Gravity Wave Dynamics in Hot Extrasolar Planet Atmospheres Chris Watkins* & James Y-K. Cho Astronomy Unit, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK. email: [email protected], [email protected] 1. Introduction 5. Effect on the Background Stably-stratified atmospheres can support internal gravity waves, which arise from the buoyancy of the fluid. As can be seen in Fig. 3 the wave propagates in a sheared Acceleration(ms−1 rotation−1) The waves are readily excited by heating and flow over topography and are a ubiquitous feature of the terrestrial c − u −300 −200 −100 0 100 200 300 environment. So, 0 = 0 is possible in some layer 7 −3 atmosphere. They have been intensely studied over the past several decades. They transport momentum and known as a critical layer. A wave encountering a crit- 10 heat within a fluid and the effects they have on the mean flow are routinely parametrized in GCMs of the ical layer is shown in Fig. 4. The wave is attenuated Earth’s atmosphere. by the encounter and the momentum it transports is de- 6 There are over 400 known extrasolar plan- c−u posited, causing the flow speed to change. In this case 0 ets. Their orbits and masses vary far more −1 Exoplanet Distribution the acceleration, ∼250 m s per rotation, is enough to 5 du /dt 0 1 than for the planets in the Solar System, −2 ) u 10 < 0.03 M p J double the flow speed at this layer in about 2 planetary 0.9 0.03 M − 0.25 M as shown in Fig. 1. Many planets are very J J rotations. In reality a spectrum of waves will encounter 0.25 M − 13 M 4 0.8 J J close to their parent star: nearly 70 have > 13 M critical layers over a range of altitudes, causing accel- J orbits of four days or less, they are as- 0.7 Solar System erations over the range. Thus the critical layers filter HD 209458 b 0.6 sumed to be spin-orbit synchronised. They 3 out waves, preventing them from propagating to high Pressure (bar) are exposed to a very large energy flux. 0.5 Height above 1 bar (H −1 altitudes. 10 Eccentricity w 0.4 There is much interest in modelling the 2 0.3 atmospheric circulations in these extreme Mercury −1 0.2 conditions (e.g., Cho, 2008; Showman et Heating Rate (K rotation ) −100 −50 0 50 100 1 0.1 Mars al., 2009). We consider the effects gravity 15 HD 209458 b Earth Venus −6 0 waves have on these planets by studying 14 10 0 100 200 300 400 500 600 700 800 900 1000 0 Period (days) 0 10 their behaviour in an atmosphere based on 13 −100 −50 0 50 100 Velocity (m s−1) 12 T Fig. 1: Known extrasolar planets. Solar System bodies the hot-Jupiter planet HD 209458 b. −5 Student Version of MATLAB 10 are shown for comparison and the hot-Jupiter 11 Fig. 4: A wave with c =600 m s−1 and dT /dt HD 209458 b is highlighted. (Data from exoplanet.eu) 10 0 ) p −4 2π=k =2500 km saturates in an encounter with 10 9 a critical layer near 5 Hp. 8 −3 2. Gravity Waves 7 10 6 Pressure (bar) Internal gravity waves have large scale effects in the terrestrial atmosphere - for example, the Quasi-Biennial Height above 1 bar (H 5 −2 Oscillation (Baldwin et al., 2001). They have also been observed in many Solar Systems atmospheres beyond 10 Gravity waves saturate when the temperature perturba- the Earth (e.g., Young et al., 1997). The atmospheres of extrasolar planets are expected to be stably stratified: 4 tions cause the atmosphere to become locally connec- 3 so, gravity waves will be important for modelling their dynamics. −1 tively unstable and the wave stops growing. This is 10 2 shown in Fig. 5. Here we show the heating the wave 1 induces on the background, roughly 75 K per rotation. 0 0 10 −50 0 50 Such heating could trigger new gravity waves and will 3. The Taylor-Goldstein Equation Temperature Perturbation (K) induce additional dynamical effects. Waves originat- Student Version of MATLAB The behaviour of a linear gravity wave is described by the Taylor-Goldstein equation which describes how the ing from deeper levels in the atmosphere have larger ef- −1 vertical velocity w~ varies with altitude: Fig. 5: A wave with c = −40 m s and fects. For example, excitation at the 30 bar level leads 2π=k = 2500 km saturates above 10 H . " # p to heating of nearly 3000 K per rotation. d2w^ N2 u00 u0 1 κ Q_ 0 0 2 −z=2Hρ 2 + 2 + + − 2 − k w^ = 2 e ; dz (c − u0) (c − u0) Hρ (c − u0) 4Hρ Hp (c − u0) Z z 6. Horizontal Transport −χ(z) dξ w^(z) = w~ e ; χ(z) = : For shorter waves it is possible for k2 to dominate the Index of Refraction (× 10−6 m−1) zb 2Hρ(ξ) index of refraction, which becomes imaginary (indicat- 0 10 20 30 40 50 60 15 Here c is the horizontal phase speed and k is the horizontal wavenumber. The mean flow is u0, primes are ing that the wave is evanescent). A wave is reflected _ 14 −6 differentiation w.r.t. z, and Q is the heating. In the square brackets is the index of refraction. at a layer where it becomes evanescent, as can be seen 10 13 The equation is solved numerically using Gaussian elimination with 3000 to 10000 equally spaced levels. in Fig. 6; the vertical group velocity vanishes at around 12 −5 As the equation is linear the wave can grow without bounds. To counter this a scheme is used to introduce the 10 Hp level, where Re(m) vanishes. Where there 10 11 saturation when the wave becomes convectively unstable. are layers that support propagation between evanescent 10 ) layers, the wave is ducted and energy is transported hor- p −4 izontally. The wave in Fig. 6 is an example of this. 9 10 Waves are often ducted in jets and the energy is trans- 8 −3 4. HD 209458 b ported at roughly the speed of the jet. 7 10 w The extrasolar planet HD 209458 b, is a hot-Jupiter planet that orbits its star at just 0.47 AU with a period of 6 g Pressure (bar) Re(m) u 3.52 days. It is expected to be tidally locked to its star. It has a radius of 1.32 R . Height above 1 bar (H g J 5 −2 10 −1 −1 4 Specific Gas Constant R 3523 J kg K Index of Refraction (× 10−6 m−1) 3 −1 −1 0 20 40 60 80 100 −1 Specific Heat at Constant Pressure cp 12300 J kg K 15 10 −2 2 g −6 Acceleration Due to Gravity 10 m s 14 10 Ω ×10−5 −1 1 Rotation Rate 2.08 s 13 0 0 10 0 100 200 300 400 500 600 12 −5 −1 10 Velocity (m s ) HD 2094568 b is a widely studied planet and Student Version of MATLAB 11 HD 209458b Model Zonal Flow and Temperature it is assumed to be typical for the purpose of 1520 −1 90 10 c = ) Fig. 6: A wave with 700 m s and our study. The physical parameters we used p −4 1500 9 10 2π=k = 1410 km. The vertical and horizontal are given in the table above. 60 1480 8 Re(m) group velocities and the real part of the index 1460 path −3 of refraction are shown. The forcing in our exploration is centred at 30 7 10 1440 Our Study Pressure (bar) 1 scale height above the 1 bar level. It takes 6 0 1420 the form of a modified Gaussian that is zero Height above 1 bar (H Latitude 5 −2 10 beyond two half-widths (∼ 0.3 Hp) from the 1400 Energy can be transported across vast distances in ducts −30 4 forcing centre. The heating rate we use is 1380 as Fig. 7 shows. With just one reflection the wave is car- 3 −1 modest at roughly 300 K per rotation (i.e. −60 1360 10 ried about two planetary radii across the surface. This 2 _ −3 −1 Q=cp = 10 K s ). 1340 will enable energy to be carried across the terminator 1 −90 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 from the day-side to the night-side. The properties of 0 Longitude 0 10 0 0.2 0.4 0.6 0.8 1 the jet will change around the planet. In places it will Distance (R ) Fig. 2: Modelled flow and temperature on HD 209458 b p become leaky and the wave will escape, via this mech- Temperature (K) Student Version of MATLAB 1300 1320 1340 1360 1380 1400 H 15 at about 1 p above the 1 bar level. The longest zonal anism heat and momentum can be transported between flow vectors are 533 m s−1. The sub-stellar point is at the Fig. 7: The path of propagation of the wave in hemispheres thus contributing to the homogenisation of 14 −6 10 centre (0◦ N, 0◦ E). Fig. 6. the atmospheric temperature. 13 12 u 0 −5 10 11 The model atmosphere used is based on simulations 10 generously shared by Thrastarson & Cho (2009), part ) p −4 7.