Lagrangian transport through surfaces in volume-preserving flows
Advective transport of scalar quantities through surfaces is of fundamental importance in many scientific applications. From the Eulerian perspective of the surface it can be quantified by the well-known integral of the flux density. The recent development of highly accurate semi-Lagrangian methods for solving scalar conservation laws and of Lagrangian approaches to coherent structures in turbulent (geophysical) fluid flows necessitate a new approach to transport from the (Lagrangian) material perspective. We present a Lagrangian framework for calculating transport of conserved quantities through a given surface in -dimensional, fully aperiodic, volume-preserving flows. Our approach does not involve any dynamical assumptions on the surface or its boundary.
The transfer of a quantity along the motion of some carrying fluid, or Lagrangian transport for short, is of fundamental importance to a broad variety of scientific fields and applications. The latter include (geophysical) fluid dynamics [30, 37, 34, 20], chemical kinetics , fluid engineering [31, 35] and plasma confinement . Existing methods for computing transport (i) have mostly been developed under certain assumptions on temporal behavior (steady or periodic time-dependence) [23, 24, 22, 33]; spatial location (regions related to invariant manifolds such as lobe dynamics and dividing surfaces in transition state theory) [23, 24, 22, 33, 25, 1]; state space dimension (2D) [23, 33, 25, 1]; or (ii) restrict to a perturbation setting . Recently, the problem of quantifying finite-time transport in aperiodic flows between distinct, arbitrary flow regions has been considered by Mosovsky et al. . They present a framework for describing and computing finite-time transport in -dimensional (chaotic) volume-preserving flows, which relies on the reduced dynamics of an ()-dimensional ‘minimal set’ of fundamental trajectories. In this paper, we present a Lagrangian approach to the complementary problem of computing transport through a codimension-one surface over a finite-time interval in volume-preserving flows. This cannot be reformulated in the setting of , since (i) initial and final positions of surface-crossing particles are a priori unknown, and (ii) particles may cross the surface several times, possibly in opposite directions, leading to a net number of surface crossings different from one, in particular including zero.
The problem of computing the flux through a surface in general flows admits a well-known solution in terms of an integral of the flux density over the surface and the time interval of interest, cf. the left-hand side of Eq. (1). We view this approach as Eulerian, in that it involves instantaneous information (flux density) at fixed locations in spacetime. Recently, two lines of research emerged that inevitably require a Lagrangian approach to the flux calculation.
The first is concerned with the numerical solution of advection equations (or, in the absence of sources, conservation laws) for conserved quantities by semi-Lagrangian methods, which enjoy geometric flexibility and the absence of Eulerian stability constraints [38, 39]. Roughly speaking, the term semi-Lagrangian refers to methods which evolve material densities on spatially fixed test volumes based on short-term material advection steps.
The second is concerned with transport by coherent vortices in oceanic flows, more precisely with the determination of the relevance of coherent transport [6, 42, 17], an open problem in physical oceanography and climate science. Coherent structures have long been studied in fluid dynamics , typically from an Eulerian point of view, i.e., by defining coherent structures as subsets of spacetime, usually as (sub-)level sets of scalar fields on spacetime [26, 32, 6, 42]. This approach, however, yields coherent structures with a priori unclear relation to actual fluid motion and hence ambiguous role in coherent transport; see transfer-operator related approaches which seek Eulerian coherent sets with minimal flux through the boundary under advection with small-scale diffusion superimposed, cf., for instance, . For these reasons, Lagrangian approaches to coherent structures have been developed over the last few years. These seek coherent structures as subsets of the material evolving under the flow [14, 15, 18, 13, 10, 11, 21, 29], building on a wide variety of mathematical principles. The location of a material structure in spacetime is fully determined by its motion.
This implicit spatial definition of material structures poses a severe challenge to determining their contribution to transport through a surface in an Eulerian manner. First, for each point on the extended surface in spacetime knowledge is required whether it is occupied by a particle originating from the material structure of interest. Second, the subset of points on the surface which are occupied by material particles of interest may be very complicated, cf. Fig. 1, especially in turbulent flows. As a consequence, an Eulerian integration of the flux density restricted to the intersection of the extended surface with the path of the material set of interest is practically infeasible.
Conversely, from the material point of view, it is intuitive that the number of surface crossings of each individual Lagrangian particle is relevant to determine its flux contribution. First steps towards a classification of Lagrangian particles with respect to net number of curve crossings have been proposed for two-dimensional flows under additional technical assumptions by Zhang [40, 41]. In this paper, we solve the following more general donating region problem [39, Def. 1.2].
For a given regular, divergence-free and time-dependent vector field , consider a conserved quantity , which satisfies the scalar conservation law in Eulerian form 222The Lagrangian form is with the material derivative. Physically, this means that the scalar does not change along particle motions.
Let be a compact, connected, embedded codimension-one surface in (configuration space) , and be a compact time interval. The problem is to find pairwise disjoint sets , indexed by , of Lagrangian particles at time , such that
where is the unit normal vector field to characterizing the direction of positive flux.
In fact, we may generalize Problem 1 and drop the assumption that be stationary over . Thus, we solve the analogous problem for smoothly moving surfaces , i.e., is everywhere transversal to time fibers .
Let and be as in Problem 1, and the extended velocity field. Let be a codimension-one surface in extended configuration space , , as defined above. The problem is to find pairwise disjoint sets , indexed by , of Lagrangian particles at time , such that
where is the unit normal vector field to at .
2 Illustrative discussion of the main result
The aim of this section is to discuss Eq. (1) in anticipation of mathematical constructions presented in Sections Section 3 and Section 4. In fact, those constructions emerge naturally when taking a differential topological view on the change from Eulerian to Lagrangian coordinates. We recall and discuss concepts related to that coordinate change in Section 3.
First, the left-hand side of Eq. (1) corresponds to the classical flux integral over the surface over time . The integrand, also called the flux density, corresponds to the normal component of the velocity field weighted with the material density . Here, the flux is viewed from the perspective of the section, which is why we refer to the left-hand side of Eq. (1) as the Eulerian flux integral.
In contrast, the right-hand side of Eq. (1) is an integral, or a sum of integrals, solely over material points or particles. As our later analysis reveals we may decompose the whole material domain (up to a measure zero set) into disjoint material subsets, each of which contains particles with a certain number of net crossings across the surface over the time interval . These material subsets are denoted by , where refers to the net number of transversal crossings, and have been coined donating regions of fluxing index by Zhang . The contribution to transport by each donating region corresponds to its respective mass (or volume in a homogeneous fluid) multiplied by its corresponding fluxing index. The overall transport through is then given by the sum of transport contributions of all donating regions. In summary, the right-hand side of Eq. (1) views the flux from the perspective of the mass-carrying particles, which is why we refer to it as the Lagrangian flux integral.
For illustration, consider an array of vortices, given by the stream function
subject to an aperiodic, spatially uniform spiraling forcing given by
The velocity field is hence
We set the section to , the time interval to , and, for simplicity, the material density . Fig. 2 shows the regions of particles with fluxing index , , and ; here, white corresponds to fluxing index . For details on how to construct the donating regions, see Section 4.4. There are no particles with other fluxing indices, and some particles attain a fluxing index by crossing twice in positive and once in negative direction, such as the sample trajectory shown in red.
For a numerical validation of Eq. (1), we compute its left-hand side by numerical quadrature, which yields a transport value of . For the right-hand side, we subtract the area of from that of , and obtain a transport value of , with relative error below . For further improvements of area calculations via spline-approximation of polygon boundaries see . Note that, in practice, the numerical integration of the Eulerian integral becomes more challenging in case the section is curved and requires a parametrization. In contrast, we will see that a curved section does not add any difficulty in the Lagrangian framework. This geometric flexibility has been one major motivation to develop semi-Lagrangian numerical schemes for solving conservation laws, cf. [38, 41].
For our main intended application, the calculation of the contribution to transport by a certain material subset —such as the ellipse in Figs. 1 and 2 (dark green, centered at with semi-major axis in -direction, and semi-minor axis in -direction)—reduces to subtracting the area of from , which yields .
At first sight, this may seem paradoxical, since the entire ellipse is launched from one side of the section. Fig. 1 now indicates that at the end of the time interval, parts of the material ended up on the other side of the section. Going from the first to the latter side corresponds to positive flux, and one would expect an overall positive transport contribution. However, a closer inspection of Fig. 1 reveals that most of the material which is advected to the other side has actually not crossed the section, but has flown around it. Moreover, much of the material which has flown around has actually crossed the section in negative direction, thus yielding an overall negative flux contribution of the ellipse.
3 Eulerian and Lagrangian coordinates
While Eulerian coordinates are assigned to spatial points in a fixed frame of reference, Lagrangian coordinates label material points and are usually taken as the Eulerian coordinates at some initial time, say, . The motion of material points is described by the flow , a mapping between initial positions at time and current positions at time , i.e., . Thus, the flow map can be interpreted as a change from Lagrangian to Eulerian coordinates. The inverse flow map corresponds then to the change from Eulerian to Lagrangian coordinates.
In fluid dynamics, there are two important characteristic curves associated with the flow , which we re-interpret in terms of Eulerian and Lagrangian coordinates.
A path line through is the time-curve of a fixed Lagrangian particle in Eulerian coordinates, i.e., . In other words, the path line is a collection of Eulerian positions that the Lagrangian particle will occupy at some time. Its time derivative, expressed in Eulerian coordinates, gives rise to the velocity field
By construction, the path line through is the solution of the initial value problem
A streak line through is the time-curve of a fixed Eulerian location in Lagrangian coordinates, i.e., . In other words, the streak line is a collection of material points that will occupy the Eulerian position at some time, see Fig. 3. Our definition of streak lines suits well the currently intended purpose and is consistent with Zhang’s use . It is more common and actually more intuitive, however, to view a streak line as the collection of material points that have passed the Eulerian position at some time. In this context, a streak line can be imagined as an instantaneous curve of Lagrangian markers, injected in the past at and passively advected by the flow, see . The time derivative of the streak line, expressed in Lagrangian coordinates, gives rise to the streak vector field
By construction, the streak line through is the solution of the initial value problem
To derive an algebraic relation between the velocity field and the streak vector field , we compute the velocity along the streak line through . Differentiation of the constant function with the shorthand notation yields
Using Eq. (3) and the invertibility of the linearized flow map with , we obtain
Equivalently, in global terms we have
Finally, we recall that the flow is volume-preserving on if and only if the velocity field (or, equivalently, ) is divergence-free .
4 Transport through surfaces
We decompose the change from Eulerian to Lagrangian coordinates (evaluated along the extended section ) into two steps. First, we map Eulerian spacetime points to their respective initial position at time , while keeping them on the same time slice, i.e.,
In extended Eulerian coordinates, trajectories (or, extended path lines) take the form ; in extended Lagrangian coordinates, trajectories are simply vertical lines, i.e., , since they track the same particle , see Fig. 5. Intersections of path lines with correspond one-to-one with intersections of with vertical lines.
In a second step, we project spacetime points to their spatial coordinates by the canonical projection and introduce . We view the image of as the initial time slice of the extended state space and therefore parametrized by Lagrangian coordinates. By construction, we have for the identity , and it is exactly the particles from that cross within one way or another, possibly multiple times.
For later reference, we emphasize the following observation. Let be the unit time-like vector field. Then we have
|On the other hand, for any space-like vector field such as, for instance, the velocity field , we have|
since in this case acts like .
4.2 Counting net crossings – the degree
We study the differentiable map as a map between the two -dimensional manifolds and . To this end, we recall notions from differential topology and interpret them in our setting.
First, regular points are those for which the differential is invertible. This is exactly the case, when the tangent space at to does not contain the vertical time direction. Non-regular points of are referred to as critical points, which are characterized by non-transversal crossings, i.e., instantaneous stagnation points , or tangential crossing/rebound.
According to Sard’s theorem, the set of critical values, i.e., images of critical points, has measure zero in , even though the set of critical points may be large in . Consistently, critical points do not contribute to the Eulerian flux integral.
In the Eulerian setting, positive/negative crossings of a Lagrangian particle through correspond to at corresponding intersections . In the Lagrangian setting, positive/negative crossings are equivalent to crossings of the vertical line over through from below/above with respect to the orientation on , see Fig. 5. This is formalized by , which geometrically means that is orientation-preserving or orientation-reversing at the respective crossings of . This is well-defined at all crossings exactly for particles which are regular values of .
The difference of positive and negative crossings, or, in other words, the net number of crossings, is given by the degree of at ,
The degree function is locally constant, and can be used to define a partition of the set of particles, up to the measure-zero set of critical values. Specifically, we define as the set of regular values of such that for . Following , we refer to the ’s as donating regions of fluxing index . By construction, each contains the particles with only transversal crossings of which there are net. In particular, all material points that do not cross within are contained in .
4.3 Area formula
The area formula is a generalization of the change of variables (in integral) formula to non-injective maps such as . Clearly, the degree measures the non-injectivity, and is equal to globally for orientation-preserving and -reversing diffeomorphisms, respectively.
Inhomogeneous conserved quantities are constant along trajectories in volume-preserving flows. Therefore, we have , and we conclude with the general area formula:
4.4 The two-dimensional case
Our results for volume-preserving flows on simplify in the two-dimensional case considered by Zhang [40, 41] as follows. It is well-known that, in our context, the degree of at some regular value equals the winding number of with respect to the closed curve around [5, Section 6.6]. The winding number counts the net number of turns of around under one counter-clockwise passage through , see Fig. 2.
In the 2D case, we find the donating regions of fluxing index as follows. The boundary of the extended section is a closed curve, just as its image under . In Figs. 5 and 2 its image corresponds to the concatenation of the section (blue), one streakline (say, yellow), the backward image of the section (purple) and the other streakline (brown). This closed curve gives rise to possibly several connected components, simple loops, through its self-intersections. Now, the winding number is constant on the interior of each simple loop, and it suffices to compute the winding number for some contained sample point, which is a standard task in computational geometry. As anticipated in Section 2, in the case of constant density the Lagrangian flux calculation reduces to the computation of the enclosed area of each component, multiplication by its winding number and final summation over the loops.
In summary, we have generalized the main result of , in which only the issue of counting net crossings was treated under additional assumptions from a different methodological perspective. Note also that for the isolated counting aspect, our characterization in terms of , or winding number in 2D, is also valid in the case of a compressible velocity field .
5 Moving surfaces
To find a Lagrangian flux integral in the moving surface case, Problem 2, we may proceed as in Section 4 with replaced by as the domain of the maps and . All constructions and arguments work just as well as before. It remains to find an expression analogous to Eq. (6) for :
which is shown in Appendix Section B.
In this work we have devised a Lagrangian approach to transport through (codimension-one) surfaces in general -dimensional, unsteady, volume-preserving flows. Studying the change from Lagrangian to Eulerian coordinates (and vice versa) alone, we have (i) discovered a striking analogy between path lines and streak lines, (ii) derived an algebraic relation between their associated velocity fields, (iii) found a natural way of determining the net number of surface crossings for individual Lagrangian particles up to a measure zero set, and, finally, (iv) transformed the Eulerian flux integral into a Lagrangian one. Thus, we have solved the donating region problem  for volume-preserving flows in arbitrary finite dimension and smoothly moving surfaces. An obvious extension of the methodology is to handle non-volume-preserving flows, and work is in progress in this direction.
Besides the theoretical insights, the Lagrangian approach to transport through surfaces is of major importance in at least two relevant fluid mechanical applications. First, it provides the missing theoretical foundation of a family of highly accurate semi-Lagrangian finite volume and interface tracking methods [38, 39]. Second, it facilitates the efficient computation of transport by Lagrangian coherent vortices in large-scale oceanic flows [18, 17].
Appendix A Proof of
We show that . To this end, fix a regular point . Then acts diffeomorphically between an open neighborhood of in and its image in . We may choose local coordinates on such that in we have an orthonormal basis, i.e., , and , with
unit volume parallelepipeds. To avoid a discussion on delicate orientation issues, the following calculations involving determinants are to be read up to sign. To avoid notational clutter, we use the -notation to denote both the spanned parallelepipeds and their volume, computed as the determinant of the matrix with columns given by the factors of the wedge product.
Appendix B Proof of
We choose a local unit volume tangent space basis as in Section A. The tangent space is then spanned by the orthogonal basis . With , we have that is a unit -volume parallelepiped in .
Next, we calculate as the change of volume under :
From the first to the second line, we have used that the denominator is normalized, and the different signs are due to Eq. (5).
On the other hand, let us calculate the -component of the extended velocity , i.e., the component normal to the section’s tangent space :
From the first to the second line, we have used determinant rules, and from the second to the third lines Laplace expansion along the time component.
Finally, the above calculation is, of course, consistent with the one given in Section A, since whenever , i.e., when the section does not move. In this case, and .
I would like to thank Simon Eugster, Florian Huhn and Dietmar Salamon for useful discussions, as well as George Haller for helpful comments on an earlier version of the manuscript. David Legland’s geom2d-library for Matlab was very useful in the loop decomposition used in Fig. 2.
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