## 1. Introduction

Prograde jets are ubiquitous in planetary fluid dynamics (Baldwin et al. 2007). These rotating stratified systems support Rossby waves, quasi-2D turbulent eddies, and their associated lateral transports of angular momentum and potential vorticity (PV), with overturning meridional circulations required to maintain large-scale balance. Various kinds of positive feedbacks exist between the waves, eddies, and mean flow, such that the prograde jets become self-reinforcing. The resulting “PV staircase” (McIntyre and Palmer 1983; Peltier and Stuhne 2002) is a natural end state of the planetary vorticity gradient itself, which may be regarded as an unstable equilibrium in the presence of Rossby waves and instabilities (Dunkerton and Scott 2008, hereafter DS’08).

While the atmospheres and oceans of planetary bodies in our solar system exhibit a rich variety of feedbacks and morphology, all are subject to a simple geometric constraint relating the strength and spacing of prograde jets (Dritschel and McIntyre 2008; DS’08). This constraint is enforced when the mean flow is, at worst, marginally stable to large-scale hydrodynamic instability, the constraint itself being entirely independent of the wave–mean flow phenomenology. Measured in units of a suitably defined “Rhines scale” depending on prograde jet maxima with speed *U*, the marginally stable spacing is given by

General staircases constructed entirely from stable segments, as might occur in a model fluid initially at rest, require larger meridional spacing of jets. In this sense, the asymptotic limit is relevant to stable atmospheres. Conversely, a strongly forced system, in which barotropic stabilization cannot keep pace with the forcing, may accommodate more jets than allowed by the asymptotic limit. The first case encompasses a class of numerical experiments in which the input of energy occurs randomly at small scales (Scott and Polvani 2007). With constant energy input the number of jets decreases very slowly in time, and marginal stability is avoided owing to these infrequent “mode transitions” involving lateral transport by transient eddies of larger scale. The second case might be imagined when dry convecting columns in a deep atmosphere attempt to organize jets in the statically stable weather layer (Heimpel et al. 2016).

In general, evaluation of marginal stability requires knowledge of thermal structure as well as of winds (Williams and Kelsall 2015). On Jupiter, for example, the cloud-top drift winds appear supercritical with respect to hydrodynamic instability considering the latitudinal profile of absolute vorticity per se, with significant reversals of zonal-mean meridional vorticity gradient (Ingersoll et al. 2004). But it is conceivable that the actual profile of PV is marginally stable in most places when the required thermal stratification is taken into account (Scott and Dunkerton 2017). The same point has been made for different reasons by Thomson and McIntyre (2016). Following DS’08, the barotropic model is reexamined here, but we anticipate that an extension to equivalent barotropic and shallow-water systems will be useful. The latter exhibits equatorial trapping of jets at small equivalent depth (Scott and Polvani 2007).

The jet strength–spacing marginal stability constraint derived by us and others several years ago was based on a beta-plane or equatorial approximation to the PV profile, which gave useful results for sphere-filling solutions, especially for the most likely value of spacing exponent (*p* = −1/4) having a flat lower envelope of retrograde parabolas instead of the “Roman arch” common to other values of *p* (DS’08, their Fig. 21). Further analytical progress on the full sphere was hindered by the complexity of the allowed jet profiles, which, although designed to be regular, led (via the angular momentum constraint) to a sum of various irrational integrals at second order that cannot be evaluated in closed form and a zeroth-order sum critical to the asymptotic analysis, for which the second-order difference between its discrete value and a continuous integral thereof must be known.

Since then, a few key steps have been navigated successfully, and it is the purpose of this paper to describe and exploit them (section 2). Of equal interest is our identification of a minimum kinetic energy state that exists in the parameter space bounded by interior and exterior coalescence points (borrowing from DS’08 terminology). That will be described herein also, as an illustration of the general concept of PV staircase “functional” (section 3).

## 2. Staircase model: Exact and asymptotic solutions

*U*:where

*j*th westerly jet and

*p*is a small constant assumed to lie in the range

*U*and a function of planetary vorticity gradient

*β*) divided by an arbitrary constant

*C*. For the regular spacing of westerly jets presupposed by (2.1), this constant is a function of the number of westerly jets spanning the sphere (or “mode index”

*n*) and spacing exponent

*p*. It was shown asymptotically in DS’08 that

To put into context the discussion that follows, it is helpful to recall the details of the simple asymptotic method outlined in DS’08. This method begins with a sphere-filling modal solution and gradually reduces the latitudinal extent of the staircase to zero. In this “equatorially trapped” limit, the latitudinal spacing of westerly jets and the central latitudes of these jets are reduced systematically to zero, along with *C* assumes an asymptotic value that is a function of mode number *n*. By taking a second limit

Here we present a different asymptotic method based on the requirement *n*. Interior and exterior branches of the sphere-filling modal solutions were discussed extensively in DS’08. The so-called interior solutions distribute westerlies and easterlies more or less uniformly across the sphere, with westerlies or weak easterlies near the pole, while exterior solutions have a strong, broad easterly jet in high latitudes and a prevalence of westerlies at low to midlatitudes. The latter solutions require long-range transport of angular momentum in their formation, between low and high latitudes, whereas the former solutions imply a more local redistribution in latitude. The exterior solutions suffer two additional disadvantages making their realization in planetary atmospheres less likely (as noted below) owing to (i) the bimodal phase-speed spectrum of disturbances that might give rise to such solutions and (ii) the high values of kinetic energy required relative to the interior solutions. With these lessons behind us, attention is now directed at a subset of the *U* has a local maximum, and (ii) the minimum value of *U* along this contour, here referred to as the exterior coalescence point, where the exterior asymptotic branch intersects. {The interior asymptotic branch shown in DS’08 corresponds to a collection of tangent points *μ*_{p} ∈ (0, 1]. The exterior asymptotic branch (not shown) is defined similarly by tangent points

### a. Revised strategy

*j*th prograde westerly jet and

*N*= (

*n*+ 1)/2 is the number of complete jets in a single hemisphere: see (2.31) and Fig. 4 of that paper. For

*n*even, the equatorial jet divided equally between hemispheres is not counted as a complete jet so as to maintain an integer value of

*N*. Asymptotic constructions for even and odd

*n*are then considered separately. A modification was introduced by DS’08, in (2.41) of that paper, to accommodate a polar region lying formally outside the staircase:for small

*U*were substituted for their exact functional dependence on

*μ*. This substitution is appropriate for the asymptotic method of DS’08, wherein

*μ*—more so as

*n*increases to infinity. Further discussion highlighting a few differences between equatorially trapped and sphere-filling asymptotics is provided in the appendix.

*N*we introduce a new definition [distinct from that of DS’08 given by (2.4a)] of midpoint

*μ*spacings associated with the westerly jets at

*N*, while the

*μ*) Taylor series may be employed, without catastrophic loss of accuracy at any latitude adjacent to the poles, beginning from the exact formulasandwhich may be combined and written (without approximation) asandwhereTerms involving

*m*part, representing piecewise linear segments of angular momentum, and a

*m*part at the most polar jet iswhere

*p*considered in DS’08. As for the Taylor series: without the factor

*ξ*(or with

*p*investigated and, similarly, the analogous expansion of the

*m*part for which the equivalent value of

*p*, the spacing between westerly jets is

*j*-discretized conservation law in (2.18). It is necessary also to retain the

*m*profile (leftmost term in sum). The second collection of terms on the lhs (in brackets) contains only one term at second order in

### b. Jet strength–spacing relationship, n = 2–23

*n*. The magnitude of westerly jets decreases with increasing mode number, approximately as

*p*extending from −0.25 (red) to +0.25 (blue) are shown, with red curves overplotted last in order to highlight what is ostensibly the most realistic value of

*p*obtained in our previous numerical results. Filled circles in Fig. 2 indicate (i) a few examples of the interior coalescence point for

*C*is minimum and the latter, by definition, where

*U*is minimum. It can be seen that the interior coalescence point converges to approximately

*n*while the exterior coalescence point converges to a larger value near

Figure 3 shows the corresponding values of near-equatorial spacing *U*, this quantity continues to decrease with increasing *C* beyond the exterior coalescence point. [The geometric constraint therefore does not apply literally to solutions in the right half of the diagram when defined in terms of the global *U*. Nevertheless, the constraint holds locally, for the same reason noted by Dritschel and McIntyre (2008): a neutral parabola, even if not retrograde absolutely, still requires enough “room” meridionally to fit between adjacent prograde jets when measured relative to a prograde reference line.] As mode index *n* become large, the logarithmic spacing between curves asymptotically approaches to a constant, as it must to accommodate the increase of *n*, noting that *O*(*e*^{−4}) for the smallest *n* (=2, 3) to *O*(*e*^{−12}) for the largest *n* (=22, 23) considered.

### c. Asymptotic analysis, p = 0

*μ*. Consideration of nonzero

*p*is deferred to the next subsection. For

*U*is

*dμ*= constant) are inconsequential at second order. When

*U*which is second order in

### d. Asymptotic analysis, p ≠ 0

*p*: (i) the spacing of jets is nonuniform in

*μ*and (ii) the summation contains irrational integrals of fractional powers of

*N*in the limit

*p*considered here, and with the latter bound, all integrals exist when

*O*(1) to begin with, because the error of this integral’s discrete evaluation must be known. When

*N*/12 on the rhs. [The arithmetic sum (2.21) can be recovered for nonzero

*p*via a transformed coordinate

With no analytic answer to this question, the alternative (a numerical method) is tricky since (i) the “exact” value of the integral is known only up to machine precision and (ii) its discretized “staircase value” is to be differenced against this exact value, then multiplied by *O*(1). We are expected, then, to take the difference between two floating point numbers at suitably high precision, the difference becoming smaller with increasing *n*, and then multiply the difference by a large number to arrive at another number in the vicinity of 1/12. As shown next, this method actually works (with double precision) to the highest *n* considered (=22, 23) for all negative *p* considered and for *p* slightly above zero. It should be noted that, while the exact value is a geometric entity (viz., the area beneath a curve) independent of its computation, its discrete approximation always depends on the details of the spacing—that is, the definition of integrand increments. In planetary FD terms, it is the piecewise constant PV staircase—a conservative rearrangement of continuous planetary vorticity linear in *μ*—that discretizes the summation and precludes Newtonian calculus from solving the problem entirely.

Shown in Fig. 4 (left) are estimates of the fraction *p*. At *p*, the values are higher by a much as a factor of 2. The sloping curves are ambiguous, especially for small positive *p*, which bend down sharply to the right. However, the most extreme values (at right) occur at the upper end of the interior branch, where *p*. We are not interested in this point for the purpose of the asymptotic analysis because the relevant values of *C* lie in the neighborhood of interior and exterior coalescence points. (Please recall that “sphere filling” refers to *C* inferred by setting (2.24) identically to zero and solving for *C*, but using discretized approximations to the integrals, with increments matching the steps of the staircase. The inferred values are shown against their actual values, and the error is zero along the diagonal line. As expected, errors diminish with increasing *n*, but for negative *p*, the errors are compact. The two cases of special interest,

Agreement with the equatorial asymptotics *n*. The equatorial limit corresponds to *C*. By contrast, (ii) the asymptote for *n* shown, the interior and exterior coalescence points are distinct and remain separated by a finite span of *C* even as their respective *U* minima converge logarithmically to zero. The fraction *p* and *C* agree best are indicated by vertical bars in Fig. 4a.

A possible explanation of the former result (i) lies in the isomorphism between the beta-plane and sphere-filling solutions, both of which have a flat bottom envelope: that is, identical retrograde parabolas. In a stretched Mercator-like coordinate, the sphere-filling solution for *y*. To appreciate the subtle variations of spacing in the DS’08 model it is eye opening to consider two values of *p* outside the range of interest: *θ* such that *η* such that *μ* but neither is germane to the staircase. The Mercator coordinate was designed for a two-manifold or “chart” of Earth’s surface so as to preserve horizontal isotropy. [For application of the Mercator coordinate to 2D Rossby wave propagation on a spherical surface, see Hoskins and Karoly (1981).] As defined for zonally averaged systems, the PV staircase is one dimensional. Nor is latitude useful for equal spacing, as it suffers contraction of area approaching the poles; the underlying conservations of angular momentum, potential vorticity, etc. are described more naturally in *μ* once the zonal average is taken. The *η* coordinate is grossly stretched whereas when *ν* coordinate is weakly stretched.

*j*)upon substitution of (5.6) of DM’08, setting

*b*is the half-width of jet spacing,

*a*are planetary rotation rate and radius in the usual notation. Approaching the pole, with

*θ*(or decrease slowly in

*μ*) while the zonal wind structure of the staircase (daisy chain of parabolas) remains isomorphic to the beta plane. These slow variations accompany ever-contracting parabolas in the limit

Needless to say, the “periodic extension” of the local PV-mixing event envisaged by DM’08 is required. Their solution so extended (5.6) contains the square root of 6 (or of 3) implied by the equatorial asymptotics, while their single-event solution in (5.5) does not. Local tangency is the essence of the beta plane and exactly what is implied in the limit

### e. Global kinetic energy and mode transition

Global KE is frame relative and requires definition of a resting state, which is problematic on the gas giants. This quantity, as it were, minimizes the moment of inertia of zonal wind about its zero reference line, when such exists. Sharp minima occur in Fig. 5 where prograde and retrograde maxima differ least from their global average. For this reason, owing to its flat bottom envelope, *C*. Such differences on Jupiter and Saturn likely arise from outcropping tangent cylinders of the respective metallic cores.)

Global kinetic energy is a PV staircase functional: a scalar measure comprising (possibly many) discrete states of local steps/risers in the staircase. A simpler functional is global angular momentum, equivalent to contours of

A leftward trajectory in the *n* and traveling upward through the interior coalescence point toward the end of the interior branch. Two types of trajectories are portrayed in Fig. 6. (i) One preserves hemispheric symmetry by linking up with the next solution trajectory at *n* to (*n* − 1), implying a near-equatorial alteration between super and subrotating states. [Such alternation is the hallmark of the quasi-biennial oscillation (QBO) of the equatorial lower stratosphere; see Dunkerton (2017) and references therein. In the QBO, vertical transports of angular momentum play an important and probably essential role.] According to Fig. 5, this type of trajectory is almost seamless at *p*, the gap is larger. The symmetry flip is quite obvious at the equator, where a new jet comes or goes. Contrary to our first impression of “waves on a choppy sea”—unlikely realizable—close inspection indicates that midlatitude jets require only a small meridional displacement during the transition.

Spontaneous jet formation in a retrograde parabola was demonstrated by Scott and Tissier (2012), a time reversal of the near-equatorial behavior in Fig. 6d, as it were. Intriguing as such trajectories are, with simple morphology in physical space, they have otherwise not yet been simulated numerically. With energy input, an alternative and seemingly preferred pathway is for adjacent prograde jets to come closer together, albeit very slowly, and eventually to merge (Scott and Polvani 2007). In this way the global staircase configuration accommodates a continuous random input of energy at high horizontal wavenumber. It is not surprising that merger depends on initial proximity, if indeed that is the lesson taught by these experiments. Long-range interaction (e.g., via Rossby waves and instabilities) is implied by the simulated merger (DS’08). The shorter the range, the faster the interaction? Nevertheless, we find the state of understanding unsatisfactory at present. Somewhat bothersome is that, in light of comments regarding global KE, any irregular spacing of jets is not a minimum energy or “ground state” of the system. Should not a natural system find itself in such a ground state eventually, if not initially?

Further numerical study is needed to address this question and to explore the two preferred types of solution trajectory identified in Figs. 5 and 6. Whether or not the examples of Fig. 6 are realizable numerically, they provide a compact depiction of possible regular staircases in the most relevant range of *C*. In lieu of global KE, a better diagnostic approach is to exploit objective Lagrangian methods (Rutherford and Dunkerton 2017, manuscript submitted to *J. Atmos. Sci.*). Eulerian kinetic energy is not objective and therefore is not optimal for theoretical analysis of energy–momentum states in fluid continua. Indeed, the physical transitions depicted in Fig. 6 (and their respective inserts showing KE) require conversions between eddy and mean energies, or equivalently, conservation of Lagrangian metrics such as pseudoenergy and Kelvin’s circulation (Andrews and McIntyre 1978a,b; Dunkerton 1980).

## 3. Conclusions

For barotropic flow in spherical geometry, the ideal potential vorticity staircase with flat steps and vertical risers exhibits a relationship between prograde jet strength and spacing such that, for regular spacing, the distance between adjacent jets is given by a suitably defined “Rhines scale” multiplied by a positive constant equal to

The staircase model of DS’08 consists of equal prograde (westerly) jets bounding parabolas of variable depth, which may be easterly or westerly at their core. In sphere-filling solutions, a small gap is allowed between the last prograde jet and the pole. Unlike the equatorial asymptotic method of DS’08, which confined the staircase proper to an ever-smaller range of latitudes, the sphere-filling asymptotic method presented herein maintains a small gap near the pole. For the largest negative value of exponent *p*.

The concept of PV staircase functional is introduced, featuring global kinetic energy and angular momentum as examples. Global KE exhibits a deep minimum between the interior and exterior coalescence points, the latter terminology borrowed from DS’08. Four examples of mode-*n* transitions are presented for a system with monotonic energy input but with angular momentum held constant. In these examples the number of jets decreases slowly with time, but migrating poleward. Such behavior is very different from simulated transitions in which adjacent jets merge on a very long time scale.

As shown by the author at the 2015 Atmospheric and Oceanic Fluid Dynamics (AOFD; Dunkerton 2015) meeting of the AMS, any perturbation to

Finally, it is suggested that the asymptotic method, which is not trivial numerically, may be useful nevertheless to explore this question and others that arise along the solution trajectory bounded below by the exterior coalescence point, and the interior branch, through the interior coalescence point to its end where the identified mode transitions occur. The utility of the method is not so much that one can derive results valid for arbitrarily high mode index *n*, but that the asymptotic results are quantitatively useful for *n* of *O*(10–20) as observed in planetary atmospheres nearby.

## Acknowledgments

TJD is supported by the National Science Foundation.

## APPENDIX

### Further Details of the Sphere-Filling Method

#### a. Differences with respect to equatorially trapped asymptotics

The asymptotic analysis of DS’08 requires that the latitudinal extent of the staircase shrink to zero; that is, *y* axis linear in *U*, rather than logarithmic as in this paper. The above limit is tantamount to *n* and increasingly independent of *p*. For all values of *p* considered, the limiting value of *C* is less than its value at the interior coalescence point (but only slightly so when *C* independent of *U*, that is, vertical lines when the *y* axis is logarithmic in *U*.

The logarithmic display gives the illusion of a contradiction with DS’08, which is resolved easily by noting that for the sphere-filling results shown here, *n*. Even though

For the purpose of functional analysis, it is helpful to keep in mind that solution trajectories *p* to second order in

#### b. Absolute vorticity, relative zonal wind, and kinetic energy

*ζ*

_{a}) segments of the staircase, derived from the piecewise linear angular momentum (

*m*) segments of the corresponding “necklace of bamboo,” areor, in dimensional terms,in the usual meteorological notation, and

*μ*, the center position

*m*is concave downward, the small zonal wind parabolas are concave upward, as shown in Fig. 22 of DS’08. The discussion after (2.57a) in that paper invoked some equatorial asymptotics (needlessly, as it turns out) and the desired sphere-filling alternative [identical in essence to (A.1a)] iswithout approximation, where

*m*deviation is evaluated about

*m*between westerly jets is exactly linear in

*μ*, so the value of

*m*at

*m*on the stair step. [The linear deviation averages to zero in each segment, explaining why the average stair step

*m*suffices for the global summation (2.2) and (2.18).] The last batch of terms contains the

*resting*profile of

*m*spanning this interval: parabolic in

*μ*itself.

*m*deviation is second order in

*n*contouring

*m*deviation with comparable maximum (westerly and easterly) amplitudes over the range of

*C*considered by DS’08. But it was shown in Fig. 5 that the kinetic energy (KE) associated with these profiles varies greatly within the same mode, affording guidance for how staircases may evolve in realistic planetary atmospheres. In the sphere-filling asymptotics, each quasi-parabolic segment of relative mean zonal wind contributes a finite meridional increment of kinetic energy as measured in the resting planetary frame:As it turns out, a global minimum KE state exists between the interior and exterior coalescence points (Fig. 5).

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