Jupiter-style Jet Stability

Studies of Jupiter’s zonal jets, facilitated by the two Voyager flybys, the Galileo entry probe, the Cassini flyby, and the Juno orbiter have led to two fundamental insights into inviscid shear stability that have implications for a wide range of large-scale fluid systems involving alternating shear instability. The first insight is that the analog of the Mach number for vorticity (Rossby and drift) waves, “Ma,” ensures shear stability via the criterion “Ma”−1 < 1, which includes both Kelvin–Arnol’d branches of stability, and is edged with a shock. A surprise is the well-studied first branch (KA-I), which includes as special cases the textbook shear stability theorems of Rayleigh, Kuo, Charney–Stern, and Fjørtoft, merely corresponds to “Ma”−1 < 0. The second insight is that Jupiter’s tropospheric jets achieve stability via a second branch (KA-II) strategy, a 3/4 layer, undulating control surface supplied by the dynamic topography of the planet’s deep jets, which maintains “Ma”−1 ≲ 1 via stretching vorticity. The deep jets are similarly stabilized by the spherical shape of the planet itself. Although Jupiter-style zonal jet stabilization is precluded by the torus geometry used in hot-plasma fusion reactors, it is directly applicable to the tube with ends geometry used in cool-plasma applications, including antimatter storage at high-energy colliders. In general, the lessons learned from analyzing Jupiter’s jets eliminate much of the guesswork from predicting and controlling inviscid shear instability.


Introduction
Inviscid shear instability is a classical fluids problem that is central to the dynamics of a wide range of large-scale, nonlinear fluid systems in science and engineering, including the meandering of jet streams in atmospheres and oceans (Dowling 2019), the formation of planets in proto-planetary disks (Manger & Klahr 2018), and the enhancement of magnetic containment of hot plasmas in fusion reactors (Diamond et al. 2005). Despite this crossing of disciplines, shear instability has yielded its secrets slowly, to the point of being called out by science historians, to wit: "Apart from the Helmholtz-Kelvin instability and Rayleigh's inflection theorem, the theoretical yield was rather modest" (Darrigol 2005, p. 217).
This article summarizes two lessons about zonal jets that have emerged from analysis of Voyager vorticity data from Jupiter's jets and vortices, which have been upheld by recent Juno and Cassini observations. Jupiter's first lesson is that once the two branches of inviscid shear stability theory are properly nondimensionalized-in hindsight, an obvious step that nevertheless was neglected throughout the twentieth century-the necessary and sufficient conditions for inviscid shear instability become apparent. The second lesson is that any alternating zonal jet profile can be made stable or unstable via boundary control of stretching vorticity.

Mach Number Portfolio
Nondimensional reasoning is standard across fluid dynamics and brings clarity to any discussion, often by reducing competing ideas to "0 versus 1." Nevertheless, there are two topics related to zonal jets for which nondimensionalization is underutilized, and which turn out to be two sides of the same coin. The first is the set of established inviscid shear stability theorems, which from the time of the Rayleigh (1880) inflection-point theorem have traditionally been expressed in dimensional form only. The second is a member of the nondimensional Mach number portfolio that is conspicuous by its absence-namely the analog of the Mach number for vorticity waves, or Rossby waves, which are analogous to drift waves in magnetized plasmas (Hasegawa & Mima 1978).
The Mach number portfolio has been the workhorse for fluid mechanics since the nineteenth century, in large measure because it provides the bookkeeping for upstream information propagation. Imagine working with compressible flow, which includes all of aeronautics, without the original Mach number, Ma, the ratio of flow speed to the upstream speed of sound waves. Likewise, imagine working with hydraulic flow without the analogous Froude number, Fr, the ratio of flow speed to the upstream speed of buoyancy waves, i.e., the waves on a lake, but also the slow, internal bouncing of a stratified fluid in a gravity field (thus often called internal waves or gravity waves). Even as simple an act as running water in a sink reveals an ever-present ring, a hydraulic jump-the shock of water waves -located at Fr=1, which is a common manifestation of information propagation that is every bit as dramatic as the event horizon of a black hole (Afanasyev & Ivanov 2019 ). In what follows, the "prodigal" Mach number for vorticity waves is denoted "Ma" in quotes and likewise "supersonic" and "subsonic" refer to the vorticity wave analogs.

Charney-Obukhov/Hasegawa-Mima Equations
In geophysical fluid dynamics, for adiabatic motions the conservation laws of mass, momentum, and thermal energy are combined into a single law, the conservation of potential vorticity, hereafter PV, with symbol q (and sometimes called "vortensity" in the astrophysical literature). The most tractable case is when both mass and momentum are expressible in terms of a single variable, which reduces the system to one equation and one unknown-the point of "balanced flow" theories (Vallis 2017). This coupling is accomplished via the Coriolis acceleration in rapidly rotating geophysical flow and via the magnetic field in a plasma tube.

Conservation of PV
For vortical systems, the mass and momentum are expressed in terms of a scalar stream function, ψ, such that the zonal (east-west) and meridional (north-south) wind components are (u−α)=−ψ y and v=ψ x , respectively (the sign convention is sometimes reversed in the engineering literature), where α is an arbitrary Galilean shift, x and y are east-west and northsouth coordinates, and subscripts denote partial differentiation. Conservation of PV is then dq/dt=0, where xx yy 0 0 2 and for horizontal motions, There are three types of vorticity combined together to make PV in Equation (1). The first is planetary vorticity, , the leading terms in its Taylor series. The second is relative vorticity (local-vertical component of the curl of the velocity as measured in the rotating frame), expressed as ψ xx +ψ yy . The third is stretching vorticity, the rightmost term in Equation (1), where g is gravitational acceleration, r á ñ and á ñ h are reference profiles of regular and thickness density, respectively, and the vertical coordinate is potential temperature, θ, an isentropic variable along whose isosurfaces adiabatic and reversible motions occur (Vallis 2017).

Stretching Vorticity
The key to how Jupiter's zonal jets are stabilized is stretching vorticity. A 2 layer model, in the argot a "1-3/4" layer model, is sufficient to demonstrate the mechanism. On top is a weather layer that models the troposphere and is free to evolve. In the planetary literature, a steady deep layer has traditionally been called a "1/2" layer whether or not it contains jets, but it has recently been rebranded as a "3/4" layer when it contains jets to emphasize the major effect the deep jets have on the PV in the weather layer via stretching (Flierl et al. 2019). In the limit of infinite depth, the deep jets are not affected by the weather layer jets (Majda & Wang 2005). If the deep jets are stable, they provide a steady-in-time but undulating-in-latitude bottom boundary condition on the weather layer, via the highs and lows associated with their anticyclonic and cyclonic shears, respectively. On Jupiter, the interface between these layers roughly corresponds to the moist/dry transition just below the water clouds, which span the 5000-8000 hPa (5-8 bar) pressure region, or two pressure scale heights below the surface ammonia clouds at 700 hPa.
Quasi-geostrophic PV in the weather layer takes the form (Ingersoll & Cuong 1981) where stretching vorticity is expressed in terms of the first baroclinic deformation length, L d , and the stream function associated with the deep jets,¯( ) y y deep . The latter is the 3/4 layer stream function. In the oceanographic literature, it is called dynamic topography instead of bottom topography to distinguish it from seamounts. The value of L d "is estimated to be ∼2000 km in Jupiter's troposphere at mid latitudes, with both the uncertainty and the natural variability probably a factor of two in each direction" (Ingersoll et al. 2004, p. 120).
Expressing dq/dt=0 solely in terms of ψ via Equation (2) applied to Equation (1) or (3) yields alternate versions of the Charney-Obukhov equation. An established and yet alwaysstriking equivalence is that a tube of plasma subject to a magnetic field is governed by similar mathematical terms, called the Hasegawa-Mima equation in plasma physics (Hasegawa & Mima 1978), where ψ is the perturbed electric potential, typically written as f, and L d is the Larmor radius, typically used to nondimensionalize lengths (Hasegawa 1985).

The "Mach" Number
Sound waves and buoyancy waves are both filtered out of quasi-geostrophic systems (Vallis 2017), hence the Mach and Froude numbers do not figure directly into shear instability. However, vorticity waves run rampant in such systems. From the nineteenth century point of view, it is therefore surprising, and telling, that "Ma" has not been in the foreground in vortical flow discussions to date.
Vorticity waves are called Rossby waves in geophysical fluid dynamics, and the analogous waves in plasma physics are called drift waves (Estrada 2019; Terry 2019). These are dispersive in general, but fortuitously, the fastest waves relative to the flow are the longest ones, and for finite deformation length, L d , in the long-wave limit, they are nondispersive (Hide 1969). Relative to the flow (before Doppler shifting), the phase speed and group velocity asymptote to the same fastest value (Dowling 2014) Having the fastest waves be nondispersive makes it straightforward to define "Ma" for information propagation purposes (the original Mach number was defined in terms of linear, or smallamplitude, sound waves, which are nondispersive). Mach-type numbers take the form −(u−α)/c, where the minus sign makes the value positive when c is in the upstream direction. This detail about the sign does not explicitly arise for sound waves or buoyancy waves, because they are both omnidirectional, and the upstream value of c is always used to calculate Ma and Fr. Consequently, the negative halves of the Mach and Froude number lines are generally not in play.
In contrast, vorticity waves are not omni-directional: they travel in one direction at a time relative to the flow, keeping high PV on the right (Equation (4)). Thus, there is no luxury to assume upstream waves are available; for vortical flow, the negative half of the "Mach" number line is in play when the waves point downstream (that is, have high PV on their right looking downstream). The general form of "Ma" is (Dowling 2014(Dowling , 2019 where k 1 2 is the first baroclinic eigenvalue connecting the eddy PV and stream function, , where a is the planetary radius (or the domain size of an ocean basin). The proceeds by the chain rule, and the sign of "Ma" is set by the sign of the correlation between stream function and PV, which can be either positive or negative.

Escalator Analogy
An analogy with people riding escalators, in place of vorticity waves riding jets, helps to interpret negative "Ma": it corresponds to impatient commuters or shoppers walking down the down escalators or up the up escalators (when the escalators are operating). Positive "Ma" describes kids messing about, heading up the down escalators or down the up escalators. Escalators are "supersonic" when they are winning in terms of altitude change, moving either commuters or half-hearted kids in the escalators' direction, corresponding to "Ma"<0 or 1<"Ma," respectively. Notice the reciprocal concatenates both into one compact "supersonic" criterion, "Ma" −1 <1. Escalators are "subsonic" when they are losing, as when kids are gaining altitude on down escalators or vice versa, corresponding to 0<"Ma"<1, which maps to 1<"Ma" −1 .

Two Stability Branches
To date, most science and engineering students of fluid mechanics are taught only one branch of shear stability theory, which traces back to Rayleigh (1880)ʼs inflection-point theorem, with no mention of the existence of a second branch. However, Kelvin recognized, also in 1880, there are two branches (Thomson 1880, p. 619): V. In the present case, clearly, though there are an infinite number of unstable steady motions, there are only two stable steady motions-those of absolute maximum and of absolute minimum energy.
In an appropriate phase space for alternating jets, for inviscid systems there is no spiraling into an equilibrium point like for viscous systems. Rather, stability implies orbits around equilibrium points, which look like closed contours on a topo map. Kelvin's point is that topo maps are ambiguous; they could either be the map of a valley or a hill (Dowling 2014, Figures 1 and 2). In 1966, 86 years later, another geometrical thinker, Arnol'd, applied nonlinear Lyapunov stability theory to inviscid shears, constructing the appropriate topo map out of a combination of perturbation energy and perturbation enstrophy (squared perturbation PV; Arnol'd 1966), now called the pseudoenergy.

Kelvin-Arnol'd First Branch
The pseudoenergy-valley case is the Kelvin-Arnol'd first stability branch (KA-I). It states an inviscid, quasi-geostrophic shear is stable if the stream function-PV correlation satisfies (Mcintyre & Shepherd 1987 which is a generalization of the zonally symmetric case The stability theorems of Fjørtoft (1950) and Ripa (1983) are logically equivalent to Equations (6) and (7). (Ripa's theorem extends from the quasi-geostrophic to the primitive system.) The stability theorem of Charney & Stern (1962),¯> q 0 y everywhere or <0 everywhere, is included as a special case; just do a Galilean shift α to satisfy the inequality. The barotropic stability theorem of Kuo (1949),b -> u 0 yy or <0, is included; just turn off the stretching vorticity in Charney-Stern. Finally, the inflection-point stability theorem of Rayleigh (1880),¯> u 0 yy or <0 is included; just turn off the planetary vorticity in Kuo. To summarize, KA-I covers all the shear stability theorems commonly found in science and engineering textbooks. But, notice none of these are presented in nondimensional form. Why not? This state of affairs is ironic when one considers that otherwise, Rayleigh was a "champion of dimensional reasoning" (Darrigol 2005, p. 290) who "never missed an opportunity to denounce its neglect" (Darrigol 2005, p. 257). One reason is nondimensionalization is not required by the mathematical proofs, tracing back to Rayleigh's inflection-point theorem.
To nondimensionalize (6), multiply by k -1 2 , which with Equation (5) yields ( ) -< KA I stable: "Ma" 0. 8 Hence, the textbook branch (KA-I) corresponds to the negative half of the "Mach" number line (see Figure 1). This change to a "Ma" point of view is arithmetically simple, yet insightful (this is common in astronomy, such as when the negative print of a star field is used to aid the search for faint objects). In this case, PV eddies are like impatient commuters, moving downstream faster than the jets are flowing. Vortical eddies that tilt downstream across alternating jets are self-damping (Dowling 2014, Figure 6), which is the gist of the physical mechanism supporting Equation (8). On the reciprocal number line, "Ma" −1 , these domains concatenate together.
To move out of them requires passing through the "supersonic" to "subsonic" transition with respect to vorticity waves, as indicated by red arrows, where a shock is to be expected (Whitham 1974).
However, the direction of Jupiter's zonal jets correlates with PV in exactly the opposite sense to KA-I, corresponding to the positive half of the "Mach" number line, as can be seen in Figure 2, where two empirically determined profiles of PV for Jupiter are compared. The top profile (the dashed curve in panel (a)) is PV in primitive shallow-water form, with q redefined to be (ζ+f )/(gh), where ζ is relative vorticity, and the gravity factor is included with the layer thickness in the denominator. The bottom profile (the solid curve in panel (b)) is PV in quasigeostrophic form of Equation (1). These are positive "Ma" PV profiles, which are handled by the Kelvin-Arnol'd second branch of shear stability theory.

Kelvin-Arnol'd Second Branch
The pseudoenergy-hill case is the Kelvin-Arnol'd second stability branch (KA-II), which states an inviscid, quasigeostrophic shear is stable if (Mcintyre & Shepherd 1987) where k 1 2 is the first baroclinic eigenvalue, as in Equation (5). If this branch is mentioned at all in textbooks, it is either done so disparagingly: "this is a less common situation for it demands that dΨ/dQ be sufficiently negative so that the (negative of the) enstrophy contribution is always larger than the energy contribution" (Vallis 2017, p. 407); or in passing without further comment: "the basic flow is stable Kif the integrand Kis either positive or negative definite in some frame of coordinates" (Drazin & Reid 1981, p. 433). In the plasma physics literature, the existence and importance of the KA-II branch has also not been widely appreciated, but with the occasional exception (O'Neil & Smith 1992).
To nondimensionalize KA-II, multiply Equation (9) by k 1 2 , which yields ( ) -< KA II stable: 1 "Ma". 10 Thus, KA-II corresponds to the "supersonic" case, as an aeronautical engineer would recognize it. Just as in the escalator analogy, KA-I and KA-II correspond to the two types of "supersonic," both of which cause vortical eddies laying across alternating jets to experience a net pivot downstream, which leads to self-damping and stability (Dowling 2014, Figure 6). As above, the reciprocal concatenates the two into one compact "supersonic" criterion (see Figure 1): In passing, notice that PV homogenization (McIntyre 2015), a related topic that has also traditionally not been nondimensionalized, corresponds to "Ma" −1 →0, which is inside the stability interval of Equation (11), not on its boundary. As anticipated, nondimensional reasoning has reduced competing vortical flow ideas to "0 versus 1" and clarified them.

Candidate for the On-Off Switch
Significantly, the boundary of Equation (11) is the "supersonic" to "subsonic" transition with respect to vorticity waves, marked by red arrows in Figure 1. This has led to the conjecture that the shock of vorticity waves in alternating shear is the on-off switch of shear instability (Stamp & Dowling 1993;Dowling 2014).

Applications
The following two planetary applications of Equation (11) serve as tests of the idea that "Ma" −1 crossing back and forth across unity dictates shear instability. In a nutshell, two different types of results with predictions arising from the same theory, based on Voyager vorticity data-one for Jupiter and one for Saturn-have been upheld by recent Juno and Cassini results.

Jupiter's Jets
Jupiter's sharp, alternating zonal jets have long provided stimulating challenges to geophysical fluid dynamics, starting with the discovery of the planet's belts in 1630 (Graney 2010). Their stability is nothing short of "remarkable" on a decadal timescale (Porco et al. 2003). Observational strategies have recently been summarized as follows  Statements (a) and (b) are correct. However, statement (c) is missing both the pre-and post-Galileo remote sensing constraints on the deep jets. Prior to Galileo, Voyager images enabled detailed analysis of vorticity (Dowling & Ingersoll 1988, 1989Dowling 1993Dowling , 1995a, which in turn led to the emergence of shear stability as a stringent top-down (atmosphere→interior) constraint on jet structure (Dowling 1995b). At the present time, proximity-orbit gravity results from Juno at Jupiter (Kaspi et al. 2018) and Cassini at Saturn (Galanti et al. 2019) are providing complementary, bottom-up constraints on the jet structure. An updated (c) thus reads: (c) below the clouds, from Voyager vorticity measurements, Galileo entry-probe in situ measurements, and Juno gravity measurements.
A brief summary of the top-down vorticity literature is given in the Appendix.
In addition to the remote sensing and in situ evidence, deep jets have proven to be necessary to model Jupiter's tropospheric jets in a stable manner, which otherwise exhibit strong shear instability (Yamazaki et al. 2004, Figures 18 and 19). The observable effect deep jets have on tropospheric jets is captured by the 1-3/4 layer model, as demonstrated by Voyager-era applications to Jupiter that assume different deepjet profiles (see Table 1). In hindsight, one can see the progression of nondimensional "0 versus 1" thinking used to motivate the various jet configurations. One pair of competing ideas came from applying "0 versus 1" directly to the zonal jets, in terms of¯ū u deep ( Table 1, third column). Another pair came from applying "0 versus 1" to the PV gradient, in terms of "Ma" −1 (fourth column). Note Equations (3) and (5) with k  -L d 1 2 2 imply the sum of the third and fourth columns in Table 1 , which does not depend onū deep and hence is the same for each row; this is a manifestation of the non-Doppler effect, which "results from the mean flow being balanced by a height gradient that alters ¶ ¶ q y" (Held 1983, p.130). In the quasi-geostrophic 1-3/4 layer model, the deep stream function is found from the deep-jet profile by integrating,ò y =u dy deep deep , whereū deep is specified by the chosen model (e.g., third column in Table 1). Similarly to a plasma tube with flat ends, the shallow Jupiter model ( Table 1, top row) has constantȳ deep , i.e., no varying 3/4 layer (no dynamic topography). In terms of "Ma" −1 , it is clear the extent to which such shallow jets cross from "supersonic" to "subsonic," and hence violate Equation (11), leading in practice to strong shear instability (Dowling 1993, Figure 7).
In contrast, all alternating deep-jet models have an undulatingȳ deep profile. Boundary control of stretching vorticity is implemented by shapingȳ deep to give the active layer's basic state the desired "Ma" −1 profile relative to Equation (11). See Dowling & Ingersoll (1989) and Dowling (1993) for initial value numerical simulations and plots of dynamic topography associated with each of the four models in Table 1. As an illustration of this boundary control, the "Ma" −1 =1 basic state (last row in Table 1) is which, via Equations (3) and (11), renders the overlying zonal wind profile,ū, marginally stable. This "Ma" −1 =1 model was motivated by, and reproduces, the Jupiter vorticity data (Dowling & Ingersoll 1988), whereas the other models do not satisfy this constraint (Dowling & Ingersoll 1989;Dowling 1995a). Furthermore, the Juno gravity (and the magnetic field) interior-jet retrievals (Kaspi et al. 2018;Kong et al. 2018) have been setting the top boundary condition for the interior jets to be the cloud-top jets (the black curve in Figure 3), which corresponds to the¯¯= u u 1 deep model ( Table 1, second row) and hence also does not satisfy the vorticity constraint. Going forward, combining the top-down and bottom-up strategies shows promise for significantly reducing nonuniqueness from interior-jet retrievals.

Saturn's Length of Day
The second planetary application of Equation (11) is the determination of Saturn's rotation period. Herschel (1794) made the first attempt, coming close to the modern System I value that applies to Saturn's super-rotating equatorial region. System II refers to the mid-latitude region of a gas giant, and System III refers to the bulk rotation period, which is the most fundamental. Normally, magnetometers flying near a gas giant establish System III by measuring the periodicity of the external magnetic field's tilted dipole, as has been done for Jupiter, Uranus, and Neptune. However, for Saturn, the net result of three flybys (Pioneer 11 and Voyager 1 and 2) and one flagship orbiter (Cassini) has been to establish the dipole tilt as not exceeding 0°. 007, which is not sufficient to establish the rotation period (Cao et al. 2019, Figure 7).
The fall-back plan during the Voyager era was the observed periodicity of the Saturn kilometric radiation. However, Cassini revealed this to be a magnetospheric period that slips like a clutch, changes its value from decade to decade, and differs between the planet's northern and southern hemispheres (Gurnett et al. 2009). In other words, Saturn's current official length of day, 10hr 39minutes 24s, is not the planet's rotation period.
Fortunately, Saturn's interior has large mass anomalies that perturb nearby wave-bearing fluid regions in a conspicuous manner-namely in its atmosphere and in its C-ring (see Table 1 1-3/4 Layer Models of Jupiter's Deep Jets

Model
Reference¯ū u deep - Figure 4). In classical physics, forces come in two varieties, surface forces and body forces, and Saturn's interior employs both, perturbing the atmosphere via pressure anomalies and the C-ring via gravity anomalies. Like Jupiter, Saturn uses its deep jets to stabilize its tropospheric jets. This is where the Galilean shift term comes into play: α corresponds to the reference frame in which the fastest (and longest) vorticity waves are stationary, after Doppler shifting by the wind (Read et al. 2009;Dowling 2014). Cassini temperature data facilitated the calculation of stretching vorticity, which completed the mapping of PV and in 2009 led to a determination of α, which yielded the rotation period 10hr 34 minutes 13±20s (Read et al. 2009; the reported uncertainty range is the 1σ value unless otherwise noted).
A decade later, in 2019, the first C-ring kronoseismology result for Saturn's rotation period was published: 10hr -+ 33.6 1.3

1.9
minutes (Mankovich et al. 2019), where the reported uncertainties are the 5%/95% quantiles, or effectively the 2σ values. The 1σ atmospheric wave uncertainty interval falls within the 2σ ring-wave uncertainty interval. Based on these wave analyses, an argument can be made that Saturn's day is about 10hr 34 minutes, and the official value is about 5 minutes too long. In addition, similar results have been obtained from independent studies that are not based on waves but on optimization of external gravity data, the planet shape, and wind data or internal density profiles. These have yielded 10hr 32 minutes 35±13s (Anderson & Schubert 2007) and 10hr 32 minutes 45±46s (Helled et al. 2015), which are somewhat shorter than the values from the wave analyses. Nevertheless, it is significant that all these modern approaches are landing in the same neighborhood.

Discussion
Following Voyager, Galileo, Cassini, and Juno, we now have in hand two key lessons regarding shear stability: (i) alternating zonal jets are stable when < -"Ma" 1, 1 (ii) and stabilization (and destabilization) of alternating zonal jets is readily achieved via boundary control of stretching vorticity.
Criterion (i) concatenates the two Kelvin-Arnol'd stability branches and corresponds to "supersonic" flow, which has led to the conjecture that "Ma" −1 crossing back and forth across unity is the long-sought on-off switch for inviscid shear instability (Dowling 2014). It is clear that the Mach number portfolio, which has been central to compressible and hydraulic fluid mechanics since the nineteenth century, is also central to vortical flow.
Having a viable candidate for the on-off switch for shear instability in hand helps to frame outstanding questions and to clarify control mechanisms. Along these lines, some ideas for future work involving zonal jets in giant planets and in plasma physics are discussed below.

Giant-planet Jets
Jet streams in Earth's atmosphere and oceans writhe like snakes-their barotropic modes are significantly "subsonic"but in contrast, zonal jets on the giant planets generally run staff-straight. Interestingly, two notable exceptions are found in Saturn's northern hemisphere, the Ribbon at 47°latitude (Sromovsky et al. 1983) and the Hexagon at 78° (Sánchez-Lavega et al. 2014).

Hypothesis for Isolated Wavy Jets
Isolated, wavy jets like Saturn's Ribbon and Hexagon motivate the question: how can a meandering jet be sandwiched indefinitely between straight jets? Thinking about the problem in terms of "Ma" −1 and boundary control of stretching vorticity leads to the following hypothesis: the basic state for a wavy zonal jet bounded by straight zonal jets is a "subsonic" (1<"Ma" −1 ) latitudinal band bounded northward and southward by "sonic-to-supersonic" ("Ma" −1 1) bands.
Such a configuration appears to hold for the Hexagon (Dowling 2019), and the hypothesis is amenable to testing in  (Porco et al. 2003). (Red curve) Deep-jet profile that yields the basic state "Ma" −1 =1, with L d =2000 km at | | =  lat 40 (Dowling 1995a, Figure1). The deep-jet profile has been interpolated across the equatorial region, | | <  lat 17 . The gray marks on the left, spanning −10°to −40°and at 6°. 5, indicate where deep jets were detected via Voyager vorticity (Dowling & Ingersoll 1988, 1989 and Galileo entry-probe data (Atkinson et al. 1997), respectively.  (Read et al. 2009) and of density and bending waves in its inner C-ring (Mankovich et al. 2019) each yield 10hr 34 minutes for the rotation period, give or take a few tens of seconds. general via observed PV fields and shallow-water and n-layer simulations.

Interior Jets
Jupiter's interior jets also appear to be stabilized by boundary control of stretching vorticity. The gyroscopic Taylor-Proudman effect suppresses motions parallel to the rotation axis, such that internal jets alternate with respect to the cylindrical radius from the rotation axis, rather than latitude, as verified in simulations (Kaspi et al. 2009;Wicht et al. 2018). The interior is a tube, albeit as short a tube as there can be, capped by hemispheres. Given the universal nature of the Mach number portfolio, it is reasonable to suppose that Equation (11) applies to Jupiter's jets from top to bottom. What changes in the interior is the orientation of vorticity waves, and the boundary that controls stretching, which is the planet's shape itself, leading to an effective β deep ≈−3β (Ingersoll & Pollard 1982) and shear stability (Kaspi & Flierl 2007).

Plasma Physics Jets
Switching the setting for zonal jets from quasi-geostrophic dynamics and Coriolis effects to plasma physics and magnetic effects is straightforward because, as alluded to above, similar jets and vortices arise in both settings (Kaladze et al. 2005) and discoveries have crisscrossed between these disciplines for the better part of a century (McIntyre 2015). There are two distinct magnetic confinement problems, the hot-plasma case, which is key to fusion electricity (Donné 2019;Li et al. 2019), where the plasma must be of high density and so is kept nearly neutral in charge; and the cool, non-neutral plasma case.

Hot-plasma Confinement
Regarding the hot-plasma case and fusion reactors, Jupiter is two orders of magnitude smaller than the smallest type of star that produces energy by fusion (a red dwarf), thus it is not fusion itself, but rather the stability of Jupiter's alternating jets that makes the planet relevant to the discussion of fusion electricity (see Figure 5). This is because zonal jets are implicated in the low-to-high confinement transition in tokamaks and stellarators (Diamond et al. 2005;Estrada 2019;Terry 2019). Considering that the ignition, energy extraction, and exhaust phases of fusion reactor operation are each likely to be optimized by different zonal jet configurations, controlling these jets in fusion reactors with Jupiter-like precision would be a boon.
The extent to which Jupiter-style jet stability applies to the fusion problem, and to plasma physics in general, hinges on a question of topology: whether one can contain the plasma in a tube with ends, or instead must resort to a torus. Present-day fusion reactor design strongly favors the latter, largely to avoid melting the ends of a tube by eliminating ends altogether.
This presents a conundrum for controlling the jets, because Jupiter favors boundary control of stretching vorticity, which corresponds to a tube with 3/4 layer topography on its ends, not a torus. (In mythology, Jupiter/Zeus disguised himself as a bull/taurus; yet when it comes to zonal jet stability, one would be correct to surmise that Jupiter is not bullish on the torus). Given the practical need for the torus geometry, various ideas for promoting beneficial zonal jets in fusion reactors without using 3/4 layers are under investigation-for example, via external manipulation of geodesic acoustic modes (Hallatschek & McKee 2012).

Cool-plasma Confinement
In contrast to the hot-plasma case, the standard confinement topology for cool-plasma applications is a simple tube with ends, for which the analog of stretching vorticity is in play (Finn et al. 1999;Hilsabeck & O'Neil 2001). Hence, Jupiterstyle jet stabilization via boundary control of stretching vorticity offers direct benefits to the cool-plasma case. One potential application is the improvement of antimatter storage at high-energy physics colliders (Danielson et al. 2015).
Going forward, Jupiter-style, 3/4 layer end-tube topography should be investigated to advance engineering control of zonal jets in the plasma physics setting. In general, across the various disciplines where alternating jets are important, Jupiter's two key lessons above should help to propel the field into the twenty-first century, by removing much of the guesswork from the prediction and control of shear instability.
The author thanks Stefan Llewellyn Smith and the scientific and local organizing committees of the 2019 IUTAM Vorticity Symposium in La Jolla, CA, where a preliminary version of this material was presented, and Geoff Stanley, Noah Hurst, and two anonymous reviewers for comments that helped improve the article.

Appendix Vorticity as a Diagnostic
The following is a literature summary related to using vorticity as a top-down diagnostic for field studies of shear stability on gas giants. The purpose is to motivate future work that will combine this with the gravity (and magnetic field) bottom-up approach to form a powerful diagnostic squeeze on the permissible parameter space of zonal jets.
Prior to the Galileo entry probe in 1995, Voyager vorticity results provided "a map of the zonal speed under the GRS [Great Red Spot] against latitude" (Rogers 1995, p. 197), which led to the first detection of deep jets on Jupiter (Thomson & McIntyre 2016, p. 1120-1121: Dowling & Ingersoll 1989ʼs evidence for deep jets remains important today because, as yet, there are no other observational constraints on the existence or nonexistence of deep jets outside the equatorial region. No such constraints are expected until, hopefully, gravitational data come in from the Juno mission in 2016. Figure 5. Zonal jets in tokamaks and stellarators circulate in the poloidal ("Bundt-cake ridge") direction, alternating as a function of the small radius. They play a key role in enhancing the magnetic confinement of the ultra-hot (10 8 K) plasma.
These vorticity observations covered −10°to −40°in latitude (Dowling & Ingersoll 1988, 1989 and yielded the basic state of "Ma" −1 ∼1 for Jupiter (Dowling 1993 (Dowling 1995a) is shown in Figure 3. This was translated into the pre-Galileo-probe prediction (Dowling 1995a, p. 439): The speed and direction of the zonal wind as a function of latitude for the model's lower layer is thus determined, which roughly indicates the circulation in Jupiter's interior below the water clouds. Predictions are that the westward jets change little with depth but that the eastward jets become stronger by 50%-100%.
The subsequent probe results revealed that Jupiter's eastward jet at latitude 6°.5 becomes stronger with depth by 70%-113% (Atkinson et al. 1997, p. 649): We determine wind speeds at the cloud tops (700mbar level) in the range 80-100 m s −1 , in agreement with the results of cloud tracking; the speed increases dramatically between 1 and 4 bar, and then remains nearly constant at ∼170 m s −1 down to the 21bar level.
The Galileo-probe team (Atkinson et al. 1997, p.650) point out "only one analysis (Dowling 1995a) predicted that the winds would increase with depth," and observe: The constancy of the wind in the 4-21 bar range implies that the fluid there is barotropic. This is expected, as sunlight does not penetrate and latent heat is not released at these levels. Convection and radiation, the other modes of heat transfer, are likely to maintain a barotropic state. Thus the -170 ms 1 winds measured by the DWE [Doppler Wind Experiment] probably extend well into the fluid interior of Jupiter. This result is qualitatively consistent with studies of cloud-top winds, whose vorticity can be used to infer the winds at depth (Dowling & Ingersoll 1989).
Juno has subsequently confirmed that Jupiter's zonal jets are thousands of kilometers deep (Kaspi et al. 2018), and Cassini has found a similar result for Saturn (Galanti et al. 2019)-in both cases, the zonal jets are approximately as deep as they are wide (Dowling 2019).