The increasing richness of exoplanet observations has motivated a variety of three-dimensional (3D) atmospheric circulation models of these planets. Under strongly irradiated conditions, models of tidally locked, short-period planets (both hot Jupiters and terrestrial planets) tend to exhibit a circulation dominated by a fast eastward, or 'superrotating', jet stream at the equator. When the radiative and advection timescales are comparable, this phenomenon can cause the hottest regions to be displaced eastward from the substellar point by tens of degrees longitude. Such an offset has been subsequently observed on HD 189733b, supporting the possibility of equatorial jets on short-period exoplanets. Despite its relevance, however, the dynamical mechanisms responsible for generating the equatorial superrotation in such models have not been identified. Here, we show that the equatorial jet results from the interaction of the mean flow with standing Rossby waves induced by the day-night thermal forcing. The strong longitudinal variations in radiative heating-namely intense dayside heating and nightside cooling-trigger the formation of standing, planetary-scale equatorial Rossby and Kelvin waves. The Rossby waves develop phase tilts that pump eastward momentum from high latitudes to the equator, thereby inducing equatorial superrotation. We present an analytic theory demonstrating this mechanism and explore its properties inmore » a hierarchy of one-layer (shallow-water) calculations and fully 3D models. The wave-mean-flow interaction produces an equatorial jet whose latitudinal width is comparable to that of the Rossby waves, namely the equatorial Rossby deformation radius modified by radiative and frictional effects. For conditions typical of synchronously rotating hot Jupiters, this length is comparable to a planetary radius, explaining the broad scale of the equatorial jet obtained in most hot-Jupiter models. Our theory illuminates the dependence of the equatorial jet speed on forcing amplitude, strength of friction, and other parameters, as well as the conditions under which jets can form at all.« less

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Equatorial superrotation on tidally locked exoplanets

Some active galactic nuclei, microquasars, and gamma ray bursts may be powered by the electromagnetic braking of a rapidly rotating black hole. We investigate this possibility via axisymmetric numerical simulations of a black hole surrounded by a magnetized plasma. The plasma is described by the equations of general relativistic magnetohydrodynamics, and the effects of radiation are neglected. The evolution is followed for $2000 G M/c^3$, and the computational domain extends from inside the event horizon to typically $40 G M/c^2$. We compare our results to two analytic steady state models, including the force-free magnetosphere of Blandford & Znajek. Along the way we present a self-contained rederivation of the Blandford-Znajek model in Kerr-Schild (horizon penetrating) coordinates. We find that (1) low density polar regions of the numerical models agree well with the Blandford-Znajek model; (2) many of our models have an outward Poynting flux on the horizon in the Kerr-Schild frame; (3) none of our models have a net outward energy flux on the horizon; and (4) one of our models, in which the initial disk has net magnetic flux, shows a net outward angular momentum flux on the horizon. We conclude with a discussion of the limitations of our model, astrophysical implications, and problems to be addressed by future numerical experiments.

https://arxiv.org/pdf/astro-ph/0404512.pdf

A Measurement of the Electromagnetic Luminosity of a Kerr Black Hole

Hydrodynamic and hydromagnetic stability

Prompt gamma-ray burst (GRB) emission requires some mechanism to dissipate an ultrarelativistic jet. Internal shocks or some form of electromagnetic dissipation are candidate mechanisms. Any mechanism needs to answer basic questions, such as what is the origin of variability, what radius does dissipation occur at, and how does efficient prompt emission occur. These mechanisms also need to be consistent with how ultrarelativistic jets form and stay baryon pure despite turbulence and electromagnetic reconnection near the compact object and despite stellar entrainment within the collapsar model. We use the latest magnetohydrodynamical models of ultrarelativistic jets to explore some of these questions in the context of electromagnetic dissipation due to the slow collisional and fast collisionless reconnection mechanisms, as often associated with Sweet-Parker and Petschek reconnection, respectively. For a highly magnetized ultrarelativistic jet and typical collapsar parameters, we find that significant electromagnetic dissipation may be avoided until it proceeds catastrophically near the jet photosphere at large radii (r {approx} 10{sup 13}-10{sup 14}cm), by which the jet obtains a high Lorentz factor ({gamma} {approx} 100-1000), has a luminosity of L{sub j} {approx} 10{sup 50}-10{sup 51} erg s{sup -1}, has observer variability timescales of order 1s (ranging from 0.001-10s), achieves {gamma}{theta}{sub j} {approx} 10-20more » (for opening half-angle {theta}{sub j}) and so is able to produce jet breaks, and has comparable energy available for both prompt and afterglow emission. A range of model parameters are investigated and simplified scaling laws are derived. This reconnection switch mechanism allows for highly efficient conversion of electromagnetic energy into prompt emission and associates the observed prompt GRB pulse temporal structure with dissipation timescales of some number of reconnecting current sheets embedded in the jet. We hope this work helps motivate the development of self-consistent radiative compressible relativistic reconnection models.« less

/pdf/a-reconnection-switch-to-trigger-gamma-ray-burst-jet-2bkbcmeu0m.pdf

A Reconnection Switch to Trigger gamma-Ray Burst Jet Dissipation

The rapid pace of extrasolar planet discovery and characterization is legitimizing the study of their atmospheres via three-dimensional numerical simulations. The complexity of atmospheric modelling and its inherent non-linearity, together with the limited amount of data available, motivate model intercomparisons and benchmark tests. In the geophysical community, the Held–Suarez test is a standard benchmark for comparing dynamical core simulations of the Earth’s atmosphere with different solvers, based on statistically averaged flow quantities. In the present study, we perform analogues of the Held–Suarez test for tidally locked exoplanets with the Geophysical Fluid Dynamics Laboratory (GFDL) Princeton Flexible Modelling System (fms) by subjecting both the spectral and finite difference dynamical cores to a suite of tests, including the standard benchmark for the Earth, a hypothetical tidally locked Earth, a ‘shallow’ hot Jupiter model and a ‘deep’ model of HD 209458b. We find qualitative and quantitative agreement between the solvers for the Earth, tidally locked Earth and shallow hot Jupiter benchmarks, but the agreement is less than satisfactory for the deep model of HD 209458b. Further investigation reveals that closer agreement may be attained by arbitrarily adjusting the values of the horizontal dissipation parameters in the two solvers, but it remains the case that the magnitude of the horizontal dissipation is not easily specified from first principles. Irrespective of radiative transfer or chemical composition considerations, our study points to limitations in our ability to accurately model hot Jupiter atmospheres with meteorological solvers at the level of 10 per cent for the temperature field and several tens of per cent for the velocity field. Direct wind measurements should thus be particularly constraining for the models. Our suite of benchmark tests also provides a reference point for researchers wishing to adapt their codes to study the atmospheric circulation regimes of tidally locked Earths/Neptunes/Jupiters.

/pdf/atmospheric-circulation-of-tidally-locked-exoplanets-a-suite-1cly9o12z1.pdf

Atmospheric circulation of tidally locked exoplanets: a suite of benchmark tests for dynamical solvers

We consider a two-parameter family of cylindrical force-free equilibria, modeled to match numerical simulations of relativistic force-free jets. We study the linear stability of these equilibria, assuming a rigid impenetrable wall at the outer cylindrical radius R{sub j}. Equilibria in which the Lorentz factor {gamma}(R) increases monotonically with increasing radius R are found to be stable. On the other hand, equilibria in which {gamma}(R) reaches a maximum value at an intermediate radius and then declines to a smaller value {gamma}{sub j} at R{sub j} are unstable. A feature of these unstable equilibria is that poloidal field line curvature plays a prominent role in maintaining transverse force balance. The most rapidly growing mode is an m = 1 kink instability which has a growth rate {approx}(0.4/{gamma} {sub j})(c/R{sub j}). The e-folding length of the equivalent convected instability is {approx}2.5{gamma} {sub j} R{sub j}. For a typical jet with an opening angle {theta}{sub j} {approx} few/{gamma}{sub j}, the mode amplitude grows only weakly with increasing distance from the base of the jet. The growth is much slower than one might expect from a naive application of the Kruskal-Shafranov stability criterion.

/pdf/stability-of-relativistic-force-free-jets-416h1vpqx8.pdf

Stability of relativistic force-free jets

Many global circulation models predict supersonic zonal winds and large vertical shears in the atmospheres of short-period Jovian exoplanets. Using linear analysis and nonlinear local simulations, we investigate hydrodynamic dissipation mechanisms to balance the thermal acceleration of these winds. The adiabatic Richardson criterion remains a good guide to linear stability, although thermal diffusion allows some modes to violate it at very long wavelengths and very low growth rates. Nonlinearly, wind speeds saturate at Mach numbers {approx}2 and Richardson numbers {approx}<1/4 for a broad range of plausible diffusivities and forcing strengths. Turbulence and vertical mixing, though accompanied by weak shocks, dominate the dissipation, which appears to be the outcome of a recurrent Kelvin-Helmholtz instability. An explicit shear viscosity, as well as thermal diffusivity, is added to ZEUS to capture dissipation outside of shocks. The wind speed is neither monotonic nor single valued for a range of shear viscosities larger than about 10{sup -3} of the sound speed times the pressure scale height. Coarsening the numerical resolution can also increase the speed. Hence global simulations that are incapable of representing vertical turbulence and shocks, either because of reduced physics or because of limited resolution, may overestimate wind speeds. We recommend that suchmore » simulations include artificial dissipation terms to control the Mach and Richardson numbers and to capture mechanical dissipation as heat.« less

Circulation and dissipation on hot jupiters

Many global circulation models predict supersonic zonal winds and large vertical shears in the atmospheres of short-period jovian exoplanets. Using linear analysis and nonlinear local simulations, we investigate hydrodynamic dissipation mechanisms to balance the thermal acceleration of these winds. The adiabatic Richardson criterion remains a good guide to linear stability, although thermal diffusion allows some modes to violate it at very long wavelengths and very low growth rates. Nonlinearly, wind speeds saturate at Mach numbers $\approx 2$ and Richardson numbers $\lesssim 1/4$ for a broad range of plausible diffusivities and forcing strengths. Turbulence and vertical mixing, though accompanied by weak shocks, dominate the dissipation, which appears to be the outcome of a recurrent Kelvin-Helmholtz instability. An explicit shear viscosity, as well as thermal diffusivity, is added to ZEUS to capture dissipation outside of shocks. The wind speed is not monotonic nor single valued for shear viscosities larger than about $10^{-3}$ of the sound speed times the pressure scale height. Coarsening the numerical resolution can also increase the speed. Hence global simulations that are incapable of representing vertical turbulence and shocks, either because of reduced physics or because of limited resolution, may overestimate wind speeds. We recommend that such simulations include artificial dissipation terms to control the Mach and Richardson numbers and to capture mechanical dissipation as heat.

Circulation and Dissipation on Hot Jupiters

We consider a two-parameter family of cylindrical force-free equilibria, modeled to match numerical simulations of relativistic force-free jets. We study the linear stability of these equilibria, assuming a rigid impenetrable wall at the outer cylindrical radius R_j. We find that equilibria in which the Lorentz factor \gamma(R) increases monotonically with increasing radius R are stable. On the other hand, equilibria in which \gamma(R) reaches a maximum value at an intermediate radius and then declines to a smaller value \gamma_j at R_j are unstable. The most rapidly growing mode is an m=1 kink instability which has a growth rate ~ (0.4 / \gamma_j) (c/R_j). The e-folding length of the equivalent convected instability is ~2.5 \gamma_j R_j. For a typical jet with an opening angle \theta_j ~ few / \gamma_j, the mode amplitude grows weakly with increasing distance from the base of the jet, much slower than one might expect from a naive application of the Kruskal-Shafranov stability criterion.

/pdf/stability-of-relativistic-force-free-jets-3496j7yatv.pdf

Jason Li

Papers

Stability of relativistic force-free jets

Circulation and dissipation on hot jupiters

Circulation and Dissipation on Hot Jupiters

Stability of Relativistic Force-Free Jets