a. Revised strategy

As in DS’08 the angular momentum–conserving staircase assumes the simple form

where is the absolute angular momentum of the jth 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 , as allowed by the equatorially trapped asymptotics, where

In the above equations, Taylor series expansions (in ) of (i) planetary angular momentum and (ii) the moment arm for relative flow U were substituted for their exact functional dependence on μ. This substitution is appropriate for the asymptotic method of DS’08, wherein , but it is unlikely to suffice for a revised method that requires a sphere-filling solution since at least some of the westerly jets then lie at finite 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.

For sphere-filling asymptotics at large N we introduce a new definition [distinct from that of DS’08 given by (2.4a)] of midpoint :

such that

The angular momentum and μ spacings associated with the westerly jets at and are then

and

Although the previous (DS’08) definition of was more accurate for the purpose of approximating terms involving —note that did not appear in that formulation—this loss of accuracy is inconsequential for the sphere-filling asymptotics because we no longer impose small (as in the equatorially trapped asymptotics) but expect rather that the alone become small in the limit of large N, while the [and ] remain finite. For the sphere-filling asymptotics, local (in μ) Taylor series may be employed, without catastrophic loss of accuracy at any latitude adjacent to the poles, beginning from the exact formulas

and

which may be combined and written (without approximation) as

and

where

Terms involving , whose values span the semiopen interval [0, 1), are retained exactly (without approximation of small ) while a local Taylor series expansion is applied; namely,

The efficiency of the local expansion is apparent, noting that

the lhs of (2.14a) being generated by the local expansion and subsequent combination of terms inside the square brackets in (2.11a) and (2.11b). The sum of higher powers is

As in (2.14a), the sum of odd powers is relegated to one higher order in .

Some care is warranted evaluating the staircase model and its Taylor series representation approaching the poles . The model is the discrete sum of a product of two parts: an m part, representing piecewise linear segments of angular momentum, and a part, representing jet spacing. Convergence of the model does not guarantee convergence of a Taylor series representation of it. Regarding the model parts, the relative m part at the most polar jet is

where is a small gap lying between the formal end of the staircase and its most polar prograde jet (Fig. 4 of DS’08). The jet spacing assigned to this last jet is

which may be written

where and . So the model itself is well posed in the limit for the range of p considered in DS’08. As for the Taylor series: without the factor ξ (or with ) the series (2.13) diverges (converges) for positive (negative) . However, the series converges in either case when , since

The expression on the lhs arises from the leading term on the upper rhs of (2.14b) and ensures that the polar singularity of inverse powers of is kept under control by the jet spacing with their ratio placed in a geometric series via (2.17). In this way, our small gap formally outside the staircase ensures convergence of the Taylor series in the range of p investigated and, similarly, the analogous expansion of the m part for which the equivalent value of .

With these considerations in mind, the sphere-filling asymptotics become

for conservation of angular momentum and

for staircase extent. Aside from small variations of spacing owing to a nonzero exponent p, the spacing between westerly jets is and, to be consistent, the strength of westerly jets is , requiring that the correction to the average angular momentum between jets be retained in the j-discretized conservation law in (2.18). It is necessary also to retain the corrections to jet spacing multiplying the resting m profile (leftmost term in sum). The second collection of terms on the lhs (in brackets) contains only one term at second order in , which is needed nevertheless to balance the planetary contribution to angular momentum between and . Equivalently, it can be shown that the bracketed quantity may be ignored and the rhs replaced by

simplifying the asymptotic analysis and ensuring it is independent (to second order) of the model gap outside .

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

Exact solutions contouring in the plane are shown in Fig. 2, extending the results of DS’08 to higher-mode indices n. The magnitude of westerly jets decreases with increasing mode number, approximately as over the range shown as noted previously by DS’08. Values of 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 , where the equatorial asymptotic branch of interior solution

intersects the contour, and (ii) a single example of the exterior coalescence point for , where the equatorial asymptotic branch of exterior solution

intersects the contour. The former intersection, by definition, occurs where C is minimum and the latter, by definition, where U is minimum. It can be seen that the interior coalescence point converges to approximately (red vertical line) for large n while the exterior coalescence point converges to a larger value near (yellow vertical line) predicted by the sphere-filling asymptotics (next subsection).

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Solution trajectories with for jet-spacing parameter C and prograde jet magnitude U (both axes are logarithmic). Mode index n increases from top to bottom. A modest range of p values is shown, corresponding to DS’08, divided among three panels for clarity. Vertical lines denote asymptotic values of (red) and (yellow) relevant to and , respectively. Red and yellow filled circles indicate examples of interior and exterior coalescence point.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0090.1

Figure 3 shows the corresponding values of near-equatorial spacing in the same format. Unlike 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 . Detailed diagnosis of each term on the lhs of (2.18) via the numerical method was done to evaluate the magnitude of error in the Taylor series expansion of total absolute angular momentum in the staircase (rhs). The results (not shown) indicate errors ranging from O(e⁻⁴) for the smallest n (=2, 3) to O(e⁻¹²) for the largest n (=22, 23) considered.

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As in Fig. 2, but for the square of near-equatorial jet spacing .

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0090.1

c. Asymptotic analysis, p = 0

The sphere-filling asymptotic analysis is relatively easiest for because the spacing of westerly jets is then constant in μ. Consideration of nonzero p is deferred to the next subsection. For and , (2.18), (2.19), and their dependencies reduce to

Noting that

implies, using (2.20) and (2.21), that

and upon equating powers of ,

Substitution of the continuous integral (equal to one-fourth of the unit circle’s area, ) is permissible since (i) U is already and (ii) small deviations of , if any, from regular spacing of integrand increments (implicit in the Newtonian calculus dμ = constant) are inconsequential at second order. When there are, in fact, no deviations from regular spacing, and one might ask why the same calculus cannot be applied to the summation of farther to the left. The answer is that this term is , unlike the integral containing U which is second order in , and it is therefore necessary to include a second-order correction representing the difference between the integral of and its discrete sum (this nuance is easily overlooked). The contribution represented by is not small relative to other terms of the same order on the lhs. Zeroth-order terms on the lhs cancel the rhs exactly, while the sum of second-order terms on the lhs (which must equal zero identically when ) yields . It will be noted that this formulation of the matched asymptotics is similar to that of DS’08 but without invoking small anywhere in the derivation.

d. Asymptotic analysis, p ≠ 0

Two complications arise for nonzero p: (i) the spacing of jets is nonuniform in μ and (ii) the summation contains irrational integrals of fractional powers of . Some effort is required to identify these integrals and all must be evaluated numerically (or if discovered in the table of integrals, would involve irrational constants and functions evaluated numerically by someone else). It proves convenient to rewrite (2.18) as a second-order summation error:

discarding the outer part and defining

recalling once again that as in (2.23b). The lhs is second order because a single power of is balanced by N in the limit . For the half-width of jet spacing, from (2.19),

The most important insight in (2.24) is that Newtonian calculus is invoked to replace discrete sums with their continuous integral equivalent, as done for one term in the case [see (2.23b)] because for these integrals only, the difference between the discrete sum and integral is small and contributes to the next (third) order of error. [For the same reason it makes no difference whether the upper bound of continuous integrals in (2.25) is or , although does not exist with the former bound when and . The cases and exist when in the entire range of p considered here, and with the latter bound, all integrals exist when .] Just as before, this strategy must be avoided for the first term in (2.24), which is O(1) to begin with, because the error of this integral’s discrete evaluation must be known. When the error is known exactly from the arithmetic sum of in (2.21): namely, the term −N/12 on the rhs. [The arithmetic sum (2.21) can be recovered for nonzero p via a transformed coordinate defined to enforce equal spacing. However, each term in the sum is then weighted by a variable amount.] What is the equivalent term when ?

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 to be included among the integral terms in (2.24), which are 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 for nonzero p. At the fraction 1/12 is predicted correctly by the numerical method in the limit , while for negative 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 comes closest to the pole for any 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 , while in the limit .) In the right panel of Fig. 4 are shown values of 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, and , agree with their asymptotic values, albeit with small uncertainties.

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(a) Estimates of effective fraction for various p, corresponding to the arithmetic result [1/12 in (2.21)] valid when (horizontal dashed line). (b) Actual vs inferred numerical values of C obtained from the asymptotic method upon setting the second-order error [see (2.24)] identically to zero. Values of corresponding to the best agreement are indicated along the right edge of (a).

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0090.1

Agreement with the equatorial asymptotics at is consistent with the ascent of the equatorial branch upward to the interior coalescence point (see DS’08, their Figs. 8 and 9), the ascent becoming nearly or exactly vertical at large n. The equatorial limit corresponds to while the sphere-filling point of intersection is directly above that (when ) or displaced to the right (when ). If it can be shown that the equatorial branch for is exactly vertical in the limit , the square root of 6 then assumes a new role as the spacing requirement for the sphere-filling solution at the interior coalescence point. All indications suggest so. Indeed, the impression given by Fig. 2 is that (i) the solution trajectory is tangent to the red vertical line at this special value of C. By contrast, (ii) the asymptote for (yellow vertical line) intersects the solution trajectory in perpendicular fashion where it is almost flat. For the range of high 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 varies along the corner of the solution trajectory because it depends on the jet spacing, which also varies along the trajectory even when p and are fixed. Ranges of values for which the actual and inferred 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 in the limit resembles a beta-plane solution extending progressively to larger 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: (latitude θ such that ) and (Mercator coordinate η such that ). Both are stretched coordinates relative to 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 the ν coordinate is weakly stretched.

Obviously a true Mercator coordinate is not needed to explain the isomorphism of the sphere-filling solution to the meridionally periodic beta plane of Dritschel and McIntyre (2008, hereafter DM’08); fortunately, a simpler line of reasoning suffices. This argument is astonishingly direct and agrees with the equatorial asymptotics also. The beta plane, whether at the equator or anywhere else, is a tangent approximation for the Coriolis parameter at a certain latitude. According to (2.1), the special value and conjecture imply that (dropping subscript j)

upon substitution of (5.6) of DM’08, setting , where b is the half-width of jet spacing, , , and and a are planetary rotation rate and radius in the usual notation. Approaching the pole, with , (dimensional) jet spacing is allowed to increase slowly in 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 , so the daisy chain is endless asymptotically (DM’08) even though the actual distance from equator to pole is finite.

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 . By the same reasoning, the equatorial asymptotic of DS’08 is a special case of beta-plane thinking, the only complication being that , instead of , plays the role of “midpoint,” as might be expected for an equatorial beta plane.

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, has the least KE of all in the left half of the diagram. Wind profiles in the right half of the plane have smaller prograde jet spacing than the asymptotic value but at the expense of much larger KE, attributable not to the local neighborhood, but to the enormous difference between interior westerlies and exterior easterlies. Such profiles resemble the cartoon character Dilbert, although the “jets” of his hairdo are retrograde. (Caricature aside, gas giants in our solar system do exhibit large differences in tropical versus extratropical jet magnitude, though lacking the contrast of interior and exterior winds at large C. Such differences on Jupiter and Saturn likely arise from outcropping tangent cylinders of the respective metallic cores.)

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As in Fig. 2, but for global kinetic energy.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0090.1

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 in the interval (0, 1]. Sphere-filling solutions are exclusively at . Such contours correspond in the above terminology to solution trajectories in the plane. States may evolve along such trajectories while conserving global angular momentum but not global kinetic energy. The parameter space between interior and exterior coalescence points is interesting because a sharp minimum of KE exists in this region. In numerical solutions forced by random inputs of energy but not angular momentum at high horizontal wavenumber, a system state with regular-spacing constraint follows a trajectory to higher KE until such a state is no longer realizable or unrealistic for other reasons.

A leftward trajectory in the ) plane is realistic—for example, departing the KE minimum for a particular 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 . For , the intersection is exact, occurring at the exterior coalescence point. For an energy impulse is required at the transition (see insert). In either case the transition takes the most poleward prograde jet closer to the pole until the next lower (even or odd) mode is formed. The equatorial state is unaffected. (ii) A second type (bottom panels) requires symmetry flip, from 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 , with a tiny gap in KE values. For other values of 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.

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Mode-n transitions obtained by following solution trajectories from the exterior coalescence point, leftward and upward along the interior branch until coming in close proximity to the next lower mode (a),(b) or (c),(d) . The latter require a symmetry flip, most obvious at the equator, where transitions between super- and subrotating states occur. Insert in lower-left corner of each panel shows the evolution of global KE from front to back. The scaling and direction of “time” are completely arbitrary, and the rate may vary.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0090.1

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).