Transport Induced by Mean-Eddy
I. Theory, and Relation to Lagrangian Lobe Dynamics
In this paper we develop a method for the estimation of Transport Induced by the Mean-Eddy interaction (TIME) in two-dimensional unsteady flows. The method is built on the dynamical systems approach and can be viewed as a hybrid combination of Lagrangian and Eulerian methods. The (Eulerian) boundaries across which we consider (Lagrangian) transport are kinematically defined by appropriately chosen streamlines of the mean flow. By evaluating the impact of the mean-eddy interaction on transport, the TIME method can be used as a diagnostic tool for transport processes that occur during a specified time interval along a specified boundary segment.
We introduce two types of TIME functions: one that quantifies the accumulation of flow properties and another that measures the displacement of the transport geometry. The spatial geometry of transport is described by the so-called pseudo-lobes, and temporal evolution of transport by their dynamics. In the case where the TIME functions are evaluated along a separatrix, the pseudo-lobes have a relationship to the lobes of Lagrangian transport theory. In fact, one of the TIME functions is identical to the Melnikov function that is used to measure the distance, at leading order in a small parameter, between the two invariant manifolds that define the Lagrangian lobes. We contrast the similarities and differences between the TIME and Lagrangian lobe dynamics in detail. An application of the TIME method is carried out for inter-gyre transport in the wind-driven oceanic circulation model and a comparison with the Lagrangian transport theory is made.
keywords:Eulerian Transport, Lagrangian Transport, Mean-Eddy Interaction, Dynamical Systems Approach, Wind-Driven Ocean Circulation
Pacs:47.10.Fg, 47.11.St, 47.27.ed, 47.51.+a, 92.05.-x, 92.10.A-, 92.10.ab, 92.10.ah 92.10.ak, 92.10.Lq, 92.10.Ty, 92.60.Bh
- 1 Introduction and motivation
- 2 Mathematical background
- 3 Transport functions for TIME
- 4 Characteristics of TIME
- 5 Application to a numerical simulation of the wind-driven double-gyre ocean circulation
- 6 Summary and concluding remarks
- A Mathematical Background on Perturbation of Trajectories
- B Relation to Lagrangian Transport
1 Introduction and motivation
Lagrangian transport methods are based on following the individual trajectories obtained by solving the original differential equation (ODE) for the particle location starting from a set of initial conditions at time :
where is the velocity field. The geometrical approach of dynamical systems theory is particularly useful when the flow field has Lagrangian coherent structures that separate the flow into distinct regions. Then Lagrangian lobe dynamics describes the transport process between these regions using stable and unstable manifolds of hyperbolic trajectories as (moving) boundaries. The Lagrangian methods have been applied successfully to a number of unsteady geophysical flow problems; for a review, see [1, 2]. If the flow is steady, i.e., , the invariant manifolds are stationary and no transport occurs between the regions.
On the contrary, Eulerian-based methods are mainly concerned with the amount of transport across stationary (Eulerian) boundaries without computing individual trajectories. An advantage of Eulerian methods is that they tend to be much less elaborate than Lagrangian methods in terms of computational implementation. The choice for the Eulerian boundaries is generally flexible, unlike the Lagrangian methods.
From the dynamical systems point of view, a parallel development of a method that computes transport across the Eulerian boundary has yet to take place. In this paper we begin the development of such a method. The method makes use of the interaction between the reference (mean) state and the unsteady variability (eddy) as the fundamental mechanism of transport. Hence we refer to it as the Transport Induced by the Mean-Eddy interaction (TIME). Using a streamline of the reference state as the boundary across which we consider transport, TIME can be thought as a hybrid of Lagrangian and Eulerian methods. Like the Eulerian method, the boundary is stationary. Like the Lagrangian method, the boundary is kinematically defined and there is no TIME in the steady flow without the unsteady eddy component in the velocity. In certain situations we are able to describe the geometrical relationship of TIME along the Eulerian boundaries with Lagrangian lobe dynamics.
We require no assumption of incompressibility in our theoretical framework. Therefore the ideas and techniques of the TIME method can be applied to two-dimensional compressible flow or three-dimensional volume-preserving flow which can be represented as special classes of two-dimensional flows, such as the shallow-water model. Remarks concerning incompressibility are provided throughout the paper as special cases. Extensions to three-dimensional flow are possible , but there is more complexity in the geometry of the transport, and this is will be the topic of a future publication.
The outline of this paper is as follows. In Section 2, we provide a brief mathematical background and introduce the notion of a kinematically-defined Eulerian boundary; readers who are familiar with elementary dynamical systems theory may omit this section without significant loss of continuity by referring back to the notation and definitions as necessary. A brief glossary is also provided in Table 1. The TIME method is defined in Section 3, along with the two types of TIME functions. These functions, along with the notion of pseudo-lobes, are further explored in Section 4. An application of the TIME method is carried out in Section 5 for the inter-gyre transport in the double-gyre ocean circulation model and a comparison with Lagrangian transport theory is presented. Appendix A.1 provides details of perturbation theory, and Appendix B compares the TIME method with the Lagrangian transport methods. ††margin: [Tab.1]
While in this paper we focus on introducing and developing the two TIME functions that estimate the amount and the geometry of transport, in the companion paper , we expand the TIME method further as a diagnostic tool for transport processes by analyzing in detail the influence of the mean-eddy interaction.
2 Mathematical background
In this section, we introduce the basic mathematical background necessary to develop the TIME method. The starting point is first expressing the velocity field (1) in the following form:
where and , respectively, correspond to the steady reference state and the unsteady fluctuation around the reference state. The choice of reference state may not be unique. We choose the time-average (mean) of the full time-dependent field as the reference state in this study because the mean-eddy decomposition is natural when the flow field is given by a data set; the TIME method itself does not require the reference state to be the mean. Many of our results will be perturbative in nature, with the (small) perturbation parameter being the amplitude of the fluctuation that is implicitly included in with respect to . Appendix A gives results on the length of time intervals on which perturbed trajectories remain close to trajectories of the reference state. These results will provide the validity of the perturbative nature of our method since the TIME functions that we derive will be of the form of integrals along perturbed trajectories and approximations that, in principle, can be analytically computed are of the form of integrals along trajectories of the reference state. The regularity assumptions required on the velocity field are minimal. Essentially, we need existence and uniqueness of fluid particle trajectories, the ability to linearize about points in space, compute Taylor expansions through second order with respect to parameters, and for certain integrals of components of the velocity field along trajectories of the reference velocity field to exist. Assuming that the velocity field is twice continuously differentiable with respect to the spatial coordinates, time, and any parameters is adequate. No further assumptions on the nature of the time dependence (e.g. time periodicity, quasiperiodicity, etc.) are required.
The use of perturbation theory in the development of the TIME method is made more transparent if we introduce an ”order parameter”, , associated with the fluctuation term as follows:
where . The introduction of in this way makes perturbation arguments more transparent. However, the TIME functions can be equally as well expressed in terms of or , but in either case the approximation is to leading order in the size of the fluctuation.
2.1 Reference state and kinematically-defined Eulerian boundary
We refer to a curve as Eulerian if it is stationary. The TIME method uses an Eulerian curve that is defined kinematically as a streamline of the reference flow. It can be given as a solution of
with an initial condition at time . For to be a physically meaningful boundary, must be a regular point of , i.e., .
A trajectory with an initial condition at time reaches at time in the reference flow. This trajectory is uniquely identified by a scalar, , because time shifts of a trajectory remain on the same trajectory in the reference flow. Throughout the paper, we interpret the flight-time coordinate variable strictly as a spatial coordinate variable along while is a temporal variable. Accordingly and
can be viewed as different parametrizations of the same trajectory, which we call the reference trajectory. The Cartesian pair of coordinates will prove to be particularly convenient for describing the TIME method. A glossary is provided in Table 1 for the principal definitions.
A hyperbolic stagnation point is a singular point. In the reference flow, it is a special trajectory called the distinguished hyperbolic trajectory (DHT)  and we denote it by , i.e., . Although itself cannot be a physically meaningful Eulerian boundary , the unstable and stable invariant manifolds that have a DHT at the starting and end point, respectively, are special types of :
where denotes the reference manifold with the superscripts and for unstable and stable invariant manifolds, respectively. In a case where and coincide, the reference streamline is called the separatrix or heteroclinic connection of the upstream DHT and the downstream DHT :
where the superscript stands for heteroclinic connection and the subscripts represent the direction of towards the corresponding DHT. In addition, if and coincide, then is called the homoclinic connection. An invariant manifold is special case of kinematically-defined because has a semi-infinite or bi-infinite range as in (6) and (7). We emphasize that the terms finite and infinite refer to the range of on , rather than the physical length of .
For the description of the transport geometry near , it is often convenient to use an orthogonal arc-length coordinate system, . Along , the arc-length and the flight-time are related by the local velocity, i.e., . Normal to , is defined to be the signed distance of a neighboring point to ; , and correspond to the left, on, and the right of with respect to the forward direction of along . A pair of orthogonal unit vectors in the tangent and normal directions to are given by
where . The transformation between the Cartesian and arc-length coordinates is area-preserving.
2.2 Unsteady flow and perturbation theory
As we have noted, the mathematical formulation of the TIME method is based on perturbation theory for a velocity field given by (2) and (3). The necessary background and results are given in Appendix A. Trajectories of the unsteady flow passing through on at are of the following form:
where is the leading-order displacement vector with . Computing the Taylor expansion of (2) and the time derivative of (9) with respect to gives to the following linear ordinary differential equation for :
Given , can be obtained by solving this linear system where the nonlinear evolution of provides us with the time-dependent coefficients and the inhomogeneous term. In Appendix A we show that perturbation theory can provide valid approximations in situations where is defined over finite, semi-infinite or bi-infinite time intervals.
It is worth noting here that for many of the most fruitful perturbation theories used in dynamical systems type analyses rarely are precise bounds available for the size of the perturbation for which the method is applicable. Nevertheless, this has not limited the insights they have provided in a variety of applications. For example, the typical statements of Melnikov’s method [6, 7, 8] indicate only that it is valid for sufficiently small. Another example is the well-known Kolmogorov-Arnold-Moser (KAM) theorem , which has been proven useful in many applications despite the fact that the bounds are generally too strict to be practically applicable. The situation with the KAM theorem is even worse since rarely are the hypotheses of the theorem even verified in applications since they require the velocity to be expressed in action-angle variables, which can rarely be achieved. This limitation also prevents one from obtaining any type of bound on the perturbation for which the theorem is valid.
3 Transport functions for TIME
Having kinematically defined the Eulerian boundary by the reference state, we now turn our attention to transport across . There are two aspects: one is concerned with the amount of flow property and the other is concerned with geometry of transport. Examples of flow properties are mass, temperature, humidity in the atmosphere, salinity in the oceans, and such. The TIME functions are developed for these two aspects, first for a finite time interval along any (Section 3.1) and then for an infinite time interval along an infinite (Section 3.2).
3.1 Derivation of the finite-time TIME functions
3.1.1 Accumulation of a flow property
We assume that the time-dependent fluctuation in the flow property distribution, denoted by , is also small
Like in (2) and (3), we introduce with . To illustare the basic idea for estimating the amount of property transport, we consider the imaginary fluid column in the flow (Figure 1c). By accumulating the flux at the moving intersection of with , we obtain the net amount of accumulation.
Up to leading order, the intersection of with at time is approximated by the reference trajectory using perturbation theory (Appendix A.1). At , the instantaneous flux of carried by the local velocity across per unit length is
where and are defined in (8). The positive value means the flux from the right to the left across with respect to the forward direction of . This formula (12) says that the instantaneous flux of across exists if has a component normal to , and that the time-dependent fluctuation contributes to the transport at the higher order. At the leading order, the instantaneous flux of penetrating across at per unit flight time is , i.e.,
|using (3), where|
We refer to as the instantaneous flux function, induced by the unsteadiness (eddy) of the velocity through the interaction with the reference (mean) flow. This is the origin of the transport induced by the mean-eddy interaction (TIME) across . The sign of indicates the direction of the instantaneous flux.
The accumulation over the interval is thus approximated by up to leading order. Because this amount is the same for any along the reference trajectory with , we obtain a general form of the accumulation
where the first pair in the arguments of the left-hand side represents the combination of spatial coordinate and time at which the net accumulation of is evaluated, while the next pair concerns the time interval on which the transport takes place. Here can be either before or after or . We refer to as the accumulation function. Characteristics of will be discussed further in Section 4.
3.1.2 Displacement distance and area
For the geometry, we consider the displacement distance of the particle starting from on . In the unsteady flow at time , the displacement of the particle from is up to leading order by (10). For particle transport and its geometry, we choose to use arc-length coordinates in the description of the displacement functions because the displacement distance has the physical dimension of length. Using (9) along a reference trajectory with the initial condition , the leading order term for the displacement distance due to particle transport at is given by
|Because can be obtained by solving (10), then so can by the direct substitution. However, a simpler formula is available by considering|
which corresponds to the displacement area per unit along as shown in Appendix A. Using for the initial condition, construction, the solution for the displacement area at is given by
reflects the compressibility of the reference flow; for an incompressible flow, .
Like the accumulation, the displacement area associated with transport over can be evaluated at where can be before, in, or after the time interval. Conceptually, this is to let obtained by (16) evolve under the reference flow over an additional time interval to take the incompressibility into account. As shown in Appendix A, the final form of the displacement functions is given by
As in the case of , the first pair in the argument represents the spatial coordinate and time at which the function is evaluated, and the next pair correspond to the time interval when transport takes place.
Accordingly over , the displacement is determined by two contributions: one is from the unsteadiness of the flow measured along through the instantaneous flux , and the other from the compressibility of the reference flow through , which may result in compression or expansion of the area. Sign of indicates the directionality of transport across .
3.2 Extension over the infinite TIME functions
We refer to the accumulation function (14) and displacement distance and displacement area functions (18) as the (finite) TIME functions because they are defined over a finite time interval and hence the finite range of along . These TIME functions can be extended over the semi-infinite and and bi-infinite , because the exponential decay of the velocity towards DHTs at the starting or(and) end point(s) guarantees the convergence conditions required for the validity of perturbation theory (Appendix A). The extension of the displacement functions over can be particularly useful since it provides a direct link to the Melnikov function which measures the leading order distance between the time-dependent unstable and stable invariant manifolds [6, 7, 8]. The Melnikov function has been used to study Lagrangian transport, mostly in incompressible flows ; also see .
The TIME functions for these special are as follows. The transport that has happened in the past across can be obtained by extending the TIME functions over a semi-infinite time interval :
|for some .|
|Similarly, the transport that will happen in the future across the stable manifold can be obtained by extending the TIME functions over the semi-infinite interval :|
Finally, the entire transport across the separatrix can be obtained by extending the TIME functions over the bi-infinite interval :
The displacement area function for is the same as the so-called Melnikov function.
4 Characteristics of TIME
4.1 Characteristics along an individual trajectory
The accumulation is obtained by following the individual reference trajectories (Section 3). This leads to the concepts of invriance and piece-wise independence.
Invariance of the accumulation function. Given a fixed time interval , the accumulation function is invariant:
for any . This invariance implies that each trajectory has perfect memory for the amount of transport. Invariance for the displacement area function:
is subject to the compressibility factor of the reference flow.
Piece-wise independence. It is clear in the definition (14) that the time interval can be broken up into an arbitrary number (say ) of pieces:
where with and . This is a temporal piece-wise independence. Using (5) along , spatial piece-wise independence follows naturally by breaking the spatial segment into pieces by with and and transforming them into temporal pieces with over . Because the independence is a characteristics defined for a fixed , both temporal independence and spatial independence hold for the displacement area function .
4.2 Coherency of transport
Geometry of particle displacement leads to the concepts associated with the coherency of transport. Because the geometrical characteristics discussed here hold for any over any time interval, we will drop from each notation for simplicity after the first appearance as indicated in ; for example, . When is taken as , there is a geometrical relation to Lagrangian transport, which we will treat separately in Appendix B.
For an illustration of transport geometry and coherency, let us consider an imaginary material curve placed initially on at time , i.e., (Figure 1d). In the reference flow, advects along without any displacement. In the unsteady flow, velocity normal to may let depart from . For the transport geometry associated with the TIME method, we define
which is the leading order approximation to .
Pseudo-primary intersection point (pseudo-PIP) sequence of and . In the unsteady flow, may intersect with to form a chain of lobe-like structures as shown schematically in Figure 1d. We denote such a ordered sequence of such zeros with
where we use the fact that the zeros of and those of are identical. Unless the zero is non-degenerate, is generally identical to the intersection sequence of up to the leading order; see  and also Appendix A.1. We call the pseudo-primary intersection point (pseudo-PIP) sequence in contrast to the Lagrangian lobe dynamics for the heteroclinic connection. The term ’primary’ is used here to emphasize the analogy of the PIP of Lagrangian lobe counterpart. A pseudo-PIP sequence can be transformed into arc-length coordinate so that . ††margin: [Fig.2]
Invariance of pseudo-PIP sequence. Because the displacement area is invariant subject to the compressibility effect from (23) and the compressibility effect will not change at given , is invariant. Each coincide with a reference trajectory, i.e.,
Pseudo-lobe seqence. We denote the lobe-like structure defined by the segments of and between a pair of two adjacent pseudo-PIPs by and call it the pseudo-lobe:
Its sequence makes the chain-like structure, which we call the pseudo-lobe sqeuence, . Using , the pseudo-lobe is the area surrounded by and between two adjacent pseudo-PIPs. It is worth emphasizing that temporal and spatial piece-wise independence (Section 4.1) holds for the displacement function of each pseudo-lobe individually.
Signed area. The size of each pseudo-lobe measures the amount of the locally coherent transport. Its signed area is given by:
If the flow is incompressible, the area of each lobe is invariant, i.e., .
Directionality of pseudo-lobes. Each pseudo-lobe represents the amount of fluid particles that go across over . Depending on whether lies to the left or right of , can be of two types, denoted respectively by or corresponding to or for transport across from right to left or left to right. Provided all intersections are transverse, alternated the types between and along , resulting in a chain structure at any given time.
The geometry associated with transport of can be established conceptually by replacing the displacement area function for with the accumulation function in arc-length coordinates. In the case for mass with where the flow is incompressible, this operation results exactly in the displacement functions of fluid particles, .
5 Application to a numerical simulation of the wind-driven double-gyre ocean circulation
In this study we apply the TIME method to the inter-gyre transport in the mid-latitude, wind-driven ocean circulation. The data set is obtained by a numerical simulation of a quasi-geostrophic (QG) 3-layer model in a rectangular basin geometry with free slip boundary conditions . Due to the latitudinal antisymmetric wind-stress curl applied at the ocean surface, the basic circulation pattern in the top layer is a double-gyre structure separated by an eastward jet shooting off from the confluence point of the southward and northward western boundary currents (Figure 3). Driven by this strong jet, the subpolar gyre circulates counterclockwise in the north and the subtropical gyre circulates clockwise in the south. Depending on the value of the parameters such as the viscosity and the wind stress curl, the ocean circulation exhibits a rich time-dependent dynamics [13, 14]. ††margin: [Fig.3]
At a wind stress curl of , the ocean dynamics is nearly periodic with dominant spectral peak at period days in a 1000 km 2000 km rectangular domain. We choose this flow since the physical interpretation for a flow field close to periodic is more simple and therefore it allows us to focus more on demonstrating our method. It is worth noting again that the method itself does not require time periodity of the flow. The velocity data set used in this study has a spatial resolution of 12.5km12.5km and is saved daily after a 30,000-day spin-up from rest. Figure 3a shows the streamfunction of the top layer at when is close to the reference state . From here on, the subscript in denotes days after the completion of the spin-up. The most significant region of unsteadiness of the flow lies in the upstream region of the eastward jet near the western boundary, while small-amplitude Rossby waves propagate westward in the entire ocean basin. The Lagrangian transport processes between the two gyres are governed by the lobe dynamics associated with the unstable invariant manifold of the upstream DHT on the western boundary and the stable invariant manifold of the downstream DHT on the eastern boundary; see Appendices A.1 and B for the definitions and more details of and . The inter-gyre transport in the top layer was carefully studied by  using Lagrangian lobe dynamics methods.
For the comparison with the Lagrangian method, we choose the Eulerian boundary of the TIME method as the bi-infinite, reference heteroclinic connection which spans over with on the western boundary and on the eastern boundary (solid line in Figure 3c). It is worth noting again that application of the TIME method is not limited to flows that possess a heteroclinic connection. For the computation of the TIME functions, we choose the location of so that is very close to and is exponentially close to zero. Along , the speed of the mean jet significantly increases starting near . Geographically is separated from only by 600m. As increases towards the downstream direction, makes a sharp turn in around in the north, followed by the second sharp turn around in the south. Measured in the flight time, , , and are 110days, 129days, and 174.5days, respectively. In the further downstream direction, extends to the east and exhibits little meandering. After about with the subscript in for the flight-time coordinate in days from , becomes extremely small. In the region near , the order of the unsteadiness relative to the reference state is small (), supporting the applicability of the TIME method (see Appendix A.2).
Figure 4 shows the instantaneous flux on as a Hovmöller diagram [16, 17] in the space for . ††margin: [Fig.4] The signals of are periodic in with the period because of the time-periodic ocean dynamics. For small , is always near zero because is exponentially small near . Also for large , is always almost zero because both