Circulation Regimes for the Hitchhiker Through the Galaxy (H2G2)

Michael Ghil Ecole Normale Supérieure, Paris, and University of California, Los Angeles

Pleasevisitthesesitesformoreinfo.h#ps://dept.atmos.ucla.edu/tcdh#p://www.environnement.ens.fr/

Diversity of Planetary Circulation Regimes, in our Solar System and Beyond

Les Houches, March 2017

A little diversity

Terrestrial planets Jovian planets

More diversity

Potentially habitable exoplanets

So how about a hitchhiker’s guide to all this diversity?

Outline•  The general circulation of Earth’s atmosphere, or what is there left to explain?

–  Three questions (still) waiting for an answer …•  The rotating annulus as a paradigm of bifurcation theory

–  The laboratory results–  Some theoretical results

•  The “barotropic annulus” and the role of topography•  Where does the Hadley cell stop and the Ferrel cell start?

–  The Held and Hou approach–  Another approach: Could we find the answer to two questions in one fell swoop?

•  Planetary atmospheres as another dynamical systems exercise–  Comparative planetology and the dynamical systems approach

•  Coda – Concluding remarks

– Bibliography

Outline•  The general circulation of Earth’s atmosphere, or what is there left to explain?

–  Three questions (still) waiting for an answer …•  The rotating annulus as a paradigm of bifurcation theory

–  The laboratory results–  Some theoretical results

•  The “barotropic annulus” and the role of topography•  Where does the Hadley cell stop and the Ferrel cell start?

–  The Held and Hou approach–  Another approach: Could we find the answer to two questions in one fell swoop?

•  Planetary atmospheres as another dynamical systems exercise–  Comparative planetology and the dynamical systems approach

•  Coda – Concluding remarks

– Bibliography

1.  Why doesn’t the Hadley cell on Earth extend to the poles, like on Venus?

2.  Why is the mid-latitude circulation more highly variable than the tropical one?

3. What is the role of topography in mid-latitude variability?

Outline•  The general circulation of Earth’s atmosphere, or what is there left to explain?

–  Three questions (still) waiting for an answer …•  The rotating annulus as a paradigm of bifurcation theory

–  The laboratory results–  Some theoretical results

•  The “barotropic annulus” and the role of topography•  Where does the Hadley cell stop and the Ferrel cell start?

–  The Held and Hou approach–  Another approach: Could we find the answer to two questions in one fell swoop?

•  Planetary atmospheres as another dynamical systems exercise–  Comparative planetology and the dynamical systems approach

•  Coda – Concluding remarks

– Bibliography

Earth’s atmosphere & the rotating, differentially heated annulus

So let’s look at this rotating annulus more closely!

Geophysical fluid dynamics (GFD) = rotation + stratification

M. Ghil, P. Read & L. Smith(Astron. & Geophys., 2010):“Geophysical flows as dynamical systems: the influence of Hide's experiments.” Thanks to Peter Read +Sue Bowler for better figures!

Equations of motion: Navier-Stokes, with Boussinesq approximation Main phenomenon: “sloping convection”

Nondimensional parameters R = αgDΔT/Ω2L2 — R. Hide number (“thermal Rossby number”) = (buoyancy force)/(Coriolis force) = 4RiRo2 T = 4Ω2L5 /ν2D — Taylor number = (Coriolis force)/(viscous dissipation)

Flow patterns and the regime diagram Successive transitions from higher to lower symmetry of the flow pattern, in space and time, as the rotation rate Ω increases: from steady-state, axisymmetric (Hadley regime), via purely periodic in space and time (steady waves, Rossby regime) and doubly-periodic vacillation (amplitude, shape), on to irregular, quasi-geostrophic (QG) turbulence.

Regime diagram – experimental For a fixed apparatus (height D, gap width L = b–a) and fluid (expansion coefficient α, viscosity ν),one can change the rotation rate Ω and the temperature difference ΔT = Tb–Ta. As Ω increases,we move along a given straight, downward-slanting diagonal, to the right and down; as ΔTincreases, we move from one diagonal to another, to the right and up. The heavy contoursrepresent sharp transitions from one regime to another one. These transitions are now associated with bifurcations.

Amplitude vacillation Tilted-trough vacillation

Lorenz was motivated by the atmospheric index cycle (Rossby, 1939; Namias, 1950; etc.), but clearly inspired by the rotating annulus results. It is the latter that he was modeling in this paper; see also Lorenz (1967).

Regime diagram – simplified Lorenz (1963b) studied a truncated model of 14 ODEs.He essentially obtained thefirst few bifurcations, up to and including the quasi-periodic, vacillation regime. Beyond that, the low-order truncation prevents one from reaching the QG turbulence.Today, such studies can be carried out on the full systemof high-resolution equations.

Outline•  The general circulation of Earth’s atmosphere, or what is there left to explain?

–  Three questions (still) waiting for an answer …•  The rotating annulus as a paradigm of bifurcation theory

–  The laboratory results–  Some theoretical results

•  The “barotropic annulus” and the role of topography•  Where does the Hadley cell stop and the Ferrel cell start?

–  The Held and Hou approach–  Another approach: Could we find the answer to two questions in one fell swoop?

•  Planetary atmospheres as another dynamical systems exercise–  Comparative planetology and the dynamical systems approach

•  Coda – Concluding remarks

– Bibliography

Transitions Between Blocked and Zonal Flows in a Rotating Annulus with Topography

Eric R. Weeks, Yudong Tian, J. S. Urbach,* Kayo Ide,

Harry L. Swinney,† Michael Ghil The mid-latitude atmosphere is dominated by westerly, nearly zonal flow. Occasionally, this flow is deflected poleward by blocking anticyclones that persist for 10 days or longer. Experiments in a rotating annulus used radial pumping to generate a zonal jet under the action of the Coriolis force. In the presence of two symmetric ridges at the bottom of the annulus, the resulting flows were nearly zonal at high forcing or blocked at low forcing. Intermittent switching between blocked and zonal patterns occurs because of the jetʼs interaction with the topography. These results shed further light on previous atmospheric observations and numerical simulations.

Zonal Flow Blocked Flow 13–22 Dec. 1978 10–19 Jan. 1963

SCIENCE, Vol. 278, 28 Nov. 1997, www.sciencemag.org

Zonal vs. Blocked streamfunction contours

Relative duration of blocked events Compare Legras & Ghil (JAS, 1985)

Outline•  The general circulation of Earth’s atmosphere, or what is there left to explain?

–  Three questions (still) waiting for an answer …•  The rotating annulus as a paradigm of bifurcation theory

–  The laboratory results–  Some theoretical results

•  The “barotropic annulus” and the role of topography•  Where does the Hadley cell stop and the Ferrel cell start?

–  The Held and Hou approach–  Another approach: Could we find the answer to two questions in one fell swoop?

•  Planetary atmospheres as another dynamical systems exercise–  Comparative planetology and the dynamical systems approach

•  Coda – Concluding remarks

– Bibliography

Dynamical systems and comparative planetology, I The atmospheres of the terrestrial (inner) planets, on the one hand, and ofthe Jovian (outer) planets, on the other, exhibit many similarities. They differ,of course, in chemical composition and other fluid properties. But aren’t therate of rotation and the pole-to-equator temperature gradient (two of) themost striking differences?

Isn’t in high time to consider “comparative planetary meteorology” as the ultimate applicationof the ideas of R. Hide and E. N. Lorenz?

Just remember that thefirst few bifurcations can take us pretty far!

Dynamical systems and comparative planetology, II

The tentative place of Earth, Mars, Venus & Titan in this scheme of things

Outline•  The general circulation of Earth’s atmosphere, or what is there left to explain?

–  Three questions (still) waiting for an answer …•  The rotating annulus as a paradigm of bifurcation theory

–  The laboratory results–  Some theoretical results

•  The “barotropic annulus” and the role of topography•  Where does the Hadley cell stop and the Ferrel cell start?

–  The Held and Hou approach–  Another approach: Could we find the answer to two questions in one fell swoop?

•  Planetary atmospheres as another dynamical systems exercise–  Comparative planetology and the dynamical systems approach

•  Coda – Concluding remarks

– Bibliography

Concludingremarks

•  H2G2 to the planetary flows in the galaxy: a unifying view via regime diagrams and bifurcation theory.

•  The general circulation of the atmosphere still has a few “details” left to explain, but hierarchical modeling and the dynamical systems approach to it can help.

•  The rotating annulus can serve as a paradigm for dealing with –  Global change in Earth’s atmospheric circulation–  Comparative planetology

Concludingremarks

•  H2G2 to the planetary flows in the galaxy: a unifying view via regime diagrams and bifurcation theory.

•  The general circulation of the atmosphere still has a few “details” left to explain, but hierarchical modeling and the dynamical systems approach to it can help.

•  The rotating annulus can serve as a paradigm for dealing with –  Global change in Earth’s atmospheric circulation–  Comparative planetology

Concludingremarks

•  H2G2 to the planetary flows in the galaxy: a unifying view via regime diagrams and bifurcation theory.

•  The general circulation of the atmosphere still has a few “details” left to explain, but hierarchical modeling and the dynamical systems approach to it can help.

•  The rotating annulus can serve as a paradigm for dealing with –  Global change in Earth’s atmospheric circulation–  Comparative planetology

Some general references Hide, R., 1953: Some experiments on thermal convection in a rotating liquid, Quart. J. Roy.

Meteorol. Soc., 79, 161.—— , and P. J. Mason, 1975: Sloping convection in a rotating fluid. Adv. Physics, 24, 47–100.Lorenz, E. N., 1963: The mechanics of vacillation. J. Atmos. Sci., 20, 448–464. —— , 1967: The Nature and Theory of the General Circulation of the Atmosphere, World

Meteorological Organization, Geneva, Switzerland, 161 pp.Roberts, P. H., and A. M. Soward (Eds.), 1978: Rotating Fluids in Geophysics, Academic Press,

London/New York/San Francisco, 551 pp.Ghil, M., and S. Childress, 1987: Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics,

Dynamo Theory and Climate Dynamics, Ch. 5, Springer-Verlag, New York, 485 pp.—— , R. Benzi, and G. Parisi (Eds.), 1985: Turbulence and Predictability in Geophysical Fluid

Dynamics and Climate Dynamics, North-Holland, 449 pp.—— , M.D. Chekroun, and E. Simonnet, 2008: Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237, 2111–2126,

doi:10.1016/j.physd.2008.03.036.—— , P. L. Read & L. A. Smith, 2010: Geophysical flows as dynamical systems: the influence of

Hide's experiments, Astron. Geophys., 51(4), 4.28-4.35, https://doi.org/10.1111/j.1468-4004.2010.51428.x.Weeks, E. R., Y. Tian, J. S. Urbach, K. Ide, H. L. Swinney, and M. Ghil, 1997: Transitions between

blocked and zonal flows in a rotating annulus with topography. Science, 278, 1598–1601.

Rotating annulus & Earth’s atmosphere

Tropics (Hadley cell) Midlatitudes

(Ferrell cell)

Or why doesn’t the Hadley cell on Earth extend to the poles,

like on Venus ?

Tropics : both ƒ (i.e., Ω)

and ∆Τ small

Midlatitudes : both Ω

and ∆Τ large