asin 2y0
Bcos y0,
for which y0=±π/2are invariant lines. Along those invariant lines, we have
x0
0|y0=±π/2k¯qt1
t0(x0)|y0=±π/2.
Therefore, the channel walls at y0=±π/2are uniform, constrained minimizers to vorticity diffusion
along which the pointwise diffusive transport of vorticity is equal to zero. In particular, the channel
walls are perfect constrained barriers to diffusive transport.
To evaluate the second necessary condition we need to check the definiteness of the matrix
L=ZM0
sign ¯qt1
t0(x0(s)) ,Ωx0
0(s)2
x0x0¯qt1
t0(x0(s)) ,Ωx0
0(s)ds.
Note that
2
x0x0¯qt1
t0(x0(s)) ,Ωx0
0(s)|y0=0 =2
x0x01
2ν(t1t0)Asin 2y0
Bcos y0|y0=0 ·Ωx0
0(s)|y0=0
=2
x0x01
2ν(t1t0)Asin 2y0
Bcos y0·B
2ν(t1t0)0
B|y0=0
=B2
4ν2(t1t0)22
x0x0 Asin 2y0
Bcos y0·0
1|y0=0
=B3
4ν2(t1t0)22
x0x0[cos y0]|y0=0
=B3
4ν2(t1t0)20 0
0cos y0|y0=0
=B3
4ν2(t1t0)20 0
0 1 ,
and
sign ¯qt1
t0(x0(s)) ,Ωx0
0(s)|y0=0 =B3
4ν2(t1t0)2[cos y0]|y0=0 =1.
16
As a consequence, we have
L=ZM0
sign ¯qt1
t0(x0(s)) ,Ωx0
0(s)2
x0x0¯qt1
t0(x0(s)) ,Ωx0
0(s)ds
=B6length (M0)
16ν4(t1t0)40 0
0 1 ,
implying
hLΩx0
0,Ωx0
0i ≤ 0
satisfying the necessary condition (5.19) for a maximizer.
In summary, our theory correctly identifies the walls and the jet core as noteworthy features
of this channel flow. While these features are all considered as inhibitors of transport in informal
descriptions of jet-type flows, our approach reveals that in strict mathematical terms, only the walls
act as diffusion minimizers. The jet core, in contrast, is a diffusion maximizer with respect to any
localized perturbation and with respect to parallel translations.
7.2 Spatially periodic recirculation cells
A spatially periodic, unsteady solution of the 2D Navier–Stokes equations is given by (cf. Majda
and Bertozzi [21])
v(x, t) = ae4π2νt sin(2πx) sin(2πy)
cos(2πx) cos(2πy),(7.6)
whose vorticity field and Jacobian are given by
ω(x, t) = 4aπe4π2νt sin(2πx) cos(2πy),
v(x, t)=2πae4π2νt cos(2πx) sin(2πy) sin(2πx) cos(2πy)
sin(2πx) cos(2πy)cos(2πx) sin(2πy),(7.7)
satisfies the advection–diffusion equation (7.2) with viscosity νand a real parameter athat controls
the overall strength of the vorticity field.
The vorticity gradient is
ω(x, t) = 82e4π2νt cos(2πx) cos(2πy)
sin(2πx) sin(2πy)=82Ωv(x, t),
whose squared norm satisfies
|ω(x, t)|2
8π2e4π2νt 2=a2cos2(2πx) cos2(2πy) + sin2(2πx) sin2(2πy)
a2.
The flow has horizontal and vertical heteroclinic orbits connecting the array of saddle-type fixed
points at (x, y) = j
4,k
4for arbitrary integers jand k. This heteroclinic network surrounds an
array of vortical recirculation regions. Even though the velocity field is unsteady, its streamline
geometry consists of steady material lines (cf. Fig. 7.2). Only the value of the vorticity changes in
time by the same factor along these material lines, just as in our previous example.
17
Figure 7.2: Unsteady vortex array with a= 1 at time t= 0.
The main observed features in the diffusive transport of vorticity in this flow are the cell bound-
aries formed by the heteroclinic orbits. Along these orbits, |ω(x, t)|2admits maximum ridges that
decay slowly in time by a uniform factor. The ridges contain global minima with |ω|2= 0 at
the hyperbolic equilibria and global maxima with |ω|2=a28π2e4π2νt 2halfway between them.
Inside the cells, |ω|2decays away from the ridge boundaries and reaches the global minimum
|ω|2= 0 again at the elliptic equilibria. All closed, periodic streamlines in the vortical region
are also perceived as features hindering the spread of high vorticity from the centers of the vortical
regions.
We now examine how our theory of constrained diffusion extremizers bears on the vorticity
field features identified above from observations. Along, say, the y= 0.25 horizontal heteroclinic
streamline, the velocity Jacobian (7.7) becomes
v((x, 0.25) , t)=2πa cos(2πx)e4π2ν t 1 0
01,
implying
0Ft
t0((x0,0.25))1=
exp hRt
t02πa cos(2πx(s))e4π2ν s dsi0
0 exp hRt
t02πa cos(2πx(s))e4π2ν s dsi
,
ωFt
t0((x0,0.25)) , t= 82sin(2πx(t))e4π2ν t 0
1.
This gives
0Ft
t0(x0)1ωFt
t0(x0), tk0
1.
and hence, for any t1> t0and for any initial point x0= (x0,0.25), we have (cf. Remark 4)
¯
qt1
t0(x0) = 1
t1t0Zt1
t00Ft
t0(x0)1ωFt
t0(x0), tdt k0
1.
We conclude that y= 0.25 horizontal heteroclinic streamline with unit normal n0(x0) = (0,1) is a
perfect transport enhancer in the sense of formula (5.13). An identical conclusion holds for all other
heteroclinic connections.
In contrast, along closed, vortical streamlines, we find the integrand of ¯
qt1
t0(x0)to align with these
streamlines due to the shearing effect of the inverse flow map 0Ft
t0(x0)1on the streamline-
normal vorticity gradient ωFt
t0(x0), t.This implies that in the t1→ ∞ limit, all closed stream-
lines become asymptotically perfect transport barriers, with their normals asymptotically aligning
with ¯
qt1
t0(x0)at the same rate at all of their points.
18
0.0 0.1 0.2 0.3 0.4 0.5
0.2
0.1
0.0
0.1
0.2
x
y
0.0 0.1 0.2 0.3 0.4 0.5
0.2
0.1
0.0
0.1
0.2
x
0.0 0.1 0.2 0.3 0.4 0.5
0.2
0.1
0.0
0.1
0.2
x
Figure 7.3: Closed material vorticity transport barriers (in light red) in the lower-left recirculation
cell of Fig. 7.2 for integration times T= 3,T= 13, and T= 23 (from left to right), on top of
vorticity level sets (yellow-green-blue).
For finite times, the exact closed transport barriers in this flow can be identified numerically by
computing the ODEs appearing in Theorem 5 for the velocity field (5). For this computation, we
set a= 1 and ν= 0.001. The results shown in Fig. 7.3 show a close match between an increasing
number of detected barriers and vorticity level sets as the integration time is extended from T= 3
(left) via T= 13 (middle) up to T= 23 (right).
8 Application to transport-barrier detection in ocean-surface
dynamics
We now illustrate our results on two different ocean surface velocity data sets. The first one is
HYCOM, a data-assimilating hybrid ocean model, whose ocean-surface velocity output is generally
not divergence-free and hence represents a compressible 2D-flow. The second data set is a two-
dimensional unsteady velocity field obtained from AVISO satellite altimetry measurements under
the geostrophic approximation. This data set is currently distributed by the Copernicus Marine and
Environment Monitoring Service (CMEMS). Due to the geostrophic approximation, this velocity
field is constructed as divergence-free.
All simulations in this section have been performed with the package CoherentStructures.jl, a
collection of implementations of objective coherent structure detection methods written in the open-
source programming language Julia. Our computations rely crucially on the ODE integration codes
provided by the DifferentialEquations.jl package [25].
8.1 Unconstrained transport-barriers in the compressible HYCOM ve-
locity data set
We use ocean surface velocity data from 2013-12-15 to 2014-01-14, i.e., 30 days, taken from the
Agulhas leakage area at the southern tip of Africa. In Fig. (8.1), we show the diagnostic DBS field
(4.8), whose features align, as expected, remarkably close with the unconstrained, uniform transport
barriers extracted as trajectories of the η+
T0field (4.9) for T0= 1.
As a second step, we now verify if these material curves (obtained from purely advective cal-
culations) indeed act as observed transport barriers for a diffusive scalar field. To this end, we
solve the advection–diffusion equation (2.1), with k(t) = f(x, t)0and D(x, t)I, in Lagrangian
coordinates for the initial concentration shown in Fig. 8.2 (left). The final density obtained from
this computation is then shown in Fig. 8.2 (right), with the same uniform transport barriers overlaid
as in Fig. (8.1). Note how the predicted barriers indeed capture detailed features in the evolving
concentration field.
19
-4 -2 0 2 4 6
- 34
- 33
- 32
- 31
- 30
- 29
- 28
DBS field
lon
lat
0
1
2
3
4
5
Figure 8.1: Open transport barriers in the HYCOM ocean surface data set. Shown in the background
is the DBS field (with tails cut for visualization purposes). Overlaid are short integral curve segments
of the η+
T0field (4.9) for T0= 1. Other values of T0produce similar results.
-4 -2 0 2 4 6
-34
-33
-32
-31
-30
-29
-28
Initial scalar concentration
lon
lat
0.25
0.50
0.75
1.00
-4 -2 0 2 4 6
-34
-33
-32
-31
-30
-29
-28
Final scalar concentration
lon
lat
0.25
0.50
0.75
1.00
Figure 8.2: Transport barriers in the HYCOM ocean surface data set. The scalar field corresponds
to the initial (left) and final (right) scalar density, evolved under the advection–diffusion equation
(2.1) in Lagrangian coordinates. The final scalar concentration field is shown with short integral
curve segments of the η+
T0field (4.9) overlaid for a small value T0= 1 of the nondimensionalized,
uniform transport density.
20
-4 -2 0 2 4 6
-34
-33
-32
-31
-30
-29
-28
Initial vorticity
lon
lat
-1.00
-0.75
-0.50
-0.25
0
0.25
0.50
0.75
1.00
-4 -2 0 2 4 6
-34
-33
-32
-31
-30
-29
-28
Final vorticity
lon
lat
-1.00
-0.75
-0.50
-0.25
0
0.25
0.50
0.75
1.00
Figure 8.3: Closed, constrained vorticity transport barriers in the AVISO ocean surface data set.
The scalar field corresponds to vorticity (truncated at ±1for visualization purposes) at the initial
(left) and the final (right) time instances in Lagrangian coordinates.
8.2 Constrained diffusion barriers in the AVISO velocity data
We now illustrate our results on two-dimensional unsteady velocity data obtained from AVISO
satellite altimetry measurements. The domain of the dataset is the Agulhas leakage in the Southern
Ocean. Under the assumption of a geostrophic flow, the sea surface height hserves as a streamfunc-
tion for the surface velocity field. In longitude–latitude coordinates (ϕ, θ), particle trajectories are
then solutions of the system
˙ϕ=g
R2f(θ) cos θθh(ϕ, θ, t),˙
θ=g
R2f(θ) cos θϕh(ϕ, θ, t),(8.1)
where gis the constant of gravity, Ris the mean radius of the Earth and f(θ):= 2Ω sin θis the
Coriolis parameter with denoting the mean angular velocity of the Earth. The computational
domain is chosen as in several other studies before (see, e.g., [12, 14, 13]), with integration time T
equal to 90 days.
In this two-dimensional flow, we wish to determine material transport barriers for the vorticity
ω(x, t), i.e., the single nonzero component of ×vnormal to the plane of the flow (8.1). Following
Remark 4, we use the exact transport vector field ¯qt1
t0, (5.8). For closed-orbit detection, we employ
a numerical scheme analogous to [14], with
1. an index-theory based preselection of elliptic-type subdomains;
2. placement of Poincaré sections in regions with appropriate index;
3. launch of integral curves from seed points, solving for the transport parameter which yields a
closed orbit at the respective seed point.
The closed material vorticity transport barriers obtained from this procedure are shown in
Fig. 8.3, on top of the initial (left) and final (right) vorticity fields in Lagrangian coordinates.
We note that closed diffusion barriers arise around all four vortical regions identified by previous
studies (see, e.g., [12, 14, 11, 28, 13]) as materially coherent. The closed region boundaries here are
optimized to be extremizers of the vorticity transport, as opposed to be outermost coherent material
curves.
9 Conclusions
We have shown how recent results on barriers to diffusive transport extend from the incompressible
case treated in [13] to compressible flows. In our present setting, we have also allowed for the presence
21
of sources and sinks in the concentration field. In addition, we have distinguished between the case
of an unknown initial concentration (unconstrained extremizers) and the case of a specifically known
initial concentration field (constrained extremizers) with respect to which diffusive transport is to
be extremized over material surfaces.
For unconstrained barriers, we have obtained results that formally coincide with those in [13],
except that the flow map here is compressible and the initial density of the fluid appears in the
transport tensor. Despite the similarity with the results in [13], the present results have required a
different derivation and additional assumptions on the selection of the most diffusion-prone initial
concentration field near each material surface in the flow. In this formulation, concentration sinks
and sources turn out to play no explicit role in the leading-order transport extremization problem.
As in the incompressible case, we have obtained explicit direction fields defining the barriers for two-
dimensional flows. For higher-dimensional flows, the barriers continue to be null-surfaces of a tensor
field, but satisfy partial differential equations. In any dimension, however, the diffusion barriers
strength (DBS) field can directly be computed from the velocity field and serves as a diagnostic to
map out the global barrier distribution.
For constrained barriers, we have sought material surfaces that block transport more than their
neighbors do under a specific initial concentration field. In this case, the equations defining the
diffusion extremizers depend explicitly on the sink or source distribution for the concentration, as
well as on the (possibly time-dependent) spontaneous concentration decay rate. Constrained barrier
surfaces also satisfy explicit differential equations in two dimensions and partial differential equations
in higher dimensions. A DBS scalar field is again available in any dimensions for diagnostic purposes.
We have found in canonical examples that some classically documented transport barriers (such as
jet cores and unstable manifolds) are, in fact, perfect enhancers of diffusive transport. Barriers
that are strict local minimizers of diffusive transport, by contrast, are rare and must completely
block transport at leading order, such as the walls of a channel flow. In two dimensions, with the
exception of perfect barriers, constrained uniform barriers are local maximizers of diffusive transport
with respect to all localized perturbations.
Finally, we have shown how the above results extend to barriers to the transport of probability
densities for particle motion in compressible, stochastic velocity fields modeled by Itô processes. This
extension enables one to locate barriers to stochastic transport from purely deterministic calculations,
as long as the diffusivities involved are small, which is generally the case for geophysical flows.
As we illustrated on two explicitly known Navier-Stokes flows, the present results on constrained
barriers enable the detection of barriers to the diffusion of vorticity in two-dimensional flows. This
follows because planar vorticity transport is governed by an (incompressible) advection–diffusion
equation whose initial condition (the initial vorticity) cannot be considered unknown or uncertain
once the velocity field is known. Barriers to vorticity diffusion in three dimensions, however, cannot
be treated by the present results, given that vorticity is an active vector field, rather than a passively
diffusing scalar field, in three-dimensional flows. More generally, the construction of barriers to the
transport of active scalar- and vector field requires new ideas relative to those in the present work.
Acknowledgment
We would like to thank Ryan Abernathey, Francisco J. Beron–Vera, Stergios Katsanoulis and
Jean-Luc Thiffeault for useful discussions, and Nate Schilling for valuable contributions to the
code as part of CoherentStructures.jl. The 1/12 deg global HYCOM+NCODA Ocean Reanaly-
sis was funded by the U.S. Navy and the Modeling and Simulation Coordination Office. Computer
time was made available by the DoD High Performance Computing Modernization Program. The
output is publicly available at http://hycom.org. The Ssalto/Duacs altimeter products were pro-
duced and distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS)
(http://www.marine.copernicus.eu). G.H. and D.K. acknowledge partial support from the Turbulent
Superstructures priority program of the German National Science Foundation (DFG).
22
A Proof of theorem 1
We start by establishing an expression for the instantaneous flux vector associated with the transport
of µthrough a material surface. We consider first the transport of µout of an arbitrary, closed
material volume V(t)to obtain an expression for the flux of µthrough ∂V (t)via the divergence
theorem. The same flux expression is then applicable to a general material surface M(t), given
that V(t)is arbitrary. Indeed, for any M(t), the set V(t)can be selected as an O()volume whose
boundary is the union of M(t), a parallel translate of M(t)by a distance O(), and a cylindrical
surface of area O(2).Taking the 0limit, one then obtains the same flux vector for M(t)as for
∂V (t).
The rate of change in the mass-based concentration of µin a closed material volume V(t) =
Ft
t0(V(t)) is, by definition,
d
dt ZV(t)
µ(x, t)ρ(x, t)dV =d
dt ZV(t0)
µ(Ft
t0(x0), t)ρ0(x0)dV0=ZV(t0)
tˆµ(x0, t)ρ0(x0)dV0
=νZV(t0)
0·Tt
t0(x0)0(ˆµ(x0, t) + b(x0, t))dV0
=νZ∂V (t0)Tt
t00(ˆµ(x0, t) + b(x0, t)) ,n0dA0,(A.1)
with the last integral denoting the surface integral over the (n1)-dimensional boundary V (t0)
of the n-dimensional volume V(t0).As discussed above, the calculation implies that in Lagrangian
coordinates, the flux vector for the field µ(x, t)through an arbitrary material surface M(t)is also
Tt
t00(ˆµ(x0, t) + b(x0, t)) ,n0,with n0(x0)denoting an oriented unit normal field to M(t0).
To proceed, we take the 0-gradient of both sides of (3.3) and integrate in time to obtain
0ˆµ(x0, t) = 0c0(x0) + νZt
t0
01
ρ0(x0)0·Ts
t0(x0)0[ˆµ(x0, s) + b(x0, s)]ds, (A.2)
where we have used the relation 0ˆµ(x0, t0) = 0c(x0), which follows from (3.1). Then with the
flux vector obtained in the last eq. (A.1) and with expression (A.2) for 0ˆµ(x0, t)and hand, the
total transport of µthrough M(t)can be written as
Σt1
t0(M0) = νZt1
t0ZM0Tt
t0(x0)0(ˆµ(x0, t) + b(x0, t)) ,n0(x0)dA0dt (A.3)
=νZt1
t0ZM0Tt
t0(0c0(x0) + 0b(x0, t)) ,n0dA0dt+
+ν2Zt1
t0ZM0Zt
t0
01
ρ0(x0)0·Ts
t0(x0)0[ˆµ(x0, s) + b(x0, s)]T
Ts
t0n0ds dA0dt.
The statement of the theorem, therefore, follows if the last term in (A.3) is of order o(ν), i.e., if
lim
ν0νZt1
t0ZM0Zt
t0
01
ρ0(x0)0·(Ts
t0(x0)0µ(x0, s) + b(x0, s)])T
Ts
t0n0ds dA0dt = 0.(A.4)
To prove (A.4), we need estimates on the solution of the initial value problem
tˆµ(x0, t) = ν1
ρ0(x0)0·Tt
t0(x0)0[ˆµ(x0, t) + b(x0, t)],(A.5)
ˆµ(x0, t0) = c0(x0).
23
Based on our initial assumptions, we have the following bounds on the entries Tij(x0, t) := Tt
t0(x0)ij
of the matrix representation of Tt
t0:
ρ0(x0)1Tij (x0, t)ρ0(y0)1Tij (y0, s)(C1|x0y0|α+C2|ts|α
2),
ρ0(x0)10Tij (x0, t)ρ0(y0)10Tij (y0, t)C3|x0y0|α,(A.6)
for some constant 0< α 1and for all x0,y0Uand t, s [t0, t1]. By the positive definiteness of
Tt
t0(x0)and the positivity of ρ0, we also have
λ|u|2u,1
ρ0(x0)Tt
t0(x0)uΛ|u|2,uRn,x0U, t [t1, t2],(A.7)
which implies the bounds
|u|2
ΛDu, ρ0(x0)Tt
t0(x0)1uE|u|2
λ, λnρ0(x0)ndet Tt
t0(x0)Λn,(A.8)
for all uRn,x0Uand t[t1, t2].
Next, we observe that (A.4) is satisfied when
sup
x0U,t[t0,t1]|0ˆµ(x0, t)0c0(x0)|=O(νq),(A.9)
holds for some q > 0, as one obtains using (A.2) and estimating the supremum norm in x0and
tusing (A.6). Using the assumption that c0C2(U),we will now show that (A.9) holds, and
hence (A.4) is indeed satisfied. In our presentation, we will utilize a scaling approach described by
Friedman [7].
Introducing the rescaled time variable τ:= ν(tt0)as well as the shifted and rescaled concen-
tration w(x0, τ ) := ˆµ(x0, t0+τ
ν)c0(x0), then setting Tν(x0, τ ) := Tt0+τ
ν
t0(x0), we can rewrite (A.5)
as (wτ=1
ρ0
0·(Tν0w) + 1
ρ0
0·(Tν0(c0+b)) ,
w(x0,0) = 0,(x0, τ )U×[0, ν(t1t0)].(A.10)
Condition (A.9) is then equivalent to
sup
x0,t[01]|0w(x0, τ )|=O(νq), τ1:= ν(t1t0),(A.11)
for some q > 0. In non-divergence form, equation (A.10) takes the form
wτ=
n
X
i,j=1
Tij
ν
ρ0
2w
∂xi
0∂xj
0
+
n
X
i=1
1
ρ0
n
X
j=1
∂T ij
ν
∂xj
0
∂w
∂xi
0
+fν,(A.12)
where we have defined
fν(x0, τ ):=1
ρ0(x0)0·(Tν(x0, τ )0(c0(x0) + b(x0, τ)) .(A.13)
Let
Z(x0, τ ;ξ, s):=
exp hx0ξ0T1
ν(ξ,s)(x0ξ)i
4(τs)
(2π)nρn
0det Tν(ξ, s)1
2(τs)n
2
,(A.14)
Zτ=ρ1
0Tν2
0Z,
24
for x0,ξand τ, s [0, τ1], denote the fundamental solution of the homogeneous, second-
order part of (A.10). For later computations, we note that with the n-dimensional volume element
dξ=1...dξn, we have the estimate
ZZ(x0, τ ;ξ, s)dξ=Z(2π)nhρn
0det T1
νi1
2(τs)n
2eDx0ξ0T1
ν(x0ξ)E
4(τs)dξ
Z(2π)nλn
2(τs)n
2e|x0ξ|2
4Λ(τs)dξ,(A.15)
where we have used the inequalities in (A.8). With the rescaled spatial variable yand the rescaled
volume form dydefined as
y= (2Λ)1
2(τs)1/2(xξ), dy= (2Λ)n
2(τs)n
2dξ,(A.16)
we define the set x0,τ,s := (2Λ)1
2(τs)1/2(x0Ω) to obtain from (A.15) the estimate
Z
Z(x0, τ ;ξ, s)dξπn
2Λ
λn
2Zx,τ,s
e−|y|2dy
πn
2Λ
λn
2ZRn
e−|y|2dy=Λ
λn
2
,
(A.17)
where we have used that R
−∞ er2dr =π. We also recall from [7, Thm. 3, p. 8], that for any
continuous function f: Ω ×[0, τ1]R, the integral
V(x0, τ ) := Zτ
0Z
Z(x0, τ ;ξ, s)f(ξ, s)dξds (A.18)
is continuously-differentiable with respect to x0and satisfies
0V(x0, τ ) = Zτ
0Z
0Z(x0, τ ;ξ, s)f(x0, s)dξds. (A.19)
As shown in [7, Thm. 9, p.21], the variation of constants formula applied to (A.10) gives its
solution in the form
w(x0, τ ) = Zτ
0Z
Z0·ρ1
0Tν0c0dξds+
Zτ
0Z
Z(x0, τ ;ξ, s)×
×Zs
0Z
Φ (ξ, s;η, σ)ρ1
0Tν(η, σ)0c0(η)dηdξds
=:W1(x0, τ ) + W2(x0, τ ),
(A.20)
for some (not explicitly known) function Φthat satisfies the estimate
|Φ (ξ, s;η, σ)| ≤ C4
1
|sσ|h0
1
|ξη|n+22h0α,(A.21)
for any constant h01α
2,1, where αis the Hölder exponent in (A.6).
To estimate the spatial gradient of W1, we use the formula for the x0-derivative of (A.20) in
(A.19) to obtain
|0W1|=
0Zτ
0Z
Z0·ρ1
0Tν0c0dξds
=
Zτ
0Z
(0Z)0·ρ1
0Tν0c0dξds
Zτ
0Z
1
2|τs|
ρ0T1
ν(ξ, s)(x0ξ)
|Z|
0·ρ1
0Tν0c0
dξds,
(A.22)
25
where we also used the definition (A.14) in evaluating 0Z. From (A.8), we obtain kρ0T1
νk=λ1,
and hence we can further write (A.22) as
|0W1| ≤ 1
λZτ
0Z
|Z|
2|τs|
0·ρ1
0Tν0c0
dξds
k0·ρ1
0Tν0c0kC0(Ω)
λZτ
0Z
1
2|τs||x0ξ| |Z|dξds
C5
ku0kC2(Ω)
λZτ
0Z
1
2|τs||x0ξ| |Z|dξds.
(A.23)
Next, as in the calculation of the integral in (A.15), we use the scaling (A.16) in (A.23) to obtain
|0W1| ≤ C5
Λku0kC2(Ω)
λZτ
0
1
τsZRn|y|e−|y|2dyds
C6
Λku0kC2(Ω)
λZτ
0
1
τsds
C7τ=Oν1
2.
(A.24)
To estimate the spatial gradient of W2in (A.20), we proceed similarly by using the growth
condition (A.21) to obtain
|0W2| ≤ Z