# Jets in Planetary Atmospheres

## Abstract and Keywords

Jet streams, “jets” for short, are remarkably coherent streams of air found in every major atmosphere. They have a profound effect on a planet’s global circulation and have been an enigma since the belts and zones of Jupiter were discovered in the 1600s. Collaborations between observers, experimentalists, computer modelers, and applied mathematicians seek to understand what processes affect jet size, strength, direction, shear stability, and predictability. Key challenges include nonlinearity, nonintuitive wave physics, nonconstant-coefficient differential equations, and the many nondimensional numbers that arise from the competing physical processes that affect jets, including gravity, pressure gradients, Coriolis accelerations, and turbulence. Fortunately, the solar system provides many examples of jets, and both laboratory and computer simulations allow for carefully controlled experiments. Jet research is multidisciplinary but is united by a common language, the conservation of potential vorticity (PV), which is an all-in-one conservation law that combines the conservation laws of mass, momentum, and thermal energy into a single expression. The leading theories of how jets emerge out of turbulence, and why they are invariably zonal (east-west orientated), reveal the importance of vorticity waves that owe their existence to conservation of PV.

Jets are observed to naturally group into equatorial, midlatitude, and polar types. Earth and Uranus have weakly retrograde equatorial jets, but most planets exhibit strongly prograde (superrotating) equatorial jets, which require eddies to transport momentum up-gradient in a manner that is not obvious but is beginning to be understood. Jupiter and Saturn exhibit multiple alternating jets spanning their midlatitudes, with deep roots that connect to their interior circulations. Polar jets universally exhibit an impressive inhibition of meridional (north-south) mixing, and the seasonal nature of the polar jets on Earth, Mars, and Titan contrasts with the permanence of those on the giant planets, including Saturn’s beautiful north-polar hexagon. Intriguingly, jets in atmospheres have strong analogies with jets in nonneutral plasmas, with practical benefits to both disciplines.

Keywords: jet, jet stream, polar vortex, potential vorticity, superrotation, zonal flow

Introduction

This article reviews jet streams in planetary atmospheres, or “jets,” which are fast-moving streams of air found in every major atmosphere. The diverse and magnificent beauty of jets is reminiscent of the diverse and magnificent beauty of stalactites and stalagmites in caves—superficially, they have similar silhouettes, but they also share the concept that the barest drip leads to the mightiest structure. The scientific literature on jets is interdisciplinary and rapidly growing; it spans observational, experimental, and theoretical geophysics, with a focus on planetary-scale, rotating, and stratified fluid systems. This is a leading topic in *geophysical fluid dynamics* (GFD).

The word “jet” derives from the French *jeter*, which comes from the Latin *jactare, “*to throw.” One way to appreciate the fundamental nature of the concept is to notice the word “derive” itself comes from the Latin *rivus*, “stream.” “Jet” can equally well refer to modern travel by airplane (or James Bond’s jet pack), to one of the gangs in the musical * Westside Story* (the Jets versus the Sharks), or to the relativistic streams that beam out along the rotation axes of active galaxies. The focus here is on atmospheric jets.

Planetary jets are an advanced topic in GFD, but the basic notion is like blowing out candles on a birthday cake—except no one can blow out candles from across a room—and so the most striking feature of planetary jets is their long-range coherence. Jets also tend to be invisible and to be suspended. A dip in a swimming pool often provides a pleasant encounter with an unexpected jet of water; perhaps one feels it first on one’s back or underfoot. There is something enchanting about discovering a jet first by touch and then exploring it without ever really seeing it.

## Observations

The first atmospheric jets to be discovered were plainly visible, at least once telescopes were invented in the 1600s; these were the belts and zones of Jupiter. Jupiter’s belts are its dark bands, dark like a belt worn around the waist, and its zones are its light bands; the detailed nomenclature is given by Peek (1958/1981). A recurring theme in the study of planetary jets is the unsurpassed flow visualization of the gas giants, especially Jupiter (longitudinal variability has been assessed on Jupiter, e.g., Johnson et al., 2018). Jupiter’s cloud-top motions reveal the most complicated zonal-jet profile in nature, and Saturn’s profile is equally byzantine (see Figure 1).

Jets figure into one of the major surprises of planetary exploration, * Suomi’s paradox*, the revelation that jet speed increases with distance from the sun (named after Verner Suomi, the father of weather satellites and a member of the

*Voyager*imaging team, who drew attention to this paradox). The trend is striking, Earth’s winds rarely exceed 50 ms

^{–1}, whereas Neptune’s winds top out over 400 ms

^{–1}. This reversed correlation suggests there is reduced dissipation away from the sun (Ingersoll, 1990), probably because of reduced turbulence. Suomi’s paradox delivers a fatal blow to the simplistic notion that jets are directly powered, or pushed, by sunlight. Instead, jets are shaped and maintained by Coriolis effects from planetary rotation, vorticity dynamics, and converging eddy fluxes.

Jets in GFD share a strong mathematical analogy with jets in magnetized, nonneutral plasmas, a connection with a long history that threads through this review and continues to be beneficial to both disciplines (e.g., Kaladze, Rogava, Tsamalashvili, & Tsiklauri, 2005; McIntyre, 2015). In atmospheres, *zonal flow* refers to east-west flow, which is the preferred orientation for jets. In *tokamaks*, the Russian acronym for “toroidal chamber with magnetic coils,” “zonal flow” is used to describe circulations in the poloidal, or Bundt-cake-ridge direction (as opposed to the toroidal, or frisbee-groove, direction), and zonal flows play a key role in the fusion confinement problem (Diamond, Itoh, Itoh, & Hahm, 2005).

Jupiter’s jets are deep, as discovered via *Voyager* vorticity data of the Great Red Spot (GRS) and White Oval BC (Dowling & Ingersoll, 1989) and confirmed via *Juno* gravity data (Kaspi et al., 2018). They run approximately 3000 km into the interior (Guillot et al., 2018), which is comparable to their horizontal widths and presents an unusual ~1:1 aspect ratio to GFD, a discipline more accustomed to ~1:50 pancake-like aspects.

Earth’s prevailing zonal winds are not everywhere in the form of jets, but they nevertheless are of great importance today, directly affecting both global weather forecasting and local commerce, such as the continuous adjustments airlines make to minimize fuel consumption and reduce flight times. They also have historical significance, for instance in the tropics starting in the 15th century, the broad surface *trade winds* that prevail westward (“easterly”) were largely responsible for advecting the Portuguese language across the Atlantic from Portugal to Brazil and for causing the word “trade” to change from its original connotation of “track” to today’s “commerce.”

Earth often has partial jets, called *jet streaks*, which strongly affect local meteorology. The entry and exit regions of jet streaks are associated with confluence and diffluence, respectively, which promotes vertical interactions. For example, upper-level jet streaks can generate lower-level jets that affect the planetary boundary layer (Uccellini, 1980), anomalously dry surface air can develop under exit regions (Kaplan, Huang, Lin, & Charney, 2008), and entry and exit circulations can interact to enhance snowfall (Uccellini & Kocin, 1987).

Earth’s planetary-scale jets are of the suspended and mostly invisible variety, and consequently they were discovered two centuries after the discovery of Jupiter’s belts and zones. There are two in each hemisphere (northern and southern), a polar jet (see Figure 2) and a subtropical jet. These writhe like snakes compared to the staff-straight jets of the gas giants. Interestingly, over the past few decades Earth’s jets have been slowly rising in altitude and moving poleward (Archer & Caldeira, 2008). They went undetected until as late as 1883, when the massive volcanic eruption of Krakatoa provided flow visualization in the form of an “equatorial smoke-stream” (Bishop, 1884). In the Orient, Wasaburo Oishi in Japan made a series of upper-air observations in the 1920s that revealed the jet stream. In the Occident, measurements of jet speed and extent were first sampled by military airplanes, before and especially during World War II. Lewis (2003) reviews the discovery and early research of Earth’s jets.

## Two Tenets of Fluid Dynamics

Jets are a topic in GFD, and GFD is a branch of fluid dynamics, so at the outset it is beneficial to review two foundational tenets of the discipline. First, equations should be nondimensional. An exception that proves the rule is found in the engineering thermal-fluid sciences, which abound with properly nondimensional parameters, like Mach numbers, Reynolds numbers, and coefficients of drag. But, every year, when the curriculum turns to open-channel flow, the dimensional Manning coefficient is encountered. Manning tried in 1890 to unify the tangle of dimensional open-channel relations of his day, but the problem was, and is, a difficult one; his formula is dimensional and is still taught and used that way today, albeit sheepishly. How long does it take to fix such a problem? The answer appears to be 100 years; to wit, the chapter “A Dimensionless Manning Type Equation” can be found in the book * Channel Flow Resistance: Centennial of Manning’s Formula* (Yen, 1991). The takeaway is that expressions about jets should be nondimensional, but it may take 100 years to meet this standard.

The second fundamental tenet of fluid dynamics relates to shocks, the fact that they always occur when crossing from supersonic to subsonic flow. This can be understood universally in terms of the theory of characteristics applied to hyperbolic partial differential equations (PDEs), which are the class of PDEs that support waves (Whitham, 1974, ch. 2). A sound-wave shock is located where the Mach number crosses unity, and a water-wave shock, called a hydraulic jump, is located where the water-wave analogue of the Mach number, the Froude number, crosses unity. Jet dynamics is governed by a third type of wave called a Rossby wave, or vorticity wave; in this article the analogous nondimensional number is labeled “*Ma*.” In a shear flow, one should expect a shock to occur where and when “*Ma*” crosses from “supersonic” to “subsonic.”

The Study of Jets: Four Challenges

Asked why the Shakers, who expected the end of the world at any moment, were

nevertheless consummate craftsmen, Thomas Merton replied:

“When you expect the world to end at any moment,

you know there is no need to hurry. You take your time, you do your work well.”—Wendell Berry,

Discipline and Hope

If the story of jets had a simple arc, which of course it does not, the beginning of the end would start with the hero figuratively riding in on a white horse. A white horse *is* trampling up and down the field, however, it has been joined by a red horse, a black horse, and a pale horse, which is to say, there are four portentous challenges that complicate the study of jets.

## The First Horseman: Nonlinearity

The white horse has brought the plague of nonlinearity to the study of jets. This plague affects all of fluid dynamics, or at least all fluid dynamicists whose job it is to find analytical solutions (laboratory and numerical modelers enjoy some immunity—their experiments operate equally well with nonlinearity as without). The point is the material time derivative itself, $d/dt$, ensures Newton’s Second Law is nonlinear for fluids, because acceleration is $\left(d/dt\right)\mathit{v}$, where $\mathit{v}$ is the vector velocity, but for a fluid, the chain rule of calculus gives $d/dt=\partial /\partial t+\left(\mathbf{v}\mathbf{\cdot}\mathbf{\nabla}\right)$ such that acceleration always contains $\left(\mathit{v}\mathbf{\cdot}\mathbf{\nabla}\right)v$, the nonlinear advection term. For example, a zonal jet streak traveling at $40{\text{m s}}^{-1}$ advects that very attribute, a local gust of $40{\text{m s}}^{-1}$, eastward at $40{\text{m s}}^{-1},$ via the term $u\partial u/\partial x$. This is self-referential, and hence nonlinear and difficult to analyze, but it is one of the reasons jet streaks are interesting in the first place.

If the nonlinear term $u\phantom{\rule{0.2em}{0ex}}\partial u/\partial x$ looks like the zonal gradient of the zonal part of the specific kinetic energy (“specific” in this context means “per mass”), $\left(\partial /\partial x\right)\left({u}^{2}/2\right)$, then the reader may want to consider changing into this field, because this is one of the underpinnings of writing “**a** = **f**/m” in vector-invariant form. In the inviscid (frictionless) case, this form has the following horizontal-component momentum-balance equations in Cartesian coordinates:

Here, *x* and *y* are longitude and latitude measured in length units from a central reference point, $\left(u,v\right)$ are the horizontal winds with (eastward, northward), also known as (zonal, meridional) components, $\zeta =\partial v/\partial x-\partial u/\partial y$ is the *relative vorticity*, relative to the planet’s rotating reference frame, $f=2\text{\Omega}\mathrm{sin}(\phi )$ is the *planetary vorticity* (Coriolis parameter) at latitude $\phi $, with planetary angular velocity $\text{\Omega}$(for Earth, $\text{\Omega}=2\pi /{24}^{h}$), and $B=\text{\Phi}+{c}_{p}T+K$ is the * Bernoulli streamfunction*, with geopotential $\text{\Phi}$, specific enthalpy ${c}_{p}T$ (written here assuming for convenience the specific heat ${c}_{p}$ is constant), temperature

*T*, and the aforementioned specific kinetic energy,

*K*. The pressure-gradient force is accounted for via

*B*, and specific enthalpy is part of

*B*because the vertical coordinate is taken here to be the isentropic variable

*potential temperature*, $\theta $, which is related to specific entropy, $s$, by $ds={c}_{p}d\mathrm{ln}\theta $.

In layers stable against buoyancy-driven convection, the usual case for jets, $\theta $ increases monotonically with altitude and is a valid vertical coordinate, in particular one that renders adiabatic motions computationally horizontal. Isentropic coordinates provide this and a host of other diagnostic advantages (Andrews, 1983; Hoskins, McIntyre, & Robertson, 1985; Hsu & Arakawa, 1990). In oceanography, the situation is more complicated because the equation of state is nonlinear such that the helicity is non-zero, but nevertheless it is beneficial to employ approximately neutral surfaces (e.g., de Szoeke & Springer, 2009). Regarding the vertical component of momentum balance, most studies of jets neglect vertical accelerations and so assume *hydrostatic balance*. In isentropic coordinates applied to atmospheres, hydrostatic balance takes the form

where $\text{\Phi}+{c}_{p}T=B-K=M$ is called the dry static energy, or the Montgomery potential. See Vallis (2017, Section 3.10) for more details on isentropic coordinates.

The traditional way to combat the first horseman, nonlinearity, is to linearize, or to assume small-amplitude perturbations about a basic state, where the latter is a solution to the original nonlinear equations. A term like $u\phantom{\rule{0.2em}{0ex}}\partial u/\partial x$ then simplifies to $\overline{u}\phantom{\rule{0.2em}{0ex}}\partial u\prime /\partial x$, which is linear in the perturbation or eddy, ${u}^{\prime}$, as measured relative to the basic state or mean, $\overline{u}$.

Some progress has been made identifying and studying weakly nonlinear phenomena associated with jets (e.g., Feldstein, 1991; Manfroi & Young, 1999) and with waves (e.g., Boyd, 2018; Whitham, 1974). An example of a fully nonlinear analysis is the work of Arnol’d (1966), who pioneered an application of Lyapunov’s nonlinear stability theory to inviscid shear flow; this is now referred to as the energy-Casimir method (Shepherd, 1990). Nonlinear stability theorems allow for transient flare-ups but ensure long-term boundedness. The first horseman is doing his job well, because when Earth’s jets become unstable, they do not maintain weak nonlinearity but rapidly develop large-amplitude meanderings to the point of breaking and shedding vortices.

## The Second Horseman: One-Way Waves

The red horse has brought civil war to the study of jets. In this context, the red horse is none other than the largest and oldest inhabitant of jets in the solar system, Jupiter’s Great Red Spot (GRS). The conflict stems from a deficiency in shear-instability theory—the fact that the on-off switch has not yet been found mathematically. At least it is a civil war, and the soldiers all speak the same language. This common framework is conservation of *potential vorticity* (PV, or symbolically *Q*)

where *h* is the thickness density (Vallis, 2017), *g* is gravity, and *p* is pressure. Conservation of PV combines the conservation laws of mass, momentum, and thermal energy into a single, all-in-one law. The concept was first developed by Rossby in a series of papers in the 1930s, culminating in Rossby (1940); the definitive mathematical-physics derivation was established by Ertel (1942). (English translations of 21 of Ertel’s papers, which were originally written in German, are provided by Schubert et al. (2004).)

That *h* in (4) is a density, the density in $\left(x,y,\theta \right)$ space, follows from the mass of an air parcel being $\rho \phantom{\rule{0.2em}{0ex}}\text{\Delta}x\phantom{\rule{0.2em}{0ex}}\text{\Delta}y\phantom{\rule{0.2em}{0ex}}\text{\Delta}z=h\phantom{\rule{0.2em}{0ex}}\text{\Delta}x\phantom{\rule{0.2em}{0ex}}\text{\Delta}y\phantom{\rule{0.2em}{0ex}}\text{\Delta}\theta $, where $\rho $ is ordinary density (mass per volume [kg m^{–3}]) and *z* is ordinary altitude. For rotating and hence oblate planets, to maintain accuracy, an extraordinary definition of altitude is employed, the geopotential height, $Z=\text{\Phi}/{g}_{0}$, where ${g}_{0}$ is a reference value for gravity. Conservation of energy is folded into (4) because air parcels move nearly adiabatically, such that their top and bottom values of $\theta $ do not change; in other words, $\theta $ serves as a label to keep track of the parcel’s vertical thickness following the motion. Conservation of mass is folded into (4) because vertical stretching of *h* implies reduction of the parcel’s horizontal area. Conservation of momentum is folded into (4) because when horizontal area retracts, there is an increase in vorticity, like when a skater pulls in her arms. Hence the numerator and denominator in (4), the absolute vorticity and thickness density, are correlated, and their ratio, *Q*, is conserved following the motion for inviscid, adiabatic, and reversible flow.

The effects of irreversibility and entropy production, which generate right-hand side source/sink terms for *Q* in (4), become important when modeling climate (e.g., Johnson, 1997) but are generally negligible in jet research. Diabatic heating generates right-hand side source/sink terms as well, and these are important when considering the genesis and maintenance of jets.

In the bare-bones inviscid case, mass and momentum fields (i.e., pressure and wind fields) are nearly balanced by Coriolis effects, and when stratification is strong, all the moving parts in (4) can be expressed asymptotically in terms of a single variable, a streamfunction, $\psi $. One then has a single equation in a single unknown—a major theoretical achievement, albeit one that still requires the solution to a nonlinear PDE. This is the realm of “nearly geostrophic” theories, which include the quasigeostrophic, semigeostrophic, and planetary geostrophic varieties (Pedlosky, 1987; Vallis, 2017). These all have noncanonical Hamiltonian structure, which provides a unifying foundation (e.g., Holm, 1996; Salmon, 1988a, 1988b; Shepherd, 1990).

Conservation of PV gives rise to the most important type of wave for jet dynamics, the large, ponderous Rossby wave (Platzman, 1968), also known as the planetary wave; in spherical coordinates it is called the Rossby-Haurwitz wave, and the more generic name *vorticity wave* is also used in this article. In Earth’s atmosphere, vorticity waves are visible in animations of the day-to-day, synoptic-scale undulations of the jet stream (the polar jet) in TV weather reports, but they are ubiquitous in rotating fluids and strongly affect jet formation, maintenance, and stability. The restoring force for vorticity waves comes from conservation of PV in the presence of an environmental PV gradient. This is similar to buoyancy waves, what GFDers call “gravity waves,” which include waves on a lake, or more generally the slow, internal bouncing of an atmosphere or ocean, and whose restoring force comes from conservation of $\theta $ in the presence of an environmental $\theta $ gradient—except, in the case of vorticity waves, there is a single time derivative in (4), rather than the more common ${\partial}^{2}/\partial {t}^{2}$ found in the equation that govern buoyancy waves, or sound waves. As a result, vorticity waves are not omni-directional like buoyancy waves, or sound waves; instead they travel in one direction at a time from a source. Their intrinsic phase speed relative to the flow is such that high PV is always to the right (Gill, 1982; Vallis, 2017).

This unidirectional wave property is not unheard of—another example is a Kelvin wave, a buoyancy “half” wave or shelf wave that travels in one direction and must lean against something to exist (Vallis, 2017). But in the curriculum of wave equations, it is more common to run across the ${\partial}^{2}/\partial {t}^{2}$ and hence ${\nu}^{2}$ type of wave, and consequently researchers occasionally find themselves directly in the path of the second horseman and his nonintuitive, unidirectional waves. For example, on Jupiter the meridional gradient of the quasigeostrophic PV, alternatively denoted ${\beta}_{e}$ or ${\overline{q}}_{y}$, is negative in the South Equatorial Belt (Rogers et al., 2016, Section 4.1), but one should not implicitly assume the square root of ${\beta}_{e}$ is taken to get the phase speed of Rossby waves, as would be the case for waves governed by ${\partial}^{2}/\partial {t}^{2}$, which associates with squared frequency, ${\nu}^{2}$. Instead, a negative value of ${\beta}_{e}$ yields an *eastward* intrinsic Rossby-wave phase speed, keeping high PV on the right, not an imaginary phase speed. Eastward propagating Rossby waves are as real and as important as westward propagating ones. Each constitutes half of the counter-propagating wave mechanism that leads to shear instability; further discussion may be found in Harnik and Heifetz (2007) and references therein.

Now recall the second tenet of fluid dynamics, the inevitability of shocks and the importance of Mach-type numbers. In engineering, where the majority of practical fluid-dynamical studies take place (e.g., plumbing, ventilation, hydraulics, irrigation, dams, boats, cars, planes, rockets, the “A” and “S” of “NASA”), the Mach number, *Ma*, is the primary tool for characterizing compressible flow, and for hydraulic flow it is the Froude number, *Fr*. Engineers are taught compressibility effects are negligible for *Ma* < 0.3. Compare this to what meteorologists are taught, which is that an air parcel’s internal pressure matches its environment’s pressure. The two disciplines are making the same point here, but whereas the engineers have not lost sight of the Mach number, the meteorologists have expunged it from their curriculum. Unfortunately, this has left them ill-prepared to take on Mars (Dowling, Bradley, Lewis, & Read, 2016). In the engineering curriculum, every upstream formula involving *Ma* has an analogous formula involving *Fr* (e.g., Munson, Young, & Okiishi, 2006). Passing from less than unity to greater than unity results in choked flow (*Ma*) or controlled flow (*Fr*), and passing from greater than unity to less than unity necessitates a shock (*Ma*) or a hydraulic jump (*Fr*). In the grand scheme of fluid dynamics, shocks and choked flow are about the only sure things, because they are governed by the universal rules of information propagation. Choked flow, which involves a disconnect between the mass flow rate at a compressible-flow choke point (the minimum-area cross-section of a converging-diverging rocket nozzle, where the flow is precisely sonic) and the hardness of the downstream vacuum, is every bit as dramatic, and of a similar nature, to the event horizon of a black hole.

Where, then, is the analogue of the Mach number for vorticity waves, and where is its shock? In practice, *Ma* and *Fr* are nondimensional ratios of the form $\u2013u/c$, where $u$ is the flow speed and $c$ is the upstream intrinsic wave speed relative to the flow (the group velocity, which is the velocity of information propagation, should be used for dispersive waves), such that $\u2013u/c\ge 0$, since $u$ and $c$ have opposite signs from the sense of “upstream.” But this freedom is not available for unidirectional waves; if the vorticity-wave intrinsic wave speed happens to be downstream, then “*Ma*” is negative (Dowling, 2014). The second horseman is using this complication to fan the flames in the study of jets. Like the Sharks versus the Jets in * Westside Story*, two opposing camps have arisen, the negative “

*Ma*” camp and the positive “

*Ma*” camp. Both sides believe they have corralled the elusive on-off switch for shear instability, but neither camp has pinpointed it yet mathematically.

Fortunately, the field of jet research advances when either camp makes a new discovery. The negative “*Ma*” camp has found in Earth’s atmosphere, and in laboratory and numerical simulations, PV mixing is implicated in jet formation and jet sharpening (McIntyre, 2015). When it is energetically favorable, inertia-gravity waves (buoyancy waves large enough to feel the Coriolis force) cause cross-jet mixing (Pierce, Fairlie, Grose, Swinbank, & O’Neill, 1994), and in Earth’s stratospheric *surf zone*, between the subtropical and polar jets, vorticity waves grow so large that meanders break off, like the formation of oxbow lakes by a meandering river, only breaking in days rather than decades. Multiple jets can form without PV-gradient reversals, corresponding to PV staircases (Baldwin, Rhines, Huang, & McIntyre, 2007; Dritschel & McIntyre, 2008). A guiding philosophy of the negative “*Ma*” camp is $\partial Q/\partial y\to 0$ is natural, and shear instability tends to remove PV-gradient reversals (e.g., Scott & Dunkerton, 2017).

The positive “*Ma*” camp considers vorticity-wave shocks to be a more complete picture of shear instability than PV mixing (Dowling, 2014). This camp argues putting the focus on PV mixing is akin to characterizing an acoustic shock, or a hydraulic jump, exclusively in terms of its mixing properties. Jupiter’s GRS is the red horse of jet research, because a vorticity analysis of the GRS wind field led to the inference that “*Ma*” is positive everywhere on Jupiter. If “*Ma*” > 0, then, because there are dozens of jets in Jupiter’s atmosphere that alternate eastward ( $+$) and westward ( $-$), the PV gradient reverses sign with each switch (e.g., Dowling, 1993).

## The Third Horseman: Non-Constant Coefficients

Compounding the challenges of nonlinearity and nonintuitive waves, jets in nature vary strongly in space, and this gives rise to PDEs with nonconstant coefficients. Analytically, it is easier to solve PDEs with constant coefficients, and so analytical jet models are skewed toward simplified cases. The black horse has brought a famine of realistic analytical models to the study of jets.

Vorticity-wave dispersion relations, the equations relating a wave’s frequency to its zonal, meridional, and vertical wavenumbers, were originally established in the special case when the meridional gradient of PV is dominated by the change in the planetary vorticity, $df/dy=\beta =2\text{\Omega}\left(\text{cos}\phantom{\rule{0.2em}{0ex}}\varphi \right)/a$, where *a* is the planetary radius (the local meridional radius of curvature for an oblate sphere), and $\beta $ is treated as a constant. This is how Rossby’s analysis proceeded, and how the subject is introduced in textbooks. For example, in the introductory text by Cushman-Roisin and Beckers (2011), a hierarchy of Rossby waves is considered from barotropic to baroclinic but always with the environmental PV gradient restricted to be just $\beta $. Interestingly, $\beta $ is positive in both the northern and southern hemispheres, zero at the poles, and strongest at the equator, such that Rossby waves figure prominently in the dynamics of equatorial jets.

The quasigeostrophic version of (4) is valid for rapid rotation and strong stratification, and is consequently quasi-nondivergent, such that the horizontal velocity components may be expressed in terms of a streamfunction, $u=-\partial \psi /dy$ and $v=\partial \psi /dx$,

where ${\nabla}^{2}\psi =\left({\partial}^{2}/\partial {x}^{2}+{\partial}^{2}/\partial {y}^{2}\right)\psi $ is the relative vorticity, ${f}_{0}$ and $\beta y$ are the first two Taylor-series terms of $f$, called the $\beta $-plane approximation, and the rightmost term in (5b) is the expansion of the $1/h$ factor in (4) in terms of reference density profiles $\u3008h\u3009$ and $\u3008\rho \u3009$ (Vallis, 2017).

Here arises the mathematical analogy alluded to in the introduction: (5) also governs magnetized plasmas. In fact, it is more accurate when applied to long columns of nonneutral plasmas in a magnetic field than to rotating atmospheres and oceans. The gist of the analogy is both the magnetic force and the Coriolis force cause the fluid to move in circles, similarly via cross-products with velocity (“Coriolis force” and “Corolis acceleration” are used interchangeably, with the appropriate sign, and “force” means per mass unless otherwise specified). In the plasma community, (5) is called the *Hasegawa-Mima* equation and in the GFD community, (5) is called the * Charney-Obukhov* equation. The latter is written in terms of a streamfunction, as in (5), and governs the dynamics of vorticity waves, whereas the former is written in terms of the perturbed plasma potential and governs the dynamics of plasma drift waves, which are mathematically analogous to vorticity waves. See Kaladze et al. (2005) for an example of both being analyzed simultaneously. With such an equivalence, it is not surprising the same two opposing “

*Ma*” camps have arisen in the plasma community (e.g., O’Neil & Smith, 1992).

Returning to the analysis of waves that govern jets, the third horseman, nonconstant coefficients, can be dodged by assuming no vertical variation at all, called the barotropic case, and the first horseman, nonlinearity, can be dodged by assuming a constant zonal-wind basic state (no jet!), $\overline{u}=U$. Then $\psi =-Uy+{\psi}^{\prime}\left(x,y,t\right),$ and the eddy-eddy or “prime-prime” terms are neglected for small-amplitude perturbations (subscripts denote partial differentiation):

This is a one-equation, one-unknown, linear, constant-coefficient PDE. Traveling sinusoidal solutions of the form ${\psi}^{\prime}=\mathrm{exp}\phantom{\rule{0.2em}{0ex}}[\text{i}\left(kx+ly-\nu t\right)]$ solve it and generate the dispersion relation $(\partial /\partial t\to -i\nu ,\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\partial /\partial x\to ik,\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\partial /\partial y\to il\phantom{\rule{0.2em}{0ex}})$

where $\nu $ is frequency [s^{–1}], and *k* and *l* are zonal and meridional wavenumber [m^{–1}], respectively. Rossby’s discovery of (7) constitutes a significant breakthrough in the history of meteorology (Platzman, 1968). Useful features of (7) include the demonstration that the intrinsic zonal-phase speed,

is unidirectional, westward in this case, and (7) makes clear the role of the Doppler-shift term, $Uk$. In addition, notice $c$ increases in magnitude as the wavenumber decreases, or equivalently as the wavelength increases—the longest waves are the fastest waves relative to the wind. Meanwhile, the shortest waves are passively advected, $\left({k}^{2}+{l}^{2}\right)\to \infty $ implies $c\to 0$. Both of these features are common to vorticity waves in general, and the latter helps to explain how in practice Jupiter’s tropospheric winds are reproducibly extracted by tracking its finest cloud features.

In the context of the strong jets of Jupiter and Saturn, (8) has non-useful features as well, in particular the association of “westward” with “Rossby wave.” This association came about as researchers responded to a coordinated attack by the first three horsemen. To dodge nonlinearity, they focused on linear waves. Then, the most important wave revealed itself to be unidirectional. Then, to dodge nonconstant coefficients, even as the physics became more sophisticated, these waves were chiseled into the curriculum and common wisdom as “westward.” For example, consider the following statement from a review of Saturn dynamics (here ${q}_{E}$ is *Q* and ${\beta}_{e}$ is the quasigeostrophic analogue of $\partial {q}_{E}/\partial y;$ the case-sensitive subscripts “E” and “e” stand for “Ertel” and “equivalent”):

The recent results using the best available velocity and thermal measurements for both Jupiter and Saturn do suggest that these reversals in $\partial {q}_{E}/\partial y$ are robust features of the circulations of both planets, at least in their tropospheres. Taken at face value, however, this would seem to imply the possibility of barotropic instability around those latitudes where the reversals occur, on the flanks of the eastward jets, while Rossby wave propagation and inhibited meridional transport is anticipated around the eastward jet peaks.

(Del Genio et al., 2009, Section 3.8)

Why are Rossby waves only anticipated around the eastward jet peaks? In other words, why are they only expected where the PV gradient is positive? The westward-propagation connotation of “Rossby wave” is the primary reason “vorticity wave” is promoted as an alternative in this article. (Some authors address the issue by using the term “pseudo-westward” to mean high PV is on the right.)

For descriptions of the many improvements to (7) since the time of Rossby, which fold in horizontal and vertical wind shear (i.e., barotropic and baroclinic wind shear) and stratification, consult Andrews, Holton, and Leovy(1987); Gill (1982); Pedlosky (1987); Salby (1996), and Vallis (2017). One version that helps dispel the “westward” connotation of Rossby waves is the barotropic dispersion relation by Karoly (1983):

where *U* and *V* are basic-state zonal and meridional winds in a Mercator projection and $\overline{\eta}=\left(\overline{\zeta +f}\right)$ is the basic-state absolute vorticity. This is a more symmetrical dispersion relation than (7) and makes the point that vorticity waves can travel in any direction relative to the flow, so long as high PV is on the right. An application of (9) to superposition and stationary waves is presented by Shaman and Tziperman (2016).

Summing to this point, jets in nature are nonlinear, they are governed by one-way-at-a-time waves, and they are described by nonconstant-coefficient PDEs, all of which make them difficult to study. And yet, the jets themselves do not have any trouble with any of this. Jupiter’s dozens of alternating jets are the most complicated in nature, and yet they are steady—one is in awe watching them fly past each other, like a flawless * Cirque du Soleil* act. In light of this, what can be done to use the full weight of the nonconstant-coefficients to at least solve the nonconstant-coefficient linear problem, like a judo master using the weight of his opponent against him?

We start with a zonally symmetric zonal jet structure in the meridional plane, $\overline{u}\left(y,\theta \right)$, and linearize (4) in the quasigeostrophic limit, which yields

where

is the eddy quasigeostrophic PV, and (10) is written with an optional Galilean reference-frame shift, $\alpha $, recognizing that $q\prime $ itself is Galilean invariant. (On an oblate spheroid with map factors, a “Galilean reference-frame shift” is a change in the planet’s angular velocity $\text{\Omega}$, which alters the planetary and relative vorticities but only in a zero-sum way such that their sum, the absolute vorticity, is invariant.) Jupiter’s jets are steady to first order, so the linear PDE under question is

The nonconstant coefficients are the $\left(\overline{u}-\alpha \right)$ and ${\overline{q}}_{y}$ factors—that they vary strongly with latitude and level, $y$ and $\theta $, is part and parcel to the subject matter. Slowly varying coefficients, which in this context means slowly varying jets, are handled with the WKB approximation (e.g., Young & Rhines, 1980), which proves in practice to be robust. However, jets in nature are not typically slowly varying, and hence the WKB approach does not attack the opponent using his full weight against him.

Now, consider the nonconstant coefficients in (12) can be kicked right out from under the PDE, judo style, if the basic state is such that ${\overline{q}}_{y}$ alternates fully in the same manner as the jets themselves,

where ${\kappa}^{2}$ is a proportionality constant with units of [m^{–2}]. The third horseman is doing his job well, because many researchers look at the left-hand side of (13) and see a constant; they struggle to conceive of ${\overline{q}}_{y}$ as being zigzaggy, like the jets on the right-hand side.

In addition to implying the numerator of the vorticity-wave dispersion relation is variable instead of constant, (13) implies a standard plot of the zonal-wind profile $\overline{u}$ is a plot of the coveted PV gradient itself, ${\overline{q}}_{y}$, which seems too much to ask. Pressing on, for *any* given zonal-wind profile, if ${\overline{q}}_{y}$ satisfies (13), then (12) reduces to

The zigzaggy nonconstant coefficients in (12) have dropped out of (14), even as they have been fully met. Apart from a derivative with respect to $x$, (14) is the standard eigenvalue problem for the generalized three-dimensional Laplacian operator in (11), which connects the eddy PV to the eddy streamfunction. In other words, the eigenfunctions of (11) are steady-state solutions to *any* zonally symmetric zonal-wind profile, so long as the PV gradient and the jets alternate together as in (13). This basic state is precisely that deduced empirically for Jupiter from a vorticity analysis of the GRS and White Oval BC, for the latitude range –10º to –40º (Dowling, 1993).

## The Fourth Horseman: 18 Orders of Magnitude

While the first three horsemen have been softening up the jet-research camps, death itself has swept in on the pale horse and let loose the wild beasts. There are 18 orders of magnitude separating the size of Jupiter, which has an equatorial radius of 7.15 × 10^{7} m, from the size of the hydrogen atom, which has a Bohr radius of 5.29 × 10^{–11} m. Roaming this vast dynamical territory are a multitude of length scales and energy scales that affect jets and a corresponding pack of nondimensional numbers. Additional such parameters are needed to handle viscous boundary layers on terrestrial planets and in laboratory experiments. Finding which parameters matter the most takes patience and persistence and constitutes the bulk of the labor in the study of jets.

Read (2011) employs nondimensional numbers to organize the dynamical regimes of the terrestrial atmospheres (Venus, Earth, Mars, and Titan) and to establish how laboratory experiments tie in. Arguably, the most fundamental length scale affecting extratropical jets is the one that demarcates between small-scale, kinetic energy dominated motions, in the “nature abhors a vacuum” and the “what goes up must come down” senses, from large-scale, potential energy dominated motions, in the “highs and lows” of a weather map and the “glass is falling” barometer senses. This is the Rossby, or Rossby-Obukhov, *deformation length*, ${L}_{d}$. The competition of forces is between the flattening tendency of gravity and the horizontal pressure-gradient force, versus the warping tendency of the Coriolis force. The compromise is *geostrophic balance*, where ${L}_{d}=c/f$ is the ratio of the buoyancy-wave (gravity-wave) speed $c$ to the planetary vorticity $f$ (Gill, 1982; Vallis, 2017; see Figure 3). In the study of magnetized plasmas, ${L}_{d}$ is analogous to the Larmor radius. Two vortices more than a few ${L}_{d}$ apart are insensitive to each other; they are shielded in the Biot-Savart sense like anomalies in a plasma. In atmospheres, ${L}_{d}$ is of order 10^{3} km. In oceans, the percentage change in density is smaller than in atmospheres, so the buoyancy-wave speed is slower and ${L}_{d}$ is correspondingly smaller, ranging in Earth’s oceans from ~200 km near the equator to ~15 km in the Antarctic Circumpolar Current.

Real atmospheres are baroclinic, with the “1/h” term in (5) and (11) playing as significant a role as the $\zeta $ and $f$ terms. In the baroclinic case, the denominator of dispersion relations like (8) and (9) generalizes to $({k}^{2}+{l}^{2}+{\lambda}_{d}^{-2})$, where ${\lambda}_{d}$ is an appropriate baroclinic eigenvalue version of ${L}_{d}$. The numerator generalizes to the full PV gradient, ${\overline{q}}_{y}$. These imply a change in the character of the waves, because if the domain is large enough, then in the long-wave limit the wavenumber factor becomes constant, since $({k}^{2}+{l}^{2}+{\lambda}_{d}^{-2})\to {\lambda}_{d}^{-2}$, such that the intrinsic vorticity-wave phase speed asymptotes to $c\to -{\overline{q}}_{y}{\lambda}_{d}^{2}$. This long-wave limit is nondispersive, such that the group and phase speeds take the same fastest value (Hide, 1969, p. 846), which makes it easier to define and employ “*Ma*” in information-propagation arguments for the giant case. Gas giants and Earth’s oceans are large enough for this to happen, since they each span dozens of deformation lengths or more. In contrast, Earth’s atmosphere is only big enough to harbor about six storms around its circumference. The governing nondimensional number is the planetary Burger number, $Bu={\left({L}_{d}/a\right)}^{2}$, where $a$ is the planetary radius (e.g., Read, 2011, Eq. 14). For a closed oceanic basin, $a$ is the corresponding span of the domain. The Burger number allows jet researchers to specify “giant” nondimensionally as $Bu<<1$ and to develop asymptotic theories using *Bu* as a small parameter.

In addition to *Bu*, four other nondimensional numbers out of the large pack have been found to be particularly significant. The thermal Rossby number, *Ro*, measures the ratio of buoyancy forces to the Coriolis force. The thermal Rhines number, *Rh*, measures the ratio of the planetary-vorticity gradient to the relative-vorticity gradient, and its square-root, ${N}_{J}=\sqrt{Rh}$, predicts the number of zonal jets that will form across an experimental channel. Frictional damping, ${F}_{f}$, measures the ratio of the viscous damping time to the period of rotation of the experimental apparatus or planet. Finally, the superrotation parameter, *S*, measures the ratio of excess specific angular momentum to the solid-body value corresponding to co-rotation with the surface (Read, 2011).

## Applying the Two Tenets

A leading open question in jet research is: What are the necessary and sufficient conditions for shear instability; in other words, where is the on-off switch? In terms of established mathematical proofs that bear on the question, there exist Arnol’d’s first and second stability theorems, which McIntyre and Shepherd (1987) apply to baroclinic jets as follows. The first theorem states a quasigeostrophic shear flow is stable in the nonlinear or bounded sense of Lyapunov if

where *c* and *C* are technical constants (not to be confused with wave speeds), and $\overline{\psi}$ and $\overline{q}$ are basic-state quasigeostrophic streamfunction and PV, respectively. This includes as special cases the theorems that light up the tables of contents of textbooks: the original inflection-point theorem for nonrotating, barotropic shears (Rayleigh, 1880), the addition of rotation (Kuo, 1949), the addition of baroclinicity (Charney & Stern, 1962), the logical equivalent of (15) established for linear (small amplitude) perturbations (Fjørtoft, 1950), and the same expanded from quasigeostrophic to primitive variables such that buoyancy waves are included as well as vorticity waves (Ripa, 1983; Shepherd, 2003). Notice by the chain rule, $d\overline{\psi}/d\overline{q}$ is a generalization of ${\overline{\psi}}_{y}/{\overline{q}}_{y}=-\left(\overline{u}-\alpha \right)/{\overline{q}}_{y}$ . Therefore, to recover Charney-Stern, which establishes stability when the PV gradient does not change sign, make a Galilean shift $\alpha $ that is sufficient to satisfy (15).

It is time to shine a light on the two fundamental tenets of fluid dynamics, starting with nondimensionality. Arnol’d’s first stability theorem (15), as well as all of its textbook special cases, are *not* nondimensional. Each term has units of [m^{2}]. They have been so for over 100 years, since the time of Rayleigh’s inflection-point theorem of 1880. Thus these venerable stability theorems have the same problem as Manning’s formula for open-channel flow.

To nondimensionalize (15), multiply by the first-baroclinic eigenvalue of (11), ${\kappa}_{1}^{2}$, which asymptotes to ${L}_{d}^{-2}$ for $Bu<<1$ and corresponds to the longest and fastest baroclinic vorticity waves (the long-wave phase and group speeds are the same), and recognize (Dowling, 2014)

such that (15) becomes (suppressing the technical constants for clarity)

It is in this sense that Arnol’d’s first stability criterion is the standard bearer for the negative “*Ma*” camp.

Arnol’d’s second theorem states a baroclinic shear flow is nonlinearly stable if

To nondimensionalize (18), multiply by ${\kappa}_{1}^{2}$ (suppressing the technical constants) to get

Arnol’d’s second stability criterion is thus the standard bearer for the positive “*Ma*” camp. The reciprocal, “*Ma*”^{–1}, actually concatenates the shear-stability regions of the two camps (Dowling, 2014),

Notice the case of homogeneous PV, which corresponds to $\u2033Ma{\u2033}^{-1}\to 0$, is inside the stable interval of (20), not on a boundary. Such is the wisdom of rendering equations to be nondimensional.

Now apply the second tenet of fluid dynamics, which says a shock must be located at the right-hand limit of (20). Shocks are associated with a sudden jump in entropy (Whitham, 1974); literally, something shocking occurs. The positive “*Ma*” camp conjectures this is the on-off switch of shear instability itself. This tenet-based assertion awaits a mathematical proof, or a counterexample. In terms of numerical stability analysis, it has been established for sinusoidal jets (see Figure 4).

Shear stability is but one of many active fronts in jet research. There are many stimulating open questions regarding genesis, zonality, meridional transport, climate, exoplanets, predictability, and more. It is an exciting time, in spite of the pestilence and wild beasts, or perhaps because of them. Given all the challenges, nonlinear numerical simulations prove in practice to be a versatile tool across all of jet research. But this is a multidisciplinary field of study involving scores of global collaborations between laboratory experimenters, planetary scientists, and applied mathematicians. To emphasize the cross-disciplinary nature of the topic, the rest of this article is organized into comparative-planetology categories. The first section highlights equatorial jets; the second and third sections highlight jets in midlatitude and polar regions, where Coriolis effects are paramount; and the last section draws attention to some lines of inquiry that might prove fruitful going forward.

Equatorial Jets

At short intervals along the moving catwalk, wide transparent tubes led down to floor level. Zaphod stepped off the catwalk into one of these and floated gently downward.

The others followed. Thinking back to this later, Arthur Dent thought it was

the single most enjoyable experience of his travels in the Galaxy.—Douglas Adams,

The Restaurant at the End of the Universe

## Hadley Cell

Near the equator, the component of the Coriolis force that couples to local-horizontal motions via $f=2\text{\Omega}\mathrm{sin}(\phi )$, as in (1) and (2), becomes negligible, nature does abhor a vacuum, even on the planetary scale, and sunlight does propel winds directly. Picture the deflating effects of turning off the magnetic field in a plasma: this is the deflation of potential energy as the dominant mode when crossing into the equatorial region. The original Hadley-cell concept from 1735 works in this region and defines its meridional boundaries. Hadley’s original idea predates an understanding of Coriolis deflections but recognizes on terrestrial planets that warm air rises at the equator, moves poleward, and descends, forming a meridional-plane cell. But, away from the equator, Coriolis deflections restrict Hadley cells. Nearly inviscid theory predicts the latitude of the poleward boundary to be approximately $15\xb0\times {\left(100\phantom{\rule{0.2em}{0ex}}Bu\phantom{\rule{0.2em}{0ex}}\text{\Delta}\theta /\theta \right)}^{1/2}$, where the Burger number is calculated at the planet’s pole and the fractional change of potential temperature is taken from the equator to the pole (Held & Hou, 1980).

Gas giants are different than terrestrial planets; they do not exhibit an equator-to-pole temperature gradient. Instead, they are affected by deep internal circulations (Ingersoll & Porco, 1978). The component of the Coriolis force that couples to local-vertical motions, called the *nontraditional component*, is proportional to $2\text{\Omega}\mathrm{cos}(\phi )$ and is often neglected for thin, spherical-shell atmospheres, although biases are known to arise from this approximation (Gerkema, Zimmerman, Maas, & van Haren, 2008). Because of their full-spherical geometry, the nontraditional component is always important in gas-giant interiors (Kaspi, Flierl, & Showman, 2009).

## Superrotation

Geographically, an equatorial region is a sump for momentum, leading to extremes in jet speed. The differences observed between the planets are remarkable, with Jupiter and Saturn each exhibiting a strong prograde equatorial jet and Neptune exhibiting a strong retrograde equatorial jet (meaning slower than the planet’s average rotation but still moving around the axis in the same direction). By comparison, the tropospheric equatorial flows of Earth and Uranus are both modestly retrograde. It is possible that in a past warm climate, with surface temperatures exceeding 33°C, Earth may have had a prograde equatorial jet (Caballero & Huber, 2010). Prograde jets are a common feature of hot Jupiters (Showman & Guillot, 2002), and Venus and Titan, the two slow rotators, both exhibit prograde equatorial jets.

Prograde equatorial jets are common, but they are not an obvious outcome if one thinks in terms of angular momentum. If an air parcel confined to a spherical shell circulates toward the equator, it moves farther from the planet’s rotation axis, so if it conserves its angular momentum about the axis, a modest retrograde wind should result. This description matches the equatorial tropospheres of Earth and Uranus, but this turns out to not be universal, because the notion utterly fails for Jupiter, Saturn, Venus, Titan, Earth’s stratosphere, and hot Jupiters. It fails because there are all manner of torques affecting air parcels, such that simple angular-momentum considerations do not capture the fluid dynamics. For this reason, fluid dynamicists instead employ conservation of a parcel’s PV following the motion.

A *superrotating* jet moves faster than the planet (i.e., prograde) and represents a maximum of momentum in the interior of the fluid, away from boundaries. A general result for parabolic (diffusive) PDEs is the *maximum principle* (John, 1982, p. 215), in which a local maximum or minimum cannot be maintained in the interior. Instead, any such local extreme melts away like a pat of butter. Without east-west varying eddies, the momentum-balance equation reduces to such a parabolic PDE in the form of an advective-diffusive system. Thus, under zonally symmetric conditions, any superrotating jet will diffuse away, a result known as Hide’s theorem (Schneider, 1977; Vallis, 2017). Hide’s (1969) original statement is in reference to Jupiter and reads:

Considerations of the general properties of thermally driven motions in a rotating spherical shell of fluid of outer radius

Rand total angular momentum divided by its moment of inertia $\text{\Omega}$ indicate that in order to account for a westerly (i.e., faster than the basic rotation) equatorial jet without appealing to sinking motions from higher levels in the atmosphere, effects due to local azimuthal pressure gradients cannot be neglected, as these gradients provide the only forces (in the absence of magneto-hydrodynamic effects) capable of increasing the angular momentum per unit mass of an individual fluid element to a value $>\text{\Omega}{R}^{2}$ .

Thus up-gradient transport of momentum by zonally varying eddies is required to maintain a superrotating jet.

A two-level model is sufficient to demonstrate that zonally asymmetric forcing can lead to superrotation. Interestingly, there can be hysteresis, such that once superrotation is established the forcing that leads to it can often be reduced (Suarez & Duffy, 1992), calling to mind the barest-drip analogy with stalactites. The behavior is strongly affected by the locations where the intrinsic wave phase speed is zero (Saravanan, 1993), called *critical lines*. Such linear-theory results have been expanded by studies of nonlinear disturbances, which show that waves can alter the winds as much as 10º to 20º in latitude away from critical lines (Randel & Held, 1991).

Rossby waves abound in equatorial regions, since $\beta =df/dy$ is largest there, and are good at accelerating a superrotating equatorial jet. Even Rossby’s original barotropic waves have their phase speeds and group velocities orientated to promote eastward flow at the wave-source region (see Figure 5). In Earth’s atmosphere, distinct zonal waveguides have been identified for the North African-Asian jet, the North Atlantic jet, the Australian jet, and the polar jet (Ambrizzi, Hoskins, & Hsu, 1995).

Early in planetary exploration, the winds of Venus were determined to be zonal across the disk (Limaye & Suomi, 1981; Schubert & Covey, 1981). Venus has an extremely slow, 243 Earth-day sidereal rotation period, which is 18 days longer than its year, and it rotates backwards compared to the other planets. Only Jupiter has an obliquity as small as Venus, both about 3º. The superrotation on Venus is in the same retrograde sense as the planet rotates, and at ~100 ms^{–1} it is just over 60 times faster than the surface and is appropriately called the four-day wind. The profile of the low-latitude zonal winds varies, sometimes being nearly constant with latitude and sometimes showing midlatitude jets. *Akatsuki* imaging exploits narrow spectral windows in the near infrared to see through to the lower to middle cloud layers and has returned evidence for an equatorial jet of 80 ms^{–1} and variability in the winds (Horinouchi et al., 2017). The leading theories for superrotation on Venus, and other slowly rotating bodies like Titan, involve interactions between Rossby and Kelvin waves (e.g., Potter, Vallis, & Mitchell, 2014; Zurita-Gotor & Held, 2018). The relevant mechanisms are likely to differ in detail from Jupiter and Saturn, which are fast rotators with deep interiors.

Hot Jupiters and super Earths that orbit in close proximity to their host stars have particularly strong zonally asymmetric forcing. This excites standing, westward planetary-scale Rossby waves and eastward equatorial Kelvin waves, which Showman and Polvani (2011) have shown organize into an eastward-pointing chevron centered on the equator, which generates a superrotating equatorial jet.

## Tides

In addition to vorticity waves and buoyancy waves, tides are an important source of planetary-scale eddy motion (tides are a type of buoyancy wave but are conveniently treated separately because of their external forcing and planetary scale). Ioannou and Lindzen (1994) show how tides raised by Jupiter’s moons can generate alternating jets, and zonal flows have been generated by tides in the laboratory (Morize, Le Bars, Le Gal, & Tilgner, 2010). On Mars, solar heating and night-side cooling are much stronger than on Earth because of the thin air and dry conditions, which results in strong thermal tides. These produce an equatorial superrotation around equinox, when nearly mirror-symmetric tides set up across the equator, especially the westward propagating diurnal tide (Lewis & Read, 2003). The details match the mechanism of Fels and Lindzen (1974), in which the mean density-weighted acceleration is in the direction opposite to the eddy phase speed of thermally excited waves. Dust loading in the Martian atmosphere strengthens the equatorial jet by amplifying the effects of solar heating.

## Equatorial Stratosphere

Every known atmosphere with a pressure greater than 100 hPa (i.e., 100 mbar; this excludes Mars) has a temperature minimum, a tropopause, at approximately 100 hPa, above which there is a stratosphere (Hanel, Conrath, Jennings, & Samuelson, 1992, Figure 7.2.5). Elementary textbooks explain Earth’s stratosphere in terms of ozone heating; however, Earth would have a tropopause and stratosphere without ozone, just a higher-altitude one (Pierrehumbert, 2010, p. 374; Thuburn & Craig, 2000). The existence of a tropopause at about 100 hPa is the result of the significant drop-off of infrared opacity that all common gases exhibit for p < 100 hPa, which causes a change in the heat-transfer mode. All stratospheres experience ultraviolet solar heating, which usually interacts with methane and its photochemical by-products and ozone for Earth. A substantial portion of the wave energy in a planet’s troposphere travels upwards into its stratosphere and beyond, and in equatorial regions this results in what some researchers think is the single most enjoyable jet structure in nature, a stack of alternating eastward and westward zonal jets that floats gently downward.

On Earth, this causes the * Quasi-Biennial Oscillation* (QBO), as reviewed by Baldwin et al. (2001). On Jupiter, it causes the

*(QQO), referring to four Earth years, which is 0.38 Jupiter years, as described by Leovy, Friedson, and Orton, (1991). On Saturn, it causes a 15-year oscillation, as described by Fouchet et al. (2008), which is about 0.5 Saturn years, so this is also a*

*Quasi-Quadrennial Oscillation**semi-annual oscillation*(SAO). Earth’s QBO actually gives way to a SAO at high altitude. On Titan, there is a notch around 20 hPa pressure where the wind speed drops to almost zero in an otherwise prograde equatorial jet (Bird et al., 2005). The other slowly rotating planet, Venus, exhibits no such notch. There is only the single

*Huygens*probe sounding for Titan to date, so it is not known if there is a QBO-like vertical descent associated with its notch, but the possibility is intriguing.

Waves generally pass through jets headed in the opposite direction but get absorbed by jets heading in the same direction. This critical-layer absorption occurs shy of the jet maximum, which leads to the downward tug. The physical mechanism of alternation of the QBO has an everyday analogy:

Imagine a father trying to retrieve two (or more) rambunctious toddlers from a tall climbing frame. He clambers up the west side to coax down tot no. 1, blocking all the other kids' access on that side in the process. Meanwhile, tot no. 2 scrambles up the east side. Dad plants tot no. 1 on the ground, and heads up the east side to retrieve tot no. 2, while tot no. 1 breezes back up the west side. When dad and tot no. 2 return to the bottom, one full cycle is complete. Of course, no self-respecting parent would put up with this nonsense, but what jet streams lack in intelligence, they make up in perseverance.

(Dowling, 2008)

Holton and Lindzen (1972) recognized that both westward and eastward waves are constantly clambering up into the stratosphere, and this is key to the QBO mechanism.

The QBO phenomenon is unusual because its period is not a cogwheel meshed to seasonal forcing but is proportional to the strength of the upward flux of waves. This may help explain why the observed periods get longer with distance from the sun—the same reduction of turbulence invoked to rationalize Suomi’s paradox may also lessen the rate of downward tugs. The QBO’s phase has a global effect and must be known to accurately predict Earth’s weather; for example it affects the Indian Monsoon in August and September (Claud & Terray, 2007). The phenomenon has been reproduced in an elegant laboratory analogue by Plumb and McEwan (1978), and in turn that analogue has been reproduced in a direct numerical simulation by Wedi and Smolarkiewicz (2006).

Midlatitude Jets

In a zero-sum game, the problem is entirely

one of distribution, not at all one of production.

—Kenneth Waltz,Man, the State, and War: A Theoretical Analysis

## Geostrophic Balance

Moving from the equatorial region into the midlatitudes is dynamically like turning on a magnetic field in a nonneutral plasma. The Coriolis force provides a strong horizontal coupling between the mass and momentum fields (pressure and wind fields) called *geostrophic balance*. This balance provides structure and clarity that is lacking in the tropics. Geostrophy comes with a bit of a learning curve, because one’s “nature abhors a vacuum” intuition, developed in childhood to maintain personal safety, must be unlearned for large-scale dynamics. Synoptic-scale cyclones, which spin in the same sense as the planet’s rotation, and anticyclones, which spin in the opposite sense, must be trusted to preserve low- and high-pressure anomalies, respectively.

Directly related to this trust, the idea that the sun pushes the jets must be unlearned. Some common wisdom does apply to the extratropics; for example, if all energy sources are removed, an atmosphere or ocean will spin down, like an undisturbed swimming pool in early morning. Solar insolation, sunlight, is the main energy source for the terrestrial-planet atmospheres, and insolation and intrinsic internal heat (above and beyond thermal balance with the sun) contribute about equally for Jupiter, Saturn, and Neptune, switching to mostly the latter for deep-interior circulations (present-day Uranus is the only gas giant lacking its own internal heat, and the only one tipped on its side, which may or may not be related).

## Alternating Jets

A key idea in the extratropics is energy sources are not being asked to push or pull the planetary jets, any more than they are being asked to push or pull the planet itself. It is not like candles on a birthday cake being blown out. In the extratropics, zonal jets are borrowed and reshaped planetary rotation. A better question to ask is how do these energy sources maintain meridional warps in the geopotential? Once the discussion of zonal jets is cast into a discussion of meridional warps of the appropriate geostrophic streamfunction (depending on the vertical coordinate employed), then the alternation of zonal jets begins to make sense—it is approximately a zero-sum game about the geoid. The balance between the gravity and Coriolis forces provides stiffness to the membrane, such that deformations can be maintained only near and above the ${L}_{d}$ scale, whereas at the large end, deformations cannot rival the planet’s oblate-spherical geoid itself, just as mountains can only be so tall. One can get an estimate of Saturn’s elusive rotation period from this zero-sum-geoid idea (e.g., Anderson & Schubert, 2007), with some uncertainty about where to draw the line for the start of the equatorial superrotation.

What happens when a closed vortex grows so big it becomes significantly larger than ${L}_{d}$ ? Jupiter’s GRS is the best example of this, with ${L}_{d}~2000$ km (estimated from various lines of evidence, including simply applying calipers to images of the annular width of the GRS’s peripheral jet), compared to a semi-minor axis of about 5000 km (the semi-major axis has been shrinking over the last several decades, but the minor axis is holding relatively steady to date). What happens is this largest of all anticyclones forms a *cyclone* in its center. This counterrotation flickers a bit but is confirmed (Liu, Wang, & Choi, 2012; Vasavada, Ingersoll, Banfield, Bell, & Gierasch, 1998). In other words, the GRS is so large it encompasses an alternating warp, just as do the zonally symmetric regions of the planet that span a similar meridional extent. This is one practical definition of “planetary scale.”

## Geostrophic Turbulence

The meridional warps in a planet’s geopotential are the net accumulation of forcing by small-scale eddies. Drip by drip, small-scale kinetic energy transforms into ${L}_{d}$ -scale potential energy. Notice there are no clamps or molds big enough to do the job directly, with the exception of orographic effects, which become extreme in ocean basins. Thus there is an upscale energy transfer, which is the opposite to the more familiar downscale energy transfer in three-dimensional turbulence, as in dissipating smoke rings or cream poured in coffee and as immortalized by Richardson’s poem: “Big whirls have little whirls, that feed on their velocity; and little whirls have lesser whirls, and so on to viscosity.”

In rapidly rotating fluids, there is a gyroscopic effect that suppresses motions in the direction of the rotation axis known as the Taylor-Proudman effect. Thus in GFD, two-dimensional-like turbulence is the norm and is usually referred to as *geostrophic turbulence*. There are differences in detail between three-dimensional geostrophic turbulence and the purely two-dimensional case, since the former still has the freedom to turn into the latter, a process called *barotropization* (Vallis, 2017).

When two like-signed, lens-shaped vortices approach each other within a few ${L}_{d}$, they can see through the fog of vorticity shielding and interact. They will often merge, and, if confined by the guiderails of alternating jets like on Jupiter and Saturn, they will merge quickly, as *Voyager* revealed. Physically, this is because where they nearly touch, the southward motion of one vortex cancels the northward motion of the other, creating a dead zone that joins with the quiescent centers of the original vortices, opening into a single vortex. This is a particularly efficient process for anticyclones (i.e., high-pressure vortices), because they always have quiescent centers (Holton & Hakim, 2013, p. 77). This is the start of the answer to the question of where jets come from, because jets are big structures, and big structures coalesce out of small ones in geostrophic turbulence.

There is an alternate point of view that makes this upscale, or inverse, energy cascade seem exotic, at least at first. In statistical physics, geostrophic turbulence is an example of a *negative-temperature* system; here “negative” means negative absolute temperature—negative [K]. This apparent oxymoron stems from the fact that the fundamental definition of absolute temperature is “upside down.” It is $1/T=\partial s/\partial u$, where $s$ and $u$ are specific entropy and internal energy (not to be confused with zonal wind), respectively (e.g., Kittel & Kroemer, 1980). Negative absolute temperature signals the system is to the right of the maximum in the plot of its entropy (i.e., the logarithm of its multiplicity), versus its internal energy, so that further addition of energy leaves fewer ways for the microscopic components to add up to the total—the system has begun to saturate. This motivates complementary bookkeeping that is reminiscent of when solid-state physicists and electrical engineers start tracking electron holes instead of electrons. See Montgomery and Joyce (1974) and Turkington, Majda, Haven, and DiBattista (2001) for articles on negative-temperature statistical physics applied to plasmas and to Jupiter, respectively, and the textbook by Majda and Wang (2006) on statistical-physics techniques applied across GFD.

Geostrophic turbulence is governed by conservation of total energy, but it also governed by conservation of total squared PV, called *potential enstrophy*. The latter is an example of a Casimir invariant, the existence of which makes geostrophic turbulence noncanonical in the Hamiltonian framework. Maintaining energy and enstrophy conservation simultaneously causes enstrophy to cascade to small scales while energy cascades to large scales, resulting in a small number of large vortices separated by stringy filaments—an apt description of Jupiter. Vallis (2017) provides an introduction to these integral constraints, and Diamond et al. (2008) examine potential-enstrophy-flux effects on jets in GFD and in magnetically confined plasmas.

## Eddy PV Flux

Hide’s theorem implies zonally varying eddies are a necessity for the formation of prograde jets, and vortical eddies are the agents of change for all manner of jets. However, many eddies can pass through a scene without changing it, like lit firecrackers flying in one train-car window and out the other: they may raise eyebrows, but they do not alter the timetable. Dunkerton (1980) reviews such *nonacceleration* conditions and gives examples. Other eddies are better coordinated to affect change, working on jets tirelessly, like cilia teasing a fluid along. When eddies are a jumbled mess, the usual case in nature, diagnostic tools are used to locate where they are accelerating or decelerating the mean flow. Tracking eddy PV flux has proven to be particularly useful, because the effects of both vorticity and form drag (on $\theta $ surfaces) are included (Vallis, 2017, ch. 10). The meridional PV flux $\overline{v\prime q\prime}$ can be equated to the divergence of a vector, $\nabla \cdot F$, where $F$ is the * Eliassen-Palm flux*. Nondivergent $F$ in the meridional plane is associated with nonacceleration in the zonal direction. Young (2012) has shown that thickness-weighted averaging can be done generally for statically (convectively) stable conditions, allowing the eddy PV flux to be tracked in both the meridional and zonal directions.

Jets are large, and large structures accumulate from small eddies in rotating atmospheres: so far, so good. Details of this process working in planetary atmospheres are starting to become clear. For example, in an analysis of *Cassini* images of Jupiter, Young and Read (2017) find evidence of a kinetic-energy inverse cascade from ${L}_{d}$ to larger scales but a downscale cascade from ${L}_{d}$ to smaller scales, with enstrophy cascading downscale at all scales. This suggests the main energy input is close to ${L}_{d}$ itself, implying baroclinic instability may be more important than thunderstorms for maintaining the jets, which is more like Earth’s oceans than its atmosphere.

## Zonal Flows

Jets are more than just big—they invariably run eastward or westward; they are zonal. Why are they not, say, meridional, spiral, checkered, or any other bold pattern? Even the sun exhibits deep bands of zonal flow, as revealed by helioseismology (e.g., Howe et al., 2000). Before the *Voyager* Uranus encounter in 1986, the ubiquitous zonal bias appreciated today had not yet been fully established for planetary atmospheres. Uranus is tipped on its side relative to the plane of the solar system, with an obliquity of 98º. Compare this to the relatively modest obliquities of Earth, Jupiter, and Saturn, which are 23º, 3º, and 27º, respectively. The zonal question was finally retired by *Voyager*, as reported by Smith et al. (1986, p. 47): “Uranus is the only planet whose poles receive more sunlight than its equator; yet, as on other planets, the winds follow circles of constant latitude as defined by the pole of rotation.”

The short explanation for the zonal bias is: rotating atmospheres, as a direct consequence of their spheroidal-shell shape, have their local vertical (outward) component of planetary vorticity change with latitude, $df/dy=\beta \ge 0$, called the $\beta $*effect*. When confined to a spherical shell, once eddies grow large enough to feel the $\beta $ effect, once they reach the * Rhines length*, ${L}_{\beta}={\left(u/\beta \right)}^{1/2}$, zonally directed vorticity waves arise and meridional motions are suppressed. The result is a dumbbell shape in spectral space (see Figure 6).

The case of small Burger number, relevant to gas giants and oceans, leads to a subset of geostrophic turbulence called *zonostrophic* turbulence. In this limit, baroclinic modes transfer energy at the scale ${L}_{d}$ into the barotropic mode, which becomes highly energetic, and “whose zonal mode alone may hold more energy than all other modes combined” (Galperin, Sukoriansky, & Dikovskaya, 2008). This theory connects specific energetic pathways from small-scale forcing to the genesis of alternating zonal jets, a process called *zonation*, and demonstrates alternating jets are an inevitable outcome of geostrophic turbulence in the $Bu<<1$ limit.

In addition to dry convection and wave dynamics, moist processes in an atmosphere are important, because the latent heat of water is large, and thunderstorms are known to occur on the gas giants (e.g., Gierasch et al., 2000). Lian and Showman (2010) apply a moist-convective parameterization and are able to generate prograde equatorial rotation for Jupiter and Saturn and retrograde equatorial rotation for Uranus and Neptune. How moist processes in Jupiter’s atmosphere interact with its approximately 3000-km deep abyssal jets, and similarly on Saturn where the jets are estimated to be 9000-km deep (Guillot et al., 2018), is an active area of research.

## Abyssal Jets

How does the third horseman, the nonconstant coefficients generated by real jets, affect the applicability of theories that place their emphasis on $\beta $ ? Where are the other two terms for ${\overline{q}}_{y}$ in (5b)? Indeed, Jupiter’s jets are so strong and so sharp, ${\overline{\zeta}}_{y}=-{\overline{u}}_{yy}$ reaches $-2\beta $ to + $3\beta $ (Ingersoll et al., 1981). In the negative “*Ma*” camp, a data analysis has been worked out that restricts Jupiter’s full PV gradient to be nonnegative everywhere (Scott & Dunkerton 2017). This may or may not be correct, but it is special pleading because it ignores the following empirical results, in reverse chronological order: Marcus and Shetty (2011) found a PV-gradient reversal for Jupiter; Read et al. (2006) and Read, Conrath, et al. (2009), used remote-sensing temperature data to calculate the “ ${(1/h)}_{y}$ term and found PV-gradient reversals where the absolute-vorticity gradient reverses, on both Jupiter and Saturn; and Dowling and Ingersoll (1989) used a shallow-water analysis to find PV-gradient reversals in Jupiter’s westward jets, in the span –10º to –40º latitude.

Something fundamental seems to be missing from the $\beta $ theories. For Jupiter, instead of ${L}_{\beta}$, one should consider the full, nonconstant-coefficient, quasigeostrophic PV gradient and think in terms of ${L}_{{\beta}_{e}}={\left(u/{q}_{y}\right)}^{1/2}$ . But this full Rhines length *equals* the deformation length, ${L}_{{\beta}_{e}}={L}_{d}$, when the basic state is $\overline{u}={\overline{q}}_{y}{L}_{d}^{2}$, which is the empirically determined basic state for Jupiter (Dowling, 1993), as well as the one that results from the judo move (13).

Interestingly, it is possible to arrange for gas-giant jets to migrate equatorward when there are no deep jets (Williams, 2003), which is not observed. The laboratory and numerical experiments do not reproduce the sharpness and steadiness of Jupiter’s jets for one reason: they do not include Jupiter’s strong abyssal jets. In the 1½ layer quasigeostrophic model for Jupiter, in which a weather layer overlies an abyssal layer that is deeply rooted such that its zonal winds are steady, the full PV gradient is (Ingersoll & Cuong, 1981)

The basic state $\overline{u}={\overline{q}}_{y}{L}_{d}^{2}$ thus corresponds to

such that the weather-layer zonal wind $\overline{u}$ cancels, called the *non-Doppler shift* property (Held, 1983), yielding a “choked PV” prescription for the abyssal zonal winds (Dowling, 1995, 2014):

This configuration is marginally stable, both theoretically (20) and in shallow-water numerical simulations initialized with Jupiter’s cloud-top jets (Dowling, 1993).

Unlike on the gas giants, Earth’s jets meander with large amplitudes, and there is evidence this amplitude increases with warming of the Arctic region (e.g., Francis & Vavrus, 2015). Gas-giant jets simply do not meander, with Saturn’s mid-northern-latitude Ribbon wave being the exception that proves the rule (Sánchez-Lavega, 2002). Thomson and McIntyre (2016) argue this lack of meandering suggests Jupiter’s atmosphere is inside the stability region (i.e., “supersonic” with “*Ma*” > 1), since their model exhibits long-wavelength meandering near marginal stability (but note their simulations are periodic in the meridional direction, which accentuates meandering compared to conditions on Jupiter). Jupiter’s winds are not likely to be overly stable, because the stable case “*Ma*” > 1 corresponds to

such that the deep jets are *stronger* in the stable case than in the marginally stable case, not weaker. The *Juno* gravity results are not yet sensitive enough to distinguish between these scenarios (Kaspi et al., 2018) but do confirm the *Voyager* vorticity results that Jupiter’s jets are deeply rooted.

## Interior Circulations

The interior of a gas giant is a different kettle of fish than its atmosphere. Interior jets experience a different geometry; there is the whole kettle rather than just the spherical shell. The gyroscopic Taylor-Proudman effect promotes cylindrical structures parallel to the rotation axis (Busse, 1976). These must follow the confines of the sphere, so Taylor columns tend to stretch as they move toward the rotation axis. Stretching is in the denominator of PV, whereas $f\approx {f}_{0}+\beta y$ is in the numerator, so this produces a change in PV opposite to the sign of $\beta $ . Ingersoll and Pollard (1982) estimate this ${\beta}_{\text{deep}}$ to be $-3$ to $-4\beta $ . The immediate implication for Jupiter’s deep jets is they are likely to be barotropically stable, or marginally stable, in such a $-3\beta $ geometry.

There are two main complications to the cylindrical-radius layout. First, density changes by four orders of magnitude in the spherical-radius direction in Jupiter’s interior. Kaspi et al. (2009) incorporate this variation into an anelastic model to simulate the deep-wind structure of the gas giants. (An anelastic model filters out sound waves, which are unnecessary to the large-scale dynamics and computationally prohibitive, while maintaining a realistic basic-state density profile.) An equatorial superrotation is produced by the collective effect of Taylor-column convective eddies aligned with the rotation axis.

The second complication comes from electrical conductivity in the interior, which generates Ohmic dissipation (Liu, Goldreich, & Stevenson, 2008). Hydrogen sits on the left-hand side of the periodic table of the elements, right above lithium; it is in the alkali metal group. Metallic hydrogen sounds exotic, but it fills the interiors of Jupiter and Saturn and is the most common planetary material in the solar system. Schneider and Liu (2009) find the concomitant Ohmic dissipation acts preferentially on extratropical flows and, together with differential solar insolation and internal heating, generates both an equatorial superrotation and alternating extratropical jets like observed. Equations of state for hydrogen-helium mixtures appropriate for gas-giant interiors are difficult to measure in the laboratory because of the megabar pressures, but theoretical calculations (*ab initio*, “from the beginning”) continue to increase in accuracy. Using an advanced equation of state, Wicht et al. (2017) model the molecular-to-metallic hydrogen transition and find zonal-jet systems in the molecular envelope similar to what is observed on Jupiter and Saturn; they also capture realistic aspects of Jupiter’s magnetic field.

One of the most interesting regions is the transition between the interior and the atmosphere of a gas giant, because there are fundamental questions about how thermal energy and momentum are transported vertically. Modelers are starting to address this region with realistic simulations (e.g., Heimpel, Gastine, & Wicht, 2015), and *Juno* is providing a wealth of new information via its Microwave Radiometer (Bolton et al., 2017). Realistic models are crucial to jet research, but they must be complemented by simple models to advance understanding (Held, 2005). The simplest model that captures both the spherical-shell and deep-interior geometries is the two- $\beta $ model, which has two active, quasigeostrophic layers, a weather layer with $\beta $, overlying an abyssal layer with $-3\beta $ . Kaspi and Flierl (2007) show alternating jets form in this model starting from random PV perturbations, even with weak vertical (baroclinic) shear. The weather-layer jets are sharp enough for the gradient of the absolute vorticity $\zeta +f$ to reverse at several latitudes as observed. The *Juno* results are beginning to clarify the dynamics of gas-giant interiors, and this is an active area of research that is benefiting from the hierarchical approach to modeling.

Polar Jets

Now the seats are all empty, let the roadies take the stage.

Pack it up and tear it down.

They're the first to come and last to leave, working for that minimum wage.

They'll set it up in another town.—Jackson Brown,

The Load Out

A planet-encircling jet is in many respects a large, annular vortex with a planet inside it. Near each pole, jets are exactly this: they form a polar vortex. These structures are fixtures on the giant planets and on Venus, whereas they are wintertime structures on Earth, Mars, and Titan. Polar jets set up effective barriers to meridional mixing and thus have a large effect on the dynamics and chemistry in the hemisphere in which they reside. Sánchez-Lavega (2011, Section 9.7.2) gives a succinct description of the polar dynamics on each planet.

## Hexagon on Saturn

Arguably the most beautiful polar jet is Saturn’s at 78º north latitude, because it has a planet-sized hexagon on it (Godfrey, 1988; see Figure 7). *Cassini* and ground-based imaging confirm both the jet and its hexagon survive Saturn’s long polar night virtually unchanged (Sánchez-Lavega et al., 2014). Between 2008 and 2014, the hexagon had a steady rotation period of ${10}^{h}{39}^{m}23.01\pm {0.01}^{s}$ . This is ${3.5}^{s}$ longer than in the 1980s and 1990s, when a large anticyclone pressed against one of its sides. Most interpretations of the hexagon are in terms of a Rossby wave, with details differing about how the wave may be trapped and/or forced (e.g., Aguiar, Read, Wordsworth, Salter, & Yamazaki, 2010; Allison, Godfrey, & Beebe, 1990; Morales-Juberias, Sayanagi, Dowling, & Ingersoll, 2011). The hexagon’s rotation period is within seconds of the Saturn Kilometric Radiation (SKR) period as measured by *Voyager 1* in November 1980 and *Voyager 2* in August 1981. However, *Cassini* discovered the SKR period varies with time and is different in the northern and southern hemispheres (Gurnett et al., 2009), hence it cannot be Saturn’s internal rotation period but rather is an external magnetospheric period tied indirectly to the planet’s rotation, like a clutch that slips. The hexagon’s phase speed is approximately $\u2013$ 20 ms^{–1} in the System IIIw reference frame, and the jet itself is approximately +60 ms^{–1} (Read, Dowling, & Schubert, 2009), implying the jet’s PV gradient is positive and large enough to make the jet “subsonic,” with “*Ma*” $\lesssim $ 0.75. Being “subsonic” with respect to vorticity waves is an unusual characteristic for a giant-planet jet, but this is an unusual jet.

In Saturn’s southern hemisphere, the corresponding polar jet is more than twice as fast as in the north, so probably is not “subsonic,” and is located significantly closer to the pole (see Figure 1). Although there is a warm, hurricane-like eye centered on each pole (Fletcher et al., 2008), there is no hexagon or other polygon around the south pole. Having northern and southern polar jets be different is not the end of the world, or rather it is, but Earth has polar bears only in the north and penguins only in the south, and no one is particularly surprised by this. When the polar jets of a gas giant turn out to be significantly different, perhaps it is surprising given the strong gyroscopic effects permeating throughout the interior, but the poles are separated by a nearly solid-body rotation of the interior, based on *Juno* gravity results for Jupiter (Guillot et al., 2018).

## Vortex Crystals on Jupiter

Jupiter’s polar regions are quite different than Saturn’s, as revealed by *Juno*. There is no polar eye or polygon in either hemisphere. Anticyclones are mostly expelled from the polar regions, and what have collected are steadfast cyclones, one near each pole, surrounded by a ring of others, all about the same size (Adriani et al., 2018; Orton et al., 2017). This stable pattern is like the “vortex crystal” configuration first described for magnetically confined electron columns (e.g., Driscoll, Jin, Schecter, Moreau, & Dubin, 2000). At the far end of the solar system, in a reanalysis of *Voyager* 2 images, Karkoschka (2015) sees a slightly off-axis polar vortex for Uranus, and Fletcher et al. (2014) see evidence of a warm polar eye on Neptune, reminiscent of Saturn.

## Set it Up and Tear it Down on Earth, Mars, and Titan

One gets accustomed to the unchanging jets on the giant planets, with the occasional exception (Sánchez-Lavega, Pérez-Hoyos, Rojas, Hueso, & French, 2003). Similar to the giant planets, Venus exhibits a permanent polar vortex at each pole, with a structure that varies and precesses over the course of 5 to 10 Earth days (Luz et al., 2011). These perpetual structures make the seasonal polar vortices on Earth, Mars, and Titan all the more fascinating. Like roadies, the eddies set up a winter-polar jet each fall, which then dissipates each spring. Charney and Drazin (1961) established that stationary Rossby waves propagate vertically in the wintertime, when the waves and wind are traveling in different directions but are cut off, or filtered, in the summertime, when both waves and wind are trying to move westward. Solomon (1999) reviews the history and chemistry of Earth’s stratospheric ozone depletion, which is tied to the dynamics of its stratospheric polar vortex. Titan’s winter polar vortex similarly interacts with trace chemicals in its atmosphere, as mapped by *Cassini* (Teanby et al., 2008).

On Mars, seasons are labeled by the solar longitude ${L}_{s}$, and there is virtually no thermal phase lag because of the thinness and dryness of the atmosphere. In practice the northern winter solstice, ${L}_{s}=270\xb0$, which corresponds to December 21 on Earth, is in the dead of winter rather than marking its beginning. The orbit of Mars is noticeably elliptical, and perihelion falls on ${L}_{s}=251\xb0$, such that southern summer gets blasted by the sun compared to northern summer. This results in homogenization of PV across the entire southern hemisphere and into the low latitudes of the northern hemisphere (Barnes & Haberle, 1996) and makes the northern-winter polar jet much stronger than the southern-winter polar jet. The polar jet on Mars has an annular PV structure (hollow over the pole), unlike on Earth. The latent heating in the polar region associated with CO_{2} condensation has been implicated in this structure (Toigo, Waugh, & Guzewich, 2016). Also, in the Mars Analysis Correction Data Assimilation atmospheric data set, the Froude number, *Fr*, drops from above critical (“shooting”) to below critical (“tranquil”) meridionally across the polar jet, implying a planetary-scale hydraulic jump, which correlates with the annular structure (Dowling, Bradley, Du, Lewis, & Read, 2017). Moreover, the polar jet exhibits transonic jet streaks (Dowling et al., 2016), whereas jets on Earth do not exceed *Ma* ~ 0.3.

## Barrier to Mixing

A polar jet sets up an impressive barrier to meridional mixing. Schoeberl, Lait, Newman, and Rosenfield (1992) describe the isolation of air inside Earth’s polar vortex, based on in situ observations from stratospheric expedition aircraft. Waves act as an erosion mechanism on the outside of the jet, but mixing rates drop by an order of magnitude or more across the jet. In the laboratory, Sommeria, Meyers, and Swinney (1989) injected red dye outside an eastward jet in a rotating water tank and demonstrated no transport across the jet for more than 500 rotations of the tank, even with a large wave (see Figure 8). A strong eastward jet eliminates wave breaking and mixing for at least two reasons: (a) there are no eastward waves with phase speeds that match its core speed, hence there are no critical lines (Bowman, 1996), and (b) there is a strong bunching together of the PV contours associated with the jet, which makes them move in unison (McIntyre, 2015, p. 35).

Discussion

Since World War II, the study of Earth’s jet streams has motivated many new discoveries in GFD and has had a significant impact on operational meteorology, while jets in planetary atmospheres have now been explored at close range for the better part of a century. Consider the first boots-on-the-ground Mars astronauts are in elementary school, and there are plans at various stages of technical maturity to fly unmanned drones in Titan’s atmosphere and hydrogen ramjets on gas giants. These endeavors require practical applications of jet research to ensure mission safety and success. The deployment of synoptic planetary observing platforms will expand as operational requirements ramp up, especially in support of human exploration of Mars, and this makes the newest developments in multiple small-satellite (e.g., cubesat) technology to support operational forecasting all the more exciting.

The confirmed deep jets on Jupiter underscore the need to better understand jet formation in gas-giant interiors. They also add weight to the idea that atmospheric jets are flexible, to the point of developing PV-gradient reversals, and perhaps to the point of being able to adjust their own deformation length. Great strides have been made in understanding the predilection for planetary jet streams to blow eastward and westward, or to be zonal. But Jupiter’s sharp westward jets and PV-gradient reversals continue to befuddle experimenters, numerical modelers, and applied mathematicians alike. In terms of rotating tanks, there is a need for jet experiments with sinusoidally varying bottom topography with respect to radius, to provide boundary conditions like the dynamic topography of Jupiter’s abyssal jets.

A top quest is to find the on-off switch for shear instability and to establish it rigorously. It has been proven that counterpropagating vorticity waves can mutually enhance each other, leading to shear instability, if they phase lock appropriately, and this phase-speed point of view has been reconciled with an alternative group-velocity point of view that considers overreflection (Harnik & Heifetz, 2007). Does this mean shear-instability theory should rest on the concept of locked eddy pairs? There is a famous precedent: the 1972 Nobel Prize in physics was awarded to the authors of the Cooper-pair theory of superconductivity, in which two electrons or other fermions (half-integer spin particles), bind at a distance to form a composite boson (whole-integer spin particle), such that multiple Cooper pairs can condense into the same quantum state, leading to superconductivity. Quantum mechanics is fundamentally simple in this sense—one satisfies a whole-integer requirement by adding two half integers—but continuum fluid dynamics would seem to require shear instability to be more a property of the shear itself than of special eddy pairs. In this sense, at a critical line the tenet that a shear crossing from “supersonic” to “subsonic” passes through a shock provides just such a property of the shear. Going forward, it is clear that planetary jets, most particularly Jupiter’s sharp westward jets, motivate a closer look at quasigeostrophic shear instability in the neighborhood $1-\epsilon \le -{\kappa}_{1}^{2}\partial \overline{\psi}/\partial \overline{q}\le 1$ .

A major realization is any alternating zonal-wind profile $\overline{u}\left(y\right)$ in the 1½ layer configuration can be made stable, and the formula for the abyssal jet required to achieve this, or equivalently the dynamic topography, is known. If the conjecture proves true that the shock of vorticity waves in shear is the necessary and sufficient condition for shear instability, then any alternating zonal-wind profile can also be made unstable—the on-off switch is found, and it is “*Ma*” $\lessgtr 1$ at a critical line. Near this configuration, the differences in the abyssal jets, or in the corresponding dynamic topography, which yield a stable shear versus an unstable one, are almost too fine to see by eye. Examples of this engineering control applied to Jupiter’s alternating jets, which are not sines and cosines, are given by Dowling (1993). To cross between disciplines and bring this level of engineering control to zonal flows in magnetized plasmas (e.g., Diamond et al., 2005) will require identification of an analogous dynamic topography for the Hasegawa-Mima equation. If this can be engineered, then fusion reactors can be designed to precisely manage zonal-wind shear stability the same way Jupiter does it (Dowling, 2003).

Acknowledgments

The author thanks Reta Beebe, Beth Bradley, Shawn Brueshaber, Nancy Chanover, Peter Davidson, Glenn Flierl, A. James Friedson, Peter Gierasch, Thomas Greathouse, Joseph Harrington, Aaron Herrnstein, Gary Hunt, Andrew Ingersoll, Donald Johnson, Yohai Kaspi, Raymond LeBeau, Conway Leovy, Stephen Lewis, Richard Lindzen, David Marshall, John Marshall, Raúl Morales-Juberías, Duane Muhleman, Mikhail Nezlin, Glenn Orton, Csaba Palotai, Joseph Pedlosky, Peter Read, Agustín Sánchez-Lavega, Kunio Sayanagi, Adam Showman, Amy Simon, Andrew Stamp, Geoff Stanley, David Stevenson, Peter Stone, Geoffrey Vallis, and Gerald Whitham for useful input.

## Further Reading

Galperin, B., & Read, P. L. (2019). *Zonal jets: Phenomenology, genesis, physics*. Cambridge, U.K.: Cambridge University Press.Find this resource:

This a compendium of review articles by leading experts.

Jets and annular structures in geophysical fluids. (2008). *Journal of the Atmospheric Sciences, 64–65*.Find this resource:

This is a special collection of 27 research papers published in 2007 and 2008.

Vallis, G. K. (2017). *Atmospheric and oceanic fluid dynamics: Fundamentals and large-scale circulation* (2nd ed.). Cambridge, U.K.: Cambridge University Press.Find this resource:

This is a readable and authoritative text covering jets.

## References

Adriani, A., Mura, A., Orton, G., Hansen, C., Altieri, F., Moriconi, M. L., . . . Amoroso, M. (2018). Clusters of cyclones encircling Jupiter’s poles. *Nature*, *555*, 216–219.Find this resource:

Aguiar, A. C. B., Read, P. L., Wordsworth, R. D., Salter, T., & Yamazaki, H. (2010). A laboratory model of Saturn’s north polar hexagon. *Icarus*, *206*, 755–763.Find this resource:

Allison, M., Godfrey, D. A., & Beebe, R. F. (1990). A wave dynamical interpretation of Saturn’s polar hexagon. *Science*, *247*, 1061–1063.Find this resource:

Ambrizzi, T., Hoskins, B. J., & Hsu, H.-H. (1995). Rossby wave propagation and teleconnection patterns in the austral winter. *Journal of Atmospheric Science*, *52*, 3661–3672.Find this resource:

Anderson, J. D., & Schubert, G. (2007). Saturn’s gravitational field, internal rotation, and interior structure. *Science*, *317*, 1384–1387.Find this resource:

Andrews, D. G. (1983). A finite-amplitude Eliassen-Palm theorem in isentropic coordinates. *Journal of Atmospheric Science*, *40*, 1877–1883.Find this resource:

Andrews, D. G., Holton, J. R., & Leovy, C. B. (1987). *Middle atmosphere dynamics*. New York, NY: Academic Press.Find this resource:

Archer, C. L., & Caldeira, K. (2008). Historical trends in the jet streams. *Geophysical Research Letters*, *35*, L08803.Find this resource:

Arnol’d, V. I. (1966). On an a priori estimate in the theory of hydrodynamic stability. *Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika*, *5*, 3–5.Find this resource:

Baldwin, M. P., Gray, L. J., Dunkerton, T. J., Hamilton, K., Haynes, P. H., Randel, W. J., . . . Takahashi, M. (2001). The quasi-biennial oscillation. *Reviews of Geophysics*, *39*, 179–230.Find this resource:

Baldwin, M. P., Rhines, P. B., Huang, H.-P., & McIntyre, M. E. (2007). The jet-stream conundrum. *Science*, *315*, 467–468.Find this resource:

Barnes, J. R., & Haberle, R. M. (1996). The Martian zonal-mean circulation: Angular momentum and potential vorticity structure in GCM simulations. *Journal of Atmospheric Science*, *53*, 3143–3156.Find this resource:

Bird, M. K., Allison, M., Asmar, S. W., Atkinson, D. H., Avruch, I. M., Dutta-Roy, R., . . . Tyler, G. L. (2005). The vertical profile of winds on Titan. *Nature*, *438*, 800–802.Find this resource:

Bishop, S. E. (1884). The equatorial smoke-stream from Krakatoa. *Hawaiian Monthly*.Find this resource:

Boyd, J. P. (2018). Nonlinear equatorial waves. In *Dynamics of the equatorial ocean* (pp. 329–404). Berlin, Germany: Springer.Find this resource:

Bolton, S. J., Adriani, A., Adumitroaie, V., Allison, M., Anderson, J., Atreya, S., . . . Wilson, R. (2017). Jupiter’s interior and deep atmosphere: The initial pole-to-pole passes with the *Juno* spacecraft. *Science*, *356*, 821–825.Find this resource:

Bowman, K. P. (1996). Rossby wave phase speeds and mixing barriers in the stratosphere. Part I: Observations. *Journal of Atmospheric Science*, *53*, 905–916.Find this resource:

Busse, F. H. (1976). A simple model of convection in the Jovian atmosphere. *Icarus*, *29*, 255–260.Find this resource:

Caballero, R., & Huber, M. (2010). Spontaneous transition to superrotation in warm climates simulated by CAM3. *Geophysical Research Letters*, *37*.Find this resource:

Charney, J. G., & Drazin, P. G. (1961). Propagation of planetary-scale disturbances from the lower into the upper atmosphere. *Journal of Geophysical Research*, *66*, 83–109.Find this resource:

Charney, J. G., & Stern, M. (1962). On the stability of internal baroclinic jets in a rotating atmosphere. *Journal of Atmospheric Science*, *19*, 159–172.Find this resource:

Claud, C., & Terray, P. (2007). Revisiting the possible links between the quasi-biennial oscillation and the Indian summer monsoon using NCEP R-2 and CMAP fields. *Journal of Climate*, *20*, 773–787.Find this resource:

Cushman-Roisin, B., & Beckers, J.-M. (2011). *Introduction to geophysical fluid dynamics*. New York, NY: Academic Press.Find this resource:

De Szoeke, R. A., & Springer, S. R. (2009). The materiality and neutrality of neutral density and orthobaric density. *Journal of Physical Oceanography*, *39*, 1779–1799.Find this resource:

Del Genio, A. D., Achterberg, R. K., Baines, K. H., Flasar, F. M., Read, P. L., Sánchez-Lavega, A., & Showman, A. P. (2009). Saturn atmospheric structure and dynamics. In M. K. Dougherty, L. W. Esposito, & S. M. Krimigis (Eds.), *Saturn from Cassini-Huygens* (pp. 113–159). Dordrecht, The Netherlands: Springer.Find this resource:

Diamond, P. H., Gurcan, O. D., Hahm, T. S., Miki, K., Kosuga, Y., & Garbet, X. (2008). Momentum theorems and the structure of atmospheric jets and zonal flows in plasmas. *Plasma Physics and Controlled Fusion*, *50*.Find this resource:

Diamond, P. H., Itoh, S.-I., Itoh, K., & Hahm, T. S. (2005). Zonal flows in plasma—a review. *Plasma Physics and Controlled Fusion*, *47*, R35–R161.Find this resource:

Dowling, T. E. (1993). A relationship between potential vorticity and zonal wind on Jupiter. *Journal of Atmospheric Science*, *50*, 14–22.Find this resource:

Dowling, T. E. (1995). Estimate of Jupiter’s deep zonal-wind profile from Shoemaker-Levy 9 data and Arnol’d’s second stability criterion. *Icarus*, *117*, 439–442.Find this resource:

Dowling, T. E. (2003). Jupiter. In H. Mark (Ed.), *Encyclopedia of space science and technology* (pp. 871–889). New York, NY: Wiley.Find this resource:

Dowling, T. E. (2008). Planetary science: Music of the stratospheres. *Nature*, *453*, 163–164.Find this resource:

Dowling, T. E. (2014). Saturn’s longitude: Rise of the second branch of shear-stability theory and fall of the first. *International Journal of Modern Physics D*, *23*, 1–32.Find this resource:

Dowling, T. E., Bradley, M. E., Du, J., Lewis, S. R., & Read, P. L. (2017). Ertel potential vorticity versus Bernoulli streamfunction on Mars. *Quarterly Journal of the Royal Meteorological Society*, *143*, 37–52.Find this resource:

Dowling, T. E., Bradley, M. E., Lewis, S. R., & Read, P. L. (2016). Planetary-scale hydraulic jumps and transonic jet-streaks in the MACDA reanalysis of the Martian atmosphere: A comparison with Earth’s lower mesosphere. *Bulletin of the American Astronomical Society*, *48*(5), 98.Find this resource:

Dowling, T. E., & Ingersoll, A. P. (1989). Jupiter’s Great Red Spot as a shallow water system. *Journal of Atmospheric Science*, *46*, 3256–3278.Find this resource:

Driscoll, C. F., Jin, D. Z., Schecter, D. A., Moreau, E. J., & Dubin, D. H. E. (2000). Dynamics, statistics and vortex crystals in the relaxation of 2D turbulence. *Physica Scripta*, *T84*, 76–80.Find this resource:

Driscoll, D. G., & McIntyre, M. E. (2008). Multiple jets as PV staircases: The Phillips effect and the resilience of eddy-transport barriers. *Journal of Atmospheric Science*, *65*, 855–874.Find this resource:

Du, J., Dowling, T. E., & Bradley, M. E (2015). Ertel potential vorticity versus Bernoulli streamfunction in earth’s extratropical atmosphere. *Journal in Advances in Modeling Earth Systems*, *7*, 437–458.Find this resource:

Ertel, H. (1942). Ein neuer hydrodynamischer Wirbelsatz. *Meteorologische Zeitschrift*, *59*, 277–281.Find this resource:

Feldstein, S. B. (1991). A comparison of the weakly nonlinear instability of westerly and easterly jets in a two-layer beta-plane model. *Journal of Atmospheric Science*, *48*(15), 1701–1717.Find this resource:

Fels, S. B., & Lindzen, R. S. (1974). The interaction of thermally excited gravity waves with mean flows. *Geophysical & Astrophysical Fluid Dynamics*, *6*, 149–191.Find this resource:

Fjørtoft, R. (1950). Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. *Geofysiske Publikasjoner*, *17*, 1–52.Find this resource:

Fletcher, L. N., de Pater, I., Orton, G. S., Hammel, H. B., Sitko, M. L., & Fry, P. G. J. (2014). Neptune at summer solstice: Zonal mean temperatures from ground based observations, 2003–2007. *Icarus*, *231*, 146–167.Find this resource:

Fletcher, L. N., Irwin, P. G., Orton, G. S., Teanby, N. A., Achterberg, R. K., Bjoraker, G. L., . . . Simon-Miller, A. A. (2008). Temperature and composition of Saturn’s polar hot spots and hexagon. *Science*, *319*, 79–81.Find this resource:

Fouchet, T., Guerlet, S., Strobel, D. F., Simon-Miller, A. A., Bézard, B., & Flasar, F. M. (2008). An equatorial oscillation in Saturn’s middle atmosphere. *Nature*, *453*, 200–202.Find this resource:

Francis, J. A., & Vavrus, S. J. (2015). Evidence for wavier jet stream in response to rapid Arctic warming. *Environmental Research Letters*, *10*.Find this resource:

Galperin, B., & Read, P. L. (2019). *Zonal jets: Phenomenology, genesis, physics*. Cambridge, U.K.: Cambridge University Press.Find this resource:

Galperin, B., Sukoriansky, S., & Dikovskaya, N. (2008). Zonostrophic turbulence. *Physica Scripta*, *112*, 132–172.Find this resource:

Gerkema, T., Zimmerman, J. T. F., Maas, L. R. M., & van Haren, H. (2008). Geophysical and astrophysical fluid dynamics beyond the traditional approximation. *Reviews of Geophysics*, *46*.Find this resource:

Gierasch, P. J., Ingersoll, A. P., Banfield, D., Ewald, S. P., Helfenstein, P., Simon-Miller, A., . . . Galileo Imaging Team. (2000). Observation of moist convection in Jupiter’s atmosphere. *Nature*, *403*, 628–630.Find this resource:

Gill, A. E. (1982). *Atmosphere-ocean dynamics*. New York, NY: Academic Press.Find this resource:

Godfrey, D. A. (1988). A hexagonal feature around Saturn’s north pole. *Icarus*, *76*, 335–356.Find this resource:

Guillot, T., Miguel, Y., Militzer, B., Hubbard, W. B., Kaspi, Y., Galanti, E., . . . Bolton, S. J. (2018). A suppression of differential rotation in Jupiter’s deep interior. *Nature*, *555*, 227–230.Find this resource:

Gurnett, D. A., Lecacheux, A., Kurth, W. S., Persoon, A. M., Groene, J. B., Lamy, L., . . . Carbary, J. F. (2009). Discovery of a north-south asymmetry in Saturn’s radio rotation period. *Geophysical Research Letters*, *36*, L16102.Find this resource:

Hanel, R. A., Conrath, B. J., Jennings, D. E., & Samuelson, R. E. (1992). *Exploration of the solar system by infrared remote sensing*. Cambridge, U.K.: Cambridge University Press.Find this resource:

Harnik, N., & Heifetz, E. (2007). Relating overreflection and wave geometry to the counterpropagating Rossby wave perspective: Toward a deeper mechanistic understanding of shear instability. *Journal of Atmospheric Science*, *64*, 2238–2261.Find this resource:

Heimpel, M., Gastine, T., & Wicht, J. (2015). Simulation of deep-seated zonal jets and shallow vortices in gas giant atmospheres. *Nature Geoscience*, *9*, 19–23.Find this resource:

Held, I. M. (1983). Stationary and quasi-stationary eddies in the extratropical troposphere: Theory. In B. J. Hoskins & R. P. Pearce (Eds.), *Large-scale dynamical processes in the atmosphere* (pp. 127–168). London, U.K.: Academic Press.Find this resource:

Held, I. M. (2005). The gap between simulation and understanding in climate modeling. *Bulletin of the American Meteorological Society*, *86*, 1609–1614.Find this resource:

Held, I. M., & Hou, A. Y. (1980). Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. *Journal of Atmospheric Science*, *37*, 515–533.Find this resource:

Hide, R. (1969). Dynamics of the atmospheres of the major planets with an appendix on the viscous boundary layer at the rigid bounding surface of an electrically conducting rotating fluid in the presence of a magnetic field. *Journal of Atmospheric Science*, *26*, 841–853.Find this resource:

Holm, D. D. (1996). Hamiltonian balance equations. *Physica D*, *98*, 379–414.Find this resource:

Holton, J. R., & Hakim, G. J. (2013). *An introduction to dynamic meteorology*. New York, NY: Academic Press.Find this resource:

Holton, J. R., & Lindzen, R. S. (1972). An updated theory of the quasi-biennial oscillation of the tropical stratosphere. *Journal of Atmospheric Science*, *29*, 1076–1080.Find this resource:

Horinouchi, T., Murakami, S., Satoh, T., Peralta, J., Ogohara, K., Kouyama, T., . . . Young, E. F. (2017). Equatorial jet in the lower to middle cloud layer of Venus revealed by Akatsuki. *Nature Geoscience*, *10*, 646–651.Find this resource:

Hoskins, B. J., McIntyre, M. E., & Robertson, A. W. (1985). On the use and significance of isentropic potential vorticity maps. *Quarterly Journal of the Royal Meteorological Society*, *111*, 877–946.Find this resource:

Howe, R., Christensen-Dalsgaard, J., Hill, F., Komm, R. W., Larsen, R.M., Schou, J., . . . Toomre, J. (2000). Deeply penetrating banded zonal flows in the solar convection zone. *The Astrophysical Journal*, *533*, L163–L166.Find this resource:

Hsu, Y.-J. G., & Arakawa, A. (1990). Numerical modeling of the atmosphere with an isentropic vertical coordinate. *Monthly Weather Review*, *118*, 1933–1959.Find this resource:

Ingersoll, A. P. (1990). Atmospheric dynamics of the outer planets. *Science*, *248*, 308–315.Find this resource:

Ingersoll, A. P., Beebe, R. F., Mitchell, J. L., Garneau, G. W., Yagi, G. M., & Müller, J.-P. (1981). Interaction of eddies and mean zonal flow on Jupiter as inferred from Voyager 1 and 2 images. *Journal of Geophysical Research*, *86*, 8733–8743.Find this resource:

Ingersoll, A. P., & Cuong, P. G. (1981). Numerical model of long-lived Jovian vortices. *Journal of Atmospheric Science*, *38*, 2067–2076.Find this resource:

Ingersoll, A. P., & Pollard, D. (1982). Motions in the interiors and atmospheres of Jupiter and Saturn: scale analysis, anelastic equations, barotropic stability criterion. *Icarus*, *52*, 62–80.Find this resource:

Ingersoll, A. P., & Porco, C. C. (1978). Solar heating and internal heat flow on Jupiter. *Icarus*, *35*, 27–43.Find this resource:

Ioannou, P. J., & Lindzen, R. S. (1994). Gravitational tides on Jupiter. III. Atmospheric response and mean flow acceleration. *Astrophysics Journal*, *424*, 1005–1013.Find this resource:

John, F. (1982). *Partial differential equations*. New York, NY: Springer-Verlag.Find this resource:

Johnson, D. R. (1997). “General coldness of climate models” and the Second Law: Implications for modeling the earth system. *Journal of Climate*, *10*, 2826–2846.Find this resource:

Johnson, P. E., Morales-Juberías, R., Simon, A., Gaulme, P., Wong, M. H., & Consentino, R. G. (2018). Longitudinal variability in Jupiter’s zonal winds derived from multi-wavelength HST observations. *Planetary and Space Science*, *155*, 2–11.Find this resource:

Kaladze, T., Rogava, J., Tsamalashvili, L., & Tsiklauri, M. (2005). Investigation and numerical resolution of initial-boundary value problem for the generalized Charney-Obukhov and Hasegawa-Mima equations. *Physics Letters A*, *343*, 199–215.Find this resource:

Kaplan, M. L., Huang, C., Lin, Y.-L., & Charney, J. J. (2008). The development of extremely dry surface air due to vertical exchanges under the exit region of a jet streak. *Meteorology and Atmospheric Physics*, *102*, 63–85.Find this resource:

Karkoschka, E. (2015). Uranus’ southern circulation revealed by Voyager 2: Unique characteristics. *Icarus*, *250*, 294–307.Find this resource:

Karoly, D. (1983). Rossby wave propagation in a barotropic atmosphere. *Dynamics of Atmospheres and Oceans*, *7*, 111–125.Find this resource:

Kaspi, Y., & Flierl, G. R. (2007). Formation of jets by baroclinic instability on gas planet atmospheres. *Journal of Atmospheric Science*, *64*, 3177–3194.Find this resource:

Kaspi, Y., Flierl, G. R., & Showman, A. P. (2009). The deep wind structure of the giant planets: Results from an anelastic general circulation model. *Icarus*, *202*, 525–542.Find this resource:

Kaspi, Y., Galanti, E., Hubbard, W. B., Stevenson, D. J., Bolton, S. J., Iess, L., . . . Wahl, S. M. (2018). Jupiter’s atmospheric jet streams extend thousands of kilometres deep. *Nature*, *555*, 223–226.Find this resource:

Kittel, C., & Kroemer, H. (1980). *Thermal physics* (2nd ed.). New York, NY: W. H. Freeman.Find this resource:

Kuo, H. L. (1949). Dynamic instability of two-dimensional non-divergent flow in a barotropic atmosphere. *Journal of Meteorology*, *6*, 105–122.Find this resource:

Leovy, C. B., Friedson, A. J., & Orton, G. S. (1991). The quasi-quadrennial oscillation of Jupiter’s equatorial stratosphere. *Nature*, *354*, 380–382.Find this resource:

Lewis, J. M. (2003). Ooishi’s observation: Viewed in the context of jet stream discovery. *Bulletin of the American Meteorological Society*, *84*, 357–369.Find this resource:

Lewis, S. R., & Read, P. L. (2003). Equatorial jets in the dusty Martian atmosphere. *Journal of Geophysical Research*, *108*.Find this resource:

Lian, Y., & Showman, A. P. (2010). Generation of equatorial jets by large-scale latent heating on the giant planets. *Icarus*, *207*, 373–393.Find this resource:

Limaye, S. S., & Suomi, V. E. (1981). Cloud motions on Venus: Global structure and organization. *Journal of Atmospheric Science*, *38*, 1220–1235.Find this resource:

Liu, J., Goldreich, P. M., & Stevenson, D. J. (2008). Constraints on deep-seated zonal winds inside Jupiter and Saturn. *Icarus*, *196*, 653–664.Find this resource:

Liu, T., Wang, B., & Choi, D. S. (2012). Flow structures of Jupiter’s Great Red Spot extracted by using optical flow method. *Physics of Fluids*, *24*.Find this resource:

Longuet-Higgins, M. S. (1964). On group velocity and energy flux in planetary wave motions. *Deep-Sea Research*, *11*, 35–42.Find this resource:

Luz, D., Berry, D. L., Piccioni, G., Drossart, P., Politi, R., Wilson, C. F., . . . Nuccilli, F. (2011). Venus’s southern polar vortex reveals precessing circulation. *Science*, *332*, 577–580.Find this resource:

Manfroi, A., & Young, W. (1999). Slow evolution of zonal jets on the beta plane. *Journal of Atmospheric Science*, *56*, 784–800.Find this resource:

Marcus, P. S., & Shetty, S. (2011). Jupiter’s zonal winds: Are they bands of homogenized potential vorticity organized as a monotonic staircase? *Philosophical Transactions of the Royal Society A*, *369*, 771–795.Find this resource:

McIntyre, M. E. (2015). The atmospheric wave-turbulence jigsaw. In P. H. Diamond (Ed.), *Rotation and momentum transport in magnetized plasmas* (pp. 1–43). Hackensack, NJ: World Scientific.Find this resource:

McIntyre, M. E., & Shepherd, T. G. (1987). An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol’d’s stability theorems. *Journal of Fluid Mechanics*, *181*, 527–565.Find this resource:

Montgomery, D., & Joyce, G. (1974). Statistical mechanics of “negative temperature” states. *Physics of Fluid*, *17*, 1139–1145.Find this resource:

Morales-Juberías, R., Sayanagi, K. M., Dowling, T. E., & Ingersoll, A. P. (2011). Emergence of polar-jet polygons from jet instabilities in a Saturn model. *Icarus*, *211*, 1284–1293.Find this resource:

Morize, C., Le Bars, M., Le Gal, P., & Tilgner, A. (2010). Experimental determination of zonal winds driven by tides. *Physical Review Letters*, *104*, 214501.Find this resource:

Munson, B. R., Young, D. F., & Okiishi, T. H. (2006). *Fundamentals of fluid mechanics* (5th ed.). New York, NY: Wiley.Find this resource:

O’Neil, T. M., & Smith, R. A. (1992). Stability theorem for off-axis states of a non-neutral plasma column. *Physics of Fluids B*, *4*, 2720–2728.Find this resource:

Orton, G. S., Hansen, C., Caplinger, M., Ravine, M., Atreya, S., Ingersoll, A. P., . . . Bolton, S. (2017). The first close-up images of Jupiter’s polar regions: Results from the *Juno* mission JunoCam instrument. *Geophysical Research Letters*, *44*.Find this resource:

Pedlosky, J. (1987). *Geophysical fluid dynamics* (2nd ed.). New York, NY: Springer-Verlag.Find this resource:

Peek, B. M. (1981). *The planet Jupiter* (2nd ed.). London, U.K.: Faber and Faber. (Original work published 1958)Find this resource:

Pierce, R. B., Fairlie, T. D., Grose, W. L., Swinbank, R., & O’Neill, A. (1994). Mixing processes within the polar night jet. *Journal of Atmospheric Science*, *51*, 2957–2972.Find this resource:

Pierrehumbert, R. (2010). *Principles of planetary climate*. Cambridge, U.K.: Cambridge University Press.Find this resource:

Platzman, G. W. (1968). The Rossby wave. *Quarterly Journal of the Royal Meteorological Society*, *94*, 225–248.Find this resource:

Plumb, R. A. (1977). The interaction of two internal waves with the mean flow: Implications for the theory of the quasi-biennial oscillation. *Journal of Atmospheric Science*, *34*, 1847–1858.Find this resource:

Plumb, R. A., & McEwan, D. (1978). The instability of a forced standing wave in a viscous stratified fluid: A laboratory analogue of the quasi-biennial oscillation. *Journal of Atmospheric Science*, *35*, 1827–1839.Find this resource:

Porco, C. C., West, R. A., McEwen, A., Del Genio, A. D., Ingersoll, A. P., Thomas, P., . . . Vasavada, A. R. (2003). Cassini imaging of Jupiter’s atmosphere, satellites, and rings. *Science*, *299*, 1541–1547.Find this resource:

Potter, S. F., Vallis, G. K., & Mitchell, J. L. (2014). Spontaneous superrotation and the role of Kelvin waves in an idealized dry GCM. *Journal of Atmospheric Science*, *71*, 596–614.Find this resource:

Randel, W. J., & Held, I. M. (1991). Phase speed spectra of transient eddy fluxes and critical layer absorption. *Journal of Atmospheric Science*, *48*, 688–697.Find this resource:

Rayleigh, L. (1880). On the stability, or instability, of certain fluid motions. *Proceedings of the London Mathematical Society*, *9*, 57–70.Find this resource:

Read, P. L. (2011). Dynamics and circulation regimes of terrestrial planets. *Planetary and Space Science*, *59*, 900–914.Find this resource:

Read, P. L., Conrath, B. J., Fletcher, L. N., Gierasch, P. J., Simon-Miller, A. A., & Zuchowski, L. C. (2009). Mapping potential vorticity dynamics on Saturn: Zonal mean circulation from Cassini and Voyager data. *Planetary and Space Science*, *5714*, 1682–1698.
Find this resource:

Read, P. L., Dowling, T. E., & Schubert, G. (2009). Saturn’s rotation period from its atmospheric planetary-wave configuration. *Nature*, *460*, 608–610.Find this resource:

Read, P. L., Gierasch, P. J., Conrath, B. J. Simon-Miller, A. A., Fouchet, T., & Yamazaki, Y. H. (2006). Mapping potential vorticity dynamics on Jupiter. I: Zonal-mean circulation from Cassini and Voyager 1 data. *Quarterly Journal of the Royal Meteorological Society*, *132*, 1577–1603.Find this resource:

Rhines, P. B., & Young, W. R. (1982). Homogenization of potential vorticity in planetary gyres. *Journal of Fluid Mechanics*, *122*, 347–367.Find this resource:

Ripa, P. (1983). General stability conditions for zonal flows in a one-layer model on the β-plane or the sphere. *Journal of Fluid Mechanics*, *126*, 463–489.
Find this resource:

Rogers, J. H., Fletcher, L. N., Adamoli, G., Jacquesson, M., Vedovato, M., & Orton, G. S. (2016). A dispersive wave pattern on Jupiter’s fastest retrograde jet at 20°S. *Icarus*, *277*, 354–369.Find this resource:

Rossby, C. G. (1940). Planetary flow patterns in the atmosphere. *Quarterly Journal of the Royal Meteorological Society*, *66*(Suppl.), 68–97.Find this resource:

Salby, M. L. (1996). *Fundamentals of atmospheric physics*. San Diego, CA: Elsevier.Find this resource:

Salmon, R. (1988a). Hamiltonian fluid mechanics. *Annual Review of Fluid Mechanics*, *20*, 225–256.Find this resource:

Salmon, R. (1988b). Semigeostrophic theory as a Dirac-bracket projection. *Journal of Fluid Mechanics*, *196*, 345–358.Find this resource:

Sánchez-Lavega, A. (2002). Observations of Saturn’s Ribbon wave 14 years after its discovery. *Icarus*, *158*, 272–275.Find this resource:

Sánchez-Lavega, A. (2011). An introduction to planetary atmospheres (587 p.). Boca Raton, FL: Taylor & Francis.Find this resource:

Sánchez-Lavega, A., del Río-Gaztelurrutia, T., Hueso, R., Pérez-Hoyos, S., García-Melendo, E., Antuñano, A., . . . Wesley, A. (2014). The long-term steady motion of Saturn’s hexagon and the stability of its enclosed jet stream under seasonal changes. *Geophysical Research Letters*, *41*, 1425–1431.Find this resource:

Sánchez-Lavega, A., Pérez-Hoyos, S., Rojas, J. F., Hueso, R., & French, R. G. (2003). A strong decrease in Saturn’s equatorial jet at cloud level. *Nature*, *423*, 623–625.Find this resource:

Sánchez-Lavega, A., Sromovsky, L. A., Showman, A. P., Del Genio, A. D., Young, R. M. B., Hueso, R., . . . Barbara, J. M. (2019). Gas giants. In B. Galperin & P. L. Read (Eds.), *Zonal jets: Phenomenology, genesis, physics* (pp. 72–103). Cambridge, U.K.: Cambridge University Press.Find this resource:

Saravanan, R. (1993). Equatorial superrotation and maintenance of the general circulation in two-level models. *Journal of Atmospheric Science*, *50*, 1211–1227.Find this resource:

Schneider, E. K. (1977). Axially symmetric steady-state models of the basic state for instability and climate studies. Part II. Nonlinear calculations. *Journal of Atmospheric Science*, *34*, 280–296.Find this resource:

Schneider, T., & Liu, J. (2009). Formation of jets and equatorial superrotation on Jupiter. *Journal of Atmospheric Science*, *66*, 579–601.Find this resource:

Schoeberl, M. R., Lait, L. R., Newman, P. A., & Rosenfield, J. E. (1992). The structure of the polar vortex. *Journal of Geophysical Research*, *97*, 7859–7882.Find this resource:

Schubert, G., & Covey, C. (1981). The atmosphere of Venus. *Scientific American*, *245*, 66–75.Find this resource:

Schubert, W., Ruprecht, E., Hertenstein, R., Ferreira, R. N., Taft, R., Rozoff, C., . . . Kuo, H.-C. (2004). English translations of twenty-one of Ertel’s papers on geophysical fluid dynamics. *Meteorologische Zeitschrift*, *13*, 527–576.Find this resource:

Scott, R. K., & Dunkerton, T. J. (2017). Vertical structure of tropospheric winds on gas giants. *Geophysical Research Letters*, *44*, 3073–3081.Find this resource:

Shaman, J., & Tziperman, E. (2016). The superposition of eastward and westward Rossby waves in response to localized forcing. *Journal of Climate*, *29*, 7547–7557.Find this resource:

Shepherd, T. G. (1990). Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. *Advances in Geophysics*, *32*, 287–339.Find this resource:

Shepherd, T. G. (2003). Ripa’s theorem and its relatives. In O. U. Velasco Fuentes, J. Sheinbaum, & J. Ochoa (Eds.), *Nonlinear processes in geophysical fluid dynamics* (pp. 1–14). New York, NY: Springer.Find this resource:

Showman, A. P., & Guillot, T. (2002). Atmospheric circulation and tides of “51 Pegasus b-like” planets. *Astronomy & Astrophysics*, *385*, 166–180.Find this resource:

Showman, A. P., & Polvani, L. M. (2011). Equatorial superrotation on tidally locked exoplanets. *The Astrophysical Journal*, *738*.Find this resource:

Smith, B. A., Soderblom, L., Beebe, R., Boyce, J. M., Brahic, A., Briggs, G. A., . . . Veverka, J. (1986). Voyager 2 in the Uranian system: Imaging science results. *Science*, *233*, 43–64.Find this resource:

Solomon, S. (1999). Stratospheric ozone depletion: A review of concepts and history. *Reviews of Geophysics*, *37*, 275–316.Find this resource:

Sommeria, J., Meyers, S. D., & Swinney, H. L. (1989). Laboratory model of a planetary eastward jet. *Nature*, *337*, 58–61.Find this resource:

Sromovsky, L. A., de Pater, I., Fry, P. M., Hammel, H. B., & Marcus, P. (2015). High s/n Keck and Gemini AO imaging of Uranus during 2012–2014: New cloud patterns, increasing activity, and improved wind measurements. *Icarus*, *258*, 192–223.Find this resource:

Stamp, A. P., & Dowling, T. E. (1993). Jupiter’s winds and Arnol’d’s second stability theorem: Slowly moving waves and neutral stability. *Journal of Geophysical Research*, *98*(8), 847–855.Find this resource:

Suarez, M. J., & Duffy, D. G. (1992). Terrestrial superrotation: A bifurcation of the general circulation. *Journal of Atmospheric Science*, *49*, 1541–1554.Find this resource:

Teanby, N. A., de Kok, R., Irwin, P. G. J., Osprey, S., Vinatier, S., Gierasch, P. J., . . . Calcutt, S. B. (2008). Titan’s winter polar vortex structure revealed by chemical tracers. *Journal of Geophysical Research*, *113*.Find this resource:

Thomson, R. (1971). Why there is an intense eastward current in the North Atlantic but not in the South Atlantic. *Journal of Physical Oceanography*, *1*, 235–237.Find this resource:

Thomson, S. I., & McIntyre, M. E. (2016). Jupiter’s unearthly jets: A new turbulent model exhibiting statistical steadiness without large-scale dissipation. *Journal of Atmospheric Science*, *73*, 1119–1141.Find this resource:

Thuburn, J., & Craig, G. C. (2000). Stratospheric influence on tropopause height: The radiative constraint. *Journal of Atmospheric Science*, *57*, 17–28.Find this resource:

Toigo, A. D., Waugh, D. W., & Guzewich, S. D. (2016). What causes Mars’ annular polar vortices? *Geophysical Research Letters*, *44*, 71–78.Find this resource:

Turkington, B., Majda, A., Haven, K., & DiBattista, M. (2001). Statistical equilibrium predictions of jets and spots on Jupiter. *Proceedings of the National Academy of Sciences of the United States of America*, *98*, 12346–12350.Find this resource:

Uccelini, L. W. (1980). On the role of upper tropospheric jet streaks and leeside cyclogenesis in the development of low-level jets in the Great Plains. *Monthly Weather Review*, *108*, 1689–1696.Find this resource:

Uccelini, L. W., & Kocin, P. J. (1987). The interaction of jet streak circulations during heavy snow events along the east coast of the United States. *Weather and Forecasting*, *2*, 289–308.Find this resource:

Vallis, G. K. (2017). *Atmospheric and oceanic fluid dynamics: Fundamentals and large-scale circulation* (2nd ed.). Cambridge, U.K.: Cambridge University Press.Find this resource:

Vallis, G. K., & Maltrud, M. E. (1993). Generation of mean flows and jets on a beta plane and over topography. *Journal of Physical Oceanography*, *23*, 1346–1362.Find this resource:

Vasavada, A. R., Ingersoll, A. P., Banfield, D., Bell, M., & Gierasch, P. J. (1998). Galileo imaging of Jupiter’s atmosphere: The Great Red Spot, Equatorial Region, and White Ovals. *Icarus*, *135*, 265–275.Find this resource:

Wedi, N. P., & Smolarkiewicz, P. K. (2006). Direct numerical simulation of the Plumb-McEwan laboratory analog of the QBO. *Journal of Atmospheric Science*, *63*, 3226–3252.Find this resource:

Whitham, G. B. (1974). *Linear and nonlinear waves*. New York, NY: Wiley.Find this resource:

Wicht, J., French, M., Stellmach, S., Nettelmann, N., Gastine, T., Duarte, L., & Redmer, R. (2017). Modeling the interior dynamics of gas planets. In H Lühr(Ed.), *Magnetic fields in the solar system* (pp. 7–81). Astrophysics and Space Science Library 448. Cham, Switzerland: Springer.Find this resource:

Williams, G. P. (2003). Jovian dynamics. Part III: Multiple, migrating, and equatorial jets. *Journal of Atmospheric Science*, *60*, 1270–1296.Find this resource:

Yen, B. C. (1991). *Channel flow resistance: Centennial of Manning’s formula*. Highlands Ranch, CO: Water Resources Publications.Find this resource:

Young, R. M. B., & Read, P. L. (2017). Forward and inverse kinetic energy cascades in Jupiter’s turbulent weather layer. *Nature Physics*, *13*, 1135–1140.Find this resource:

Young, W. R. (2012). An exact thickness-weighted average formulation of the Boussinesq equations. *Journal of Physical Oceanography*, *42*, 692–707.Find this resource:

Young, W. R., & Rhines, P. B. (1980). Rossby wave action, enstrophy and energy in forced mean flows. *Geophysical & Astrophysical Fluid Dynamics*, *15*, 39–52.Find this resource:

Zurita-Gotor, P., & Held, I. M. (2018). The finite amplitude evolution of mixed Kelvin-Rossby wave instability and equatorial superrotation in a shallow water model and an idealized GCM. *Journal of Atmospheric Science*, *75*, 2299–2316.Find this resource: