## 1. Introduction

Chaotic flows in stratified, rotating fluid systems like planetary atmospheres and oceans are often called “turbulent.” However, in such systems there is no such thing as turbulence without waves, a point well brought out in the celebrated paper of Rhines (1975). One way to appreciate the point is to note that such systems always have background gradients of potential vorticity (PV) and then consider the implications for the momentum and angular momentum budgets. As will be illustrated here, those budgets make no sense at all, in realistic cases, unless one considers the wavelike and turbulent aspects together.

As well as short-range turbulent momentum transports of the *austausch* or mixing-length kind, there are long-range momentum transports due to Rossby waves and other wave types—“radiation stresses” in the language of physics. Indeed, the ubiquity and importance of such stresses illustrates one of the grand themes of physics, the “dynamical organization of fluctuations” with systematic mean effects (also important in, e.g., biological molecular motors). By its very nature, a wave propagation mechanism such as the Rossby wave mechanism will *organize* the fluctuating fields, no matter how chaotic they may seem, in the sense of inducing systematic correlations among them. The correlations are shaped by the waves’ polarization relations and usually give rise to long-range stresses. They may produce phenomena like planetary equatorial superrotation, or gyroscopically pumped global-scale circulations such as the Brewer–Dobson circulation of the terrestrial stratosphere.^{1} The range of such stresses is not limited to mixing lengths, but can reach out as far as waves can propagate. And, crucially, there is a strong dynamical interplay between the more wavelike and the more turbulent aspects, not unlike the wave–turbulence interplay and stress divergence that give rise to alongshore currents in an ocean beach surf zone.

Among the consequences of such interplay, in stratified, rotating systems, are three interrelated phenomena on which this review will focus: first the spatial inhomogeneity of PV mixing by layerwise-two-dimensional turbulence, second the common occurrence of “antifrictional” or upgradient horizontal stresses

The three phenomena are all illustrated by the typical jet-sharpening scenario sketched in Fig. 1. The sketch was originally made to help understand the behavior of the terrestrial winter stratospheric polar-night jet, or polar-vortex edge, during a so-called minor warming. However, with appropriate rescaling it applies equally well to other cases such as that of the subtropical tropopause jet in the late stages of a nonlinear baroclinic wave life cycle of type 1 (LC1; e.g., Thorncroft et al. 1993). In both cases the PV is strongly mixed on the equatorward flank of the jet (Fig. 1a), reshaping the large-scale PV distribution and causing the velocity profile to sharpen, in the sense that its lateral scale becomes narrower (Fig. 1b), with concomitant changes in the angular momentum distribution. Fundamentally similar jet-sharpening processes have often been observed in laboratory and numerical experiments, with PV mixing on one or both flanks of the jet, going back to the pioneering work of Fultz et al. (1959), Hide (1958), and N. A. Phillips (1956).^{2} More recent work clearly showing jet sharpening in its simplest, barotropic form includes, for instance, the beautiful laboratory experiments of Sommeria et al. (1989, 1991), and many numerical experiments on the related phenomenon of vortex erosion or stripping (e.g., Juckes and McIntyre 1987; Legras et al. 2001). The reader may gain entry to the most recent literature from, to pick a very few, Vallis and Maltrud (1993), Rhines (1994), Nozawa and Yoden (1997), Hughes (1996), Manfroi and Young (1999), Huang et al. (2001), Robinson (2006), McWilliams (2006, his chapter 5), Greenslade and Haynes (2008), Scott and Polvani (2007), Sukoriansky et al. (2007), and Thompson and Young (2007), as well as from the other papers in this special collection.

The jet-sharpening and angular-momentum changes are tied to the PV mixing via PV inversion (e.g., Hoskins et al. 1985; Robinson 1988). Specific illustrations will be given in Eqs. (5.1)–(7.2) below. PV inversion is usually robust for jets. Invertibility depends only on the jets behaving as balanced flows. The imbalance represented by inertia–gravity and sound waves is negligible in many cases, or unimportant after averaging (e.g., Viúdez and Dritschel 2004). It follows that the link between PV mixing and jet sharpening is very tight. During mixing, if it occurs, the stress pattern has no choice but to fit in with the angular momentum changes dictated by inversion. Conversely, if there is no way to set up a suitable stress pattern, then PV mixing may be modified or inhibited.

In quasigeostrophic models, with the Rossby wave mechanism as the only available wave mechanism, the way the stress pattern fits in is described by the well-known *Taylor identity* relating stress divergences to eddy PV fluxes, recalled in the appendix. The stresses themselves extend between regions of PV mixing and, in general, outside them as well. As already noted, in principle they can extend as far as waves can propagate. The stress pattern associated with the example in Fig. 1 (e.g., Edmon et al. 1980; Dunkerton et al. 1981) corresponds mainly to Rossby waves propagating or diffracting up the jet axis from below with their characteristic upward–westward phase tilt, then refracting equatorward and breaking in the mixing region indicated in Fig. 1a. That region can therefore be thought of as a “Rossby wave surf zone.” A characteristic feature of the stress pattern is the well-known antifrictional westward–equatorward tilt of phase lines between the jet and the adjacent surf zone, signaling positive values of

Figure 1, then, can be taken as a reminder of the main concepts needed to solve that enigma, indeed to solve it in a remarkably robust and simple way. They are (i) the PV invertibility principle and its corollaries (which include making sense of the Rossby wave mechanism itself), (ii) the fact that breaking Rossby waves mix PV (though usually imperfectly), and (iii) the Taylor identity, Eq. (A.3) below, helping to show how the whole wave–turbulence jigsaw fits together.

Historically—perhaps because concept (i) was not put together with scenarios like Fig. 1a—the solution emerged only rather tortuously, over the second half of the past century, in several parallel strands of research that gradually broke away from the competing turbulence-theoretic austausch paradigms of the first half called “momentum transfer theory” and “vorticity transfer theory.” Jeffreys, early in the century, was the first to establish from the angular momentum principle that an antifrictional stress

There was first a long line of linear baroclinic instability studies showing that the fastest-growing modes on broad jets usually exhibit antifrictional

In hindsight (e.g., Edmon et al. 1980; Thorncroft et al. 1993; Hughes 1996) we can now recognize this third idea as the most important of the three, as a contribution to mechanistic understanding of the simplest, most robust, and most basic kind. It appears from Green (1970) that the idea first occurred to Eady, though it seems that Dickinson (1969) arrived at it independently while working in Starr’s group at the Massachusetts Institute of Technology and was the first to publish it. Then at Woods Hole Thompson (1971) proposed it again, it seems, independently, and Whitehead (1975) demonstrated Rossby wave-induced *nonlinear* Rossby wave critical layers (Stewartson 1978; Warn and Warn 1978; Killworth and McIntyre 1985; Haynes 1989). Such “layers” are finite, though narrow, surf zones. Those models provided mechanistically clear and explicit illustrations of the precise way in which PV mixing can influence the wave-induced

Complementing all this was yet another parallel strand of history starting with the theoretical work of Rhines (1977). It used weakly inhomogeneous turbulence theory, assuming a scale separation between large-scale mean fields and small-scale turbulent eddies and Rossby waves, to build a model of the wave–turbulence interplay within that limitation but again recognizing the insight furnished by the Taylor identity. The scale separation facilitated the inclusion of cases with nontrivial zonal as well as meridional structure.

A typical feature of PV mixing scenarios like that of Fig. 1 is, however, the strong inhomogeneity of the mixing, which precludes scale separation (e.g., McIntyre and Palmer 1983; Juckes and McIntyre 1987; Riese et al. 2002). Today’s remote sensing techniques have made this inhomogeneity more conspicuous than ever, in the case of the real terrestrial winter stratosphere, by showing in remarkable detail many examples of breaking Rossby waves with both their turbulent and their wavelike aspects visible in sharply contrasted adjacent regions. One typically sees the wavelike edge of the polar vortex with its steep isentropic gradients of PV colocated with the polar-night jet axis, immediately adjacent to the surrounding surf zone, reminding us that Fig. 1 gives only a blurred view of reality.^{3} And PV-mixing scenarios of just this kind have become increasingly familiar from high-resolution chemical–dynamical modeling motivated by the stratospheric ozone problem (e.g., within a vast literature; Manney et al. 1994; Simmons et al. 2005; and the review by Haynes 2005).

In the next section of this review we argue that the extreme inhomogeneity is itself a generic consequence of the wave–turbulence interplay, in some circumstances at least. Thus we widen the scope to include not only classic cases like Fig. 1 but also cases like Jupiter, in which a stratified layer is stirred in what is almost certainly a different way, namely by buoyant convection^{4} and associated radiation stresses. The central idea is simply that of *positive feedback,* an unstable competition between wavelike and turbulent dynamics. It leads us to expect, for instance, the spontaneous emergence of quasi-permanent jets when an initially homogeneous system is stirred homogeneously.

Part of the feedback is due simply to the nature of the Rossby wave restoring mechanism, which depends on background PV gradients. To that extent it is like the “Phillips effect” (O. M. Phillips 1972), the spontaneously inhomogeneous vertical mixing of the background buoyancy gradient in a stably stratified fluid. However, in section 2 we argue as in Juckes and McIntyre (1987) that jet *shear,* producing shear straining of vortical disturbances, can greatly enhance the feedback. The arguments are tested against some nonlinear numerical experiments in sections 3 and 4. The experiments provide further mechanistic insight not only into the terrestrial but also, arguably, into aspects of the Jovian case.

Despite the different stirring mechanism or mechanisms, the Jovian case is like that of the terrestrial winter stratosphere to the extent that the PV-mixing inhomogeneity appears to be at an extreme, with staircase-like PV profiles of many steps in the Jovian case as compared with a single step in the terrestrial case. Strong arguments for a Jovian staircase have been made by Marcus (1993), from the observed vortex-interaction phenomenology. Sections 5 and 6 discuss the implied scaling laws for step width, that is, for jet spacing. Section 7 briefly reviews the way PV inversion dictates angular-momentum changes, also pointing out that channel and doubly periodic beta-plane models may have artificial constraints on their angular momentum that conceal those changes. Section 8 offers a few concluding remarks including a new suggestion about Jupiter. For completeness the Taylor identity is reviewed in a brief appendix.

## 2. The Phillips effect for buoyancy and PV, and the formation of eddy-transport barriers

The Phillips effect for vertical buoyancy mixing was clearly demonstrated in the classic experiment of Ruddick et al. (1989), in which stirring by an array of vertical rods, setting no vertical scale, produced layering in a stratified tank. See also, within an extensive literature, the original discussion by OMPhillips (1972) and the elegant mathematical modeling studies by Barenblatt et al. (1993) and Balmforth et al. (1998). Those studies were, however, limited by vertical austausch or local flux-gradient assumptions. In a more general way, not dependent on austausch assumptions, one may argue heuristically that the observed layering results simply from a positive feedback process. The background buoyancy gradient and gravity wave elasticity are weakened in a mixing layer, facilitating further mixing across stratification surfaces. Conversely, the gravity wave elasticity is strengthened wherever interfaces are forming between the mixed layers, inhibiting mixing across the interfaces.

In the case of the quasi-horizontal, lateral mixing of PV *along* stratification surfaces, one can make a similar positive-feedback argument in terms of the Rossby-wave restoring mechanism, that is, the quasi-elasticity of sideways undulating PV contours. The background gradient and Rossby elasticity are weakened in a mixing region, facilitating further mixing, and conversely are strengthened wherever PV contours are bunched together between mixed regions (e.g., McIntyre 1994, p. 308). Thus, other things being equal, large-scale PV gradients are liable to be weakened in some regions and sharpened in others, making them more turbulent and more wavelike respectively. The extreme, staircase-like instances of this include the terrestrial stratosphere where the wavelike regions of sharpened gradients have been clearly shown, through a wealth of chemical data, to be able to shut off the mixing almost completely. That is, the wavelike regions with their associated eastward jets act as “eddy-transport barriers” (e.g., Juckes and McIntyre 1987; Nakamura 1996; Haynes and Shuckburgh 2000; Marshall et al. 2006; Haynes et al. 2007). Other such examples include the laboratory experiments of Sommeria et al. (1989, 1991)—confirming the barrier and staircase structure both by dye injection and by PV measurements using velocimetry—and the classic demonstration by Danielsen (1968) of a sharp barrier at a tropopause-jet axis, strikingly revealed by airborne observations of nuclear test debris. Marcus (1993) argues persuasively for barriers and staircases on Jupiter from model experiments showing generically realistic vortex-interaction behavior. In particular, the experiments show the eastward jets acting as barriers against vortex excursions. All these would appear to be extreme cases of the positive feedback or PV Phillips effect in action.

However, Rossby elasticity is by no means the same thing as the static stability or buoyancy elasticity acting in the Ruddick et al. experiment. Furthermore, there is no counterpart of PV inversion in the buoyancy case. That is, buoyancy layering does not necessarily cause jet formation and indeed does not, in fact, cause it in the Ruddick et al. (1989) experiments. And, as noted by Juckes and McIntyre (1987), in the PV case jet shear can act to enhance the barrier effect, hence the positive feedback effect, in an important way. In their words—referring to an early high-resolution model of stratospheric Rossby wave breaking, jet sharpening, and barrier formation in a scenario like that of Fig. 1—the Rossby elasticity “works most effectively on the largest spatial scales.” This is connected with the way Rossby elasticity depends on PV inversion, which, in contrast with the buoyancy case, is a nonlocal functional relation. Thus Rossby elasticity “cannot explain the remarkable imperviousness of the main vortex even to small-scale incursions.” That “remarkable imperviousness,” a clear-cut case of the barrier effect, is also, they go on to say, “related . . . to the existence of strong horizontal shear.”

At small enough scales, differential advection by the shear overwhelms Rossby wave elasticity in the well-known way (e.g., Yamagata 1976), leading to the passive-tracer-like behavior of weak small-scale PV anomalies, conspicuous in the form of the filamentation seen in Juckes and McIntyre’s model and in many other high-resolution numerical experiments including, most strikingly, experiments at infinite Reynolds number using accurate contour advection without “surgery” (e.g., Dritschel 1989). Such behavior is no more than would be expected from the standard Kelvin–Orr theory of “sheared disturbances” or “shear straining” (Thomson 1887; Orr 1907; Yamagata 1976). But the Rossby wave and Kelvin–Orr theories are both *linear*, as is the more recent “shear-sheltering” theory of Hunt and Durbin (1999).

For all those theories can tell us, real nonlinear surf-zone turbulence might make the vortex edge, for instance, not barrier-like but sieve-like in the sense of allowing the free exchange of material between the vortex interior and exterior on scales too small to feel Rossby elasticity. In any case, the effectiveness or otherwise of eddy transport barriers is clearly a fundamental problem in dynamics as well as chemistry and, in particular, intimately part of the whole question of why the positive feedback seems so effective, why jets are ubiquitous, and why, for instance, Jovian vortices behave as they do. To our knowledge, there has never been a case of a real Jovian vortex crossing an eastward jet (Rogers 1995; Rogers 2006, personal communication). In the next section, therefore, we revisit the eddy-transport-barrier problem in a way that focuses on its nonlinear aspects.

## 3. Barrier-penetration experiments

In this section we illustrate by simple numerical experiments at near-infinite Reynolds number how the barrier effect is crucially enhanced by the shear on the flank of a jet, even in a highly nonlinear disturbance regime. The standard quasigeostrophic shallow-water model is used, with Lagrangian contour-dynamics methods in a polar tangent plane approximation, with the winter stratosphere in mind. We focus on realistic values 1000–2000 km of the Rossby length *L _{D}*, as judged by nonlinear shallow-water behavior that best mimics the real stratosphere (Norton 1994). Thus we avoid rigidly bounded models, that is, nondivergent barotropic dynamics,

*L*= ∞. Rigid upper boundaries are of course unrealistic for atmospheres. And, though realistic for some aspects of ocean dynamics, models with

_{D}*L*= ∞ are not closely relevant to oceanic layerwise-two-dimensional turbulence with bottom topography, influenced as it is much more by baroclinic

_{D}*L*values.

_{D}*q*=

*f*+

*ψ*), the quasigeostrophic PV, with constant Coriolis parameter

*f*and with

*ψ*) = (∇

^{2}−

*L*

^{−2}

_{D})

*ψ*. The domain is horizontally unbounded, so that the PV inversion to find the streamfunction

*ψ*is

*K*

_{0}(·) is the modified Bessel function decaying exponentially for large argument. The two-dimensional velocity field is (

*u*,

*υ*) = (−∂

*ψ*/∂

*y*, ∂

*ψ*/∂

*x*), so that the material derivative

*D*/

*Dt*= ∂/∂

*t*+

*u*∂/∂

*x*+

*υ*∂/∂

*y*, to leading order.

We take an idealized version of the winter stratosphere with a perfectly developed “staircase step,” that is, with a sharp-edged polar vortex. For simplicity it is taken to be a large vortex patch of uniform PV, surrounded by an idealized surf zone of infinite extent having a smaller uniform PV value. This corresponds to an exaggerated version of the heavy curves in Fig. 1, with a discontinuous jump in PV on the left and a slope discontinuity in zonal velocity on the right, like one of the sharply peaked jets in Fig. 7 below. We take *L _{D}* = 2000 km and polar-vortex radius 3000 km in the examples shown. Values

*L*= 1000 km and

_{D}*L*= 1500 km were also used, giving qualitatively similar results. The PV contrast between the polar vortex and surf zone is denoted by Δ

_{D}*q*

_{barrier}. We have found that straightening out the vortex edge, making it into a channel-model jet with the same PV contrast, makes little difference to the behavior.

We test the resilience of the barrier at the vortex edge by bombarding it with the most powerful of coherent vortex structures, namely fast-propagating vortex pairs. A sufficiently strong vortex pair will punch through any given barrier, and a sufficiently weak one will not. Many experiments were done with vortex pairs of various strengths and sizes, incident from various directions. In all cases where the incident vortex pair gets anywhere near the barrier—which requires the vortex pair to be sent in an upshear direction—the resulting behavior can reasonably be called shear-induced disruption of the vortex pair, a kind of “divide and rule.” The experiment shown in Fig. 2 illustrates this behavior.

The polar vortex is central in the top left panel. A large, tightly spaced, fast-propagating vortex pair is incident upshear at 225°. Its members have relative strengths Δ*q*_{vortex}/Δ*q*_{barrier} = ±1.0; thus the strength of the cyclonic member matches that of the barrier. This is a strongly nonlinear situation in which the entire polar vortex is violently disturbed. Nevertheless, the shear outside the barrier quickly separates the two members of the vortex pair, disrupting its ability to propagate. Consequently—even though the barrier suffers a very large Rossby wave distortion, comparable to that in a minor stratospheric warming—only modest amounts of vortex-pair and other surf-zone material penetrate well inside. Figure 3 enlarges the ninth and twelfth panels of Fig. 2 to show the detail more clearly. The grayscaling distinguishes material from, in order from lightest to darkest, the ambient surf zone, the incident anticyclone, the incident cyclone, and the polar vortex.

The shear-induced disruption of the incident vortex pair involves more than just pulling it apart. The anticyclonic member survives intact, staying well outside the barrier and corotating with the ambient shear in the manner familiar from classic vortex-interaction studies (e.g., Kida 1981; Bell 1990). By contrast, most of the material of the cyclonic member of the pair is sheared out into a long filament, and most of the filament gets wrapped round the edge of the polar vortex, itself cyclonic. Some of the cyclonic material is promptly ejected back into the surf zone, in a familiar and typical kind of Rossby wave-breaking or barrier-erosion event (e.g., Juckes and McIntyre 1987; Polvani and Plumb 1992). That event is seen in the bottom row of Fig. 2. Of the remaining incident-cyclone material, some of it stays near the polar-vortex edge, though a modest amount ends up deeper inside.

If we make the size of the incident vortex pair smaller, keeping its relative strength at ±1.0, then it propagates more slowly, is disrupted sooner, and penetrates less. The same occurs if the relative strength is reduced below 1.0. Even for large, fast vortex pairs like that of Fig. 2, the barrier is very resilient against deep penetration for relative strengths ≲0.9. By contrast, for relative strengths ≳1.1, substantial amounts of material from the incident cyclone (though never from the anticyclone in the cases considered) penetrate deeply into the polar vortex. Figure 4 shows the most violent case considered, like that of Figs. 2 –3 except that the incident vortex pair has relative strength ±1.4. Then a large portion of the incident cyclone penetrates all the way into the polar vortex. The incident anticyclone again remains isolated. Similar results were obtained for *L _{D}* = 1000 km and

*L*= 1500 km.

_{D}*L*, that the transition from very little penetration to substantial penetration corresponds to

_{D}*L*, the thresholds are somewhat higher. Also, we have found following Bell (1990) that even without the boost from vortex-pair propagation, lone cyclones can also approach and penetrate the barrier, even a straight barrier, through a much slower and more subtle wave–vortex interaction related to classic vortex-merging dynamics. The thresholds themselves are hardly changed, except that the threshold values of |Δ

_{D}*q*

_{vortex}/Δ

*q*

_{barrier}| become somewhat higher, typically by factors of 2–3, in the peripheral cases with core sizes substantially different from

*L*. These are preliminary results only, for the purposes of illustration, and we hope to present a more systematic study in due course.

_{D}It might be asked why naturally occurring vortices in staircases are usually below threshold. At least, vortices are seldom if ever seen to cross an eastward jet either on Jupiter or in the terrestrial winter stratosphere. The answer could be that if such crossing events had ever been common they would already have broken some of the barriers, and thus widened the steps of the staircase until the barrier strengths, as measured by their PV contrasts, became comparable to or greater than those of the strongest vortices. The vortex strengths will of course be subject to some limitation arising from whatever stirring mechanism is in operation. If that mechanism, with accompanying radiation stresses as needed, is strong enough and effective enough to have created the staircase in the first place—arguably the situation on Jupiter—then the range of available vortex strengths will influence the step size such that the strongest vortices are only just below the threshold. Of course if the staircase is formed by Rossby wave breaking alone, then its strongest vortices initially have the same strengths as the material eroded from the barrier, which is automatically at or below the threshold.

## 4. Some decay experiments that illustrate the PV Phillips effect

To the extent that the PV Phillips effect is generic, it ought to be possible to demonstrate the resulting tendency toward inhomogeneous PV mixing in turbulence-decay experiments, with initial conditions statistically uniform in *y*.

Of course many such experiments have been done in the past, including the original Rhines (1975) experiments, but for various reasons (including early limitations on numerical resolution, and the privileging of power-spectral diagnostics over, e.g., PV maps) there has not always been a clear distinction between persistent jets produced by inhomogeneous PV mixing on the one hand and transient, migrating jets in the form of zonally long Rossby waves on the other. Also, both computer power and new numerical techniques now provide an opportunity to carry out such experiments at Reynolds numbers and effective resolutions vastly higher than before.

To our knowledge, however, a systematic study has yet to be completed at the level of today’s state of the art. One reason is the difficulty of getting beyond the simple positive-feedback heuristic and of quantifying the wave–turbulence interplay in this situation, including the role of Rossby wave radiation stresses. In the present review, therefore, we limit ourselves to just one illustrative example that shows the PV Phillips effect particularly clearly. The example is taken from an ongoing series of numerical experiments to be reported elsewhere. These simulate freely decaying quasigeostrophic shallow-water turbulence in a beta-plane channel, starting with random vortices on an approximately uniform background PV gradient *β*. The experiments use a very accurate “contour-advective semi-Lagrangian” (CASL) algorithm (Dritschel and Ambaum 1997; Dritschel et al. 1999) able to simulate extraordinarily complex ultrahigh Reynolds number flows with great efficiency.

Not all such experiments provide equally clear illustrations of the PV Phillips effect. The tendency for PV mixing to be inhomogeneous is always seen, but has been found to range from very weak to very strong depending upon parameter values. The dependence is not only upon the background PV gradient and initial turbulent energy but also, more sensitively than we had expected, upon the Rossby length *L _{D}*. This may well be bound up with variations in the radiation-stress field related to variations in Rossby wave excitation and dispersion.

The experiment shown here is conducted in a channel of nondimensional width and length 2*π* with *L _{D}* = 1. Free-slip boundary conditions apply at

*y*= ±

*π*, and the flow is periodic in

*x*(see Benilov et al. 2004 for the use of the CASL algorithm in this geometry). The initial quasigeostrophic PV field

*q*is built from a random anomaly field

*q*′ of maximum amplitude |

*q*′|

_{max}= 4

*π*superposed on a linear background PV gradient

*βy*, with

*β*= 2

*π*. The PV anomaly is correlated over a length ≈0.28; more precisely, the autocorrelation function of the PV-anomaly distribution has a roughly Gaussian shape with a full width at half maximum of 0.28. The full PV (background plus anomaly) is discretized into steps with an initial mean step width Δ

*y*=

*π*/16 ≈ 0.196 ≈

*L*/5, corresponding to a small PV jump Δ

_{D}*q*

_{discrete}= (2

*π*)

^{2}/31 ≈ 1.27 across each contour [a tiny fraction of the total planetary PV contrast across the width of the channel, 2

*πβ*= (2

*π*)

^{2}≈ 39.5]. The Eulerian grid resolution used to invert the PV to find the velocity field is 256 × 257; however, the PV is represented in a Lagrangian way, as contours, to permit highly accurate advection.

To deal with the turbulent dissipation of fine-grain PV gradients, surgery is applied at a twentieth of the Eulerian grid size (i.e., at the scale *δ* ≈ 0.00123). This apparent mismatch in the smallest scales used to represent the velocity and PV fields is justified (Dritschel and Ambaum 1997; Dritschel et al. 1999; Dritschel and Viúdez 2003, and references therein) by the steeper spectral decay of velocity fluctuations with decreasing scale.

Figure 5 shows the PV field at several stages in the flow evolution. The grayscale shading is sawtooth linear to make the detailed structure more visible. The flow rapidly grows in complexity, as measured by the total number of nodes on the contours. The complexity peaks at dimensionless time *t* = 14 (the unit of time being defined as 4*π*/|*q*′|_{max}). At that time, there are over seven million contour nodes. The complexity thereafter gradually decays as the flow becomes increasingly zonal. The PV Phillips effect is plainly in evidence in this case. Despite the statistical homogeneity of the initial conditions, the background PV field has been mixed in a highly inhomogeneous way.

This is seen even more clearly in Fig. 6, whose left-hand panel shows the time evolution of the mean *y* position of those PV contours that wrap the domain periodically in the *x* direction. As expected from the positive-feedback heuristic, the contours are pushed apart in some regions and squeezed together in others. The corresponding jets are evident in the right-hand panel of Fig. 6, which shows the zonally averaged zonal velocity * u*(

*y*,

*t*) at the initial and final times

*t*= 0, 84.

## 5. Staircase inversions and jet spacing

The jet profiles in the right-hand panel of Fig. 6 may be compared, and contrasted, with the theoretical velocity profiles *u*(*y*) for perfect PV staircases on an unbounded beta-plane. The theoretical expressions, obtained by PV inversion, are given in (5.1) and the following text and in (5.3), (5.5), and (5.6). Some examples are plotted in Fig. 7. For instance, the second curve from the left shows the *u*(*y*) profile for a single step cut into the uniform background PV gradient, in the form of a perfectly mixed zone |*y*| ≤ *b* of width 2*b*, in the case *b* = *L _{D}*. The leftmost curve shows the corresponding mass shift expressed as the surface elevation change

*h*(

*y*) =

*f*

_{0}

*ψ*/

*g*, where

*g*is gravity and

*f*

_{0}a constant representative value of the Coriolis parameter. This mass shift is dictated by geostrophic balance with

*u*(

*y*) = −∂

*ψ*/∂

*y*. Notice that a smoothed version of the

*u*(

*y*) curve qualitatively resembles the difference between the two velocity curves in Fig. 1b.

*L*, the

_{D}*u*(

*y*) and

*h*(

*y*) profiles within the single mixed zone |

*y*| ≤

*b*are given by

*y*|/

*L*) on each side |

_{D}*y*| >

*b*(Fig. 7b) and, with

*H*

_{0}=

*f*

^{2}

_{0}

*L*

^{2}

_{D}/

*g*, the undisturbed layer depth,

*y*|/

*L*) in |

_{D}*y*| >

*b*(Fig. 7a). The last three curves in Fig. 7 are the

*u*(

*y*) profiles for staircases of two, three, and an infinite number of steps or mixed zones, constructed by the appropriate daisychaining of (5.1), that is, by superposition of laterally shifted copies of (5.1) and its side tails. Superposition is allowed because quasigeostrophic

*β*-plane PV inversion (3.2), with

*f*now variable but

*L*still constant, is a linear operation. The limiting case of the perfect periodic staircase, uniquely determined by construction as an infinite daisy chain, has profiles that are readily shown to be

_{D}*y*| ≤

*b*, representing (5.1) and (5.2) plus infinite sums of side tails from the other zones. The rightmost curve in Fig. 7 is (5.3) shifted laterally by a distance

*b*.

*L*/

_{D}*b*→ ∞, (5.1) and (5.3) become, respectively, with fractional error

*O*(

*b*/

*L*)

_{D}^{3},

*L*/

_{D}*b*→ ∞. The reason is that the side tails of the single-zone solution (5.5) are infinitely weak and infinitely broad but contain finite relative angular momentum. We discuss the corresponding absolute angular-momentum changes in section 7 below. The staircase found by Danilov and Gurarie (2004) and Danilov and Gryanik (2004) in their rigidly bounded barotropic turbulence model produces (5.6) almost exactly, apart from an additive constant.

For all values of *L _{D}*/

*b*these examples show the generic feature of sharply peaked eastward jets with broad westward flows in between, the more so in cases with small

*L*/

_{D}*b*. This asymmetry has often been remarked on. From (5.1) and (5.3) it is evident that for small

*L*/

_{D}*b*the eastward jets are thinner and still more sharply peaked, with width scale

*L*, and the intervening westward flows relatively still more broad.

_{D}The inversions (5.1) and (5.3) assume straight jets. Recent work on shallow-water flows with a latitudinal variation of *L _{D}* like that expected on Jupiter (e.g., Theiss 2004; Scott and Polvani 2007), following ideas of Salmon (1982), suggest that for the smaller values of

*L*, as in modeling Jupiter’s high latitudes, the jets not only become thinner but meander strongly as well, while keeping themselves sharp. This is also reminiscent of many terrestrial ocean currents (e.g., Niiler et al. 2003). It is probably related to the relative “sloppiness” of vortex interactions at small

_{D}*L*(e.g., Waugh and Dritschel 1991) and to the smooth, stable behavior of large-amplitude long Rossby waves on jets like (5.1) or (5.3) at small

_{D}*L*(Nycander et al. 1993). Of course in some cases the meanders may be due to, or increased by, baroclinic instability, as with atmospheric tropopause jets, taking us outside the scope of barotropic shallow-water modeling.

_{D}It is worth emphasizing that the Rossby elasticity and hence the Rhines and Phillips effects cannot be expected to be negligible even in the limit *L _{D}* → 0. The wave–turbulence interplay is still inescapable. Rossby waves on thin jets have phase speeds of the same order as the jet velocity scale. Therefore, in the staircase scenario with vortex cores of size ∼

*L*, jet widths ∼

_{D}*L*, and eddy velocities of the same order as jet velocities, the order-of-magnitude regime is just that usually associated with the Rhines effect understood as “cascade arrest” or “cascade retardation” in wavenumber space, or simply as “Rossby elasticity significant” or “waves and turbulence both significant,” at the length scale

_{D}*L*. For instance in the thin-jet limit

_{D}*L*→ 0 with

_{D}*b*finite it is well known, and readily shown, that Rossby waves on the jet (5.3) near

*y*=

*b*, say, all have phase speeds

*c*lying between maximum

*u*and minimum

*u*. This means not only that Rossby elasticity is significant but also that there are critical layers in the jet flanks—implying a two-sided version of Fig. 1—hence, in reality, surf zones that would mix PV there had it not been mixed already and the jet already sharpened. Hughes (1996) discusses this further. Once again we have turbulence intimately and inescapably associated with waves, the essence of the Rhines effect.

## 6. Rhines scales versus jet spacing

Even a cursory perusal of the literature shows that the term “Rhines scale” is used with a plethora of meanings; see, for instance, Dunkerton and Scott (2008), who consider three different length scales, which they call “dynamical Rhines scale,” “spectral Rhines scale,” and “geometric Rhines scale.” Some authors even define Rhines scale such that it can “fail to exist” as a real number. Others appear to assume that it always means the jet spacing.

Without presuming to say what the term “should” mean, in a given dynamical context, we choose to define the Rhines scale here in its simplest possible sense. If a velocity scale *U* is given, or arises naturally (as in the staircase problem), then dimensional analysis tells us that one of the length scales in the problem must be (*U*/*β*)^{1/2}, where *β* is the planetary absolute vorticity gradient. We call this the Rhines scale based on *U*, and use the symbol *L*_{Rh}(*U*) to denote simply that length scale, by definition, recognizing of course that different velocity scales *U* will arise and that the length scale *L*_{Rh}(*U*) may or may not turn out to have a visible role in one or another dynamical regime, viewed spectrally or in any other way.

*L*

_{Rh}(

*U*

_{vortex}) and

*L*

_{Rh}(

*U*

_{jet}) based respectively on vortex peak velocity

*U*

_{vortex}and jet velocity

*U*

_{jet}= (max

*u*− min

*u*) are accordingly defined by

*L*, and assuming

_{D}*b*≳

*L*such that there is room for the vortices between the jets, the threshold relation (3.3) tells us that Δ

_{D}*q*

_{vortex}∼ Δ

*q*

_{barrier}∼

*βb*. The inversions (5.1) and (5.3) tell us that the jet velocity-profile width scale is

*L*in this case. It follows that

_{D}*U*

_{jet}∼

*L*Δ

_{D}*q*

_{barrier}∼

*L*∼

_{D}βb*L*Δ

_{D}*q*

_{vortex}∼

*U*

_{vortex}, and therefore

*L*

_{Rh}(

*U*

_{jet}) ∼

*L*

_{Rh}(U

_{vortex}) and

*L*/

_{D}*b*∼ 1.

*L*, the scaling is less simple. This is both because the thresholds are then somewhat higher than (3.3) and because, for vortices of given strength Δ

_{D}*q*

_{vortex}but core size ≪

*L*, the scaling

_{D}*U*

_{vortex}∼ Δ

*q*

_{vortex}

*L*is replaced by

_{D}*U*

_{vortex}∼ Δ

*q*

_{vortex}× core size. However, we still have a simple relation between

*b*,

*L*, and

_{D}*L*

_{Rh}(

*U*

_{jet}) across the whole range of parameter conditions. By putting (6.1) together with (5.3) we get

*b*and

*L*

_{Rh}(

*U*

_{jet}). The left-hand portion of the graph gives us the standard result

*b*∼

*L*

_{Rh}(

*U*

_{jet}) for

*L*/

_{D}*b*≳ 1, while the right-hand portion recovers the behavior (6.2) for

*L*/

_{D}*b*≲ 1.

## 7. Angular momentum changes due to PV mixing

The PV invertibility principle tells us that a PV mixing event localized in *y* will produce a definite and unambiguous total angular momentum change *δM*. The single-zone inversion (5.1)–(5.2) illustrates that fact very clearly, as will be shown next. It is one way of seeing that radiation stresses—most directly and simply those associated with Rossby wave radiation—must inevitably be associated with PV mixing as also shown by the Taylor identity (see appendix). It also suggests that PV mixing may be catalyzed by the stresses due to other wave types.

Consider a conservative PV mixing event like that required to create a single staircase step or surf zone, in which the PV is changed by *δq*(*y*) where ∫*δq*(*y*) *dy* = 0. For the case of a perfectly mixed step created from a uniform background PV gradient *β*, the profile *δq*(*y*) is N-shaped with slope −*β*. Inverting it gives the *u*(*y*) profile (5.1) and the *h*(*y*) profile (5.2). The associated angular momentum change, *δM*, can be shown as follows to be proportional to ∫*yδq*(*y*) *dy*, which is generically nonzero [e.g., −⅔ *β**b*^{3} for the N-shaped *δq*(*y*)].

*δM*as the change in absolute zonal momentum per unit

*x*distance. For our shallow-water layer of undisturbed depth

*H*

_{0}and, say, constant mass density

*ρ*

_{0}, the absolute zonal momentum per unit horizontal area is

*ρ*

_{0}[

*H*

_{0}+

*h*(

*y*)][

*u*(

*y*) −

*f*

_{0}

*y*], which to quasigeostrophic accuracy is changed by

*ρ*

_{0}

*H*

_{0}

*δu*(

*y*) −

*ρ*

_{0}

*f*

_{0}

*yδh*(

*y*) when

*u*changes by

*δu*(

*y*) and

*h*by

*δh*(

*y*) so that, with

*Kelvin impulse*for the quasigeostrophic system. For its general conservation relation see, for example, Bühler and McIntyre (2005, section 8).

*δu*= −∂(

*δψ*)/∂

*y*, which integrates to zero by virtue of the evanescence of the PV inversion-operator kernel,

*K*

_{0}. That evanescence is reflected, for instance, in the exponential decay of the side tails of

*h*(

*y*) =

*f*

_{0}

*ψ*/

*g*associated with (5.2). The fact that all the angular momentum change is in the mass shift is one of the peculiarities of quasigeostrophic theory. So a check on the correctness of (5.2) is to multiply it by

*y*and integrate (including the side tails), whereupon one finds that all the terms in

*L*cancel, as they must for consistency with (7.2) and the fact that the N-shaped

_{D}*δq*(

*y*) profile is independent of

*L*. Specifically,

_{D}*y*and

*t*across the PV mixing event.

We note in passing that channel and *y*-periodic numerical beta-plane models may have artificial constraints on their absolute momentum, which may suppress or modify the globally integrated *δM* changes just discussed. However, such artificiality is a price often worth paying for numerical power and convenience.

## 8. Concluding remarks

The PV staircase, an extreme case of the inhomogeneity encouraged by the PV Phillips effect, can arise when stirring is strong provided also that it is accompanied by a suitable radiation-stress field.

One of the simplest examples of such a stress field is that described in connection with Fig. 1, where Rossby waves propagate upward on the jet axis, setting up a vertical Eliassen–Palm (EP) flux or form stress (A.2) (see note 1), and then refract and break to one or both sides of the jet causing PV mixing on its flanks. That is one way to satisfy the Taylor identity, (A.3) below, illustrating how breaking Rossby waves tend to be efficient at PV mixing and to produce the familiar antifrictional

On Jupiter, we may speculate that the eastward jets may similarly carry upward-propagating Rossby waves as a significant part of the whole wave–turbulence jigsaw, helping the stirring and PV mixing on either side to be efficient, and keeping the associated eddy-transport barriers, acting as Rossby waveguides, as tight as the observations suggest they are in reality. The eastward jets may well have significant roots in the thermally convecting interior, by virtue of the Taylor–Proudman effect (expanding Rossby height as the static stability evanesces beneath the stably stratified cloud layers). Jostling or nudging by large, ponderous convective eddies, beneath the clouds, could be a significant source of upward-propagating Rossby waves on those jets and could take the place of baroclinic instability (see note 4) as the excitation mechanism whose radiation stress catalyzes the PV mixing aloft.

Many terrestrial ocean jets can similarly be excited, and thus self-sharpen, even when not baroclinically unstable (e.g., Hughes 1996). This is because such jets often feel the bottom topography.

Complementing this picture are the barrier-penetration experiments of section 3, for established staircases, and the decaying-turbulence experiments of section 4 and Figs. 5 –6 to investigate the Phillips effect. Both are in their infancy, and we make no pretence to having done a definitive study in either case. We have done no more than describe a few examples in the hope of illustrating and pulling together some generic points that are more or less well known. The issues thus highlighted are, however, a challenge for future work. The results are suggestive, particularly the simple barrier-penetration threshold (3.3) that is found for the most effective vortices and vortex pairs, with core sizes on the order of the Rossby length *L _{D}*. In three-dimensional, fully stratified problems, such vortices correspond to those having core aspect ratios on the order of Prandtl’s ratio

*f*/

*N*, where

*N*is the buoyancy frequency of the stratification. It is precisely such vortices that are the most robust and stable according to a number of recent studies (e.g., Reinaud et al. 2003 and references therein).

In the experiments of section 4 we found a large range of cases from weakly to strongly inhomogeneous PV mixing, the latter illustrated by Fig. 6. Progress in understanding them is bound to depend on finding ways to analyze and quantify the Rossby wave radiation-stress field that must coexist with the decaying turbulence and catalyze PV mixing to varying degrees—another challenge for the future, especially as we extend the work prior to attempting realistic forced experiments that try to establish quasi-Jovian PV staircases.

*L*, along with stirring by the most effective vortices (core sizes ∼

_{D}*L*), the threshold (3.3) and the PV inversion results of section 5 together imply that the “turbulent” Rhines scale

_{D}*L*

_{Rh}(

*U*

_{vortex}) is related to the jet spacing or staircase step width 2

*b*by

## Acknowledgments

We thank the organizers for inviting us to such an interesting Chapman Conference, in which the problems of multiple jets and annular modes were looked at from so many different angles, for the various terrestrial and extraterrestrial cases. A number of colleagues have helped us with stimulating comments on this work and in other ways, in particular Michael Allison, Pavel Berloff, Sergey Danilov, Huw Davies, Timothy Dunkerton, Boris Galperin, Peter Haynes, Christopher Hughes, John Lewis, Nili Harnik, Philip Marcus, Nikolai Maximenko, Robb McDonald, George Platzman, Peter Rhines, John Rogers, Richard Scott, Sushil Shetty, Emily Shuckburgh, Roger Smith, Jürgen Theiss, Geoffrey Vallis, John M. Wallace, William R. Young, and the reviewers of this paper.

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## APPENDIX

### The Taylor Identity

*u*′,

*υ*′) = (−∂

*ψ*′/∂

*y*, ∂

*ψ*′/∂

*x*) and

*q*′ =

*ψ*′) = (∇

^{2}−

*L*

^{−2}

_{D})

*ψ*′, the quasigeostrophic shallow-water PV anomaly. The term in

*L*averages to zero. The original version noted and used by Taylor (1915) and Kuo (1951) was for the limit

_{D}*L*→ ∞, that is, for rigidly bounded, barotropic nondivergent vortex dynamics.

_{D}*y*, and vertical,

*z*, components, with the standard sign convention (such that flux directions correspond to Rossby wave group velocities when applicable), are

*θ*is the buoyancy acceleration,

*N*the buoyancy frequency,

*ρ*

_{0}(

*z*) a background density ∝ exp(−

*z*/

*H*

_{scale}), and

*f*

_{0}a constant nominal value of the Coriolis parameter as before. The vertical component

*G*is the form stress. Then the Taylor identity becomes

*u*′,

*υ*′,

*θ*′) = (−∂

*ψ*′/∂

*y*, ∂

*ψ*′/∂

*x*,

*f*

_{0}∂

*ψ*′/∂

*z*). The first-moment Eq. (7.2) for the absolute momentum or Kelvin impulse change continues to hold provided that, in the three-dimensional case, one replaces

*ρ*

_{0}

*H*

_{0}by

*ρ*

_{0}(

*z*)

*dz*and integrates vertically as well as meridionally across the zone of PV mixing.

Consistently with (A.3), *G*/*ρ*_{0} at a flat boundary representing an oceanic upper or atmospheric lower surface can be interpreted (Bretherton 1966a) as a boundary delta-function contribution to *G* is set to zero on the boundary. Since at the boundary *θ* is approximately a material invariant, it is subject to mixing in the same way as the PV above the boundary, with the same dynamical significance for the wave–turbulence jigsaw and crucial, for instance, to the workings of linear and nonlinear baroclinic instability including the upward launch of Rossby waves in the first nonlinear stage of an LC1 baroclinic wave life cycle. The elegant work of Plumb and Ferrari (2005) extends the above to cases outside the scope of standard quasigeostrophic theory, able to encompass steeply sloping stratification surfaces in, for instance, oceanic mixed layers and atmospheric boundary layers. The appropriate set of generalized Taylor identities are presented in their Eqs. (22)–(27).