Geophysical Fluid Dynamics: A Laboratory for Statistical Physics Peter B. Weichman, BAE Systems IGERT Summer Institute Brandeis University July 27-28, 2015 Jupiter Saturn Neptune Earth (S pole hexagon) (Tasmania Chl-a) Global Outline 1. Statistical Mechanics, Hydrodynamics, and Geophysical Flows (introduction & overview) 2. Statistical mechanics of the Euler equation (technical details & some generalizations) 3. Survey of some other interesting problems (shallow water dynamics, magneto- hydrodynamics, turbulence in ocean internal wave systems) General Theme: Seeking beautiful physics in idealized models (And hoping that it still teaches you something practical!) Part 1: Statistical Mechanics, Hydrodynamics, and Geophysical Flows http://nssdc.gsfc.nasa.gov/image/planetary/jupiter/gal_redspot_960822.jpg Outline (Part 1) 1. The Great Red Spot and geophysical simulations 2. Euler’s equation and conservation laws 3. Relation to 2D turbulence: inverse energy cascade 4. Thermodynamics and statistical mechanics 5. Equilibrium solutions 6. Laboratory experimental realizations: Guiding center plasmas 7. Geophysical comparisons: Jovian and Earth flows Target Name: Jupiter Spacecraft: Voyager Produced by: NASA Cross Reference: CMP 346 Date Released: 1990 http://www.solarviews.com/cap/jup/vjupitr3.htm HUBBLE VIEWS ANCIENT STORM IN THE ATMOSPHERE OF JUPITER When 17th-century astronomers first turned their telescopes to Jupiter, they noted a conspicuous reddish spot on the giant planet. This Great Red Spot is still present in Jupiter's atmosphere, more than 300 years later. It is now known that it is a vast storm, spinning like a cyclone. Unlike a low-pressure hurricane in the Caribbean Sea, however, the Red Spot rotates in a counterclockwise direction in the southern hemisphere, showing that it is a high-pressure system. Winds inside this Jovian storm reach speeds of about 270 mph. The Red Spot is the largest known storm in the Solar System. With a diameter of 15,400 miles, it is almost twice the size of the entire Earth and one-sixth the diameter of Jupiter itself. The long lifetime of the Red Spot may be due to the fact that Jupiter is mainly a gaseous planet. It possibly has liquid layers, but lacks a solid surface, which would dissipate the storm's energy, much as happens when a hurricane makes landfall on the Earth. However, the Red Spot does change its shape, size, and color, sometimes dramatically. Such changes are demonstrated in high- resolution Wide Field and Planetary Cameras 1 & 2 images of Jupiter obtained by NASA's Hubble Space Telescope, and presented here by the Hubble Heritage Project team. The mosaic presents a series of pictures of the Red Spot obtained by Hubble between 1992 and 1999. Astronomers study weather phenomena on other planets in order to gain a greater understanding of our own Earth's climate. Lacking a solid surface, Jupiter provides us with a laboratory experiment for observing weather phenomena under very different conditions than those prevailing on Earth. This knowledge can also be applied to places in the Earth's atmosphere that are over deep oceans, making them more similar to Jupiter's deep atmosphere. Image Credit: Hubble Heritage Team (STScI/AURA/NASA) and Amy Simon (Cornell U.). http://nssdc.gsfc.nasa.gov/photo_gallery/photogallery-jupiter.html Voyager 2 (1989) images of Neptune’s Great Dark Spot, with its bright white companion, slightly to the left of center. The small bright Scooter is below and to the left, and the second dark spot with its bright core is below the Scooter. Strong eastward winds -- up to 400 mph -- cause the second dark spot to overtake and pass the larger one every five days. The spacecraft was 6.1 million kilometers (3.8 million miles) from the planet at the time of camera shuttering. http://nssdc.gsfc.nasa.gov/photo_gallery/photogallery-neptune.html Jupiter’s Great Red Spot A theorist’s/simulator’s cartoon -plane approximation: • Shear boundary conditions • Coriolis force • Weather bands MODEL: (Marcus, Ingersol,…) Two-dimensional inviscid Euler equation (Why? Why not!) P. Marcus simulations: dipole initial condition http://www.me.berkeley.edu/cfd/videos/dipole/dipole.htm http://www.me.berkeley.edu/cfd/videos/dipole/dipole.htm Initial condition: _ Two blobs of opposite vorticity, O+ and O - Turbulent + cascade t Final condition: O+ blob survives, appears stable O- blob disperses Basic question: Dynamical Stability ? Statistical equilibrium Vortex Hamiltonian? Ergodicity? YES! Sometimes! P. Marcus simulations: perturbed ring initial condition http://www.me.berkeley.edu/cfd/videos/ring/ring.htm http://www.me.berkeley.edu/cfd/videos/ring/ring.htm The Euler equation Coriolis force Basic Dtvr(,) 2 Driving and inviscid p(,)()(,)(,)(,)r t f r zˆ v r t v r t f r t dissipation Euler Dt D vr(,)t - Convective derivative Dt t vr(,)t - Velocity field pt(,)r - Pressure field 1 E dr( r , t ) | v ( r , t ) |2 - Kinetic energy 2 ω(,)(,) rtt v r - Vorticity field (scalar in d=2) f (r ) 2EL sin( ) - Coriolis parameter (rotating coordinate system) - Viscosity Fr(,)t - Driving force, often stochastic Constraints and Conservation Laws (a) Incompressibility: Determines pressure field p(x,t) Implies existence of stream function: v(r,t) 0 v(r,t) (r,t) ( , ) [(r,t) const.] y x (r,t) 2 (r,t) (b) Angular momentum: (axially symmetric domains) 1 L drr v(r,t) dr r 2(r,t) (boundary term) 2 (c) Energy: 1 E dr| v(r,t) |2 2 2D Coulomb 1 Green function dr dr'(r,t)G(r,r')(r',t) 2 2G(r,r') (r r') ( bdyconds) 1 | r r'| Analogy: Vorticity ↔ Charge density G(r,r') ln , | r r'| 0 2 R0 More constraints and Conservation Laws (d) → () All powers of the vorticity! More generally: Dv(r,t) Vorticity is p(r,t) D(r,t) Dt 0 freely self- f dr f [(r,t)] Dt advecting v(r,t) 0 - Conserved for any function f() d D(r,t) dr(r,t)n n dr(r,t)n1 0 n dt Dt All conserved integrals Convenient parametrization: may now be expressed 0, x 0 G( ) dr[ (r,t)], (x) in terms of g(): 1, x 0 d n g( ) Alternate form: n All “charge dG( ) d f ( )g( ) g( ) dr[ (r,t)] species” are f d independently g( )d fractional area on which conserved (r,t) d Simple example: single charged Dynamics fully specified by area species (charge density q) A (t) rV :(r,t) q, | A (t) |V occupying fractional area . q q Aq (t 0) G( ) ( q) (1) ( ) g( ) ( q) (1) ( ) Aq (t ) ( 0.3, q 1.2) Infinitely folded fractal structure: Statistics? Relation to 2D turbulent cascade Dynamical viewpoint on the formation of large-scale stucture: The inverse energy cascade 1 dk Energy : E | v(k) |2 2 (People) 2 (2 ) dk “Random” Enstrophy: | k |2| v(k) |2 2 (2 )2 (turbulent) initial ($$$) condition Energy Phase space: natural tendency for Driving, f flux “diffusion” to large k (Birth and Conservation laws: constraints on Grants) energy flow (absent in 3D due to vortex line stretching bending, etc.) Final Enstrophy Exists also in other systems, steady flux e.g., ocean waves state Economic analogy: under “free” capitalistic dynamics (total people & $$$ conserved), people and money go in opposite System scale L directions: an egalitarian/socialist initial Dissipation, Grid scale li state is unstable towards one with a few (Death and Taxes) rich people and lots of poor people. Statistical Mechanics 1 | r r | L. Onsager, “Statistical hydrodynamics”, N point E ln i j Nuovo Cimento Suppl. 6, 279 (1949). i j vortices 2 i j a Low E: Raise E: Raise E further: “Kosterlitz-Thouless” Momentumless “neutral Macroscopic charge dipole gas phase plasma” phase segregation Standard Coulomb Entropy Macroscopic vortices energetics: picture effectively require: 1 S T E T > 0 i.e., E→-E, or T < 0! Why are T < 0 states physical? 1 2 Hydrodynamic flow energy E dr| v(r,t) | 2 Expect energy density E /V O(1) Claim: All states with must have i.e., , = O(1) E > E , T < 0 in order to overcome screening 1 | r r | 4 i j Discrete version: a → 0 E a i j ln 2 i j a Well known fact: neutral Coulomb gas at T > 0 has E / a4 N (#sites) but : N V / a2 E /V Na 4 /V a2 0 ! Any T > 0 state has E/V = 0, hence all flows are microscopic: vmacro 0 E /V 0 requires E/a 4 N 2 T 0 Hydrodynamic states have “Super-extensive” lattice energy REALITY intrudes: Hydrodynamics is not in equilibrium with molecular T < 0 scales, which always have T > 0. Viscosity Communication between hydrodynamics and molecular > 0 dynamics: T < 0 state must eventually decay away. Pious For << 1, there will exist a time scale tmolec << t << tvisc Hope over which equilibrium hydrodynamic description is valid For now assume inviscid Euler equation to exact on all length scales. Is the theory at least self-consistent? YES! Statistical Formalism Boltzmann/Gibbs eT H ( 1/ ) 1 Free Energy Fe ln tr H V 1 H dr d r' ( r ) G ( r , r ') ( r ') 2 dhr()() r r Proper care and feeding of conservation laws: Lagrange drr[ ( )] multiplier/chemical potential for each one. 1 h()r r23 r Angular momentum multiplier 2 -plane/Coriolis potential term n Taylor coefficients correspond to () n n multipliers for vorticity powers n Continuous spin Ising model! E.g., Energy/enstrophy theory (Kraichnan,…): “Exchange” G(r,r’) 2 “Magnetic field” h(r) ( ) 2 Gaussian theory “Spin weighting factor” () Back to Jupiter for a moment: Why is only one sign of vortex blob stable? rmin/L r0/L 1 h(r) r 2 r 3 seeks minimum h(r) 2 seeks maximum h(r) 0, 0 r0 2 / 3 Balance between angular momentum and Coriolis force produces an effective potential minimum Exact mean field theory J.