Weak Jets and Strong Cyclones: Shallow-Water Modeling of Giant Planet Polar Caps

APRIL 2016 O ’ N E I L L E T A L . 1841

MORGAN EO’NEILL,KERRY A. EMANUEL, AND GLENN R. FLIERL Program in Atmospheres, Oceans and Climate, Massachusetts Institute of Technology, Cambridge, Massachusetts

(Manuscript received 16 October 2015, in ﬁnal form 14 February 2016)

ABSTRACT

Giant planet tropospheres lack a solid, frictional bottom boundary. The troposphere instead smoothly transitions to a denser fluid interior below. However, Saturn exhibits a hot, symmetric cyclone centered directly on each pole, bearing many similarities to terrestrial hurricanes. Transient cyclonic features are observed at Neptune’s South Pole as well. The wind-induced surface heat exchange mechanism for tropical cyclones on Earth requires energy flux from a surface, so another mechanism must be responsible for the polar accumulation of cyclonic vorticity on giant planets. Here it is argued that the vortical hot tower mechanism, claimed by Montgomery et al. and others to be essential for tropical cyclone formation, is the key ingredient responsible for Saturn’s polar vortices. A 2.5-layer polar shallow-water model, introduced by O’Neill et al., is employed and described in detail. The authors first explore freely evolving behavior and then forced- dissipative behavior. It is demonstrated that local, intense vertical mass fluxes, representing baroclinic moist convective thunderstorms, can become vertically aligned and accumulate cyclonic vorticity at the pole. A scaling is found for the energy density of the model as a function of control parameters. Here it is shown that, for a fixed planetary radius and deformation radius, total energy density is the primary predictor of whether a strong polar vortex forms. Further, multiple very weak jets are formed in simulations that are not conducive to polar cyclones.

1. Introduction In 2006, a second South Polar survey was taken and reported by Dyudina et al. (2009). They identiﬁed In 2004, the Keck Observatory discovered a warm region multiple polar vortex features, including an analog of situated on Saturn’s South Pole (Orton and Yanamandra- hurricane eyewalls: the vortex has concentric annuli of Fisher 2005), which implied the presence of a warm-core tall convective clouds, dropping a shadow over a deep polar vortex. This discovery was followed in 2007 by in- and largely clear, anomalously warm eye. The inner frared imaging of both polar hemispheres by the NASA eyewall has a horizontal wavenumber-2 distortion, and Cassini mission, currently in orbit around Saturn. A sur- the outer wall is azimuthally symmetric with an esti- prising result was that both poles exhibited very localized mated height of 30–40 km. The South Polar eye itself is tropospheric hot spots, though in 2007 one pole was re- consistently the warmest place imaged on the entire ceiving constant summer sunlight and the other was not planet. Predicted by Fletcher et al. (2008), a very similar (Fletcher et al. 2008). These hot spots were 6–8 K warmer hot cyclone has been observed on Saturn’s North Pole as than ﬂuid that is just 108 from the pole. The warm anom- well (Baines et al. 2009) and appears even more sym- alies extended downward at least as far as the lower limit of metrical. The polar vortex eyewalls, rapid circular jets, the observations at 1 bar. The Cassini mission took high- and deep clear eyes are reminiscent of hurricanes; yet resolution images of the southern hot spot and showed that the thermodynamic mechanism must be fundamentally it was the center of a vast, strong cyclone (Vasavada et al. different because there is no sea surface1 below. None- 2006). Sánchez-Lavega et al. (2006) found a peak tangential 2 theless, there are a remarkable number of similarities be- velocity of 160 6 10 m s 1 at 878 (3000 km from the pole). tween terrestrial hurricanes and Saturn’s polar cyclones,

Corresponding author address: Morgan E O’Neill, Weizmann Institute of Science, 234 Herzl St., Rehovot 7610001, Israel. 1 More precisely, there is no known discontinuity in enthalpy, be E-mail: [email protected] it solid or ﬂuid.

DOI: 10.1175/JAS-D-15-0314.1

Ó 2016 American Meteorological Society Unauthenticated | Downloaded 09/27/21 07:08 AM UTC 1842 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 73 detailed thoroughly in Dyudina et al. (2009). Multiple (Melander et al. 1987; Montgomery and Kallenbach observations of each pole since the cyclones’ discovery 1997), and nonlinear anomalies will additionally expe- suggests that they are ‘‘present at all epochs’’ seasonally rience beta drift. A TC is a natural laboratory for vortex (Fletcher et al. 2015). Rossby waves. TCs are small enough that the change Polar vortices have been observed on Earth, Venus, in Coriolis parameter f acrosstheirdiameterissmall, and Neptune, but they are all transient and cold or only but their significant primary circulation can provide a slightly warmer than their immediate surroundings. sufficient vorticity gradient to sustain vortex Rossby Saturn’s polar cyclones, in contrast, are remarkably waves (Montgomery and Kallenbach 1997; Möller and steady and hot. Saturn’s polar cyclones are the longest- Montgomery 1999; McWilliams et al. 2003) and poten- lived cyclones ever observed in the solar system (Jupiter’s tially beta drift. Great Red Spot is an anticyclone). While Saturn boasts Anomalies of great interest in the largely symmetric the most comprehensive observations of any giant planet, hurricane environment are small but severe thunder- the range of masses, compositions, and rotation rates of storms. The hurricane community has not resolved the Jupiter, Uranus, and Neptune provide multiple bench- role of local, deep convective thunderstorms in hurri- marks for a polar cyclone theory. cane formation and maintenance. Some authors argue In this paper, we expand the work of O’Neill et al. that azimuthally symmetric fluxes over the sea surface (2015, hereafter OEF), which proposes a predictive are sufficient for hurricane growth (e.g., Emanuel 1986) theory for polar cyclones on giant planets. In section 2, and that strong eddies are deleterious for intensification we provide a literature review of relevant terrestrial [e.g., in the idealized study of Nolan and Grasso (2003)]. tropical cyclone theory and describe its potential for Others maintain that deep convective towers pump application to gas giants. The details of the shallow- vorticity into the mean flow (Montgomery et al. 2006; water model used are described in sections 3 and 4.In Persing et al. 2013). This latter hypothesis may be rele- section 5, results are provided from freely decaying vant to a moist convective giant planet atmosphere. single-storm simulations to confirm that the model can These terrestrial deep convective towers converge high- reproduce well understood single-vortex behavior. Sec- angular-momentum air at their base, creating a positive tion 6 provides a derivation of the governing energy pa- vorticity anomaly. Moist air rises rapidly through the rameter that controls qualitative behavior for both freely cumulonimbus cloud, releases latent heat, and diverges decaying and forced-dissipative simulations. Section 7 just below the tropopause, leading to a negative vorticity modifies this energy parameter and describes steady- anomaly. Montgomery et al. (2006) contend that these state behavior for forced-dissipative simulations. Sec- anomalies can react with the environment and strengthen tion 8 discusses ambiguity in model interpretation, the a hurricane by transporting vorticity toward the hurricane model’s applicability, and future work. Appendix A eye. Idealized experiments with a basic-state parent vor- provides the model energy equations, and appendix B tex confirm that small vortical anomalies are axisymme- discusses the domain setup and numerical considerations. trized (Melander et al. 1987) and that positive anomalies indeed merge (e.g., Montgomery et al. 2006; Hendricks et al. 2014). The wind-induced surface heat exchange 2. Beta drift on Earth and giant planets mechanism, which is an important feedback for the pri- The polar cyclone hypothesis proposed in OEF bor- mary energy source of hurricanes, is precluded because rows from the understanding of terrestrial tropical cy- there is no surface at the base of giant planet tropospheres. clones (TCs). TCs in the tropics tend to move westward There is consensus that moist convection is occurring and poleward because of the varying Coriolis parameter. in the weather layers of at least Jupiter and Saturn, and This motion is called beta drift (Adem 1956) and is due potentially Uranus and Neptune as well (e.g., Gierasch to the nonlinear interaction between a strong cyclone et al. 2000; Vasavada et al. 2006); this may even be the and the vorticity gradient of the background state. This energy source for the jets (Ingersoll et al. 2000; Lian and behavior is observed in numerical simulations (e.g., Showman 2010). For Saturn’s South Polar region, Schecter and Dubin 1999) without any background flow Dyudina et al. (2009) measured the mean winds of the as well as in observations of real hurricanes. surrounding flow as well as the local rotation of many On a spinning planet, a vorticity gradient is provided puffy cloudy features that are advected around the by the variation of the Coriolis frequency with latitude vortex. They find that the flow surrounding the vortex is in a stationary atmosphere. On an f plane, a similar dominated by small anticyclonic vortices advected by gradient can be achieved if a vortex itself provides the the nearly irrotational flow outside the outer eyewall. background state. Small, linearized anomalies added to OEF assume that a fraction of the anticyclonic vorti- this background state will induce vortex Rossby waves ces in the South Polar region are the tops of tropospheric

Unauthenticated | Downloaded 09/27/21 07:08 AM UTC APRIL 2016 O ’ N E I L L E T A L . 1843 convective towers, rooted in or below the water cloud. PV assumes that perturbations of the background fluid conservation would imply that their anticyclonicity is height are comparatively small, h0 H. The QG system balanced by a cyclonic anomaly near the cloud base, is inappropriate for oceanic warm-core rings (Flierl which may react to the Coriolis gradient of the planet and 1984) and may also be a poor fit for Saturn’ polar vor- move upgradient, or poleward. Over time, a large-enough tices, because their centers are the deepest, hottest poleward flux of cyclonic vorticity may be sufficient to places observed on the entire planet, representing a create and maintain a polar cyclone. Baines et al. (2009) significant deviation in geopotential. speculate that the small cloud features may deliver energy We employ a 2.5-layer shallow-water system centered to the North Polar cyclone on Saturn, if they are indeed of on the pole for this study. The two active layers repre- moist convective origin, though they did not provide a sent the troposphere, which extends from the base of the mechanism. Other authors (e.g., Ingersoll et al. 2000) have water cloud to the temperature inversion that indicates envisioned exactly this mechanism for the jets but not for the tropopause. The troposphere is expected to be the polar cyclones. We find that beta drift of sufficient statically stable and follow a moist adiabat, much like cyclonic vorticity can drive an equivalent-barotropic polar Earth’s troposphere. The abyssal layer below the two cyclone without the need for a frictional surface. active layers is a representation of the deep and (possi- A few idealized modeling studies and laboratory ex- bly) neutrally stable convecting interior. periments have specifically studied the dynamics of giant Since the feature of interest in this study is in the planets’ thin weather layers. They provide evidence that immediate polar region, it is unnecessary to model a the drift of small positive anomalies is indeed sufficient spherical shell. On Saturn, which provides the primary to create a polar cyclone. Scott (2011) employed a polar motivation for the simulations, the polar caps are beta plane to study the motion of patches of cyclones enclosed by a jet at 758 in the Southern Hemisphere and anticyclones on a pole. His quasigeostrophic (QG) (Vasavada et al. 2006) and 748 in the Northern Hemi- single-layer simulations demonstrate the genesis of a sphere, which likely act as strong barriers to mixing circumpolar cyclone due to beta drift. The cyclone tends (Dritschel and Mcintyre 2008). In this work, we model to orbit the pole in equilibrium, strongly mixing the rest polar caps and assume that the outer radial limit plays of the fluid in the polar cap. Scott and Polvani (2007) the role of a jet barrier. use a forced-dissipative model with full spherical ge- The pole is a unique place where the planetary vorticity ometry and find a polar cyclone that swims around the gradient reaches zero. A traditional beta plane is not ap- pole, constrained by the poleward-most jet. Schneider propriate here because of the quadratic nature of the and Liu (2009) and Liu and Schneider (2010) also find a Coriolis frequency. We use the polar beta plane (Leblond broad polar cyclone that precesses around the pole in a 1964; Bridger and Stevens 1980) to approximate the plan- spherical shell that extends in pressure to 3 bars. The etary vorticity gradient near the pole. The Coriolis fre- 2 2 2 2 2 model used in the present work is the first to force a quency f 5 f0 2 br ,wherer 5 x 1 y ,andb 5 f0/(2a ) polar domain with small thermal perturbations that has units of per square meters per second. All sign con- mimic observed moist convection in size, strength, and ventions are consistent with Northern Hemisphere flows to duration, motivated by abundant observations of in- reduce potential confusion; however, we avoid the term tense moist convection on the gas giants (Little et al. ‘‘counterclockwise’’ and only refer to cyclonicity of flows. 1999; Gierasch et al. 2000; Li et al. 2004). Physical forcing and dissipation OEF (see their methods section) use a baroclinic 3. The model forcing to drive an equivalent-barotropic polar cyclone.

We use the shallow-water (SW) system as a laboratory The storm forcing function Sst simulates a moist con- for simple, parameterized convection in a thin-shell vective environment with localized deep convective weather layer. Several aspects of polar dynamics may towers, which are shaped as truncated Gaussians in render the commonly used QG system too restrictive for space, with a boxcar function for time behavior (step- our purposes. Most notably, the Rossby number of ping from zero to a fixed value over the storm duration Saturn’s polar vortices may be as high as 1 (Dyudina and then back to zero). The storms occur in batches. et al. 2009). The Rossby number suggests a cyclone in Gravity waves are not strongly triggered because the gradient wind balance, and the QG system is only valid peak vertical forcing velocity is generally very low. for motions in approximate geostrophic balance. Also, Convection is simulated by locally thinning the lower the SW system assumes that the typical fluid height is active layer and thickening the upper active layer im- much smaller than the typical horizontal length scale, mediately above (Fig. 1), as if a storm is fluxing mass H L; the QG system goes one step further and from the lower to upper layer. Simultaneously, the layer

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FIG. 1. (left) Mass forcing scheme. (right) After the transient mass forcing (typically a fraction of a day), the flow approaches geostrophic balance by establishing a baroclinic dipole in the vertical. interface is everywhere else slightly and uniformly thickness perturbations. The estimated radiative time raised to conserve layer mass at each time step and scale trad on Saturn is quite high; approximately 9 represent subsidence.2 Of course, this forcing is actually Earth years (Conrath et al. 1990). To reduce compu- dry, but the vertical velocities and areal extents of the tational expense, values used in this study range from convecting regions parameterize the behavior of a moist, 150 to 400 days, but still trad tst,wheretst is the statically stable atmosphere. This forcing adds available storm lifetime. The radiative time scale is the same for potential energy (APE), which can convert to kinetic each layer. This last choice is more for simplicity than energy (KE) as motions approach geostrophic balance. based on physical intuition, and future work should See Showman (2007, hereafter S07) for a very similar explore more complicated layer radiative functions. forcing for 1.5-layer midlatitude simulations of Jupiter This thermal damping is more physically motivated [other examples are Li et al. (2004) and Smith (2004)]. on giant planets than energy removal by mechanical An additional degree of freedom that we do not add to damping of the winds. For example, hypoviscosity, this model is the ability of storms to advect according to with no clear physical interpretation, has been used some appropriately weighted mean wind speed while by Scott and Polvani (2007); they also consider linear they grow. One could argue that deeply rooted storms Rayleigh drag. Liu and Schneider (2010, 2011)and either move negligibly during their brief active lifespan Schneider and Liu (2009) also use Rayleigh drag. or move with the upper winds of the neutrally buoyant In those papers, the stated motivation for linear drag abyssal layer (which would have an independent is the cumulative effect of magnetohydrodynamic streamfunction that we do not provide in this model). In drag that occurs much deeper. any case, fixing the storm location has not impeded the As in OEF, our 2.5-layer model assumes an inert, potential for polar cyclogenesis in our simulations, so abyssal bottom layer, which rules out a barotropic mode that advance is left for future work. [see Achterberg and Ingersoll (1989) for a normal-mode In this model, there is no large-scale forcing in the QG model that allows an abyssal flow and therefore a momentum equations. There is also only minimal nu- barotropic mode]. Because the system is nonlinear and merical dissipation in order to more realistically simu- divergent, the baroclinic modes are coupled, and a sum late an essentially inviscid weather layer. See appendix of modal energies will not include the energy in gravity B for a description of numerical friction and hypervis- waves; yet they provide us with more physically relevant cosity for stability. Radiative relaxation is the only large- deformation radii. We normalize lengths in our model scale energy removal in the model [as in S07 and Scott by the internal deformation radius LD2. and Polvani (2008)], and it removes APE by damping 4. Nondimensional system

2 Downdrafts in the vicinity of deep convection may be similarly Nondimensionalization in textbooks and well- localized and nearly as intense as others have found in observations behaved physical models commonly involves a ve- (Gierasch et al. 2000) and Jovian simulations (Lian and Showman locity scaling to arrive at a Rossby number or a 2010). We omit that complexity here, assuming instead a constant subsidence in the remainder of the domain. This may be a partic- Froude number. In such cases, it is appropriate to ﬁx ularly poor assumption for the polar region, where the deformation nondimensional parameters in this way, because the radius reaches a minimum and may conﬁne downdrafts. system is further constrained by a forcing chosen in

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TABLE 1. Nondimensional parameters and their deﬁnitions. Note that LD2 is the second baroclinic deformation radius, equal to the second baroclinic gravity wave speed ce2 divided by f0.

Symbol Parameter Range Meaning ~2 2 2 c1 c1/ce2 4–10 Scaled ﬁrst-layer GW speed ~2 2 2 c2 c1/ce2 3–9 Scaled second-layer GW speed H1/H2 — 0.5–1.5 Vertical aspect ratio 2 2 Br2 LD2/Rst 0.25–4 Second baroclinic Burger number ~ 2 2 b LD2/(2a ) 25 Scaled b 25 22 Roconv Wst/(H1f0)23 10 –2 3 10 Convective Rossby number

r1/r2 — 0.95 Layer stratification ~tst tstf0 2–15 Scaled storm duration ~tstper tstperf0 1~tst–4~tst Scaled storm period 3 ~trad tradf0 33~tst–1:3 3 10 ~tst Scaled radiative time scale # — 1–822 Simultaneous storm number 3 3 4 Re ce2LD2/n 5 10 Reynolds number 5 Pe ce2LD2/k 1 3 10 Peclet number advance to result in particular behavior. However, The Burger number appears in the forcing term be- we want to avoid scaling that assumes a ‘‘typical’’ cause the specified storm radius is an additional length horizontal velocity. It is possible that this system will scale. The stratification parameter r1/r2 and number of demonstrate different flow regimes in its very large storms forced simultaneously # remain the same since parameter space, and we do not know a priori even they were nondimensional to begin with. We normalize the order of magnitude of typical velocities in the horizontal lengths by LD2, thicknesses by H1, and time 21 system. Therefore, the system is scaled by functions by f0 . The model solves for layer velocities and heights of the control parameters only, and we consider the at each time step. Subscripts 1 and 2 refer to the upper Rossby and Froude numbers to be descriptions of the and lower layers, respectively. Our nondimensional behavior at statistical equilibrium. We still can define a shallow-water system is convective Rossby number, Roconv 5 Wst/( f0H1)(Kaspi et al. 2009), since its components are all control pa- ›u 1 52(1 2 b~x2 1 z )k^ 3 u rameters, but this should not be confused with the ›t 1 1 global Rossby number that results from horizontal ve- 1 2 2 locity and length scales. The 13 nondimensional parame- 2 = c~2h 1 c~2h 1 ju j 2 Re 1=4u , (1) 1 1 2 2 2 1 1 ters are listed in Table 1. The upper- and lower-layer ›u thicknesses are H1 and H2, respectively; Rst is the di- 2 52 2 ~ 2 1 ^ 3 (1 bx z2)k u2 mensional storm radius; a is the dimensional planetary ›t radius; Wst is the dimensional peak vertical velocity of the 1 2 2 = gc~2h 1 c~2h 1 ju j2 2 Re 1=4u , (2) convective mass flux; tst is the dimensional storm lifetime; 1 1 2 2 2 2 2 t is the dimensional time between the beginning of one stper › 2 2 h1 h1 1 21 2 batch of storms and the next batch such that tstper tst is 52= (u h ) 1 S 2 1 Pe = h , (3) ›t 1 1 st ~t 1 the duration between the end of one batch and the be- rad ginning of the next; and trad is the dimensional radiative ›h H h 2 1 2 2 52= (u h ) 2 1 S 2 2 1 Pe 1=2h , (4) relaxation time scale. Table 1 is reproduced here, in part 2 2 st ~ 2 ›t H2 trad from the supplementary information in OEF.

8 2 3 > # 2 2 <> 6 (x xj) 7 å Ro exp42B 5 1 subsidence, if ~t # ~t conv r2 : clock st S 5 j51 0 36 (5) st > :> ~ , ~ # ~ 0, if tst tclock tstper

~ for a tclock that resets to zero every time it equals ~tstper. et al. 2003). The factor of 0.36 in the denominator of the ~2 ~2 The g term is equal to (r1/r2)(c2/c1)(H1/H2)(Simonnet Gaussian exponent adjusts the vortex width such that Rst

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FIG. 2. Case of a single, strong baroclinic storm. The white contours indicate upper-layer potential vorticity; the maximum is at the pole in the center of the domain. The colors indicate lower-layer potential vorticity. A single pulsed storm is vertically sheared, and the upper- layer anticyclone moves equatorward while the lower-layer cyclone moves poleward. In this case, the lower-layer potential vorticity maximum is at the center of the very strong pulsed storm and not at the pole. is approximately the radius at which the amplitude is j , f (r)], which is wrapped around and poleward of the

Wst/2 (full-width half-max). strong cyclone, moves directly over the pole as the cy- The nondimensional Coriolis parameter b~ provides a clone moves poleward. The low-PV air remains there measure of the Coriolis gradient with respect to the unless or until the cyclone itself makes it to the pole. second baroclinic deformation radius LD2 in the non- This can be seen in Fig. 2; as the cyclone advects high-PV dimensional model. For constant f0, a planet with large air equatorward, the highest-latitude region entrains ~ 2 2 b 5 L /(2a ) will have small values of a/LD2: around 20 low-PV air. Meanwhile, the anticyclone moves toward D2 ~ or 30. A planet with small b has a/LD2 $ 40. In this pa- the equatorward boundary. In both cases, each layer’s per, we generally use a/LD2 for ease of interpretation. vortex begins to spin up a vertically aligned like-signed vortex in the opposite layer. 5. Freely evolving, single-storm simulations Energy transfer and vertical alignment The first set of experiments explores freely evolving Our simulated storms are baroclinic. When a single flows, wherein one storm is initially forced and then storm is initially forced and allowed to evolve freely, the allowed to decay. These experiments serve to bench- anomaly in each layer reacts with the local Coriolis mark the model and confirm predictable behavior. The gradient, and each layer evolves nearly independently of background state has zero mean flow, for both these the other for early times. However, a vertical energy experiments and the forced-dissipative set in following cascade to the gravest mode (which, in this simple sys- sections. For these decaying experiments in particular, a tem, is just from the second to the first baroclinic mode) zero background flow allows us to isolate the nonlinear begins almost immediately and results in increasing dynamics due to the vortex only, as in Chan and Williams vertical alignment of the flow. This is the mechanism by (1987). No radiative relaxation is imposed. The storms which baroclinic storm forcing can ultimately force decay more quickly than they would in a real, nearly in- equivalent-barotropic cyclones [see Polvani (1991) for viscid atmosphere, but slowly enough to observe the an exploration of vertical vortex–vortex alignment] and nonlinear interaction with the planetary vorticity gradient. is more efficient for higher local Coriolis gradient When a single equivalent-barotropic cyclone is ini- (Venaille et al. 2012). Its effect can be seen in Fig. 3. tially forced and then allowed to evolve, it induces a In Fig. 3, the contours measure the magnitude of the Rossby wave by wrapping high-PV fluid from the pole to equivalent barotropic component of the fluid and would its west and low-PV fluid from more equatorward re- not appear for purely baroclinic motions. We can see gions to its east. For sufficiently strong cyclones, this that the lower-level cyclone is able to spin up more happens quickly and nonlinearly, and creates b gyres upper-layer fluid, more quickly, than its counterpart on (Chan and Williams 1987) that also propel the cyclone the large planet because of its higher Coriolis gradient. poleward. The inverse case occurs for anticyclonic vor- We can also see that, while the cyclone moves poleward tices: a Rossby wave always forms, and the strongest on the small a/LD2 planet, it temporarily advects nega- anticyclones self-advect equatorward. tive PV fluid over the pole in both layers before it rea- In the case of a single, strong baroclinic storm, in ches the pole. The cyclone on the large a/LD2 planet has which an anticyclone sits exactly above a cyclone, the the same size and intensity but is unable to self-advect a stack begins to shear apart immediately after the brief significant distance poleward because of a significantly forcing period ends (Fig. 2). In the lower layer, the cy- weaker beta drift effect. Nonetheless, nonlinear effects clone moves poleward. Low-PV air [relative vorticity can still be seen in the advection of low-PV air poleward

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FIG. 3. Vertical alignment of the ﬂuid is faster and more effective for larger beta. The colors show the azimuthally h 0 i ^ averaged perturbation PV q2 of the lower layer, where a cyclonic storm is initialized—on (left) a small-b (large ^ h 0 i 5 a/LD2) planet and (right) a large-b (small a/LD2) planet. White is q2 0, and red colors denote positive perturbation PV. Gray contours indicate positive and black contours indicate negative depth-integrated perturbation PV. The contour intervals are the same for each graph, denoting an increase or decrease of 5 3 1024. Not shown is the anticyclone that forms in the upper layer. The depth integration shown by the contours indicates the extent to which the vorticity is vertically aligned; for a purely second baroclinic structure, the depth integration would be zero.

of the original cyclone. Strong storms on small a/LD2 The thickness perturbation due to one storm, over the area planets appear the most promising candidates for polar of the storm, scales as cyclone forcing, and the forthcoming forced-dissipative Dh 5 Ro ~t and (6) experiments suggest that strong vortex–vortex interac- 1st conv st H tion does not change this outcome. D 52 1 D h2st h1st . (7) H2 6. Deriving an energy parameter The areal fraction covered by one storm ar is a function In the course of running hundreds of simulations, we of the Burger number: found that increasing any parameter that increases 5 p energy density has a similar impact on the qualitative ar 2 . (8) Br Ldom nature of the equilibrium flow. Energy density (depth- 2 2 integrated total energy per LD2) seems to be a governing This expression neglects the Gaussian shape of the parameter for the flow. Therefore, it would be conve- storms and instead assumes cylinders or ‘‘top hats’’ nient to be able to predict the energy density as a with a radius equal to the location of its full-width at function of control parameters without having to run the half-maximum amplitude of the vortex). This approxi- simulation for every single parameter combination. mation should introduce a negligible error in the S07 derives an ‘‘energy parameter’’ that scales as the energy scale. equilibrated APE value. We generalize this parameter An additional source or sink of APE can come from to our 2.5-layer model with variable stratification and the subsiding regions in the rest of the domain (1 2 ar). layer thicknesses for our particular constraint of instan- The domain-wide storm forcing and subsidence are not taneous mass conservation in each layer (O’Neill 2015). necessarily a source of APE at any given instant. APE is D 0 In this section, an energy parameter Ep is derived for the only increased where hi is the same sign as hi. In this single-storm experiments described above, and in section instance of a single storm being initialized as thickness 7 it is modulated for forced-dissipative simulations. perturbations to two otherwise quiescent layers, the The available potential energy induced by one storm entire domain contributes APE.

Ast affects the entire domain as a result of mass-conserving Because of layer mass conservation, imposed at every subsidence outside of the storm environment. To find a time step, the total volume of the storm forcing is al- scaling for Ast, we first look at the modified height fields in ways equal to the total volume of the subsidence else- each layer. The top layer experiences an increase in where in the domain. We can relate a perturbation thickness Dh1st within the boundaries of the storm; and the thickness due to subsidence to the perturbation thick- rest of the domain experiences constant subsidence Dh1sub. ness due to storms:

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a Dh 52 r Dh . (9) isub 2 ist 1 ar

To ﬁnd a scale Asc for the APE of the entire domain, we consider the contributions from both the storm and subsidence regions, using the above substitutions to express each height in terms of Dh1st: ð " r H 5 1 1 1 ~2D 2 AscdAr ar c1 h1st storm 2 r2 H2 # H 2 H 1 1 ~2 1 D 2 2 ~2 1 D 2 c2 h1st gc1 h1st . (10) 2 H2 H2

Summing these contributions yields Asc as a function of our scaled Dh1st and other nondimensional parameters, which we designate as Ep: 1 r 1 H H a E 5 1 c~2 1 1 c~2 2 gc~2 1 (Ro ~t )2 r . p 1 2 1 conv st 2 2 r2 2 H2 H2 1 ar (11)

It is entirely composed of control parameters and does not require any guess about equilibrium wind speeds. This parameter should scale with the total equilibrated 2 potential energy per deformation radius area LD2 of each simulation. The energy parameter is not a function of the hyperviscosity or diffusion parameters, though in reality the model’s high dissipation must impact the FIG. 4. (top) The energy parameter Ep and (bottom) the modi- dynamics to some extent. However, our choice of ﬁxing ^ nondimensional viscosity across the models and consis- ﬁed Ep, with the 1:1 line for reference. The size of the circle scales linearly with the Burger number; big circles indicate large Burger tently resolving LD2 allows us to largely ignore this issue, numbers (small storms). The difference is small, but the correction because the impact should then be uniform among the removes a systematic bias. simulations.

A modiﬁed Ep Burger number simulations the energy parameter does

The parameter Ep is evidently a scaling only for po- overestimate APE. To correct the bias, one can divide Ep tential energy, given its basis in a forcing of height per- by the Burger number (black circles in Fig. 4). We define ^ 5 turbations only. Depending on the nature of the flow at Ep Ep/Br2 as our primary energy scaling. The rms ve- equilibrium for the forced-dissipative cases, it is possible locityq isffiffiffiffiffiffiffiffi also predicted very well by the energy parameter; 5 ^ that much of the energy could be in kinetic form. The U 2Ep (O’Neill 2015). 5 2 2 internal Burger number Br2 LD2/Rst can serve as a . ratio of potential to kinetic energy. When Br2 1, 7. Forced-dissipative multistorm simulations storms are smaller than LD2, and much of the energy is converted to kinetic energy of horizontal winds. A The primary set of experiments constantly forces the 3 Burger number less than 1 allows a large storm forcing to model with APE via randomly placed storms and maintain interface deviations and store potential energy. Our energy parameter does not include a term for kinetic 3 energy, because Br2 only appears in Ep as a scaled storm The storms are random in space but not in time. They are either area, without any connection to balanced flow. We can all ‘‘on’’ or all ‘‘off.’’ This does cause a sawtooth variation in the total model energy, but it is small, and the steady-state polar cy- expect that Ep in simulations with large Br overestimates 2 clones do not exhibit this sawtooth oscillation for short time scales APE, because much of the energy becomes kinetic. This is relevant to moist convection. In fact, once a polar cyclone develops, indeed the case, as shown in Fig. 4. The size of the markers it is difficult to observe small storms at all, because their magnitude scales linearly with the Burger number, and for the largest and size are much smaller.

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slowly; and a longer period between storms ~tstper will add energy more slowly. Thus, in the forced-dissipative case, ^ ^ we update our energy scale Ep 5 Ep~trad/~tstper. For the freely evolving cases above, in which a single storm is briefly forced in otherwise equal-thickness layers, it is appropriate to consider the contribution of APE due to subsidence as well as the storm itself. Since each round of storms (they are introduced in sets) is randomly placed, subsidence tends to damp locations that previously held small storms. Simulations with large ar corroborate this, and we can therefore neglect the (1 2 ar) factor in Eq. (11). Additionally, the square root 1/2 5 of the Burger number, Br2 LD2/Rst, provides a mar- ginally better match to the measured energy density of the forced-dissipative simulations than just the Burger number. These modifications slightly improve the en- ^ FIG. 5. The energy parameter Ep (graycircles)andthe1:1line ergy parameter reported in EOF. for reference. The size of the circle scales linearly with the Our new energy parameter is Burger number; big circles indicate large Burger numbers (small storms). 1 r 1 H H ~t a E^ 5 1 c~2 1 1 c~2 2 gc~2 1 (Ro ~t )2 rad pffiffiffiffiffiffiffiffir . p r 1 H 2 1 H conv st ~t 2 2 2 2 2 stper Br2 removes APE through damping thickness perturba- (12) tions in each layer. Each of the 11 nondimensional parameters was varied individually across a range of The various regimes cannot be completely flattened to observation-based values when possible. Extreme ca- two dimensions. This is apparent, for example, when we ses (e.g., a storm areal fraction ar as high as 40%) were change r1/r2 or H1/H2 by 50%. However, for nominal simulated to test efficacy of the energy parameter. values of the control parameters, the outcome is quite ^ Empirically, the energy density in a given simulation predictable. The resulting Ep gives a reasonable esti- appears to strongly affect its equilibrium behavior, and mate of the energy density in steady state for scores of it is desirable to predict the energy density based on simulations (Fig. 5). In Fig. 6, a subset of simulations is control parameters. Increasing the size, number, strength, depicted where only 2 of the 11 control parameters are ~ ^ or duration of the baroclinic storm forcing consistently varied: b and Ep. Other parameters are held constant ~2 5 ~2 5 5 5 5 : ~ 5 results in a higher likelihood of a large-scale equivalent- (c1 4, c2 3, H1/H2 1, Br2 1, r1/r2 0 95, tst 6, barotropic cyclone in the domain. An increase in the ~tstper 5 15, ~trad 5 2000, and ar 5 15%). These simulations storm frequency or the radiative time scale is also exhibit a representative range of behaviors. conducive to a polar cyclone. a. Large b~ (low a/L ) planets Statistically steady states of the forced-dissipative D2 ^ models exhibit a broad spectrum of behavior, from Low-Ep simulations are wavelike, with multiple very ^ very low-energy wave- and jet-dominated domains to weak jets. As Ep is increased, cyclonic eddies move very intense polar cyclones. OEF find that these states poleward and a broad, transient region of positive per- are controlled principally by two control quantities: the turbation PV collects on the pole, possibly similar to the ~ ^ strength of a nondimensional beta parameter b (which is transient polar hotspots observed on Neptune. As Ep is proportional to the inverse of a/LD2) and the energy further increased, the transient region becomes much ^ ^ density Ep due to convection of the shallow atmosphere. stronger and more symmetric. Low-a/LD2, moderate-Ep This energy parameter does a good job of capturing planets most closely resemble polar observations of forced-dissipative simulation energy density, and here Saturn. Finally, more energy causes the polar cyclone to we further modify it for a better fit to the simulations. orbit/precess at greater distances from the pole. In the In the case of the freely evolving single storms, there is bottom row of Fig. 6, note that the most energetic/right- no imposed radiative relaxation, and the forcing is only most simulation has a polar vortex with a very tight pulsed once in the beginning. For a constantly forced circulation (the mean wind peaks close to center of domain, energy is constantly injected and removed. We vortex, shown by the dashed line). However, because of modify Eq. (11) by two corresponding time scales. A the relatively large distance of the vortex’s center from longer radiative time scale ~trad will remove energy more the pole (gray line), the pole-centered mean winds are

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~ ^ FIG. 6. Nondimensional mean zonal winds are shown for a range of planet sizes b and energies Ep (changing Roconv as a proxy). Zonal winds are shown both with respect to the pole (solid lines) and with respect to the dominant cyclone center (dashed lines). (bottom)–(top) The effective Coriolis parameter decreases, and (left)–(right) energy increases. Vortex-centered profiles are not provided for the left column because these simulations do not produce a dominant vortex. smeared in the time averaging and appear much weaker increasingly equivalent-barotropic jets. These jets are (black line). initially fully baroclinic, in response to the purely baroclinic storm forcing. As time progresses, the jets ulti- b. Small b~ (high a/L ) planets D2 mately align in the vertical. They are so weak as to be A much weaker effective Coriolis gradient allows completely imperceptible in movies, as the wind field in initially wavelike behavior to create and then merge these cases is dominated by small, storm-induced vorti- ^ multiple coherent vortices away from the pole as Ep is ces. They can be seen quite clearly in a Hovmöller plot increased. Beyond the short deformation radius, a lim- of polar radius and time (Fig. 7, top-left panel). ited range of interaction and weakly nonlinear beta drift These weak jets may help us better understand ob- keeps low-energy behavior local. Stronger forcing can servations of Jupiter’s high latitudes (the poles them- cause one or more strong circumpolar cyclones, but they selves will be imaged for the first time in 2016, when the will not be stable near the pole, instead directly tran- NASA Juno mission reaches Jupiter). While snapshots sitioning to a polar orbit or nearly random motion. of Jupiter’s high latitudes show an abundance of coherent cyclones and anticyclones, a movie4 taken by WEAK JETS AND STRONG CYCLONES Cassini’s narrow-angle camera over 70 days in 2000 The strong polar cyclone is by far the most dramatic ^ outcome of the parameter sweep across a/LD2 and Ep. However, in weakly energetic atmospheres, or on large 4 Available online at http://saturn.jpl.nasa.gov/video/videodetails/ ~ b planets, another persistent feature is very weak, ?videoID5224; credit: NASA/JPL/Southwest Research Institute.

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FIG. 7. Contours indicate nondimensional mean zonal winds in the upper layer; shading indicates nondimensional mean zonal winds in the lower layer. Maximum and minimum values are provided on the right-hand side of each ^ ^ panel. (top) Large planets relative to LD2; (left) low Ep leads to multiple very weak jets, and (right) high Ep results in one or several strong cyclones that swim around in the polar region. (bottom) Small planets relative to LD2; (left) ^ ^ low Ep leads to a couple broad and very weak jets, and (right) high Ep results in one strong cyclone that stays very close to the pole. showed, surprisingly, that these vortices move zonally in This model is adequate for a study of horizontal very weak but well-defined narrow jets. This is similar to shallow motions. Shortcomings in the shallow-water ^ what is observed in the low-Ep, high-a/LD2 simulations. system and our geometry limit the applicability of our Whether they are due to the same mechanism can be results to polar cyclones with Rossby number less than 1; answered by a momentum flux convergence budget and for Rossby number much greater than 1, cyclostrophic comparison to pending detailed Jovian observations. vortices have vanishing layer depths. Both poles of Saturn appear to be depleted of phosphine gas, which varies as a function of both vertical mixing and photo- 8. Discussion chemical destruction. This implies significant subsidence A shallow-water model, customized for a polar cap right at the poles (Fletcher et al. 2008), so we should that lacks a static bottom boundary, is able to achieve a seek a model that would permit this. The shallow-water strong, equivalent-barotropic polar cyclone under cer- model does not permit overturning circulations, though tain conditions. The model forcing emulates small, fast, it can parameterize an overturning circulation through energetic convective storms, and the radiative relaxa- radiative relaxation. However, understanding whether tion represents energy loss to space. The vortical hot that proxy is physically relevant or unphysically imposed tower mechanism of Montgomery et al. (2006) seems to remains to be seen in this case. Additionally, the most be sufficient for cyclone spinup and maintenance. energetic simulations exhibit a peculiar instability, in

Unauthenticated | Downloaded 09/27/21 07:08 AM UTC 1852 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 73 which the strongest cyclone spontaneously switches include any mass exchange between the troposphere layers, leaving a weaker and annulus-shaped vortex in and the abyssal layer and instead assume that implicit the former layer. Understanding and characterizing this mass exchange at the lower layer–abyss interface in- instability is left for future work, and results for these duces convergence as warmed, unstable weather-layer extreme cases are not shown in this paper. parcels rise convectively. It would be interesting to ex- While our most stable polar cyclones are similar to plore this assumption with a plume model—it is possible those of Saturn, there is one major difference. Saturn’s that the entrainment of moist tropospheric air that we polar cyclones have never been observed away from assume here is insufficient to drive sufficient horizontal their positions exactly over the poles, though admittedly convergence. In this case, the divergence at upper levels the observations are extremely sparse by terrestrial would be greater than the lower-layer convergence, standards. In the present simulations, no polar cyclone since some of the mass originated in the interior. Ex- has been simulated that is as stationary. The most stable periments in which H1/H2 was varied explored the cyclones still wobble slightly over the pole, though they possibility of a deep lower layer to emulate a very stray no farther than 1LD2 radially. Perhaps, on Saturn, gradual transition to the neutral interior. These experi- there is a complex coupling between a fully developed ments, in which the lower layer was at least twice as thick weather-layer cyclone and the interior below. For ex- as the upper layer, consistently developed a polar cy- ample, if the forcing is large enough, it may spin up an clone under similar conditions to the other simulations. equivalent-barotropic cyclone in the dry interior below This is despite a much lower maximum amplitude of that acts to stabilize the lateral motion of the weather- lower-layer cyclonic vorticity compared to the anticy- layer cyclone. clonic vorticity above. Interaction of the deep interior convection with the If we did include abyssal mass input, we would also upper weather layer on giant planets is very uncon- include abyssal mass removal—just as in our two-layer strained and an active area of research. It is commonly exchange, these would result in sources of anticyclonic assumed that the interior just below the water cloud is and cyclonic PV, respectively. This suggests an impor- fully convective and therefore adiabatic (e.g., Achterberg tant role for downdrafts as a source of cyclonic PV, in and Ingersoll 1989; Li and Ingersoll 2015). However, the event that lower-layer convergence is negligible. plume structures may allow local static stability (Lindzen This is an intriguing possibility because of the difficulty 1977), and tidal forcing considerations suggest that below in motivating lower-layer convergence without a fric- the cloud deck Jupiter may be (slightly) stably stratified tional boundary layer. Such a model would effectively (Ioannou and Lindzen 1994)5. This will affect the inter- be 1.5 layers, as in S07. pretation of our results. We assume that the deep interior If the interior just below the weather layer is, in fact, is neutrally stable, which precludes a deformation radius. statically stable, the present 2.5-layer model can be The water cloud may act as a waveguide (Ingersoll and interpreted very differently. The upper layer can repre- Kanamori 1994) only if the stratification changes rapidly sent the entire weather layer; the lower layer can repre- below it, and the only in situ evidence we have6 suggests sent the upper, slightly stratified portion of the deep that this may not be true (Showman and Ingersoll 1998) interior; and the abyssal layer can represent an even (assuming that Jupiter and Saturn have similar upper deeper neutral region. Under this interpretation, the first stratifications). (external) baroclinic radius is actually the barotropic We additionally assume that there is sufficient heat mode of the weather layer. The model’s second baroclinic available in the lower layer to induce mass convergence radius would then be the weather layer’s external mode. and therefore cyclonic vorticity anomalies. On giant This would address one shortcoming of our results: planets, it is unlikely that insolation reaches the lower a factor-of-almost-2 difference between Saturn’s ob- troposphere, so the source of heat must be from the hot served weather-layer deformation radius [1000–1500 km interior fluid. However, warming by the interior likely for the first baroclinic mode (Read et al. 2009)] and the involves entrainment of interior parcels into the lower simulated LD2 that produces qualitatively similar polar troposphere (our lower layer). In this work we do not behavior, which is in the 1800–2800-km range. A first

baroclinic deformation radius LD1 in our model is generally twice LD2, which would be up to 4 times larger than estimates for Saturn. However, in our alternate 5 There may even be a radiative zone just below the weather interpretation above, it is possible that our LD2 actually layer, which would require an entire reassessment of our abyssal- layer assumptions (Guillot et al. 1994). acts like a ﬁrst baroclinic mode, in which case the sim- 6 The Galileo probe unfortunately pierced a ‘‘hot spot,’’ which is ulations are just a factor of 2 different than observa- likely quite anomalous (Niemann et al. 1998). tional estimates.

Unauthenticated | Downloaded 09/27/21 07:08 AM UTC APRIL 2016 O ’ N E I L L E T A L . 1853 Saturn’s polar cyclones have a peak in radial velocity ›K 1 r H 1 52= h u 1 1ju j2 approximately 3000 km from the pole. For our most 1 1 1 ›t 2 r2 H2 Saturn-like simulations (Fig. 6, third column, middle and 1 r H r H 2 bottom rows), the radius of maximum winds is 2L 1 u h = 1 1ju j2 2 1 1 h u Re 1=4u , D2 1 1 2 r H 1 r H 1 1 1 from the vortex center. For the lower end of simulated 2 2 2 2 (A1) LD2 values that produce a relatively stable polar cyclone, this is roughly the same size as the real cyclones. Perhaps ›K 1 2 52= h u ju j2 there is some merit to the second interpretation? Be- ›t 2 22 2 cause there is very little theoretical understanding of the 1 2 transition from the troposphere to the interior, and be- 1 u h = ju j2 2 h u Re 1=4u , (A2) 2 2 2 2 2 2 2 cause the deformation radius estimates also cannot ›P r H benefit from more vertical sounding data, for now this is 1 52= h u 1 1 (c~2h 1 c~2h ) the realm of speculation. Also, given the large number ›t 1 1r H 1 1 2 2 2 2 of poorly constrained free parameters (layer gravity 1 r H 2 u h = 1 1ju j2 wave speeds, convective storm abundance, etc.), it is 1 1 2 r H 1 difficult to tell whether we are in the right ballpark for 2 2 2 ^ 1 r H 2 h 1 2 2 Ep, even as we use observationally motivated values for 1 1 1ju j S 2 1 1 Pe 1= h , (A3) 2 r H 1 st ~t 1 each parameter whenever possible. 2 2 rad OEF predict that, if polar cyclones are indeed shallow ›P 1 2 52= [h u (gc~2h 1 c~2h )] 2 u h = ju j2 features, and if measurements of Saturn’s and Jupiter’s ›t 2 2 1 1 2 2 2 2 2 2 Rossby radii are sufficiently accurate, then NASA’s 1 H h 2 1 2 Juno mission may not observe polar cyclones on Jupiter— 1 ju j2 1 S 2 2 1 Pe 1=2h , 2 2 H st ~t 2 at least not ones of comparable strength and stability to 2 rad those on Saturn. Another testable component of our (A4) theory is a cold temperature anomaly near the bottom of where total kinetic energy K, total potential energy P (to the troposphere, consistent with the warm upper tropo- some constant), and total available potential energy A sphere and the equivalent-barotropic PV structure of the are, respectively, cyclone. 1 r H K 5 1 1 h ju j2 1 h ju j2 , (A5) Acknowledgments. The authors wish to thank Adam 1 1 2 2 2 r2 H2 Showman and James Cho for insightful conversations 1 r H 1 and feedback. Andy Ingersoll, Mike Montgomery, and P 5 1 1c~2h2 1 c~2h2 1 gc~2h h , (A6) 2 r H 1 1 2 2 2 1 1 2 one anonymous reviewer provided thorough reviews 2 2 r H that improved the manuscript. M.E.O. was supported by 5 1 1 1 ~2 02 1 1~2 02 1 ~2 0 0 A c1h1 c2h2 gc1h1h2 . (A7) an NSF Graduate Research Fellowship, as well as NSF 2 r2 H2 2 ATM-0850639, NSF AGS-1032244, NSF AGS-1136480, and ONR N00014-14-1-0062. Some portions of this pa- It can be shown that while the final term on the rhs of the per are published in thesis form in fulfillment of the expression for A is often negative, total A is always requirements for the Ph.D. (O’Neill 2015). positive, as we expect.

APPENDIX B APPENDIX A Numerical Considerations Energy Equations As in OEF (methods section), the Cartesian grid is a To derive nondimensional energy equations, multiply the staggered Arakawa C grid. The time-stepping scheme upper-layer momentum equation by (r1/r2)(H1/H2)h1u1 is a second-order Adams–Bashforth algorithm. Early and the lower-layer momentum equation by h2u2. Likewise, tests showed that this provided dynamics nearly identi- the upper mass conservation equation is multiplied by cal to the third-order Adams–Bashforth scheme. Hori- j j2 =4 (1/2)(r1/r2)(H1/H2) u1 ; and the lower mass conservation zontal hyperviscosity is used instead of viscosity to 2 equation is multiplied by (1/2)ju2j . reduce its impact on the dynamics, which at upper levels Expressions for layer kinetic and potential energy on giant planets is virtually inviscid. The Reynolds and conservation are, respectively, Peclet numbers are ﬁxed at the highest values that

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