2.1 Flow barriers: unsteadiness and finite-time issues

This subsection establishes some insights into flow barriers, to help explain the development in the next subsection. A simple transport barrier for a steady flow is illustrated in figure 1(a), which shows the streamlines of the classical Kelvin–Stuart cats-eyes flow (Stuart Reference Stuart1967; Sharma & McKeon Reference Sharma and McKeon2013). The distinguished red/green curves clearly represent the barriers between the vortex and the upper and lower jets, all three of which can be interpreted as coherent in the sense of Lagrangian particle trajectories of the flow $\dot{\boldsymbol{x}}=\boldsymbol{v}(\boldsymbol{x})$ , where $\boldsymbol{v}$ is the steady Eulerian velocity field. Manipulating these flow barriers, which are internal to the fluid, in a specified time-varying way will therefore be a method for controlling transport between the coherent structures.

Figure 1. A well-defined internal flow barrier for an unsteady flow is shown in (a) and its counterpart (at a general time  $t$ ) in an unsteady situation in (b).

Unfortunately, the straightforward flow-barrier interpretation for the steady flow of figure 1(a) becomes less clear in an unsteady situation. However, barriers can be rationalised in terms of extremal stretching of fluid elements, as can be argued from the steady picture of figure 1(a). As the blue fluid element at time $t=0$ evolves, it approaches the stagnation point $\boldsymbol{a}$ and gets pulled apart in the direction of the red arrows while compressing in to $\boldsymbol{a}$ . The green curve it straddles is the stable manifold of $\boldsymbol{a}$ , comprising fluid particles which approach $\boldsymbol{a}$ in the limit $t\rightarrow \infty$ , while the red curve which influences its stretching is the unstable manifold of $\boldsymbol{a}$ . Stable and unstable manifolds are now well established in the dual role of being associated with flow barriers and yet engendering transport between coherent structures through their potential tangling (Rom-Kedar, Leonard & Wiggins Reference Rom-Kedar, Leonard and Wiggins1990; Wiggins Reference Wiggins1992; Franco et al. Reference Franco, Pekarek, Peng and Dabiri2007; Haller Reference Haller2015; Balasuriya Reference Balasuriya2016a ). Red and green will respectively be used for unstable and stable manifolds throughout this paper; the fact that the flow barriers are coloured with both red and green in figure 1(a) is a consequence of the fact that, for example, the unstable manifold of $\boldsymbol{a}$ coincides with the stable manifold of $\boldsymbol{b}$ , thereby forming an indubitable flow barrier. The stretching occurs at an exponential rate as a consequence of the fact that flow near the saddle stagnation point $\boldsymbol{a}$ is approximated by $\dot{\boldsymbol{x}}=\unicode[STIX]{x1D735}\boldsymbol{v}(\boldsymbol{a})\boldsymbol{x}$ , with the eigenvectors corresponding to the negative and positive eigenvalues of $\unicode[STIX]{x1D735}\boldsymbol{v}(\boldsymbol{a})$ defining the local stable and unstable manifold emanation directions respectively. Thus, fluid elements straddling a stable manifold will stretch exponentially in forward time, and the two ends of the elongated fluid element will get pushed rapidly into the two different coherent structures. Similarly, fluid elements straddling an unstable manifold will stretch exponentially in backward time. Put another way, stable manifolds are flow barriers in forward time, while unstable manifolds are flow barriers in backward time.

The advantage of the above statement (in contrast with some other properties of stable/unstable manifolds) is that it remains legitimate for unsteady flows, as illustrated in figure 1(b), which shows stable (green) and unstable (red) manifolds emanating from two anchoring points $\boldsymbol{a}(t)$ and $\boldsymbol{b}(t)$ , at an instance in time  $t$ . Once again, the fate of the blue fluid element straddling the stable manifold must be similar to before: it approaches $\boldsymbol{a}(t)$ as time increases and gets stretched out by the unstable manifold. This is irrespective of the nature of the intersection pattern; for example in figure 1(b) it is irrelevant that this element also intersects the unstable manifold of $\boldsymbol{b}(t)$ . An important difference from the steady situation pictured in figure 1(a), however, is the fact that neither the manifolds nor the anchor points are stationary. (The time-varying anchor points are actually specialised trajectories – hyperbolic trajectories – of the flow, whose control has recently been pursued (Balasuriya & Padberg-Gehle Reference Balasuriya and Padberg-Gehle2013, Reference Balasuriya and Padberg-Gehle2014a ).) In spite of this, the forward- and backward-time separating properties of the unsteady stable and unstable manifolds remain true. This is why forward-time finite-time Lyapunov exponent (FTLE) fields (Shadden, Lekien & Marsden Reference Shadden, Lekien and Marsden2005; Senatore & Ross Reference Senatore and Ross2008; Tallapragada & Ross Reference Tallapragada and Ross2013; Kucala & Biringen Reference Kucala and Biringen2014; Raben, Ross & Vlachos Reference Raben, Ross and Vlachos2014; Allshouse & Peacock Reference Allshouse and Peacock2015b ; BozorgMagham, Ross & Schmale Reference BozorgMagham, Ross and Schmale2015; Haller Reference Haller2015; Balasuriya et al. Reference Balasuriya, Kalampattel and Ouellette2016), which measure exponential stretching, reveal strong ridges along stable manifolds and backward-time FTLE fields along unstable manifolds. In spite of the intersections possibilities between the manifolds, the following characteristic remains true for unsteady flows: stable manifolds are flow barriers in forward time, while unstable manifolds are flow barriers in backward time. Thus, having these move in ways which are prescribed is a first step in controlling fluid transport between coherent structures.

Defining stable and unstable manifolds, however, requires knowing the Eulerian velocity field over infinite times. This is clearly unfeasible in any realistic finite-time situation. Different diagnostic methods (of which FTLEs is one) seek entities which generalise stable/unstable manifolds in some way, or possess characteristics which are similar to those. While giving an extensive review on these is beyond the scope of this paper, since the idea here is to control such finite-time flow barriers, a discussion on some of these diagnostic methods is given in appendix A.

For this work, the following understanding suffices for stable/unstable manifold proxies (i.e. flow barriers) for general flows

in two dimensions defined over a finite-time interval $[-T,T]$ . Necessary (but not sufficient) conditions for a time-varying curve $\unicode[STIX]{x1D6EC}(t)$ to be a stable manifold proxy (SMP) for (2.1) for times $t\in (-T,T)$ are:

1. (i) the definition of $\unicode[STIX]{x1D6EC}(t^{\star })$ will depend on behaviour of (2.3) for times $t\in [t^{\star },T]$ ;

2. (ii) there is a neighbourhood around $\unicode[STIX]{x1D6EC}(t^{\star })$ such that if a fluid element $e$ were chosen in this neighbourhood, the following is true: the distance between the Lagrangian evolution of both $e$ and $\unicode[STIX]{x1D6EC}(t^{\star })$ from time $t^{\star }$ onwards, increases with time (i.e.  $\unicode[STIX]{x1D6EC}(t^{\star })$ is repelling);

3. (iii) the above property is true at all choices of times $t^{\star }$ in $(-T,T)$ , implying that $\unicode[STIX]{x1D6EC}(t)$ is a genuinely unsteady entity in this interval (and not just defined at a fixed time  $t^{\star }$ );

4. (iv) moreover, $\unicode[STIX]{x1D6EC}(t)$ is Lagrangian; it is a material curve of (2.1).

Beyond these conditions, a specific proxy will not be chosen. However, it is noted that these conditions are shared by FTLEs and hyperbolic LCSs. Similarly, necessary (but not sufficient) conditions for a time-varying curve $\unicode[STIX]{x1D6EC}(t)$ to be a unstable manifold proxy (UMP) for (2.1) for times $t\in (-T,T)$ are:

1. (i) the definition of $\unicode[STIX]{x1D6EC}(t^{\star })$ will depend on behaviour of (2.3) for times $t\in [-T,t^{\star }]$ ;

2. (ii) there is a neighbourhood around $\unicode[STIX]{x1D6EC}(t^{\star })$ such that if a fluid element $e$ were chosen in this neighbourhood, the following is true: the distance between the Lagrangian evolution of both $e$ and $\unicode[STIX]{x1D6EC}(t^{\star })$ from time $t^{\star }$ onwards, decreases in time (i.e. $\unicode[STIX]{x1D6EC}(t^{\star })$ is attracting);

3. (iii) the above property is true at all choices of times $t^{\star }$ in $(-T,T)$ , implying that $\unicode[STIX]{x1D6EC}(t)$ is a genuinely unsteady entity in this interval (and not just defined at a fixed time  $t^{\star }$ );

4. (iv) moreover, $\unicode[STIX]{x1D6EC}(t)$ is Lagrangian; it is a material curve of (2.1).

2.2 Control velocity

The control velocity problem and its solution can now be stated. Suppose two-dimensional Eulerian velocity data are available on a spatial grid, at discrete times, over a time interval $(-T,T)$ . The data will typically be in the form $\boldsymbol{u}_{ij}=\boldsymbol{u}(\boldsymbol{x}_{i},t_{j})$ , with the $i$ representing the grid points and the $j$ the temporal sampling. Lagrangian trajectories therefore obey

whose implementation will require appropriate interpolation of the data. It will be assumed that the system (2.2) has a flow barrier – either a SMP or a UMP – $\tilde{\unicode[STIX]{x1D6E4}}(t)$ for each $t$ in the time domain. If a SMP, it could for example be a strong ridge of the forward-time FTLE field generated by beginning at time $t$ , or a curve from which there is locally maximal repulsion (if using the concept of hyperbolic LCSs of Haller (Reference Haller2015)). The question is the following: given a prescribed flow barrier $\unicode[STIX]{x1D6E4}(t)$ which is close to $\tilde{\unicode[STIX]{x1D6E4}}(t)$ , what is the Eulerian velocity adjustment $\boldsymbol{c}(x,t)$ that needs to be added to (2.2) in order for the system to generate $\unicode[STIX]{x1D6E4}(t)$ as a Lagrangian flow barrier? Putting it another way: is it possible to determine the control velocity $\boldsymbol{c}$ in

such that (2.3) possesses the time varying $\unicode[STIX]{x1D6E4}(t)$ as a flow barrier?

This question can be answered under certain conditions. The first is that (2.2) is a flow in which the unsteadiness is not too dominant: $\boldsymbol{u}_{ij}$ changes little with $j$ . Define the steady velocity

by simply averaging over time, and the corresponding steady flow

Since (2.5) is steady, it can be considered a dynamical system for all (infinite) times. Suppose that (2.5) has a flow barrier (in this case precisely a stable or an unstable manifold) $\unicode[STIX]{x1D6E4}$ . It will be necessary that the flow barriers $\unicode[STIX]{x1D6E4}$ , $\tilde{\unicode[STIX]{x1D6E4}}(t)$ and $\unicode[STIX]{x1D6E4}(t)$ are all of the same type (i.e. either all SMPs or all UMPs). The specification of $\unicode[STIX]{x1D6E4}(t)$ shall be assumed to be given as in figure 2. At any point $\boldsymbol{x}\in \unicode[STIX]{x1D6E4}$ , $\bar{\boldsymbol{u}}(\boldsymbol{x})$ is tangential to $\unicode[STIX]{x1D6E4}$ . By defining the $90^{\circ }$ -rotation operator $(x_{1},x_{2})^{\bot }:=(-x_{2},x_{1})$ componentwise for vectors in $\mathbb{R}^{2}$ , it is clear that a unit normal vector to $\unicode[STIX]{x1D6E4}$ at $\boldsymbol{x}$ is given by $\hat{\boldsymbol{n}}(\boldsymbol{x}):=\bar{\boldsymbol{u}}(\boldsymbol{x})^{\bot }/|\bar{\boldsymbol{u}}(\boldsymbol{x})|$ . The required time-varying manifold $\unicode[STIX]{x1D6E4}(t)$ is defined by specifying the displacement $r(\boldsymbol{x},t)$ along the direction $\hat{\boldsymbol{n}}(\boldsymbol{x})$ at each location $\boldsymbol{x}$ on $\unicode[STIX]{x1D6E4}$ , and at each time $t$ , as shown in figure 2. The prescribed $r$ shall be assumed to be small and suitably smooth adjacent to $\unicode[STIX]{x1D6E4}$ . A specification of the required manifold is thus given by

valid in whatever part of $\unicode[STIX]{x1D6E4}$ , and for whatever times $t$ , required. This is a ‘normal mappability’ condition from $\unicode[STIX]{x1D6E4}(t)$ to $\unicode[STIX]{x1D6E4}$ at all times, and a similar mappability is assumed between $\tilde{\unicode[STIX]{x1D6E4}}(t)$ and $\unicode[STIX]{x1D6E4}$ .

Figure 2. Original flow barrier $\tilde{\unicode[STIX]{x1D6E4}}(t)$ (black) of (2.2), steady manifold $\unicode[STIX]{x1D6E4}$ (magenta) of (2.5) and required unsteady flow barrier $\unicode[STIX]{x1D6E4}(t)$ (blue) of (2.3) at a time instance $t$ ; $r(\boldsymbol{x},t)$ is specified at each location $\boldsymbol{x}\in \unicode[STIX]{x1D6E4}$ as the displacement in the direction  $\hat{\boldsymbol{n}}$ .

It is shown in appendix B that the control velocity needs to satisfy the condition

on $\unicode[STIX]{x1D6E4}$ . To achieve $\unicode[STIX]{x1D6E4}(t^{\star })$ at a time $t^{\star }$ , it is necessary that (2.7) be satisfied for $t\geqslant t^{\star }$ if $\unicode[STIX]{x1D6E4}(t)$ is a SMP or for $t\leqslant t^{\star }$ if a UMP. The accumulated effect of the control velocity $\boldsymbol{c}$ over time results in the appropriate manifold proxy $\unicode[STIX]{x1D6E4}(t^{\star })$ at time $t^{\star }$ . To achieve $\unicode[STIX]{x1D6E4}(t)$ for $t\in [-T,T]$ , then, (2.7) needs to be obeyed for all $t\in [-T,T]$ . Therefore, flow barriers varying continuously according to a specification can be achieved.

With regards to the spatial variation, the derivation in appendix B indicates that (2.7) needs to be satisfied on $\unicode[STIX]{x1D6E4}$ , and be extended continuously in space. An easily stated approach would be define $\boldsymbol{c}(\boldsymbol{x},t)$ specifically by the formula (2.7) in a strip around $\unicode[STIX]{x1D6E4}$ (this might be the easiest approach if using for data postprocessing). Another would be to have (2.7) define $\boldsymbol{c}(\unicode[STIX]{x1D6E4},t)$ , but to extend this into a strip in a continuous way while ensuring that any other required conditions are satisfied. For example, incompressibility can be satisfied since the condition is on $\boldsymbol{c}(\unicode[STIX]{x1D6E4},t)$ at any time $t$ , and thus this imposes conditions on the tangential (to $\unicode[STIX]{x1D6E4}$ ) and normal components of $\boldsymbol{c}$ , and also the tangential derivative of the tangential component. An extension can then be performed in the normal direction by insisting that the normal derivative of the normal component is negative the tangential derivative of the tangential component, thereby ensuring that incompressibility is satisfied. Any alternative extension – subject to a required physical constraint or based on realisability, say – is also possible. Having extended $\boldsymbol{c}$ to a strip in whatever way, it can be clipped to zero outside the strip; smoothness is only required near $\unicode[STIX]{x1D6E4}$ . Also note that several manifold segments in a flow can be simultaneously manipulated by applying the velocity (2.7) in local strips around each such manifold.

Some intuition on the terms appearing in (2.7) may be welcome. The first two terms in the second row of (2.7) give the material derivative of $\boldsymbol{r}$ with respect to (2.2); this captures the fact that as time progresses, particles on $\unicode[STIX]{x1D6E4}$ will travel along it according to (2.2), and thus $r$ will be different at such locations. The term associated with the first bracket in (2.7) is guaranteed to be a velocity in the direction normal to $\unicode[STIX]{x1D6E4}$ . Thus, in achieving this normal displacement, the second bracket term reveals that a tangential component of velocity might be needed. (Intuitively, the terms in these second brackets capture the curvature of  $\unicode[STIX]{x1D6E4}$ .) The presence of such a tangential term is not surprising, since purely normal Eulerian velocities give tangential displacements of Lagrangian manifolds (Balasuriya Reference Balasuriya2011). The present context is the inverse of this: a purely normal Lagrangian manifold displacement requiring a tangential Eulerian velocity component.

While the development in appendix B is formal in nature, it relies on properties of SMPs/UMPs, the displacement $|r|$ and the unsteadiness $|\boldsymbol{u}-\bar{\boldsymbol{u}}|$ being small, and is valid in regions and times where both $\tilde{\unicode[STIX]{x1D6E4}}(t)$ and $\unicode[STIX]{x1D6E4}(t)$ are normally mappable to $\unicode[STIX]{x1D6E4}$ . Based on rigorous results (Balasuriya & Padberg-Gehle Reference Balasuriya and Padberg-Gehle2014b ) from a different (and not easily implementable) perspective, the error in using (2.7) to obtain $\unicode[STIX]{x1D6E4}(t)$ is expected to be of second order in these quantities. Note moreover that the details of the flow barrier $\tilde{\unicode[STIX]{x1D6E4}}(t)$ of the original flow are not required in defining (2.7); its existence is all that is needed.

The intersection pattern between stable and unstable manifolds at a given instance in time is well known to strongly influence fluid mixing (Rom-Kedar et al. Reference Rom-Kedar, Leonard and Wiggins1990; Franco et al. Reference Franco, Pekarek, Peng and Dabiri2007; Balasuriya Reference Balasuriya2016a ). A classical method of imparting mixing is to apply an unsteady velocity perturbation to a steady situation such as figure 1(a), which results in the coincident stable and unstable manifolds splitting apart (for example like figure 1 b). An interesting question in attempting to control the transport across such a broken connection would therefore be whether it was possible to separately manipulate the stable and unstable manifolds. So, for example, transport between the inside of the vortex and the upper region in figure 1(a) might be controlled by manipulating the locations of the upper red (unstable) and green (stable) curves in figure 1(b), at some time $t^{\star }$ . Since (2.7) would need to apply for $t>t^{\star }$ for the stable manifold and $t<t^{\star }$ for the unstable manifold, the relevant velocities can be separately specified to achieve this.

3.1 Kelvin–Stuart cats-eyes flow

The accuracy of the method is first illustrated for the Kelvin–Stuart cats-eyes flow pictured in figure 1(a). The steady Kelvin–Stuart flow – an exact solution to the Euler equations – is given by the velocity field (Stuart Reference Stuart1967; Sharma & McKeon Reference Sharma and McKeon2013; Balasuriya Reference Balasuriya2016a )

where $b>1$ is a parameter and $\boldsymbol{x}=(x,y)$ . The initial focus will be on the unstable and stable manifolds emanating from $(0,0)$ into the first and fourth quadrants in figure 1(a), which are given by $y=\pm \cosh ^{-1}[1+\sqrt{b^{2}-1}(1-\cos x)/b]$ , and are shown by thin red and green curves respectively in figure 3. Suppose a time-varying bulge is to applied in the normal direction to the unstable manifold, given by the expression

where $\unicode[STIX]{x1D700}$ is a small parameter and $\unicode[STIX]{x1D7D9}$ is the indicator function (taking the value one when the argument is in the indicated domain, zero otherwise). Simultaneously, suppose that the stable manifold is desired to be moved according to the time-aperiodic specification

It is now straightforward to compute the values of the control velocity $\boldsymbol{c}$ associated with each of these manifold deviations using (2.7). In particular, since the original flow is steady, $\boldsymbol{u}(\boldsymbol{x},t)=\bar{\boldsymbol{u}}(\boldsymbol{x})$ and the terms in the first line disappear. In this example, suppose the expression given specifically by (2.7) is used to define $\boldsymbol{c}(\boldsymbol{x},t)$ in regions around the relevant manifold as demarcated by the thick curves in figure 3. Care should be taken to ensure that the required manifold deviations remain within these regions; thus the width of the regions should be large enough depending on the value of $\unicode[STIX]{x1D700}$ . This enables the control velocity to be defined for all times, and numerical experimentation is possible.

Figure 3. The regions (heavy curve boundaries) in which the control velocity (2.7) is to be applied with respect to moving the unstable (red) and stable (green) manifolds.

The process of comparing the results to the desired manifold variation at any time $t$ is as follows. The full velocity (Kelvin–Stuart plus the control velocity as computed above) is used for numerical particle advection beginning at time  $t$ . Backward FTLE fields are computed within the domain shown in figure 3; the strong ridges of these, adjacent to the thin red curve in figure 3, will numerically determine the UMP at that instance in time. The time of advection used in this process is an important parameter. Longer segments of the unstable manifold emanating from $(0,0)$ can be captured by increasing this time of advection, but this means that other unstable manifolds will also emerge as ridges of the backward FTLE field. For example, the green curve in figure 3 is a SMP of $(0,0)$ , but since it is also a UMP of the hyperbolic trajectory near $(2\unicode[STIX]{x03C0},0)$ this will also emerge as a ridge of the backward FTLE field, but will be connected to the right hyperbolic trajectory. On the other hand, computing forward FTLE fields (by advecting particles forward in time from  $t$ ) will reveal SMPs.

Figure 4. Backward (a,c,e) and forward (b,d,f) FTLE fields associated with the controlled Kelvin–Stuart flow with $b=1.5$ and $\unicode[STIX]{x1D700}=0.2$ , at $t=-3$ (a,b), 0 (c,d), and 3 (e,f). The required flow barriers are superimposed as thin red (unstable) and green (stable) curves.

Figure 4 shows the results of this comparison where the manifold deviation parameter has been chosen to be $\unicode[STIX]{x1D700}=0.2$ and a time of advection of 12 is used throughout. Forward and backward FTLE fields are shown at several different times, and the desired flow barrier locations at those times, as given by (3.2) and (3.3), are indicated by the thin red (UMP) and green (SMP) curves superimposed on the same figures. The results are excellent. Tests at other $t$ values also indicated very good agreement. Even better agreement results if using smaller $\unicode[STIX]{x1D700}$ (not shown). Several time-varying flow-barrier deviations which are different to (3.2) and (3.3) were analysed; usage of the relevant control velocity using (2.7) once again gave excellent results (not shown).

Figure 5. Backward (a) and forward (b) FTLE fields at $t=0$ associated with splitting the lower coincident stable/unstable manifold into non-intersecting curves to enable transport from the lower jet to the eddy, with parameters $b=1.5$ and $\unicode[STIX]{x1D700}=0.5$ .

Next, consider forcing transport into the cats-eyes eddy from the lower jet, by splitting the lower coincident stable/unstable manifold in figure 1. Thus, the same SMP as considered in the previous numerical experiments shall be used, with deviation given by $\tilde{r}_{s}=r_{s}$ as given by (3.3). The unperturbed unstable manifold is the same entity, but now suppose that it is required to deviate this by $\tilde{r}_{u}(x,y,t)=-r_{s}(2\unicode[STIX]{x03C0}-x,y,t)$ . This sign ensures that the SMP and UMP at a given time $t$ split in opposite directions, thereby creating a channel along which fluid will transport from the lower jet to the eddy. Once again, the relevant control velocity can be computed using (2.7), and now it is sufficient to use the green region from figure 3 since the action is only in this region. The resulting FTLE fields at time 0 with $\unicode[STIX]{x1D700}=0.5$ is shown in figure 5. The agreement between the UMPs (red) and SMPs (green) with the relevant ridge of the backward (a) and forward (b) FTLE fields is excellent, even at this significant value of $\unicode[STIX]{x1D700}$ . The ‘other’ manifold proxy is also shown in each figure to show that the required splitting has been achieved, and since the flow in the region in between is from right to left, fluid will become entrained into the eddy from the lower jet. This is a uni-directional transport caused by the SMP and UMP not intersecting at time zero. (A time-dependent transport quantification of this form is possible for any intersecting/non-intersecting pattern between such manifolds (Balasuriya Reference Balasuriya2016a ,Reference Balasuriya c ), which generalises the classical lobe dynamics approach (Rom-Kedar et al. Reference Rom-Kedar, Leonard and Wiggins1990; Franco et al. Reference Franco, Pekarek, Peng and Dabiri2007) to arbitrary time variation with or without lobes.) This behaviour has been created by design, showing the potential of the control methodology to control transport between coherent structures. Note that this required separate specifications of the control velocity in the times $t<0$ and $t>0$ , to respectively control the unstable and the stable manifolds. Controlling both the stable and unstable manifolds over a range of time does not seem possible, since if controlling at a time $t^{\star }$ , separate specifications for $t<t^{\star }$ and $t>t^{\star }$ would be necessary. If control is to be also achieved at a different $t^{\star }$ (call it $\tilde{t}<t^{\star }$ ), this will result in ambiguity of the control velocity definitions in the range $[\tilde{t},t^{\star }]$ . It may, however, be possible to control the transport as a function of time, even if the individual manifolds cannot.

3.2 Controlling the unsteady double gyre

Next, an example with unsteady data is considered. In keeping with a system whose behaviour is well understood, the unsteady double gyre introduced by Shadden et al. (Reference Shadden, Lekien and Marsden2005) will be used to produce the data for the uncontrolled system. This obeys

where the function $\unicode[STIX]{x1D719}$ is defined by

The parameter $\unicode[STIX]{x1D6FF}$ is usually considered small, since if $\unicode[STIX]{x1D6FF}=0$ , the system (3.4) consists of two counter-rotating gyres in the cells $[0,1]\times [0,1]$ and $[1,2]\times [0,1]$ . The flow barrier is then along $x=1$ , being simultaneously the stable manifold of the point $(1,0)$ and the unstable manifold of $(1,1)$ . When $\unicode[STIX]{x1D6FF}\neq 0$ , this barrier breaks apart into separate stable and unstable manifolds, which intersect infinitely often leading to chaotic transport between the left and the right gyre. Finite-time simulations of (3.4) are common using most of the proxies discussed in this paper for stable/unstable manifolds, and they all succeed in revealing the expected structure of the flow barriers (Shadden et al. Reference Shadden, Lekien and Marsden2005; Froyland & Padberg Reference Froyland and Padberg2009; Allshouse & Peacock Reference Allshouse and Peacock2015b ; Balasuriya Reference Balasuriya2016a ).

Figure 6. Forward-time FTLE fields for the uncontrolled (a,c,e) and controlled (b,d,f) double gyre subject to the prescription (3.6) at different times, with the desired SMP shown by the green curve. Here, $\unicode[STIX]{x1D6FF}=0.2$ , $\unicode[STIX]{x1D700}=0.1$ and $\unicode[STIX]{x1D702}=0.2$ .

Since the manifolds of (3.4) when $\unicode[STIX]{x1D6FF}\neq 0$ have infinitely many folds, and re-enter the boundary of the usual domain of definition $[0,2]\times [1,0]$ infinitely closely, controlling the global extent of these manifolds is a daunting task. This section will focus on controlling the part of the stable manifold emanating from near $(1,0)$ . In the absence of explicit expressions for the flow barriers, (analytical approximations for small $\unicode[STIX]{x1D6FF}$ in the vicinity of $x=1$ are available even when $\sin (\unicode[STIX]{x1D714}t)$ in (3.5) is replaced by an arbitrary bounded time-varying function (Balasuriya Reference Balasuriya2016a )) a SMP will serve to help identify this manifold. The strongest yellow ridge of the forward-time FTLE fields shown in figure 6(a,c,e) indicate the beginnings of this SMP. The ridge has many subsequent folds which the FTLE struggles to capture at this particular choice of system and plotting parameters ( $A=1$ and $\unicode[STIX]{x1D714}=15$ are chosen throughout). The foot of the SMP is moving around near $(1,0)$ ; this represents the hyperbolic trajectory to which the SMP is attached. In all the FTLE calculations to be described in this section, a spatial grid of $200\times 100$ points is used, with time steps of size 0.01 used for a time duration of $3(2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714})$ units. Forward advection is performed using a third-order Runge–Kutta scheme.

The intention is to force the beginning part of the flow barrier to follow a prescribed time-varying behaviour, which will be specified by writing it in the form

for small $|\unicode[STIX]{x1D700}|$ . Going towards values close to $y=1$ is clearly impossible since the flow barrier stretches and folds globally. Thus (3.6) is the prescribed SMP $\unicode[STIX]{x1D6E4}(t)$ , which in this case is pinned at $(1,0)$ and swivels from pointing into the left gyre to the right gyre as time progresses. Figure 6(a,c,e) indicates that the unsteady double-gyre flow of (3.4) clearly does not conform to this behaviour, and therefore a control mechanism must be applied.

To apply (2.7), the nearby steady velocity $\bar{\boldsymbol{u}}$ must be determined in the vicinity of $x=1$ . Now, the $\unicode[STIX]{x1D6FF}=0$ steady flow seems like the obvious candidate, but this turns out to not be the flow arising from taking the time average (except when $\unicode[STIX]{x1D6FF}=0$ ). While the time average of the data at each point $(x,y)$ of (3.4) can be computed, in this case an exact expression for the time average of (3.4) near $x=1$ over a time period $2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}$ (the period of the flow) is given by

where $\text{J}_{0}$ is the Bessel function of the first kind (see appendix C). It was checked that the numerically averaging the data in a strip near $x=1$ did indeed converge to (3.7) for a suitably small strip and in the limit as the time step went to zero. A specific observation – either from the data or from (3.7) – is that ${\dot{x}}=0$ on $x=1$ , which then is $\unicode[STIX]{x1D6E4}$ , the stable manifold for the averaged steady system. For any given $r(y,t)$ , it is shown in appendix C that the quantity $\boldsymbol{c}$ in (2.7) should be chosen to be

on $x=1$ . In the specific case (3.6), $r$ is given by

which can be inserted into (3.8). This velocity (3.8) will be applied uniformly in $x$ , within the strip $x\in [1-\unicode[STIX]{x1D702},1+\unicode[STIX]{x1D702}]$ . To obtain a SMP at each given time $t^{\star }$ , (3.8) needs to apply for all times $t\geqslant t^{\star }$ . Only a finite time is being considered here, and thus the requirement needs to be met for times $t\in [t^{\star },t^{\star }+3(2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714})]$ .

Figure 7. Backward-time FTLE fields for the uncontrolled (a,c,e) and controlled (b,d,f) double gyre subject to the prescription (3.10) at different times, with the desired UMP shown by the red curve. Here, $\unicode[STIX]{x1D6FF}=0.4$ , $\unicode[STIX]{x1D700}=0.3$ and $\unicode[STIX]{x1D702}=0.35$ .

The controlled system is therefore (2.3) subject to the above identification of functions. Figure 6(b,d,f) indicates the forward-time FTLE fields obtained from the data associated with (3.9), with the choice $\unicode[STIX]{x1D700}=0.1$ and $\unicode[STIX]{x1D702}=0.2$ (i.e. the control operates in the full strip shown in figure 6). There is a strong FTLE ridge lining up along the desired SMP (green curve). Comparing with (a,c,e), it is clear that the method has been outstandingly successful in moving the SMP in the desired time-varying fashion.

The parameter $\unicode[STIX]{x1D6FF}$ measures the level of unsteadiness present in the uncontrolled flow, and the method as developed assumed small $\unicode[STIX]{x1D6FF}$ and small deviation $\unicode[STIX]{x1D700}$ . To push the method beyond its originally designed framework, suppose $\unicode[STIX]{x1D6FF}=0.4$ and $\unicode[STIX]{x1D700}=0.3$ . Think of the unstable manifold emanating from near $(1,1)$ this time, and suppose it were required to pin it as

For this instance, the relevant $r$ obeys

This value can be substituted into (3.8) to determine $\boldsymbol{c}$ . While the bracketed term in (3.8) has no time dependence in this case, the presence of the $\bar{\boldsymbol{u}}-\boldsymbol{u}$ indicates that an unsteady control velocity applies in this instance in which a steady flow barrier is to be achieved. Moreover, in this instance if the required UMP is to be obtained at time $t^{\star }$ , (3.8) must be obeyed at times $t\in [t^{\star }-3(2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}),t^{\star }]$ .

With these new parameter choices in mind, figure 7 displays the comparison between the uncontrolled flow and the controlled one, in terms of backward-time FTLE fields, at several different choices of time. The required curve (red) in the left panels is seen to be very different from the uncontrolled unsteady double gyre. The controlled flow on the right indicates that there is a dominant FTLE ridge lying along the required red curve. In this controlled situation, (3.8) has resulted in many more (smaller) FTLE ridges than in the uncontrolled system, yet the dominant one clearly follows the required behaviour. Even with relatively large unsteadiness in the data, and required deviation, very good results are obtained.