Solute mixing in fluids is enhanced significantly by chaotic advection, the phenomenon in which fluid pathlines completely fill the spatial domain explored by a laminar flow. Steady groundwater flows are, in general, not well conditioned for this phenomenon because Darcy's law confines them spatially to nonintersecting stream surfaces. Unsteady groundwater flows, however, may, in principle, induce chaotic solute advection if their time dependence is periodic and produces frequent reorientation of pathlines, an effect connected closely to the development of high solute mixing efficiencies. In this paper, a simple, two-dimensional model groundwater system is studied theoretically to evaluate the possibility of inducing chaotic solute advection in the flow field near a recirculation well whose pumping behavior is time periodic. In the vernacular of dynamical systems theory the model studied is an example of a Hamiltonian system, thus allowing a mathematical formulation that facilitates consideration of whether chaotic advection can occur. When the model system is driven time periodically by alternate operation of the production and injection components of the recirculation well, numerical simulations of the fluid pathlines indicate that regions of chaotic solute advection will, indeed, develop in the flow field near the well, with expected major improvement in plume spreading.
 The early phase of solute plume spreading by groundwater is not governed by local dispersion processes, but is instead dominated by the advective stretching and folding of material fluid elements, a process that has been termed “mixing” [Ottino, 1989, section 1.4, 1990]. During mixing, plume boundaries evolve to become greatly elongated and highly irregular. Weeks and Sposito  modeled passive solute mixing in two-dimensional groundwater flow through a layered sand aquifer whose hydraulic conductivity varied significantly along the vertical direction. Their calculations showed mixing to be enhanced whenever solute pathlines traversed zones of high contrast in the hydraulic conductivity, or when they were stretched downward by time-dependent surges in the z coordinate of the specific discharge vector. Both of these flow phenomena produced a strong reorientation of the solute pathlines, an effect well known to yield good mixing [Ottino, 1989, chap. 4, 1990].
 Plume spreading by mixing could be especially effective if solute advection by groundwater flows were to be driven chaotic, meaning that the solute pathlines completely fill the spatial domain of flow while the stretching of material filaments exhibits asymptotic (long time) exponential growth [Ottino, 1989, chap. 5, 1990]. Chaotic advection was precluded in the model groundwater flows studied by Weeks and Sposito [1998, Figure 8], mainly because solute pathlines were confined spatially to the level surfaces of a stream function and therefore could not explore the entire domain of flow; but many examples of chaotic advection by laminar fluid flows are known and applied in engineering design [Stremler et al., 2004]. A common feature of these fluid flows is periodicity in the velocity field, which provides the frequent reorientation of solute pathlines so crucial to inducing chaotic advection [Ottino, 1989, chap. 5; Aref, 2002; Stremler et al., 2004].
where denotes the instantaneous position of a moving spatial point by which a passive scalar field (e.g., solute concentration) is tracked during advection by the velocity field . Although the velocity field may be quite regular in its dependence on space and time, the pathlines associated with equation (1) may be quite irregular, in some circumstances becoming chaotic in the space-filling sense defined above [Aref, 1990, 2002].
 The present paper is a heuristic examination of this possibility for unsteady groundwater flows. Technical details are kept to a minimum by considering a very simple model of flow near a well in two spatial dimensions, for which chaotic advection always requires imposition of unsteady conditions [Ottino, 1989, chap. 7; Aref, 1990, 2002; Stremler et al., 2004]. The time dependence of the flow is selected to provide strong periodic reorientations of solute pathlines, with the result that stroboscopic illumination of these pathlines will reveal a chaotic sea of moving spatial points in certain regions of the flow domain. The guiding paradigm for this study has been stated in an especially cogent fashion by Stremler et al. : (1) the intended application is matched to a fluid flow known to generate chaotic advection, (2) a mixing device is designed that meets the application requirements, and (3) the device is modeled and its diagnostics are computed to verify the occurrence of chaotic advection. The results of the present paper are intended only as the nascent step in this paradigm.
2. Dipole Flow
 Vertical circulation wells and, more generally, recirculation wells have found wide application in scenarios for aquifer characterization and groundwater remediation (see the reviews by Zlotnik and Ledder  and Cirpka and Kitanidis ). For the former application, a vertical well is constructed with two perforated sections separated by a central packer through which groundwater can be transferred from one chamber to another by a submersible pump [Kabala, 1993; Zlotnik et al., 2001; Zlotnik and Zurbuchen, 2003]. With this arrangement, shown schematically in Figure 1, the hydraulic conductivity tensor of an aquifer can be determined on spatial scales probed by the induced flow field, after due consideration is given to important mensuration issues, such as support volume and averaging mechanism [Zlotnik et al., 2001; Zlotnik and Zurbuchen, 2003].
 The operation of a vertical circulation well to explore aquifer hydraulic conductivity requires extracting groundwater into the upper chamber, then pumping the extracted water into the lower chamber, where it is injected back into the aquifer [Kabala, 1993; Zlotnik and Zurbuchen, 2003]. The flowfield induced by this operation thus involves no net withdrawal of groundwater and, in the absence of an ambient flow, it is characterized by streamlines that emanate from the lower chamber and disappear into the upper chamber of the well [Zlotnik and Ledder, 1996]. Although transient operation of a vertical circulation well is possible and has been evaluated, steady state conditions in the induced flowfield are readily achieved, with large enough pressure changes developing in the two chambers to provide for a reliable measurement of hydraulic conductivity [Zlotnik et al., 2001; Zlotnik and Zurbuchen, 2003]. Under steady state conditions, the induced flowfield scales with the well recirculation rate Q and, in an incompressible aquifer, this flow field is solenoidal.
Kabala , who pioneered the analytical modeling of the flowfield induced by a vertical circulation well, termed the operation just described a “dipole flow test,” by analogy with the well-known solenoidal flowfield around a point source and sink of equal strength embedded in an incompressible fluid [Milne-Thompson, 1996, section 16.26]. This latter axisymmetric flowfield is an example of dipole flow in fluid mechanics. It can be pictured as a limiting case of the steady state flow field induced by a vertical circulation well when chamber size and the well radius are negligibly small relative to chamber separation, the latter itself being taken negligibly small relative to the distance from the central packer in the well to the aquifer boundary [Zlotnik and Ledder, 1996]. The steady state flow field induced in a homogeneous aquifer by a vertical circulation well whose chamber size is not neglected can be described analytically by a Stokes stream function [Zlotnik and Ledder, 1996; Zlotnik et al., 2001]. If the aquifer is isotropic, and in the limit of negligible chamber size, this stream function becomes identical with that for the dipole flow of an incompressible fluid.
 Perhaps the simplest mathematical model of an isolated production or injection well represents it as a point sink or point source in an unbounded two-dimensional domain [Strack, 1989, section 19]. Generalization of this representation to involve a three-dimensional (axisymmetric) flowfield, and to take into account finite well size and aquifer boundaries, is straightforward [Zlotnik and Ledder, 1996; Sutton et al., 2000], but not necessary for a heuristic examination of the onset of chaotic advection. A description of the dipole flow test using the language of dynamical systems theory necessary to this examination thus can take advantage of an expedient mathematical simplification.
3. Dynamical Systems Formulation
 It is well known that the flow of groundwater out of a point source in two dimensions admits a Lagrange stream function if both the groundwater and the isotropic, homogeneous aquifer matrix through which it flows are assumed undeformable [Bear, 1988, section 6.5.2]. As discussed by Sposito and Weeks , the existence of a Lagrange stream function implies that the associated groundwater flow is a Hamiltonian dynamical system. This means that solute advection is guided by a form of Hamilton's equations for a system with one degree of freedom [Ottino, 1989, chap. 6]. Aref [1990, 2002] has emphasized the important point that the Hamiltonian description in this case arises solely from kinematics (i.e., an undeformable aquifer with groundwater flow subject to mass balance). The phase space for the Hamiltonian description is simply the configuration space of the flow field.
3.1. Injection Well
 Following on the correspondence between a Lagrange stream function and a Hamiltonian, one can express the latter for a point source embedded in a two-dimensional, unbounded domain on the Cartesian y axis at (0, −L) by the equation:
where m is the total (areal) discharge across any closed curve surrounding the source. Equation (2) reflects the well-known fact that the stream function for a point source in an unbounded domain is proportional to the angle between the Cartesian x axis and a ray emanating from the source, with this angle being swept out in a counterclockwise direction by the ray [Bear, 1988, section 7.8.3]. A point source provides a simple model of an injection well in an isotropic, homogeneous aquifer. Hamilton's equations for the system then follow as [Goldstein, 1980, chap. 8]:
where the time-derivative in each equation follows from known results for the coordinates of the discharge vector at any point in the flow field [Bear, 1988, section 7.8.3]. Comparison of equations (3) with the standard definitions of generalized position and momentum coordinates [Goldstein, 1980, section 8-1] leads to the identifications: z ↔ q, x ↔ p, where q is a position coordinate and p is a momentum coordinate locating the system in phase space. Therefore equation (2) can be rewritten in the form:
after renormalizing the generalized position coordinate relative to the z coordinate at the source: q ≡ z + L. This renormalization has no impact on the right side of equation (3a).
 The level curves of H(p, q) are deduced by setting the left side of equation (4) equal to a constant, h:
from which one derives readily the relationship:
showing that the level curves in phase space are straight lines emanating from the source, with slopes determined by the value of h. Evidently h decreases from zero, for the ray that is congruent with the positive x axis, to −m/4, for the ray that is congruent with the positive z axis, to −m/2, for the ray that is congruent with the negative x axis, and so on. Each level curve (i.e., straight line) is thus identified with a certain fixed value of the Hamiltonian. These results are, of course, a dynamical systems picture of the well-known outwardly directed rays that constitute the streamlines for flow out of a point source [Bear, 1988, Figure 7.8.6; Strack, 1989, Figure 3.7].
 Hamilton's equations are solved for q(t) and p(t) by finding a canonical transformation of the phase-space coordinates, q, p,
which, along with the correspondingly transformed equations (3), renders the simple time dependence:
where α and β are constants to be determined by initial conditions [Goldstein, 1980, chap. 9]. A well-established procedure exists for finding canonical transformations [Goldstein, 1980, chap. 10], but suffice it to say here that
will do. The “canonical” nature of equations (8) can be demonstrated in various ways, one of which involves the Poisson bracket [Goldstein, 1980, section 9-4]:
Clearly, [q, p] = 1 and, if the same result holds for some Q, P defined as in equation (6), the transformation from q, p to Q, P is said to be canonical. It is straightforward to show that equations (8) indeed do lead to [Q, P] = 1.
 The transformed Hamiltonian corresponding to equations (8) is simply
as confirmed easily by reference to equation (4). The corresponding Hamiltonian equations of motion are
from which follow equations (7), as expected. (Note that the second of these equations is just a statement of the constancy of H(Q, P) = H(q, p), which always must be true if the Hamiltonian has no explicit time dependence [Goldstein, 1980, chap. 8].) Combination of equations (7) and (8) then yields the motion of the system:
where both α and β are determined by setting t = 0.
 Returning to configuration space, one can rewrite equations (12) more transparently:
where r2 ≡ x2 + (z + L)2 and θ = tan−1 [(z + L)/x]. Note that α in equation (12b) is the same as h in equation (5a). Equations (13a) and (13b) confirm that a solute is advected by the steady outflow from a point source such that the square of its distance r(t) from the source is a linear function of time, while the angle θ its path makes with the x axis is constant [Jones and Aref, 1988]. The corresponding result for a point sink is found by changing the sign of m in equation (13a).
3.2. Steady State Dipole Flow
 The stream function and therefore the Hamiltonian, for a dipole well system represented by a point source and sink (each with the same (areal) discharge, m) embedded in an infinite two-dimensional domain is proportional to the difference between the angles made by rays that define the linear streamlines of the source and sink when in isolation (principle of superposition). In Cartesian coordinates like those used in equation (2), the Hamiltonian (Lagrange stream function) for this system may be expressed [Bear, 1988, section 7.8.3]
for a source at (0, −L) and a sink at (0, L), as indicated in Figure 1. Generalized position and momentum coordinates are defined by the same identifications as were used for an injection well, but with the origin of the q axis not shifted in this case:
 Level curves of H(p, q) are deduced by setting the left side of equation (15) equal to h, from which follows the relationship [Bear, 1988, section 7.8.3; Strack, 1989, section 19]
The level curves in phase space are thus circles of radius R0, where R0 is given by equation (17), that are centered at (−cot(h)L, 0). When h ranges from 0 to m/2, the centers of these circles migrate from −∞ to +∞ along the p axis, corresponding to the same movement along the x axis in configuration space (Figure 2). As is well known, they are the streamlines [Strack, 1989, section 19]. In between these extremes, the circles are centered at (−L, 0) when h = m/8; the origin when h = m/4; and (L, 0) when h = 3m/8. The two extremes at ±∞ correspond to fluid flow along the q axis (z axis in configuration space [Strack, 1989, section 19]) between the source and the sink. The associated value of R0 is then infinite.
 The motion governed by the Hamiltonian in equation (15) can be found by a canonical transformation, the details of which are not required here. Both q(t) and p′(t) ≡ p(t) + cot (h)L, a momentum coordinate referenced to the center of a circular level curve (streamline), are found to have a sinusoidal dependence on time, such that
as expected for circular motion. Therefore a solute advected by steady state dipole flow moves from the source to the sink along a circular streamline whose radius depends on the value of the Hamiltonian in equation (15). Sutton et al.  have studied this advection process for an organic groundwater tracer injected into an unconfined, anisotropic, heterogeneous sand aquifer to which equation (18) has a qualitative relationship.
4. Chaotic Advection in Pulsed Dipole Flow
 A pulsed dipole flow can be induced by operating a vertical circulation well so as to have groundwater extracted into the upper chamber during a time period T while no injection into the aquifer is allowed, this being followed by a time period T during which injection of groundwater from the lower chamber occurs but no extraction is allowed. Groundwater extracted into the upper chamber is sent at once to the lower chamber for injection back into the aquifer. During an extraction “stroke” of this system, solute will be advected a distance equal to (mT/π) according to equation (13a) in the simple two-dimensional model adopted here. This distance is a characteristic length scale for the pulsed flow, as is L, the distance from the origin of coordinates to the sink [Jones and Aref, 1988]. The advection distance λ ≡ (mT/π) also defines the radius of the circular patch around the sink within which all solute particles will be extracted from the aquifer during the time T. This groundwater is sent to the source immediately for injection during the next “stroke” of the system. Equation (13a) thus applies to both “strokes,” with a change in the sign of m introduced whenever a sink is in operation. Given equation (5b), the path taken by a solute advected during a “stroke” is along a ray whose origin is a source or a sink, and the trajectory of the solute will be a sequence of straightline segments as it zigzags in the flow field advecting it from the source to the sink [Jones and Aref, 1988, Figure 2]. This picture, of course, is an idealization that ignores the time delay that must occur when the direction of a real groundwater flow reorients in response to alternating operation of the production and injection wells.
 If the duration of a “stroke” is quite small (say, T < πL2/10 m), then the trajectory of solute advected by the pulsed groundwater flow should approximate the circular streamline which characterizes steady state dipole flow (equations (16) and (18)). Jones and Aref [1988, Figure 3(a)] have demonstrated this limiting case by numerical simulation of passive scalar advection in a pulsed dipole flow with a “stroke” duration T = 0.1 π L2/m under a transfer protocol between the sink and the source which is constrained by the condition that the square of the distance from the sink to any advected solute particle it absorbs during the time T, plus the square of the distance traveled by the solute after injection during the next “stroke,” also of duration T, always equals mT/π. This protocol, an idealized realization of the well operation termed a “recirculation mode” by Sutton et al. , requires “instantaneous” transfer between the sink and the source, with the first solute to leave the former being also the first to be injected back into the aquifer [Jones and Aref, 1988]. Modifications of this constraint to provide a more realizable transfer protocol, however, do not affect the central issue of whether chaotic advection will occur [Stremler et al., 2004].
with n = 0, 1, 2, … denotes the “mixing protocol” [Aref, 1984], in the present case a “stroke” cycle of duration 2T. For (n + 1)T ≤ t ≤ (n + 2) T, equations (19) reduce to equations (3). Equations (19a) and (19b) constitute a set of nonautonomous, coupled, nonlinear ordinary differential equations that describe the pathlines in solute advection by a pulsed dipole flow.
Figure 3 (M. A. Stremler, personal communication, 2005) shows a simulated stroboscopic illumination of these pathlines, obtained using numerical analysis and the protocol developed by Jones and Aref , for the case T = πL2/m (i.e., the areal extraction (or injection) rate around the sink (or source) is set equal to πL2/T). This stroboscopic representation is known technically as a Poincaré map [Jordan and Smith, 1999, section 13.1]. It is obtained by plotting in configuration space the position of a moving spatial point after every “stroke” cycle of duration 2T. Three regions in Figure 3 may be distinguished. One is far from the source-sink pair (note that the pathlines in Figure 3 correspond mainly to those on the left side of Figure 2). It comprises zigzag circular pathlines whose stroboscopic traces are reminiscent of those found in steady state dipole flow (Figure 2). Another region very close to the source-sink pair comprises “islands” of regular (i.e., nonchaotic) motion in which solute is advected along the same closed curve after each “stroke” cycle [Stremler et al., 2004].
 Between these two regions of regular motion there is a sizable domain of chaotic motion, signaled by scattered, disorganized points in the Poincaré map. These passive motions can be shown to produce positive Liapunov exponents, a well-known signature of chaos [Ottino, 1989, section 5.8; Weeks and Sposito, 1998] that provides a quantitative measure of the asymptotic (large time) relative rate of line stretching in a fluid. Numerical simulation [Jones and Aref, 1988; Stremler et al., 2004] indicates that the chaotic “sea” of advected points represented in Figure 3 is associated with frequent extraction and injection of solute by the sink and source, respectively, which then produces upward jumps in the relative rate of line stretching. By contrast, solute advection in the region far from the source-sink pair is associated with a relative rate of line stretching that decreases to zero asymptotically, and therefore it is not chaotic.
Ottino [1989, section 4.6] has emphasized the importance of reorientation sequences to achieving chaotic pathlines and consequent good mixing during solute advection. In the case of pulsed dipole flow, the pathlines are reoriented after each “stroke” of duration T when the advected solute is forced to jump from a streamline of the source to that of the sink, and vice versa. This behavior may be contrasted with that of a solute advected by a steady state dipole flow, in which material filaments are destined always to remain “trapped” between two proximate level curves (i.e., the circular streamlines in Figure 2) of the Hamiltonian [Ottino, 1989, section 4.7]. Similar behavior was noted by Weeks and Sposito , who showed in their model calculation that, although high contrast in the spatial heterogeneity of the hydraulic conductivity in a sand aquifer forced reorientation of the pathlines, thereby greatly improving mixing, the existence of a stream function precluded chaotic advection. Stremler et al.  and McQuain et al.  have discussed the engineering design of pulsed dipole flows that improve significantly on the single-dipole system discussed here, in particular, by including two or more recirculation wells [cf. Cirpka and Kitanidis, 2001] whose pulsed operation can destroy the nonchaotic “island” structures near the source-sink pair in Figure 3.
 Gratitude is expressed to Mark A. Stremler for his great generosity in providing a copy of Figure 3 and to Vitaly A. Zlotnik for making available important published and unpublished material related to the modeling and implementation of the dipole flow test. Thanks also go to three anonymous referees for useful comments that improved the presentation in this paper and to Angela Zabel for excellent preparation of the typescript as well as Figures 1 and 2.