Effects of Initial Flow on Close-In Planet Atmospheric Circulation
EEffects of Initial Flow on Close-In Planet Atmospheric Circulation
Heidar Th. Thrastarson and James Y-K. Cho Astronomy Unit, School of Mathematical Sciences, Queen Mary University of London, Mile End Road,London E1 4NS, UK
[email protected]; [email protected]
ABSTRACT
We use a general circulation model to study the three-dimensional (3-D) flow and temper-ature distributions of atmospheres on tidally synchronized extrasolar planets. In this work, wefocus on the sensitivity of the evolution to the initial flow state, which has not received muchattention in 3-D modeling studies. We find that different initial states lead to markedly differentdistributions—even under the application of strong forcing (large day-night temperature differ-ence with a short “thermal drag time”) that may be representative of close-in planets. This is incontrast with the results or assumptions of many published studies. In general, coherent jets andvortices (and their associated temperature distributions) characterize the flow, and they evolvedifferently in time, depending on the initial condition. If the coherent structures reach a quasi-stationary state, their spatial locations still vary. The result underlines the fact that circulationmodels are currently unsuitable for making quantitative predictions (e.g., location and size of a“hot spot”) without better constrained, and well posed, initial conditions.
Subject headings: hydrodynamics — planets and satellites: general — turbulence — waves
1. Introduction
Understanding the flow dynamics of atmospheres is crucial for characterizing extrasolar planets. Dy-namics strongly influence the temperature distribution as well as the spectral behavior. An essential toolfor studying dynamics on the large-scale is a global hydrodynamics model. Many studies have used sucha model (e.g., Showman & Guillot 2002; Cho et al. 2003; Cooper & Showman 2005; Langton & Laughlin2007; Cho et al. 2008; Dobbs-Dixon & Lin 2008; Showman et al. 2008; Menou & Rauscher 2009). Themodels in these studies numerically solve a set of non-linear partial differential equations for the evolutionof a fluid on a rotating sphere. Hence, the initial condition (as well as the boundary conditions) needs to bespecified. Visiting scientist, Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, DC 20015, USA a r X i v : . [ a s t r o - ph . E P ] A p r arenot available and dominant dynamical processes, such as baroclinic instability and geostrophic turbulence,are not yet understood for the extrasolar planets (Cho et al. 2003, 2008; Showman et al. 2008; Cho 2008).Concerning initialization, there is a long history of research in geophysical fluid dynamics and numericalweather prediction, and it is still a subject of active research—even for the Earth (Holton 2004).In most simulations of close-in planets performed so far, the initial state is either at rest or with asmall, randomly perturbed wind field to break the flow symmetry (Cooper & Showman 2005; Langton &Laughlin 2007; Dobbs-Dixon & Lin 2008; Showman et al. 2008; Menou & Rauscher 2009). Cho et al.(2008) initialize their two-dimensional simulations with random eddies, and variations of the initial velocitydistributions are studied. They find significant differences in the flow evolution, depending on the vigor ofthe eddies. On the other hand, Cooper & Showman (2005) report on a three-dimensional (3-D) simulation,set up with an initial retrograde equatorial jet, and find no qualitative difference, compared with one startingfrom a rest state. Showman et al. (2008) and Cho (2008) give summaries of the various results.In this work, we present runs from an advanced 3-D general circulation model. As in Cooper &Showman (2005), as well as in Showman et al. (2008) and Menou & Rauscher (2009), the model used inthis work solves the full primitive equations (Pedlosky 1987). However, there are some important assetsin the model used in this work (see section 2), compared with most models used so far. For example, ituses a parallel pseudospectral algorithm (Orszag 1970; Eliasen et al. 1970; Canuto et al. 1988) with better-controlled, less invasive numerical viscosity. In this regard, our model is similar to the one used by Menou& Rauscher (2009).With our model, we focus on the sensitivity of the flow evolution to the initial state. The sensitivity hasnot been much emphasized in previous studies, particularly in those using 3-D circulation models. In orderto unambiguously delineate the sensitivity effect, we set up the simulations in a manner similar to previousstudies, apply idealized forcing (in many cases unencumbered by a vertical variation), and compare runswith all parameters identical—except for the initial condition.The basic plan of the paper is as follows. We describe the model and its setup for our simulations insection 2, where we endeavor to provide enough details to facilitate reproduction of the results. In section 3we present the results of simulations initialized with different organized large-scale flow patterns, includingthe rest state. In this section, we also show how sensitive the flow is to small perturbations in the initialwind field. We conclude in section 4, summarizing this work and discussing its implications for close-inextrasolar planet circulation modeling work. self-consistent set of fields which does not lead to excessive noise and deviations from accurate prediction
2. Method2.1. Governing Equations
The global dynamics of a shallow, 3-D atmospheric layer is governed by the primitive equations (e.g.,Pedlosky 1987; Holton 2004). Here, by “shallow” we mean the thickness of the atmosphere under consid-eration is small compared to the planetary radius R p . In atmospheric studies, pressure p is commonly usedas the vertical coordinate. In the pressure coordinate system, these equations read:D v D t + (cid:18) uR p tan φ (cid:19) k × v = − ∇ p Φ − f k × v + D v (1a) ∂ Φ ∂ p = − ρ (1b) ∂ω∂ p = − ∇ p · v (1c)D T D t − ωρ c p = ˙ q net c p + D T , (1d)where DD t = ∂∂ t + v · ∇ p + ω ∂∂ p . In the above equations, v ( x , t ) = ( u , v ) is the (eastward, northward) velocity in a frame rotating with Ω ,the planetary rotation rate; Φ = gz is the geopotential, where g is the gravitational acceleration and z is thedistance above the planetary radius R p ; k is the unit vector in the local vertical direction; f = 2 Ω sin φ is theCoriolis parameter, the projection of the planetary vorticity vector 2 Ω onto k , with φ the latitude; ∇ p is thehorizontal gradient on a constant p -surface; ω ≡ D p / D t is the vertical velocity; ρ is the density; D v = − ν ∇ v represents the momentum dissipation, with ν the constant viscosity coefficient; T is the temperature; c p isthe specific heat at constant pressure; ˙ q net is the net diabatic heating rate; and, D T = − ν ∇ T represents thetemperature dissipation.Equations (1) are closed with the ideal gas law, p = ρ RT , as the equation of state, with R the specificgas constant. A suitable set of boundary conditions, used in this work, is D p / D t = 0 at the top and bottom p -surfaces. Hence, the boundaries are material surfaces and no mass flow is allowed to cross the bound-aries. With these boundary conditions, the equations admit the full range of motions for a stably-stratifiedatmosphere—except for sound waves. For a discussion of the various aspects of the primitive equations(including superviscosity) and their use for extrasolar planet application, the reader is referred to Cho et al.(2003), Cho et al. (2008) and Cho & Gulsen (in preparation). In this work, as described below, equations (1)are actually solved in a more general coordinate system . which is useful when variations in the bottom boundary, caused by static or dynamic conditions, are not small To solve equations (1) in the spherical geometry, we use the Community Atmosphere Model (CAM3.0). CAM is a well-tested, highly-accurate pseudospectral hydrodynamics model developed by the NationalCenter for Atmospheric Research (NCAR) for the atmospheric research community (Collins et al. 2004).For hydrodynamics problems not involving sharp discontinuities (e.g., shocks) and irregular geometry, thepseudospectral method is superior to the standard grid and particle methods (e.g., Canuto et al. 1988).As in many pseudospectral formulations of the algorithm, CAM solves the equations in the vorticity-divergence form in the horizontal direction, where ζ = ∇ × v is the vorticity and δ = ∇ · v is the divergence.In the vertical direction, CAM uses the generalized p -coordinate: p ( λ, φ, η, t ) = A ( η ) p r + B ( η ) p s ( λ, φ, t ) , (2)where λ is the longitude, φ is the latitude, η is the generalized vertical coordinate, p r is a constant referencepressure, p s ( λ, φ, t ) is a deformable pressure surface at the bottom boundary, and A , B ∈ [0 , ∇ operators), as well as a small Robert-Asselin time filter (cid:15) (Robert 1966; Asselin 1972), are applied at every timestep in each layer to stabilize theintegration. The timestepping is done using a semi-implicit, second-order leapfrog scheme. Note that effectsof various numerical dissipation are often subtle and can be significant on the integration, particularly overlong times (e.g., Dritschel et al. 2007). Further details of the model and the effects of numerical viscosityon the flow evolution will be described elsewhere. In all the simulations discussed in this paper, the physical parameters chosen are based on the close-inextrasolar planet, HD209458b. The basic result presented—that the evolution depends on the initial flowstate—does not change for a different close-in planet. The physical parameters for the model HD209458bplanet are listed in Table 1.CAM is able to include radiatively-active species and their coupling to the dynamics. However, wedo not include them in the present work so that the effects discussed are not obfuscated by complicationsunrelated to the essential result. Our principle motivation is to study the dependence on the initial flow in themost unambiguous way possible. To this end, the flow is forced using the simple Newtonian drag formalism,as in many previous studies of extrasolar planet atmospheres (e.g., Cooper & Showman 2005; Langton &Laughlin 2007; Showman et al. 2008; Menou & Rauscher 2009). This drag is a simple representation of thenet heating term in equation (1d): ˙ q net c p = − τ th ( T − T e ) , (3)where T e = T e ( λ, φ, η, t ) is the “equilibrium” temperature distribution and τ th is the thermal drag time con-stant. 5 –In this work, both T e and τ th are prescribed and barotropic ( ∂/∂η = 0) and steady ( ∂/∂ t = 0), althoughsimulations relaxing these restrictions have been run to verify robustness of our results. In general, both T e and τ th are (as are R and c p ) complicated functions of space and time (Cho 2008). Here, T e = T m + ∆ T e cos φ cos λ , (4)where T m = ( T D + T N ) / ∆ T e = ( T D − T N ) / T D and T N are the maximum and minimum temperaturesat the day and night sides, respectively. Most of the simulations described in this paper have T D = 1900 K, T N = 900 K, and τ th = 3 HD 209458 b planet days (where τ p ≡ π/ Ω is 1 planet day). Note that we havevaried the timescale of the forcing by using a τ th value in the range from 0.01 to 10 planet days, as well asletting the timescale to decrease with height. The main result does not change for values ∼ > . τ th in all past studies using the Newtonian drag formalism.The spectral resolution in the horizontal direction for most of the runs described in the paper is T42,which corresponds to 128 ×
64 grid points in physical space . We have performed runs with resolutionsvarying from T21 (64 ×
32) to T85 (256 × The pressure atthe bottom η boundary is initially 1 bar, but the value of the pressure changes in time. This range of pressureis chosen because it encompasses the region where current observations are likely to be probing and wheremost of the circulation modeling studies have thus far directed their attention. We have also performed sim-ulations in which the domain extends down to 100 bars and again verified that the basic behavior describedin this paper is not affected. The entire domain is initialized with an isothermal temperature distribution, T m = 1400 K.
3. Results3.1. Basic Dependence: Jets
To examine the robustness of evolved flow states to organized initial flow configurations, we haveperformed simulations with a wide range of initial conditions. The conditions from four of those runs(labeled RUN1–RUN4) are shown in Figure 1. In all the runs presented, the setup is identical—except forthe initial flow configuration. The physical and numerical parameters/conditions are given in Tables 1 and2, respectively.RUN1 is initialized with a small, random perturbation introduced in the flow. Specifically, valuesof u and v are drawn from a Gaussian random distribution centered on zero with a standard deviation of Note that, because of the higher order accuracy of the spectral method, this essentially corresponds to a finite differenceresolution of over 420 × Table 3 in the Appendix gives the positions of all the model levels (layer interfaces). − . RUN2 is initialized with a zonally-symmetric, eastward equatorial jet of the following form: u ( φ ) = U exp (cid:26) ( φ − φ ) σ (cid:27) , (5)where u ( t = 0) = u , U = 1000 m s − , φ = 0, and σ = π/
12. RUN3 is initialized with a westward equatorialjet described by equation (5), with U = − − , φ = 0, and σ = π/
12. RUN4 is initialized with a flowcontaining three jets. Note that the condition for RUN4 is very similar to the zonal average of the wind fieldof RUN1 at 50 planetary rotations. The jet profiles presented in Figure 1 are independent of height, as wellas longitude.Figure 2 shows the temperature and flow fields of the four runs at t /τ p = 40 (or t /τ th ≈ η level-surfaces of our model are func-tions of pressure, as described by equation (2).] The figure illustrates the major point of this paper: givendifferent initial states, there are clear, qualitative (as well as quantitative) differences between the differentruns. Qualitatively, there are some common features. For example, most of the runs exhibit a coherentquadrupole flow structure—two large cyclonic and anti-cyclonic vortex-pairs straddling the equator. How-ever, the location of an individual vortex is different in the runs—as is the temperature pattern. In RUN3,a distinct quadrupole pattern is not present but there are more vortices in this run compared to the otherruns. The temperature distributions are different because they are strongly linked to the flow. Consequently,the minimum-to-maximum temperature ranges vary from a moderately large 550 K (RUN4) to only about200 K (RUN3) in the figure.The behavior just described is not restricted to a single altitude. Figure 3 shows the fields correspond-ing to those presented in Figure 2, but at a higher altitude ( p ≈
85 mbar pressure level). Comparison ofFigures 2 and 3 illustrates the structural differences in 3-D (vertical), as well as in 2-D (horizontal). InRUN2 and RUN4, the large-scale vortices are strongly aligned, forming columns through most of the heightextent of the modeled atmosphere; that is, the flow is strongly barotropic. In the other two runs, the flow isnot vertically aligned throughout in large parts of the modeled atmosphere—and, therefore, the flow is baro-clinic. The two figures also point to the corresponding strong difference in 3-D temperature distributions,associated with the flow structures.This is more clearly seen in Figure 4. The figure shows the vertical (height-latitude) cross-section ofthe temperature at 0 degrees (sub-stellar) longitude from the runs presented in Figures 2 and 3. In Figure 4,the hottest and coldest regions are at different locations in all the runs. Near the equator, RUN1 exhibitsa strong temperature inversion , while RUN2 and RUN4 do not. In addition, RUN2 and RUN4 exhibitgenerally strong decreases in temperature with height, while the others do not. As can be seen, the degree of Here, and in other figures, streamlines are shown. Streamlines are obtained by smoothly following the flow; they are tangentto the instantaneous velocity vectors at each grid point. The cyclonicity of a vortex is defined by the sign of ζ · Ω : it is positive for a cyclone and negative for an anticyclone. See Burrows et al. (2007) and Knutson et al. (2008) for discussion of thermal inversion in the context of close-in extrasolargiant planets. ∼
200 K contrast (RUN3)to ∼
500 K contrast (RUN4). In general, the vertical structure is of low order, containing usually a singleinversion. In our study, some form of inversion appears to be a generic feature.Preliminary steady state analysis of the primitive equations suggests that the basic behavior describedabove is due to the way in which the applied, ( s , n ) = (1 , s is the zonal wavenumber and n is the total (sectoral) wavenumber of thespectral harmonics. In particular, a normal mode decomposition of the atmosphere into the vertical structureand Hough functions (e.g., Chapman & Lindzen 1970; Longuet-Higgins 1967) indicates that the forcingprojects mostly onto low-order baroclinic modes, when the initial state is at rest. In contrast, when theinitial state contains large-scale jets, the forcing projects more strongly on the barotropic mode, comparedto the runs started from rest. Similar behavior has been observed in studies of the Earth’s troposphere undertropical forcing (e.g., Geisler & Stevens 1982; Lim & Chang 1983). A more detailed study of couplingbetween forcing and normal modes is currently being performed and will be described elsewhere.Furthermore, it is important to note that all of the above features, both dynamical and thermal, canvary in time. All of these features are important for observations (e.g., Knutson et al. 2007) and spectralmodeling (e.g., Tinetti et al. 2007). A thorough study of the long-time evolution (over 1000 planetaryrotations, or more than 330 τ th ) of the runs reveals a fundamental difference in their temporal behavior aswell. For example, the flow pattern in RUN2 is characterized by two vertically aligned vortex columns ineach hemisphere that translate longitudinally around the poles. The temperature in the upper altitude regionis more strongly coupled to the flow than it is in the lower altitude regions. The flow pattern in RUN4 is alsoa set of vertically aligned vortex columns, but the columns oscillate in the east-west direction. The patternsin RUN1 and RUN3 are more complex, exhibiting a mixture of vortex splitting and merger and stationarystates at different altitude levels. Figure 5, which shows a time series of the total kinetic energy, gives aquantitative measure of the different temporal behavior of the simulations.When the flow field is time-averaged over a long period, the time-mean state is also significantly sensi-tive to the initial flow. This can be seen in Figure 6, which shows temperature cross-sections at an arbitrarylongitude ( λ = 135 ◦ ), averaged over 450 planetary rotations (planet day 300 to 750). The figure clearlyshows that the variability observed is not simply a result of a phase shift in a quasi-periodic evolution. Theflow and temperature structures in each run are fundamentally different from one another.Interestingly, the full range of flow and temperature behavior described above has been previouslycaptured qualitatively, using the one-layer equivalent-barotropic model (Cho et al. 2008). The equivalent-barotropic equations are a reduced, vertically-integrated version of the full primitive equations used in thisstudy (Salby 1989). In many situations, the reduced model can be fruitfully used to study the dynamics ofthe full model by varying the Rossby deformation radius to represent different heights (or temperatures) ofthe full multi-layer model (e.g., Cho et al. 2008; Scott & Polvani 2008), and this also appears to be the casefor hot extrasolar planets. 8 – As might be expected from general nonlinear dynamics theory, in fact the evolution can be stronglysensitive to small differences in the initial flow state. This is illustrated in Figure 7. There, two simulationsare presented (panels a and b), which are identical in all respects except for a minute difference in the initialflow. The simulation in the top panel (RUN5) is started from rest. In contrast, the simulation in the bottompanel (RUN1) is started with a small perturbation: the initial values of u and v at each grid point are setto a Gaussian random distribution, centered on zero with a standard deviation of 0.05 m s − . Note that themaximum initial wind perturbation magnitude is only about 0.02% of the typical root mean square flowspeed in the frames shown ( ∼
500 m s − ).The two panels in Figure 7 show temperature and flow distributions after t /τ p = 1000 (or t /τ th ≈ p ∼
420 mbar level. The distributions in the two panels are clearly different. At times they may lookmore similar than shown here, but in general that is not the case. The two runs generally show a differenttemporal behavior. Note that in Figure 7a, there is a high degree of hemispheric symmetry, particularly in thenorth-south direction. In contrast, Figure 7b shows a clear asymmetry in both the north-south and east-westdirection. The small asymmetry in Figure 7a is entirely due to machine precision and is not physical, sincethere is no way to break the symmetry in the setup of the run. Therefore, not surprisingly, some mechanismfor inducing a noticeable symmetry breaking is necessary. The salient point here is, however, that even atiny perturbation can lead to a marked difference in the flow and temperature distributions, even at relativelyearly integration times.
It is important to understand that the dependence on the initial flow state is robust and the behavior isnot limited only to the parameters, and the ranges, discussed thus far. The dependence has been verifiedfor numerous model parameters and ranges. For example, Figure 8 shows that the strong dependence onthe initial wind exists for much shorter τ th , despite the strong forcing such drag times entail. In this case, τ th /τ p = 0 .
5. The upper panel is from RUN6 and the lower panel is from RUN7. The former run is initializedwith only small stirring and no organized jet. In contrast, the latter run is initialized with a westwardequatorial jet, identical to the setup of RUN3. At the shown time and height, there are clear differencesbetween the flow and temperature patterns of the two simulations. The coldest area is advected east of theanti-stellar point in RUN6, but west of the anti-stellar point in RUN7. Furthermore, all the vortices havedifferent locations. In RUN7, there is a fairly zonally symmetric jet at high latitudes, leading to a muchmore homogenized temperature distribution above the mid-latitudes than in RUN6.While a weaker difference might be expected in this case, based on other studies (e.g., Cooper &Showman 2005), the sensitivity is unabated in our simulations. And, this holds for the long time averagebehavior as well. For a time mean over 300 rotations (e.g., planet days 1200 to 1500), at the level shown inFigure 8, the location of the coldest region differs by 40 degrees in longitude between the two simulationsshown in the figure. Figure 9 shows a time series of the total kinetic energy in the two simulations, revealing 9 –their different evolution. Even after integrating for a very long time (15,000 rotations), we have checkedthat the differences between RUN6 and RUN7 are still present. Note that this is a much longer integrationtime than that reported in any published studies of close-in planet circulation thus far. However, the flow atsuch long times is inexorably affected by cumulative numerical dissipation and phase errors and the resultobtained should not be taken too literally (e.g., Canuto et al. 1988).In addition, we have studied the dependence when the initial jet contains a vertical shear–i.e., the jetdistribution is baroclinic. Simulations have been initialized with an eastward jet, which has zero magnitudeat the bottom and increases linearly with height so that the lateral flow distribution in the top layer of themodel is identical to that in RUN2. In the baroclinic case, vortex columns evolve that extend throughoutmost of the atmosphere, as in RUN2. However, unlike in RUN2, the temperature structure here is alsostrongly barotropic. This appears to be related to the way the forcing projects onto the free modes of thesystem, as described in section 3.1.As alluded to in section 2.1, the timescale of the fastest motions admitted by equations (1) with thegiven boundary condition is ∼
40 minutes, for the planetary physical parameters used in this work. Hence,a forcing with timescales of the same order or smaller is not entirely physically self-consistent with the useof equations (1). Notably, such forcing introduces numerical difficulties. For the physically unrealistic dragtime of τ th = 1 hour, for which excessive numerical dissipation is required to prevent the code from blowingup or being inundated with numerical noise, the temperature evolution is only mildly sensitive to the initialcondition, at long-time integrations. However, when such short drag times are only applied over a limitedrange in pressure—i.e., τ th is allowed to vary with η — the sensitivity to the initial condition is present evenafter long time integrations (7000 planet days).
4. Conclusion
In this paper, we have shown that in a generic general circulation simulation of a tidally synchronizedplanet, the flow and temperature distributions depend strongly on the initial state. In all simulations initial-ized with a different jet configuration, large scale coherent vortices are formed; but, their location, size, andnumber varies, depending on the initial wind. The temperature distribution is relatively homogenized bythe flow, compared to the large temperature contrast in the forcing equilibrium temperature profile. But thedegree of mixing, as well as locations of temperature extremes vary between differently initialized simula-tions. The time variability of the atmosphere—i.e. how vortices and associated temperature patterns movearound the planet or whether they stay at fixed positions—varies depending on the initial wind.The Newtonian drag scheme used in this study is idealized and the “correct” parameters to use in theset-up are unknown, with many choices possible. Explorations of different parts of the parameter spacewill be presented elsewhere. Here, we have chosen the set-up to capture, as cleanly as possible, the effectsof the initial flow in a regime of parameter space plausibly relevant for hot Jupiters with strong zonally- Recall that the flow distribution in this run is barotropic.
10 –asymmetric forcing. We have found that the strong dependence on the initial wind is valid for a wide rangeof thermal drag times ( τ th = 0.5–10 planet days) and with T21, T42, and T85 resolutions. Some reductionin the sensitivity is sometimes observed for the very small τ th and long time integration simulations, whichrequire large numerical dissipation to prevent the fields from being dominated by small scale noise. Thissituation is, however, not physically realistic and numerically suspect.The strong dependence on initial wind has implications for the use of general circulation models forinterpretation of observations of extrasolar planet atmospheres. These results underline that, while numer-ical circulation models of the kind employed here are useful for studying plausible mechanisms and flowregimes, they are currently unsuitable for making “hard” predictions—such as exact locations of temperatureextremes on a given planet.H.Þ.Þ. is supported by the EU Fellowship. J.Y-K.C. is supported by the NASA NNG04GN82G andSTFC PP/E001858/1 grants. The authors thank O. M. Umurhan and Chris Watkins for helpful discussions,and the anonymous referee for helpful suggestions. 11 – A. Appendix
Table 3 presents the A and B coefficients of equation (2) in §2.2. Note that, as defined, each η -surfacecan span across a range of p -surfaces. REFERENCES
Asselin, R. 1972, Mon. Wea. Rev., 100, 487Burrows, A., Hubeny, I., Budaj, J., Knutson, H. A., and Charbonneau, D. 2007, ApJ, 668, L171Canuto, C., Hussaini, M.Y., Qarteroni, A., and Zang, T.A. 1988, Spectral Methods in Fluid Dynamics (NewYork, NY: Springer)Chapman, S. and Lindzen, R.S. 1970, Atmospheric Tides (New York, NY: Gordon and Breach)Cho, J. Y-K., Menou, K., Hansen, B. M. S., and Seager, S. 2003, ApJ, 587, L117Cho, J.Y-K., Menou, K., Hansen, B.M.S., and Seager, S. 2008, ApJ, 675, 817Cho, J.Y-K. 2008, Phil. Trans. R. Soc. A, 366, 4477Collins, W.D. et al. 2004, NCAR/TN-464+STRCooper, C.S. and Showman, A.P. 2005, ApJ, 629, 45LDobbs-Dixon, I. and Lin, D.N.C. 2008, ApJ, 673, 513Dritschel, D.G., Tran, C.V. and Scott, R.K. 2007, J. Fluid Mech., 591, 379Geisler, J.E. and Stevens, D.E. 1982, Quart. J. Roy. Meteor. Soc., 108, 87Eliasen, E., Mechenhauer, B., and Rasmussen, E. 1970, Copenhagen Univ., Inst. Teoretisk Meteorologi,Tech. Rep. 2Holton, J. R. 2004, Introduction to Dynamical Meteorology (Burlington, MA: Academic Press)Knutson, H., et al., 2007, Nature, 447, 183Knutson, H. A., Charbonneau, D., Allen, L. E., Burrows, A., and Megeath, S. T. 2008, ApJ, 673, 526Langton, J. and Laughlin, G. 2007, ApJ, 657, 113LLim, H. and Chang, C.P. 1983, J. Atmos. Sci., 40, 1897Longuet-Higgins, M.S. 1967, Phil. Trans. R. Soc. A, 269, 511Menou, K. and Rauscher, E. 2009, ApJ, 700, 887 12 –Orszag, A. 1970, J. Atmos. Sci., 27, 890Pedlosky, J. 1987, Geophysical Fluid Dynamics (New York, NY: Springer-Verlag)Robert, A. 1966, J. Meteorol. Soc. Japan, 44, 237Salby, M. L., 1989, Tellus, 41A, 48Scott, R. K. and Polvani, L.M. 2008, Geophys. Res. Lett., 35, L24202Showman, A. P. and Guillot, T. 2002, A&A, 385, 166Showman, A.P., Cooper, C.S., Fortney, J.J., and Marley, M.S. 2008, ApJ, 682, 559Showman, A.P., Menou, K., and Cho, J. Y-K. 2008, in ASP Conf. Ser. 398, Extreme Solar Systems, ed. D.Fischer et al. (San Francisco, CA: ASP)Tinetti, G. et al., 2007, Nature, 448, 169
This preprint was prepared with the AAS L A TEX macros v5.2.
13 –Table 1. Physical ParametersPlanetary rotation rate Ω × − s − Planetary radius R p mGravity g
10 m s − Specific heat at constant pressure c p × J kg − K − Specific gas constant R × J kg − K − Mean equilibrium temperature T m T D T N
900 KInitial temperature T τ th [2 π/ Ω ] N ν [10 m s − ]1 small noise 3 42 12 eastward jet 3 42 13 westward jet 3 42 14 three jets 3 42 15 zero winds 3 42 16 small noise 0.5 21 107 westward jet 0.5 21 10Note. — τ th is the thermal drag timescale, N is the spectral truncation wavenumber, and ν is the supervis-cosity coefficient. All the runs have Robert-Asselin filter coefficient (cid:15) = 0.06. In the T42 runs the timestepis ∆ t = 120 s, but for T21 resolution ∆ t = 240 s. 14 –Fig. 1.— Initial conditions for simulations RUN1 (—), RUN2 (- - -), RUN3 ( − · − ), and RUN4 ( · · · ). Theheight-independent, zonally-symmetric, eastward velocities, u ( λ, φ, η,
0) = u ( φ ), are shown. 15 –Fig. 2.— Temperature (color map) and flow (streamlines) fields at t = 40 τ p , near the p ∼
900 mbar level, forRUN1 (a), RUN2 (b), RUN3 (c), and RUN4 (d). The fields are shown in cylindrical-equidistant projectioncentered at the equator. The four simulations are set up identically, except for the initial wind field. Thelocation and size of vortices, and the associated temperature patterns, strongly depend on the initial windconfiguration.
16 –Fig. 3.— Temperature (color map) and flow (streamlines) fields at t = 40 τ p , near the p ≈
85 mbar level,for the same four simulations as shown in Figure 2: RUN1 (a), RUN2 (b), RUN3 (c) and RUN4 (d). Thesensitivity to the initial wind state is present throughout all heights in the atmosphere. 17 –Fig. 4.— Snapshots of temperature vertical cross-section at the sub-stellar longitude, at t = 40 τ p , for the foursimulations presented in Figure 2: RUN1 (a), RUN2 (b), RUN3 (c) and RUN4 (d). The vertical temperaturestructure is strongly sensitive to the initial flow state, and usually contains an inversion. 18 –Fig. 5.— Time series of total kinetic energy, integrated over the domain, for RUN1 (black —), RUN2 (red– – –), RUN3 (blue − · − ), and RUN4 (green - - -). 19 –Fig. 6.— Temperature vertical cross-section at an arbitrary longitude ( λ = 135 ◦ ), averaged over 450 plane-tary rotations (150 thermal drag times), for the four simulations presented in Figure 2: RUN1 (a), RUN2 (b),RUN3 (c) and RUN4 (d). The difference in the temperature structure is independent of long time-averagingand is not due to a phase shift in a quasi-periodic evolution. 20 –Fig. 7.— Temperature (color map) and flow (streamlines) fields after 1000 planetary rotations, at the p ∼
420 mbar level, for two simulations differing only by small deviations in the initial wind. The top panelshows the result of a simulation started from rest, while the bottom panel is from a simulation started witha small perturbation. Note the clear asymmetry in the north-south direction, which only appears when thesmall perturbations are present in the initial state. 21 –Fig. 8.— Snapshots of temperature fields (color coded) with streamlines overlaid. Fields at the p ∼ a Level Surface Index A × B × a The pressure at each point is given by equation (1) in the text, with p rr