Abstract

The interplay between dynamics and transport in two-dimensional flows is examined by comparing the transport and mixing in a kinematic flow in which the velocity field is imposed as a given function of time with that in an analogous dynamically consistent flow in which the advected vorticity field controls the flow evolution. In both cases the variation of the transport and mixing behavior with a parameter ε governing the strength of the time dependence is considered. It is shown that dynamical consistency has the effect of (i) postponing the breaking of a central transport barrier as ε increases and (ii) removing the property of the kinematic flow that, for a large range of ε, a weakly permeable central barrier persists. The first effect is associated with the development of a strong vorticity gradient and the associated jet along the central transport barrier. The second effect is associated with the fact that, in the dynamically consistent flow, the breaking of the central barrier is accompanied by a drastic change in the vorticity field and hence in the structure of the flow.

The relation between the vorticity field and transport barriers is further examined using a range of simple kinematic and dynamically consistent models. Implications for formulation of predictive models that represent the interactions between dynamics, transport, and mixing (and might be suggested as a basis for parameterizing eddies in flows that form multiple jets) are discussed.

1. Introduction

The transport and mixing properties of atmospheric and oceanic flows have been extensively studied over the last 10 years or so (e.g., Pierrehumbert and Yang 1993; Koh and Plumb 2000; Joseph and Legras 2002; Wiggins 2005), motivated by the need for a better understanding of the transport and mixing of chemical and biological tracers. Progress has been aided by improved numerical simulations and by the rapid growth of observational data on chemical and biological species. Study of transport and mixing is also relevant to the underlying dynamics of large-scale atmospheric and oceanic flows, since the dynamical equations can be expressed to good approximation as the conservation of potential vorticity (PV) following the fluid motion (with nonconservation associated with frictional or diabatic effects), together with an invertibility relation that determines all other flow quantities instantaneously from the potential vorticity (e.g., Hoskins et al. 1985).

Atmospheric and oceanic studies have benefited significantly from the study of transport and mixing from the mathematical point of view, through the development of chaotic advection as an application of dynamical systems theory (e.g., Wiggins 1992). By chaotic advection we mean the phenomenon (in a velocity field that is a relatively simple function of space and time) that particle paths can be chaotic, that is, sensitive to initial conditions so that particles that are initially close together rapidly separate and hence lead to complex patterns of particles or of an advected tracer.

Many aspects of chaotic advection have been explored in so-called kinematic models in which the velocity field is imposed as a given function of space and time and the resulting transport and mixing properties of the flow are calculated (e.g., by following fluid particles). The flows considered are often taken to be time periodic, in which case the advection problem reduces to the repeated application of a map. An ubiquitous feature of such flows (or the corresponding maps) is that the transport and mixing structure is highly inhomogeneous with chaotic regions, within which there is rapid separation of nearby particles and hence effective mixing, coexisting with and separated by barrier regions, across which there is no transport and within which particle separation is slow (e.g., see Meiss 1992).

A characteristic feature of many observed flows in the atmosphere and in the ocean and many numerical simulations of analogous flows is similar inhomogeneous transport and mixing behavior, with regions of strong mixing separated by transport barriers. Examples include the stratospheric polar night jet (e.g., Juckes and McIntyre 1987; Norton 1994; Waugh and Plumb 1994; Chen 1994) that acts as a barrier with mixing regions inside, in the polar vortex interior, and outside, in the so-called stratospheric surf zone, the subtropical jet (e.g., Chen 1995; Haynes and Shuckburgh 2000a, b) with adjoining mixing regions in the tropical upper troposphere, the mid- and high-latitude lower stratosphere, the Gulf Stream, and other oceanic current systems (e.g., Bower et al. 1985; Marshall et al. 2006).

The significance of the similarity between the inhomogeneous transport and mixing structure seen in the kinematic time-periodic flows on the one hand and in observed or realistic atmospheric and oceanic flows on the other is not yet completely clear. The fact is that in the former the flow is imposed in advance while in the latter the flow is the self-consistent solution of a set of dynamical equations. Indeed, for flows where the dynamics are governed by material conservation of PV, it might be argued that if particle trajectories are chaotic, the PV field will naturally assume a spatially complex, time-varying configuration and applying an inversion operator to this flow will almost always not give the flow field that has been assumed, resulting in what might be described as a “dynamical consistency” problem. Brown and Samelson (1994) argue, for example, that for time-periodic two-dimensional PV conserving flows dynamical consistency implies that chaotic particle trajectories are forbidden, unless the PV is piecewise constant. One argument for the relevance of the kinematic models is that the potential vorticity field arising from the transport and stirring is in many cases dominated by small spatial scales and the smoothing nature of the inversion operator is such that the dynamical effect is weak. However it is difficult to see that the dynamical effect will always be weak and it seems likely that in some aspects of transport and mixing behavior (e.g., the persistence versus destruction of transport barriers as perturbation amplitude is increased) the dynamical effect will be significant. It seems extremely difficult to formulate any analytic or semianalytic model that allows an interesting combination of dynamical consistency and chaotic transport. One approach to addressing the dynamical consistency problem has been to chose special flows in which advection of potential vorticity has, for some reason, no implications for the dynamics (del-Castillo-Negrete and Morrison 1993; Ngan and Shepherd 1997). Unfortunately these special flows can almost by construction give no insight into flows for which there is a nontrivial interaction between mixing and transport and dynamics.

Thus, an important question is how the predictions of kinematic chaotic advection studies carry over to the dynamically consistent case, where potential vorticity is materially conserved and, as it evolves in time, determines the flow through inversion. A refinement of this question is whether there are robust guiding principles that can be used to gain an understanding of such flows, in numerical simulation or a naturally occurring example.

Given that any interesting interaction between chaotic mixing and transport and dynamics is by nature complex and unlikely to yield to any systematic approximation procedure, the approach that we take in this paper is to report results from a combination of kinematic studies and numerical simulations and attempt to draw general conclusions from these. All these studies are based on two-dimensional flows or maps in the (x, y) plane. Bearing in mind atmospheric applications we regard x as a longitudinal coordinate and y as a latitudinal coordinate.

First, in section 2 we note that in kinematic studies in which transport becomes increasingly more chaotic as a parameter is increased it is almost always the case that the last barrier to break coincides with the axis of a jet. Second, in section 3 we carefully compare a typical kinematic problem and a corresponding dynamically consistent problem. We show through numerical simulation that dynamical consistency raises the threshold level of perturbation required to break a transport barrier. In this sense dynamical consistency strengthens transport barriers. However, we also show that once the threshold amplitude for a strong barrier to be broken is past, then the breaking of the barrier in the dynamically consistent case is much more catastrophic than in the kinematic case.

Finally in section 4 we consider the well-known self-organizing property of forced two-dimensional turbulent flow on a β plane into persistent jets. Such self-organization might be thought of as resulting from the fact that where potential vorticity gradients are strong, mixing is inhibited and where they are weak, mixing is favored. One aspect of this is that strong gradients in potential vorticity inhibit lateral displacement of fluid parcels through the Rossby restoration mechanism, in loose analogy with the inhibition of vertical displacements of fluid parcels by strong buoyancy gradients. A second aspect is that sharp gradients of potential vorticity are inevitably associated with jets (which in section 2 we have noted tend to act naturally as barriers in kinematic studies).

By comparing the transport of tracers under the full velocity field, and under a modified velocity field from which the longitudinal mean has been removed, we show that the second aspect above is an essential part of the transport barrier mechanism in these flows. Thus, any attempt to derive a simple parameterization of transport as a function of a potential vorticity gradient needs to take this into account.

2. Kinematic studies of jets

A traditional approach in kinematic studies has been to follow particle advection in a time-periodic flow. One simplifying feature of time-periodic flows is that the advection problem can be reduced to considering a map that represents the action of advection over one time period of the flow.

In this paper we are primarily concerned with the issue of transport barriers and how they change as the parameters of the flow change. One much studied map is the “standard map” defined on the unit square, extended periodically in both directions, which is the product of a shear in the x direction and displacement in the y direction with sinusoidal dependence on x. In this system it is natural to consider the changes in transport that occur as the amplitude of the sinusoidal displacement increases. For zero amplitude the particles are mapped in the x direction and each line y = constant is a barrier to transport in the y direction. When the amplitude is nonzero but small there are still barriers to transport in the y direction (i.e., unbounded particle displacements in the y direction are not possible). At a critical value of amplitude the last barrier breaks and for larger amplitudes particle displacements in the y direction may be unbounded. There has been a great deal of mathematical study of the breaking of the last barrier, which hinges on number-theoretic properties of the “frequency” of each barrier (i.e., the average rate of crossing the unit square along that barrier; e.g., Meiss 1992). But it is not clear that this tells us anything useful about transport properties of most atmospheric and oceanic flows, not least because the standard map (and hence any underlying flow that it represents) has nonvanishing shear. In any flow containing a jet, and hence in any map that might be used to represent the advection properties of that flow, the shear changes sign at the axis of the jet. Therefore, insights gained from the standard map may not be at all relevant.

Del-Castillo-Negrete and Morrison (1993), del-Castillo-Negrete (2000), Yang (1998), and others have examined the transport properties of jetlike flows, both using continuous-time time-periodic flows and also using suitably defined maps (which del-Castillo-Negrete and Morrison call the “standard nontwist map”; see also del-Castillo-Negrete et al. 1996). They use various heuristic methods well known in dynamical systems theory to give a simple theoretical description of the behavior. One of these is the so-called resonance overlap criterion that for two superposed waves of different phase speeds considers the cat’s-eye streamlines surrounding the critical lines of the two waves and predicts chaos when the two sets of cat’s-eyes intersect. This might suggest that whether or not the jet axis is the last barrier depends crucially on the phase speeds of the disturbances relative to the flow speed at the jet axis. A similar argument has been used by Bowman (1993a, b) to argue that the effectiveness of the stratospheric jet as a barrier to transport depends on the fact that the phase speeds of the wave disturbances do not match the jet speed.

Here we simply emphasize the point that the “last barrier” seems invariably to correspond to the axis of the jet, whatever the relation between the flow speed at the jet maximum and the phase speeds of the wave disturbances.

We will consider the following map:

 
formula
 
formula

where C is a constant. This is somewhat different to that considered by del-Castillo-Negrete and Morrison (1993), but is a member of a family of maps that allows for the investigation of the effect of changing the location where the shear is zero and also changing the absolute value of the velocity at the zero-shear location. The map is defined on a domain that is unbounded in y and 2π periodic in x. We focus on the behavior in 0 < y < 2π, shown in Fig. 1. For the map above the shear is independent of the constant C, and for all C there are two values of y in 0 < y < 2π for which the shear vanishes.

Fig. 1.

(b),(d),(f) Poincaré sections for the map defined by (1) and (2). A small number of points in the (x, y) plane are used as initial conditions for the map and successive iterations are plotted. In chaotic regions the iterates of the map fill out a region of finite area. In barrier regions the iterates of the map fill out a curve. In each case the value of ε is chosen to be close to, but less than, the threshold for the “last barrier” (to transport in the y direction) to disappear. Values of (C, ε) are (0, 1.5) in (b), and (1, 1.75) in (d), (2, 0.5) in (f). (a),(c),(e) The corresponding longitudinal displacement (solid) and longitudinal shear (dashed) profiles. In each of cases (b), (d), and (f) the remaining barrier region is clear as the region where the density of points is low. A rough correspondence is clear between the zero shear locations in (a), (c), and (e) and the remaining barrier regions in (b), (d), and (f). The correspondence is made precise by assigning a longitudinal velocity defined by the average rate that iterates of the map move in the x direction to each of the curves mapped out in (b), (d), or (f) (or corresponding curves that could be mapped by starting with different initial conditions). Once a longitudinal velocity has been assigned then it is straightforward to determine the longitudinal shear.

Fig. 1.

(b),(d),(f) Poincaré sections for the map defined by (1) and (2). A small number of points in the (x, y) plane are used as initial conditions for the map and successive iterations are plotted. In chaotic regions the iterates of the map fill out a region of finite area. In barrier regions the iterates of the map fill out a curve. In each case the value of ε is chosen to be close to, but less than, the threshold for the “last barrier” (to transport in the y direction) to disappear. Values of (C, ε) are (0, 1.5) in (b), and (1, 1.75) in (d), (2, 0.5) in (f). (a),(c),(e) The corresponding longitudinal displacement (solid) and longitudinal shear (dashed) profiles. In each of cases (b), (d), and (f) the remaining barrier region is clear as the region where the density of points is low. A rough correspondence is clear between the zero shear locations in (a), (c), and (e) and the remaining barrier regions in (b), (d), and (f). The correspondence is made precise by assigning a longitudinal velocity defined by the average rate that iterates of the map move in the x direction to each of the curves mapped out in (b), (d), or (f) (or corresponding curves that could be mapped by starting with different initial conditions). Once a longitudinal velocity has been assigned then it is straightforward to determine the longitudinal shear.

As noted by del-Castillo-Negrete and Morrison (1993), the continuous time flow corresponding to such a map is one where the x-dependent part of the flow varies in time as a repeating delta function. The corresponding Fourier series in time has coefficients that are independent of frequency (i.e., the x-dependent part of the flow may be regarded as an infinite superposition of waves) with uniform separation in phase speed between each wave, and with each wave having the same amplitude.

Analogous to the other maps and flows discussed above, there is a regime for small ε where large particle displacements in the y direction, specifically from y = 0 to y = 2π, are forbidden by the presence of barriers. The threshold value of ε at which the last barrier breaks depends on the constant C. The cases shown in Fig. 1 are all for values of ε slightly lower than the threshold value. The important result that may be seen from Fig. 1 is that the location of the last barrier does not seem to depend on the value of C, as it might do if the phase speeds of the disturbing waves were crucial.

3. Comparison between kinematic and dynamically consistent models

A much studied kinematic model is the flow in a channel comprising two waves propagating with different frequencies. This has variously been presented as a model for the Rayleigh–Benard convection (Weiss and Knobloch 1989) and as a model for Rossby waves (Pierrehumbert 1991). In the rest frame of one of the waves the flow might be described as a meandering jet with an imposed time-periodic perturbation. The amplitude of the time-periodic perturbation may be represented by the parameter ε. When ε = 0 the flow is steady and particles move along streamlines. For ε greater than zero there are thin chaotic regions, separated by barriers, and as ε is increased these chaotic regions enlarge and merge, but with at least one barrier region remaining to prevent transport across the channel. An important transition takes place when ε reaches a critical value at which the last barrier to transport across the channel breaks and the topology of the barriers and mixing regions changes. For larger values of ε there may be barriers, but there is no barrier that divides upper and lower or northern and southern regions of the channel. This behavior has been described by several previous authors. Pierrehumbert (1991) has gone further by considering a similar dynamically consistent model, but in this paper he tends to focus on the morphology of the mixing (the role of heteroclinic points etc.) rather than the last barrier transition.

Here we compare a particular kinematic model, in which the flow is given by the streamfunction:

 
formula

representing a time-periodic flow with period 2π/0.3, with a corresponding dynamical model, based on the two-dimensional equations on a β plane with topographic forcing:

 
formula

where the total streamfunction ψ = −0.5y + ψ̃ and β has been taken to be 0.5. The topographic forcing h (x, y, t) is given by

 
formula

The flow is taken to be 2π periodic in the x direction and to lie between rigid boundaries at y = 0 and y = π. The diffusivity κ was chosen to be as small as possible for numerical stability (typically 10−4). Note that the particular values of the constants in (4) are chosen for illustrative processes. Extensive investigation has shown that the sort of behavior reported below does not depend on the particular choices for those constants.

The two models are comparable in the sense that if the wave amplitudes are small and the initial ψ̃ is chosen to match (3) at t = 0, then the flow in each will be the same. For finite wave amplitudes what happens in practice is that (for nonzero ε) the flow evolves during the first few time periods to a different quasi-periodic state from (3). An example is shown in Fig. 2 for ε = 0.05. In this case, since ε is small, the quasi-periodic state attained after the initial adjustment period is only very slightly different from that implied by the initial condition. In the description of transport and mixing properties later, all diagnostics are applied in the quasi-periodic regime after the initial adjustment.

Fig. 2.

PV field for the case ε = 0.05 at times (a) t = 0, (b) t = 0.5π/0.3, (c) t = π/0.3, and (d) t = 16π/0.3. Note that 2π/0.3 is the period of the imposed topographic forcing in (5). The thin curves highlight PV contours chosen to coincide with a heteroclinic streamline (joining stagnation points) in the initial flow. It is only in the region of this streamline that time dependence in the PV field is visible. The thick curves show instantaneous streamlines.

Fig. 2.

PV field for the case ε = 0.05 at times (a) t = 0, (b) t = 0.5π/0.3, (c) t = π/0.3, and (d) t = 16π/0.3. Note that 2π/0.3 is the period of the imposed topographic forcing in (5). The thin curves highlight PV contours chosen to coincide with a heteroclinic streamline (joining stagnation points) in the initial flow. It is only in the region of this streamline that time dependence in the PV field is visible. The thick curves show instantaneous streamlines.

The transport and mixing properties of the kinematic model described have already been extensively studied (e.g., in the papers previously cited). For the particular choice of parameters above, the critical value of ε at which the last central barrier breaks is about 0.34.

Several different methods have been previously used to identify and quantify barriers in this sort of kinematic model (see, e.g., Shuckburgh and Haynes 2003). Here we show results from a particle-based method in which particles are released in a region on one side of the central barrier (the “source” region) and the time for them to reach a second region on the other side of the central barrier (the “receptor” region) is calculated. Poet (2004) has carefully shown that this method is robust in the sense that the calculated time to cross the barrier is insensitive to the precise choice of source and receptor regions. Applying this method to the kinematic problem for different values of ε gives the results shown in Fig. 3, which depicts the time for crossing the central barrier as a function of ε, shown by the dashed line. The time is essentially infinite for ε < 0.34 and for ε > 0.34 is a slowly decreasing function of ε. The slowness of the decrease for ε > 0.34, indicating an extended partial-barrier regime may result from the existence of “cantori”—partial barriers that are left when the last absolute barrier breaks—or from the stickiness of smaller barrier regions that are left once the main central barrier has broken (e.g., Meiss 1992), but the precise explanation is not particularly important for the present discussion.

Fig. 3.

Average transit times to cross the central transport barrier as a function of ε in the kinematic model (dashed line) and the dynamically consistent model (solid line). See text for discussion.

Fig. 3.

Average transit times to cross the central transport barrier as a function of ε in the kinematic model (dashed line) and the dynamically consistent model (solid line). See text for discussion.

The first question to ask is whether there is an analog of the central barrier for the dynamically consistent problem. The solid line in Fig. 3 shows the corresponding time to cross the central barrier in the dynamically consistent problem, again as a function of ε. There are some important differences between the behavior for the dynamically consistent problem and that for the kinematic problem. The value of the crossing time is generally smaller for a given value of ε for the dynamically consistent model versus the kinematic model. However, in the dynamically consistent case the precise flow that is realized depends, for example, on the initial conditions, and it is difficult to set up conditions where any transport barrier is perfect. The relatively large, but not infinite, values of the crossing time for ε < 0.44 in the dynamically consistent case are therefore interpreted as implying the presence of a strong central barrier that breaks at ε ≃ 0.44. Under this interpretation the dynamical consistency has strengthened the central barrier, allowing it to persist to ε ≃ 0.44 rather than ε ≃ 0.34. (A further striking difference between the dynamically consistent and kinematic cases is that in the former for ε > 0.44 the crossing time drops abruptly to small values, rather than decreasing slowly, i.e., there is no partial-barrier regime. This will be discussed further below.)

It is interesting to consider the PV field that results for different values of ε. Figure 4 shows the PV field in the quasi-periodic stage of the evolution for different values of ε, including one value for which the central barrier has broken. For values of ε where the central barrier persists, there are significant spatial variations in the value of the PV. As ε increases, the PV field on each side of the barrier become more homogeneous, consistent with the picture obtained from kinematic investigations that individual chaotic regions on one side of the central barrier tend to merge as ε increases. Thus, for ε = 0.4, just less than the threshold required for the central barrier to break, the PV field is essentially homogeneous on either side of the barrier. Also shown in Fig. 4 are the instantaneous streamlines. While the transport and mixing properties cannot be reliable inferred from the instantaneous streamline patterns, some modest changes in the structure of the streamlines are visible as ε increases from 0 to 0.4, with, for example, at the particular time shown, a narrowing of the central jet in the right-hand part of the domain. For ε = 0.5 the central jet is much less pronounced.

Fig. 4.

PV field at times t = 200π/0.3; 2π/0.3 is the period of the imposed topographic forcing in (5) for (a) ε = 0, (b) ε = 0.05, (c) ε = 0.1, (d) ε = 0.3, (e) ε = 0.4, and (f) ε = 0.5. Note that for ε = 0.5 the central barrier breaks and the PV therefore essentially homogenizes over the domain. Thick curves show instantaneous streamlines.

Fig. 4.

PV field at times t = 200π/0.3; 2π/0.3 is the period of the imposed topographic forcing in (5) for (a) ε = 0, (b) ε = 0.05, (c) ε = 0.1, (d) ε = 0.3, (e) ε = 0.4, and (f) ε = 0.5. Note that for ε = 0.5 the central barrier breaks and the PV therefore essentially homogenizes over the domain. Thick curves show instantaneous streamlines.

The conclusion from the results presented above that the dynamical consistency has the effect of strengthening the barrier depends on the assumption that the parameter ε may be considered to have a precisely equivalent role in the two models. There is no completely rigorous argument underlying this assumption, but on the other hand the general applicability of the conclusion (and undermining of any argument that it is primarily a consequence of the interpretation of ε) is supported by the results of other experiments formulated in different ways. For example, in initial-value experiments where there is no topographic forcing and the flow takes the two-wave form in (3) in the initial condition and parameters are chosen so that each of those waves is individually a freely evolving solutions of the dynamical equation, it is found that that there is an “upscaling” in the dynamical evolution that tends to increase energy in the larger-scale wave at the expense of that in the smaller-scale wave. The result is that a central transport barrier (in the flow that results from the upscaling process) persists for values of ε greater than about 0.8. So the conclusion again is that dynamical consistency strengthens the barrier relative to the kinematic case.

A further crucial difference between kinematic and dynamically consistent models, already noted above, seems to arise as ε increases through the threshold value for the barrier to break. In the kinematic case the velocity field and hence the transport and mixing properties are a continuous function of ε. In the dynamically consistent case on the other hand, the PV homogenizes across the whole domain as the central barrier breaks. The time evolution of this process is shown in Fig. 5 for ε = 0.5. (Contrast the time evolution shown in Fig. 2 for ε = 0.05.) The result is that the flow, and therefore the transport and mixing properties, change almost discontinuously as a function of ε, with the jetlike character of the flow weakening significantly.

Fig. 5.

PV field for the case ε = 0.5 at times (a) t = 0, (b) t = 2π/0.3, (c) t = 4π/0.3, (d) t = 8π/0.3, (e) t = 16π/0.3, and (f) t = 200π/0.3. Note that 2π/0.3 is the period of the imposed forcing in (5). Thick curves show instantaneous streamlines.

Fig. 5.

PV field for the case ε = 0.5 at times (a) t = 0, (b) t = 2π/0.3, (c) t = 4π/0.3, (d) t = 8π/0.3, (e) t = 16π/0.3, and (f) t = 200π/0.3. Note that 2π/0.3 is the period of the imposed forcing in (5). Thick curves show instantaneous streamlines.

The assumption of homogenization of the PV implies that

 
formula

where the constant on the right-hand side is determined by the average PV set by the initial condition. Recall that in the simulations shown here β is taken to be 0.5. Given the time-periodic topographic forcing defined by (5), this defines a new kinematic problem in which the flow is time periodic and given by the solution of (6) subject to the boundary conditions. Analysis of the transport and mixing properties in this new kinematic problem show that a very large proportion of the domain is filled by a single chaotic mixing region, with one or two very small regions isolated by transport barriers. This is shown explicitly by the Poincaré section for this new kinematic problem shown in Fig. 6. The homogenization assumption is therefore self-consistent to good approximation.

Fig. 6.

Poincaré section arising from the flow defined by (6) and (5), generated by three initial conditions, two of which lie on barriers and third of which maps out the large chaotic mixing region that covers most of the domain.

Fig. 6.

Poincaré section arising from the flow defined by (6) and (5), generated by three initial conditions, two of which lie on barriers and third of which maps out the large chaotic mixing region that covers most of the domain.

The conclusion is therefore that the dynamically consistent flow does not show an extended partial-barrier regime characteristic of the kinematic case because once the central barrier is broken the PV is stirred throughout the flow domain and the character of the flow changes drastically as a result.

4. Transport and mixing in 2D β-plane turbulence

We now move on to consider some aspects of transport and mixing behavior in forced-dissipative 2D β-plane turbulence. It is well known that homogeneous isotropic small-scale random forcing on a doubly periodic domain gives rise to an upscale energy cascade and to the formation of persistent narrow eastward longitudinal jets, whose separation is roughly (U/β)1/2, where U is a typical velocity (at the scale of the jets) (Rhines 1975; Maltrud and Vallis 1991; Vallis and Maltrud 1993; Danilov and Gurarie 2004). The eastward jets are associated with strong gradients of potential vorticity. The regions between the jets tend to have relatively weak gradients of potential vorticity (e.g., Danilov and Gurarie 2004, their Fig. 1). (At the 2006 Chapman conference on jets that prompted the Journal of the Atmospheric Sciences special collection of which this paper is a part, the term “PV staircase” was frequently used to describe this sort of flow.)

An attractive explanation/description of the formation of these jets is based on the transport and mixing of potential vorticity, if it is accepted that potential vorticity gradients tend to inhibit mixing. If some small-amplitude modulation is applied to a system where large-scale PV gradients are uniform, then where the gradients are relatively weak, the mixing is expected to be relatively strong and where the gradients are relatively strong, the mixing is expected to be relatively weak. The effect will be to reduce gradients where they are relatively weak and, since strong gradient regions sit between weak gradient regions, to enhance gradients where they are relatively strong. Thus, one might naturally realize a state where there are large-amplitude variations in the potential vorticity gradient, with a limiting form where the flow is partitioned into regions within which potential vorticity is almost constant, separated by narrow regions where the potential vorticity gradient is very large. The corresponding flow, by PV inversion, naturally has narrow eastward jets centered on the sharp gradient regions (see Dritschel and McIntyre 2007 for further discussion).

There is a useful analogy between the idea that horizontal mixing of PV is inhibited by strong horizontal gradients of PV and the idea, applied to density stratified fluids, that vertical mixing is inhibited by strong vertical gradients of density. The formation of constant density layers in stratified fluids in which turbulence is driven by some kind of stirring (Phillips 1972) has been explained by something like the argument given in the preceding paragraph. Explicit mathematical models have been devised to give a firmer quantitative basis for this mechanism (Phillips 1972; Posmeinter 1977; Balmforth et al. 1998). Take the Balmforth et al. model as an example. Along with other models it represents density transport through a diffusive turbulent eddy flux and therefore requires prediction of eddy diffusivity, which is taken to be the product of the turbulent kinetic energy density and a mixing length, the latter depending on the forcing length scale, the kinetic energy, and the buoyancy gradient. The important point here is that the formation of the density structure occurs because the turbulent eddy diffusivity is small where the buoyancy gradient is large and large where the buoyancy gradient is small.

Could a similar model be formulated to describe the formation of β-plane jets?

We consider transport and mixing in a simulation that is typical of many that have been reported in the literature. The relevant equations are

 
formula

The terms on the right-hand side represent forcing (taken to be a stochastic function of time), linear damping, and fourth-order hyperdiffusion, respectively. The forcing term is prescribed in the same way as several previous investigations, for example, as described by Smith et al. (2002): (k, t + δt) = cF̂(k, t) + (1 − c2) f1/2e(k), where (k, t) is the Fourier transform of F(x, y, t), k is wavenumber, δt is the model time step, f is a constant that sets the magnitude of the forcing, c is a constant controlling the correlation time for the forcing, and ϕ(k) is a phase chosen randomly from one time step to the next. If c = 1 − τ−1δt then, for δt small enough, the autocorrelation of the forcing decays exponentially in time at rate τ−1. For reference, in the simulation to be reported, β = 5, α = 1, κ = 10−3, the forcing is applied for wavenumbers k such that 44 ≤ |k| ≤ 46 and τ is chosen to be 1. The constant f is chosen such that the equilibrated kinetic energy integrated over the domain is about 0.3. The numerical model is based on a spectral scheme with 256 × 256 wavenumbers.

This simulation, along with many others, gives rise to a clear set of longitudinal jets, shown in Fig. 7a as a shading plot of mean longitudinal velocity u as a function of y and t. The flow has been integrated from a state of rest at t = 0. In this case there are, during the period shown in Fig. 7, five distinct jets, one of which might be described as a double jet at the beginning of the period shown and that appears to be narrowing to a single jet as the simulation proceeds.

Fig. 7.

(a) The mean longitudinal velocity u as a function of y and t, (b) as function of y and t, (c) κeff calculated from tracer advected with full velocity field, and (d) κeff calculated from tracer advected with u removed. In (b), (c), and (d) the 0.04 contour for u is overlaid.

Fig. 7.

(a) The mean longitudinal velocity u as a function of y and t, (b) as function of y and t, (c) κeff calculated from tracer advected with full velocity field, and (d) κeff calculated from tracer advected with u removed. In (b), (c), and (d) the 0.04 contour for u is overlaid.

To quantify the transport and mixing structure in this flow we use the effective diffusivity diagnostic (e.g., Nakamura 1996; Shuckburgh and Haynes 2003) calculated from an advected tracer field. The effective diffusivity, which is small in barrier regions and large in mixing regions, is a function of an equivalent y coordinate and is shown in Fig. 7c as a function of equivalent y and t. For technical reasons it is most straightforward to calculate the effective diffusivity in a smaller domain that omits regions close to the limits in y of the full domain. This subdomain contains only three of the five jets, one of which is the “double jet.” The effective diffusivity shows clear minima at the location of the eastward jets, consistent with the notion of the jets as transport barriers. (Note that the double jet is seen as a double transport barrier.)

One might try to explain these barriers, indicated by minima in the effective diffusivity, as minima of a turbulent eddy diffusivity, along the lines of the Balmforth et al. (1998) model for density stratified flows. The turbulent eddy diffusivity encodes the transport effect of the longitudinally varying part of the flow. We may consider this transport effect by following the evolution of a second tracer that is advected by a modified flow obtained by removing the longitudinal mean part from the full flow [a similar study has been conducted using velocity fields from the surface Southern Ocean by Marshall et al. (2006)]. The effective diffusivity calculated from this second tracer is shown in Fig. 7d. This is strikingly different from the effective diffusivity for the first tracer. In the second case the eastward jets correspond to maxima in the effective diffusivity, not minima [and a similar difference was observed in the Marshall et al. (2006) calculations]. Part of the explanation for the difference observed here comes from the structure of 2, where υ is the y component of velocity, shown as a function of y and t in Fig. 7b. It may be seen that the largest amplitudes of 2 tend to be centered on the eastward jets, perhaps because these act as a kind of waveguide for Rossby wave activity. It is therefore not surprising that the most effective transport of the tracer also occurs in regions centered on these jets. Note that the “double” jet appears as a single maximum in the effective diffusivity, but that this is consistent with the lack of clear separation between the two maxima in 2 associated with the two parts of this double jet.

The conclusion from the above is that the barrier effect of the jets is not simply a result of the modulation by the jets of the transport properties of the longitudinally varying part of the flow, as would be implicit in any attempt to model those transport properties along the lines proposed by Balmforth et al. (1998) for the density stratified problem. The jets themselves must play a direct role in the transport. There is clearly some relation of this to the argument of Juckes and McIntyre (1987; see also Dritschel and McIntyre 2007), that background shear plays an important role in the vortex-edge transport barrier, with the rider that, as argued previously in section 2, there is evidence that it is the location of zero shear (between shears of opposite sign) that is the preferred location of the barrier. There seems to be little close relation with the “shear-shielding” mechanism described by Hunt and Durbin (1999), which emphasizes the role of isolated vortex sheets in which vorticity is strong and single-signed.

5. Summary

In this paper we have considered transport and mixing in PV conserving dynamically consistent flows. We have made no attempt to present any precise theory, in part because all routes to such a theory seem to require some assumption (e.g., scale separation between eddies and jets) that clearly does not apply to flows of interest. Instead we have presented a set of results from different models, each of which seems to have some relevance.

Our main conclusions are as follows:

  1. In kinematic models of jetlike flows, with velocity field imposed as a function of space and time, the “last” barrier (last remaining as the amplitude of a perturbation is increased) invariably corresponds to the axis of a jet. This suggests that while shear is an important ingredient of the barrier effect, it is the zero-shear location, between two regions of opposite-signed shear, that is the strongest barrier (rather than any location determined by maximum difference between background flow and phase speed of disurbances).

  2. Dynamically consistent flows show some common transport and mixing behavior with corresponding kinematic models, in particular a jetlike flow that is weakly perturbed from a steady state shows a strong central transport barrier in both cases and this barrier breaks as the perturbation increases beyond a threshold value. However, as illustrated in section 3, dynamical consistency significantly increases this threshold value. (While this latter conclusion might be argued to depend on the particular manner in which dynamically consistent and kinematic models are quantitatively compared, it holds in different sorts of flows, both forced and unforced.) Furthermore the nature of the transport is very different beyond the threshold value. In particular the extended partial-barrier regime seen in the kinematic models is absent in the dynamically consistent case.

  3. The previously mentioned increase in the threshold value for breaking the barrier is concrete evidence for the role of dynamical consistency in PV-conserving flows in enhancing the transport barrier effect. Thus, while strong gradients of PV in simulated or observed flows may in principle be a passive result of strong stirring regions separated by a barrier region, we have clear evidence that in the flows considered here those strong gradients tend to enhance the effectiveness of the barrier.

  4. The association of stronger PV gradients with enhanced barriers to transport (and reduced PV gradients with enhanced mixing) gives a natural explanation for the spontaneous appearance of eastward jets in the β–plane turbulence. However, this effect is not simply described by reduced turbulent eddy amplitudes in the regions of strong PV gradient. The eastward jets associated (via PV inversion) with the strong PV gradients play a direct role in the enhanced barrier effect, as suggested by the first point (above). Any attempt to formulate a predictive model of the jet formation would have to incorporate this effect, by combining some representation of the turbulent eddies with the effect of the background flow to give an estimate of effective diffusivity.

Acknowledgments

DAP was supported by the U.K. Natural Environment Research Council through a Ph.D. studentship. The authors have benefitted from useful discussions with M. E. McIntyre, R. S. McKay, and W. R. Young. Helpful comments on a first version of the paper were given by T.-Y. Koh and an anonymous referee. The channel-model code on which the work reported in section 3 was based was originally provided by P. D. Killworth.

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Footnotes

Corresponding author address: Dr. P. Haynes, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd., Cambridge CB3 0WA, United Kingdom. Email: phh@damtp.cam.ac.uk

This article included in the Jets and Annular Structures in Geophysical Fluids (Jets) special collection.