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)n1  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)  rV :(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. Miller, “Statistical mechanics of Eulers equation in two dimensions”, Phys. Rev. Lett. 65, 2137 (1990). This model can be solved exactly! J. Miller, P. B. Weichman and M. C. Hint from critical phenomena: Phase transitions in Cross, “Statistical mechanics, Euler’s models with long-ranged interactions are mean-field like. equation, and Jupiter’s Red Spot”, 1 Phys. Rev. A 45, 2328 (1992). E  dr dr'(r)G(r,r')(r') 2   Energy is dominated by mutual sweeping of distant vortices: r close to r’ gives negligible contribution to E. Nearby vortices are essentially noninteracting (except for “hard core” exclusion). F  E TS, S  Local entropy of mixing of noninteracting gas of vortices; different species , different chemical potential () In terms of stream function : Details 1 E dr|  ()|, r2 S  d r W [() r  h ()] r Tomorrow! 2   W ( ) ( ) W( ) ln d e[   (  )] e ~ Laplacetransformof e    After integrating out the small scale fluctuations, the continuum limit yields an exact saddle point evaluation of F that controls the remaining large scale fluctuations. Mean field equations

 F 2 Highly nonlinear  0  0 (r)    0 (r)  d n0 (r, )  (r)  PDE   [ (r)h(r)] ( ) e 0 Probability density for vortex of n0 (r, )  charge density  at r eW [ 0 (r)h(r)]  (r)  (r) 0 “Order Parameter”

 0 (r)   (r) “Coarse-grained” stream function To be solved with constraints: F dr g( )    n (r, ) Determines ()  ( )  V 0 for given g()

Example: Point vortex limit: g( )  (  q)  (1) ( ), h(r)  0 q  ,   0, Q  qV fixed 2 1    0 (r)   (r) 1 e [q 0 (r) ] 1 e - 2 (r)  0  (r) Hard-core → Fermi-like function Q  dre dr q   2 (r) An exact solution in this case predicts  0 V collapse to a point at T = -1/8 0, r  r  0 Numerical solutions Q  /10 0 (r)   (Gauss's Law) q, r  r0 2 (r0 / L) 1

T  0 T  0

Point vortices

T  

0 (r)  q

T  0 T  0 2 (r1 / L) 

q, r  r1 0 (r)   0, r  r1 More complex initial conditions, Verification of agreement between the with large number of vorticity Monte Carlo result and the direct levels  (e.g., for comparison with solution for a case where the latter can numerical simulations): Discretize be obtained: volume onto a grid, and find equilibrium via Monte Carlo simulations (Monte Carlo move corresponds to permutation of grid elements, thereby automatically enforcing conservation laws).

We have done comparisons with the Marcus dipole and ring initial conditions, and find good quantitative agreement with his long-time states. Experimental Realization: Guiding Center Plasmas Some beautiful experiments: Guiding center plasmas

Indivual electrons oscillate rapidly up and down the column, but the projected charge density L n (r)  dz n(r, z) proj  0 Obeys the 2D Euler equation! Euler dynamics arises from the Lorentz force.

Nonneutral Plasma Group, Department of Physics, UC San Diego http://sdphca.ucsd.edu/ “Measurements of Symmetric Vortex Merger”, K.S. There exists some theoretical work as well: P. Chen and M. C. Cross: “Statistical two- Fine, C.F. Driscoll, J.H.Malmberg and T.B. Mitchell; vortex equilibrium and vortex merger”, Phys. Phys. Rev. Lett. 67, 588 (1991). Rev. E 53, R3032 (1996). Also, more Jupiter simulations by Marcus. K. S. Fine, A. C. Cass, W. G. Flynn and C. F. Driscoll, “Relaxation of 2D turbulence to vortex crystals,” Phys. Rev. Lett. 75, 3277 (1995) Some More Quantitative Comparisons with Geophysical Flows Great Red Spot: Quantitative Comparisons

Observation data (Voyager) (Dowling & Ingersol, 1988)

Statistical equilibrium (best fit to simple two-level model) (Bouchet & Sommeria, 2002) Jovian Vortex Shapes

Brown Barges (Jupiter northern hemisphere)

Great Red Spot and White Ovals

Vortex-jet phase transition line

Bouchet & Sommeria, JFM (2002) Phase diagram: energy vs. size in a confining weather band (analogous to squeezed bubble surface tension effect) Ocean Equilibria A number of vortex eddy dynamical features in the oceans can be semi- quantitatively explained • Appearance of meso-scale coherent structures (rings and jets) • Westward drift speed of vortex rings • Poleward drift of cyclones • Equatorward drift of anticyclones Venaille & Bouchet, JPO (2011)

Rings

Equilibrium Jets prediction

Westward drift speed of vortex rings

Hallberg et. al, JPO (2006) Chelton et. al, GRL (2007) Atmospheric Blocking Event: NE Pacific, Feb. 1-21, 1989

Signature of a near- steady state: 푞 ≈ 퐹(휓)

Ek & Swaters, J. Atmos. Sci. (1994) End of Part 1 Part 2: Statistical mechanics of the Euler equation (technical details & some generalizations) Outline (Part 2)

1. Derivation of the Euler equation equilibrium equations 2. Generalization to the quasigeostrophic equation (first incorporation of global wave dynamics) 3. Higher dimensional example: Collisionless Boltzmann equation for gravitating systems 4. Nonequilibrium statistical mechanics: weakly driven systems 5. Ergodicity and equilibration (some notable failures) Derivation of the Variational Equations Partition Function and Free Energy Hamiltonian functional 퐻 휔 = 퐸 휔 − 퐶 휔 − 푃[휔] 휇 (expressed in terms of vorticity) 1 Fluid kinetic energy 퐸 휔 = 푑2푟 푑2푟′휔 퐫′ 퐺 퐫, 퐫′ 휔(퐫′) 2 1 퐫 − 퐫′ 퐺 퐫, 퐫′ ≈ − ln 2휋 푅0 2 퐶휇 휔 = ∫ 푑 푟 휇[휔 퐫 ] Conservation of vorticity integrals 1 Conservation of angular 푃 휔 = ∫ 푑2푟 ℎ 퐫 휔(퐫) ℎ 퐫 = 훼푟2 + 훾푟3 2 momentum, and Coriolis force

Grand canonical partition function: Invariant phase space measure (Liouville theorem): 푍(훽, 휇, ℎ) = ∫ 퐷 휔 푒−훽퐻[휔] ∞ 푑휔 ∫ 퐷 휔 = lim 𝑖 Free energy: 푎→0 푞0 𝑖 −∞ 1 퐹(훽, 휇, ℎ) = − ln(푍) Independent integral over vorticity 훽 level at each point in space Macro- vs. Micro-scale • Main barrier to straightforward evaluation of 푙-cell partition function 푍: Highly 푎-cell nonlocal interaction 퐺(퐫, 퐫′) • Solution (“asymptotic freedom”): recognize that interaction is dominated by large scales, so integrate out small scales first, where 퐺 is negligible (local ideal gas of vortices), and then consider large scales • Variational principle emerges here • Mathematical approach: consider scales 퐿 ≫ 푙 ≫ 푎, and take the limits 퐿 푎 → 0, 푙 → 0, but in such a way that 푙/푎 → ∞ Neglecting interactions within an 푙-cell, partition function contribution becomes an 푎-cell permutation count Microscale vortex entropy

Let 푛푙(휎푘) define the number of 푎-cells with vorticity level 휎푘 in cell 푙 Permutation factor: number of distinct ways of rearranging vorticity within a given 푙-cell (automatically preserves all conservation laws)

푁푙! 푀 ∼ 푒− 푘=1 푛푙 𝜎푘 ln [푛푙 𝜎푘 /푁푙] 푛푙 휎1 ! 푛푙(휎2)! … 푛푙(휎푀)!

In the continuum limit, 푎 → 0, taking the limit of continuous set of vorticity levels as well:

푛푙 휎푘 → 푛0(퐫, 휎) Vorticity distribution at position 퐫

2 푆 푛0 /푎 2 퐷[휔] = 퐷 푛0 푒 푆 푛0 = − 푑 푟 푑휎 푛0 퐫, 휎 ln [푞0푛0(퐫, 휎)]

Microscale configurational entropy density Remaining integral over macroscale assignment of the microscale distribution function • Depends only the intermediate scale 푙 • All fluctuations below this scale have been integrated out, accounted for in 푆[푛0] Reformulation in terms of 푛0 퐫, 휎

Express everything in terms of 푛0 퐫, 휎 in order to complete the partition function integral

Constraints: Equilibrium vorticity 푑휎 푛0 퐫, 휎 = 1 Normalization 휔0 퐫 = 푑휎 휎 푛0(푟, 휎)

2 2 푑 푟 푛0 퐫, 휎 = 푑 푟훿[휎 − 휔 퐫 ] = 푔(휎)

Global vorticity conservation Can replace 휔 by 휔 for any 2 0 퐶휇 푛0 = 푑 푟 푑휎 휇 휎 푛0(퐫, 휎) smoothly varying interaction: 1 퐸 푛 = 푑2푟 푑2푟′휔 퐫′ 퐺 퐫, 퐫′ 휔 (퐫′) Additional Lagrange multiplier for 0 2 0 0 normalization constraint

2 2 푁휈 푛0 = 푑휎 푑 푟 휈 퐫 푛0(퐫, 휎) 푃 푛0 = 푑 푟 ℎ 퐫 휔0(퐫) Macroscale thermodynamics

−훽퐺[푛0] 푍(훽, 휇, 휈, 훼) = 퐷 푛0 푒

G 푛0 = 퐸 푛0 − 퐶휇 푛0 − 푃[푛0] − 푁휈 푛0 − 푇 푆[푛0] 1 푇 Key observation: Nontrivial balance between energy and 푇 = 2 = 2 훽푎 푎 entropy requires the combination 훽 = 훽푎2 to remain finite in the continuum limit 1 훽 = → ∞ Since 훽 = 훽 /푎2 → ∞, the partition function integral is 푇 푎2 dominated by the maximum of 퐺[푛0]

훿퐺 = 0 훿푛0 퐫, 휎 Variational Equations

훿퐺 푊[Ψ0(퐫)−ℎ(퐫)] −훽 𝜎[Ψ0 퐫 −ℎ 퐫 −휇(𝜎)} = 0 ⇒ 푛0 푟, 휎 = 푒 푒 훿푛0 퐫, 휎

푑휎 푊 휏 = −ln 푒훽 [휇 𝜎 −𝜎휏] From normalization condition 푞0

2 ′ Ψ0 퐫 = 푑 푟 퐺 퐫, 퐫 휔0(퐫′) Equilibrium stream function

Closed equation for the stream function

2 휔0 퐫 = −∇ Ψ0 퐫 = 푑휎 휎 푛0(퐫, 휎) = 푇 푊′[Ψ0(퐫) − ℎ(퐫)]

Variational equation obtained by minimizing the free energy fucntional 1 퐹[Ψ ] = 푑2푟 ∇Ψ (퐫) 2 − 푇 푊[Ψ (퐫) − ℎ(퐫)] 0 2 0 0 Kinetic energy Grand canonical entropy Generalizations to other Fluid Equations Quasigeostrophic (QG) Equations System of nonlinear Rossby waves Large-scale, hydrostatic (neglect gravity waves) approximation to the shallow water equations 2 Potential vorticity (PV) 푄(퐫) = 휔(퐫) + 푘푅휓 퐫 + 푓(퐫)

Rossby radius of deformation 푅0 = 1/푘푅 = 푐퐾/푓

Kelvin wave speed 푐퐾 (speed of short wavelength inertia-gravity waves – quantifies gravitational restoring force for surface height fluctuations)

Coriolis parameter (Earth rotational force): 푓 = 2Ω퐸 sin(휃퐿)

2 2 휕푡 −∇ + 푘푅 휓 + 퐯 ⋅ ∇휔 + 훽휕푥휓 = 0 “Beta parameter” 훽 = 휕푦푓

Can be written in the form 퐷푄 푄 is advectively conserved in the same way that 휔 is = 0 퐷푡 for the Euler equation

훽푘푥 휔 = − 2 2 Rossby wave dispersion relation (linearized dynamics) 푘 + 푘푥 QG Equilibria Energy function: Stream function follows surface height: 휓(퐫) ∝ 훿ℎ(퐫) 1 퐸 = 푑2푟 ∇휓(퐫) 2 + 푘2휓(퐫)2 = 푑2푟 푄 퐫 − 푓 퐫 퐺 퐫, 퐫′ [푄 퐫′ − 푓 퐫′ ] 푅 2 푄

(−∇2+푘2)퐺 퐫, 퐫′ = 훿(퐫 − 퐫′) • Logarithmic singularity at the origin, but 푅 푄 ′ exponential decay ∼ 푒−|퐫−퐫 |/푅0 at large 1 separation. 퐺 퐫, 퐫′ = − 퐾 ( 퐫 − 퐫′ /푅 ) 푄 2휋 0 0 • Rossby radius provides a vortex screening length (hydrostatic height response screens the vortex-vortex interaction) Integrating out the small-scale fluctuations produces the identical entropy term

2 푆 푛0 = − 푑 푟 푑휎 푛0 퐫, 휎 ln [푞0푛0(퐫, 휎)] Here 휎 now denotes the values of 푄

Equilibrium equations are derived by minimizing the functional: 1 1 퐹[Ψ] = 푑2푟 훻Ψ 2 + 푘2Ψ2 + 푓Ψ − 푇 푊 Ψ − ℎ 2 2 푅 QG Equilibirum Vortex

Two level system example: • Beautiful analogy with two Δ휎(푇 ) phase system, with phase separation below a critical temperature |푇 | < 푇 푐 • Vortex may be thought of as a droplet of one phase

Σ(푇) inside the other • Finite Rossby radius ⇒ Finite width interface between phases, with PV difference Δ휎(푇 ) and surface tension Σ(푇 )

|푇 |/푇 푐 • Presence of Coriolis parameter 푓 푦 produces the equivalent of a gravitational field • Droplets are then unstable, and instead the denser phase coalesces below the less dense phase, with a flat, narrow interface between ⇒ “jet” solution • Droplets in a more complex confining potential produce squeezed bubbles (Jupiter “barges”) Procedure for General Scalar Field Equilibria

휕푡푄 + 퐯 ⋅ ∇푄 = 0 Some vorticity-like field 푄(퐫, 푡) that is advectively conserved 퐸[푄] Existence of a conserved energy functional (not necessarily quadratic) • Assumed sufficiently smooth in space that 퐸 푄 = 퐸 푄 ≡ 퐸[푄0] 훿퐸 Relation to stream function 휓, from P. B. Weichman, Equilibrium theory 휓 퐫 = of coherent vortex and zonal jet 훿푄 퐫 which velocity 퐯 = ∇ × 휓 is derived formation in a system of nonlinear Rossby waves, Phys. Rev. E 73. 036313 (2006) 퐿 휓 = 푑2푟휓 퐫 푄(퐫) − 퐸[푄] Convert to function of 휓 via Legendre transform Integration over small scale fluctuations produces the identical entropy 푆 푛 = − 푑2푟 푑휎 푛 퐫, 휎 ln [푞 푛 (퐫, 휎)] 0 0 0 0 contribution, expressed in terms of the 푄-level distribution function 푛0 퐫, 휎 Exact variational condition for large scale structure produces the identical relation:

푊[Ψ0(퐫)−ℎ(퐫)] −훽 𝜎[Ψ0 퐫 −ℎ 퐫 −휇(𝜎)} 푛0 퐫, 휎 = 푒 푒 Equilibrium equations are then derived by minimizing the free energy functional: 푑휎 퐹 Ψ = 퐿 Ψ − 푇 푑2푟푊[Ψ − ℎ] 푊 휏 = −ln 푒훽 [휇 𝜎 −𝜎휏] 푞0 Higher Dimensional Example The collisionless Boltzmann equation: Flow equation for phase space probability density 푓(퐫, 퐩) 휕푡푓 + 퐫 ⋅ ∇푟푓 + 퐩 ⋅ ∇푝푓 = 0 Debye-Hückel Newton’s laws provide 퐫 , 퐩 : 퐫 = 퐩/푚 퐩 = 퐅(퐫) theory of electrolytes 퐹 퐫 = −∇휙 퐫 푑 푑 ′ ′ provides another 휙 퐫 = 푑 푟 푑 푝 푉 퐫, 퐫 푓(퐫 , 퐩) example! Energy functional: 퐩 2 1 퐸 = 푑푑푟 푑푑푝 푓(퐫, 퐩) + 푑푑푟 푑푑푝 푑푑푟′ 푑푑푝′푓 퐫, 퐩 푉 퐫, 퐫′ 푓(퐫′, 퐩′) 2푚 2

• For particles with long-ranged interactions, such as the Coulomb interaction, exact integration of small-scale fluctuations is again permitted • Equilibrium equations are derived for the particle density: 푛 퐫 ≡ −∇2 Ψ 퐫 = 푑푑푝 푓(퐫, 퐩) 1 퐹 Ψ = 푑푑푟 ∇Ψ 2 − 푇 푑푑푟 푑푑푝 푊[Ψ 퐫 − |퐩|2/2푚] 2푑 2 푇 = 푇/푎

These mean field equations for self gravitating systems, in the context of equilibration of star clusters, were derived and studied in the 1960’s! [But were found to produce unphysical solutions, likely due to absence of collisions] D. Lynden-Bell & R. Wood, Mon. Not. R. Astron. Soc., 1968 Near-Equilibrium Systems: Weakly Driven & Dissipated Generalizations to Weakly Driven Systems Dv 1   p  2vf  Dt  Near-equilibrium dynamics: • Can one derive a nonequilibrium statistical mechanics formalism for steady states in the presence of small viscosity and weak driving? • Which equilibrium state is selected for given forcing pattern? Possible tools from classic NESM: • Response functions, Kubo formulae, Kinetic equations,…? • Required formal theoretical tools exist (Poisson bracket, invariant phase space measure,…) 푖 ′ ′ ′ ′ 휒 퐫, 퐫′, 푡 − 푡′ = 〈 퐴 퐫, 푡 , 퐵 퐫′, 푡 〉 훿퐴 퐫, 푡 = 푑퐫 휒퐴퐵 퐫, 퐫 ; 푡 − 푡 ℎ퐵(퐫 , 푡) 퐴퐵 2

Thermodynamic response of density 퐴 to field ℎ퐵 conjugate to density 퐵, governed by dynamic response function 휒퐴퐵

Formalism possibly useful for treating evolution of ocean currents without massive computational effort (predictability problem) Weakly driven 2D Euler Equation

Bouchet, Simonnet, Phys. Rev. Lett. 2009

Simulations of stochastically driven transitions between near-equilibrium states • Close to an equilibrium phase transition between jet and vortex solutions • Very sensitive to slight changes in system dimensions

See also recent kinetic equation approaches: • Nardini, Gupta, Ruffo, Dauxois, Bouchet, J. Stat. Mech. 2012 • Bouchet, Nardini, Tangarife, J. Stat. Phys. 2013 Some Investigations of Ergodicity and Equilibration Ergodicity Failure: Multiple solutions

Off-center

푀 = 0.05 푀 = 0.0373 single vortex

Symmetric single vortex

Double vortex

• Vortex separation decreases with decreasing angular • Entropy comparison for locally momentum 푀 stable states with the same total • Two vortex solution disappears vorticity 푄 = 0.2, angular below a critical separation momentum 푀, and energy 퐸(푀), • Generally consistent with • Largest entropy state is the global numerically observed free energy minimum dynamical merger instability

Chen & Cross, PRE 1996 Steady State Failure 푡 = 0 푡 = 4, 40, 400, 4000

High resolution numerical simulations: • Spherical geometry blocks full equilibration, leaving an oscillating pattern of four compact vortices, plus a population of small-scale vortices • Stat. Mech. would predict a unique pattern (depending on initial condition) of exactly four stationary vortices

Quadrupolar pattern time series 푊(푡) Dritschell, Qi, Marston, JFM (2015) End of Part 2 Part 3: Survey of some other interesting problems Outline (Part 3)

1. Shallow water equilibria – Interaction between eddy and wave systems 2. Magnetohydrodynamic equilibria – Solar tachocline – Interaction between flow and electrodynamics 3. Ocean internal wave turbulence – Example of a strongly nonequilibrium system, but still amenable to simple theoretical treatment Multicomponent Equilibria (With advective conservation of some subset of components) Shallow Water Equations Shallow Water Equations

h 휌 (hv)   Coupled equations of 1 t   motion for height and Dv velocity fields 휌2  gh  Dt  (also a model for compressible potential + kinetic energy: flow: h →, g → ) 1 1 E  dr h | v |2  g dr (h  h )2 2  2  0 C  dr h( / h)n  n   Conserved for  C  dr h f ( / h) all n, f f   There now exist gravity wave excitations

  ck, c  gh0 in addition to vortical excitations Acoustic turbulence: broad spectrum of interacting shallow water or sound waves: direct energy cascade (shock waves in some models). Finite energy is lost (like in 3D) at small scales even Basic question: Is there a nontrivial final without viscosity. state? Or is all vortical energy eventually “emitted” as waves? P. B. Weichman and D. M. Petrich, “Statistical equilibrium solutions of the shallow water Answer: YES! macroscopic vortices equations”, Phys. Rev. Lett. 86, 1761 (2001). survive. Shallow Water Equilibria Free energy functional: ∇Ψ(퐫) 2 1 퐹 Ψ, ℎ = 푑2푟 − 푔ℎ 퐫 2 − 푇 ℎ(퐫)푊[Ψ(퐫)] 2ℎ(퐫) 2

Equilibrium variational equations: 1 1 ′ 1 −∇ ⋅ 훻Ψ0 퐫 = 푇ℎ0(퐫)푊 [Ψ0(퐫)] 퐯0 = ∇ × Ψ0 휔0 = −∇ ⋅ ∇Ψ0 ℎ0 퐫 ℎ0 ℎ0 ∇Ψ 퐫 2 0 2 = −푇W Ψ0 퐫 − 푔ℎ0(퐫) Additional hydrostatic balance requirement ℎ0 퐫

In equilibrium one must therefore have ∇ ⋅ ℎ0퐯0 = 0 • Existence of sensible equilibria requires the disappearance of compressive (gravity wave) motions • E.g., forward cascade of wave energy to small scales, at which they are rapidly dissipated, leaving only the large scale eddy dynamics • This is a physical assumption, not a mathematical result More recent thoughts on this problem: Renaud, Venaille, Bouchet, JFM 2015 Nontrivial equilibrium between interacting large scale negative temperature and small scale positive temperature states is not possible Magnetohydrodynamic Equilibria Ideal Magnetohydrodynamic Equations

Ideal MHD: 휕푡퐯 + 퐯 ⋅ ∇ 퐯 + 퐟 × 퐯 = −∇푃 + 푱 × 푩 휕 퐁 = ∇ × (퐯 × 퐁) 푡 Lorentz force acting on electric current passing through a fluid element • Fluid is approximated as perfectly conducting Closure equations: • Electric fields are negligibly small Quasistatic 퐉 = ∇ × 퐁 Ampere law: Advection of magnetic field by velocity field • Magnetic field lines may be stretched and tangled, but are otherwise attached to a ∇ ⋅ 퐯 = 0 Incompressibility: given fluid parcel ∇ ⋅ 퐁 = 0 2D MHD In certain physical systems a 2D approximation is valid • E.g., solar tachocline • Sharp boundary between rigidly rotating inner radiation zone and differentially rotating outer convection zone • Large-scale organized structures here would have strong implications for angular moment transport between the two zones • 퐯, 퐁 are horizontal ⇒ 퐽, 휔 are normal to the plane, and can be treated as scalars.

Resulting pair of scalar equations • Potential vorticity no longer 휕 휔 + 푓 + 퐯 ⋅ ∇ 휔 + 푓 = 퐁 ⋅ ∇퐽 푡 advectively conserved • Replaced by advective conservation 휕푡퐴 + 퐯 ⋅ ∇A = 0 of vector potential! Second derivative no longer controlled • Microscopic fields much less regular! • Leads to very different equilibria, with much stronger “subgrid” energetics

1 퐯 = ∇ × 휓 퐸 = 푑2푟[ 퐯(퐫) 2 + 퐁(퐫) 2] Stream function & 2 vector potential 퐁 = ∇ × 퐴 Conserved kinetic + EM energy 2D MHD Equilibrium Equations Two sets of conserved integrals:

푗 휎 = 푑2푟훿[휎 − 퐴 퐫 ] 푘 휎 = 푑2푟[휔 퐫 + 푓 퐫 ]훿[휎 − 퐴 퐫 ]

Controlled by Lagrange multipliers 휇 휎 , 휈(휎) P. B. Weichman, “Long-Range Correlations and Coherent Structures in Magnetohydrodynamic Equilibrium free energy functional: Equilibria”, PRL 109, 235002 (2012) 1 1 퐹 퐴, Ψ = 푑2푟[ 훻퐴(퐫) 2 + 훻Ψ(퐫) 2 − 휈′ 퐴 훻A 퐫 ⋅ 훻Ψ 퐫 + ∇ℎ(퐫) ⋅ ∇Ψ(퐫) 2 2

−휇(퐴 퐫 ) − 푓(퐫)휈 퐴 퐫 ] + 푊fluct[퐴]

Microscopic fluctuation free energy Physics is that of two coupled elastic membranes! 푊fluct[퐴] is computed from a Gaussian • Generates long-range correlations fluctuation Hamiltonian: • External localizing potential provided by 휇, 휈 1 퐻 퐴 = 푑2푟 훻훿퐴(퐫) 2 + 훻훿Ψ(퐫) 2 − 2휈′(퐴 퐫 )훻훿A(퐫) ⋅ 훻훿Ψ(퐫) fluct 2

• Quantifies the effects of microscale magnetic and velocity fluctuations (no longer controlled by the conservation laws) • Gaussian fluctuation entropy replaces Euler equation hard-core ideal gas entropy term 푊(휏) • Generates fluctuation corrections to the 퐴-membrane surface tension • Energy is no longer large scale: fluctuation contribution may dominate mean flow contribution 2D MHD Equilibria

• Jet and vortex-type equilibrium solutions continue to exist • 2D Magnetic field lines follow contours of constant vector potential 퐴0 Ocean Internal Wave Turbulence OCTS Images of Chlorophyll-a

Strong Imprint of ocean eddies; East of Honshu Island, Japan C2CS Chl-a

Tasmania SeaWIFS Chl-a

Agulhas current region, south of Africa, 1998 Chl-a 1D spectra

Agulhas region Honshu region

1/푘

1/푘3

1/푘

휆 ≈ 600 km 휆 ≈ 60 km 휆 ≈ 6 km Peak features may be due to tidal period resonances • Cholorphyll concentration field is freely advected by the fluid flow – “passive tracer” • The flow leaves an imprint of the turbulence on the spatial pattern • Slow 1/푘 decay is characteristic prediction for the forward enstrophy cascade of 2D eddy turbulence OCTS Chl-a

Nova Scotia

Cape Cod

Gulf Stream

Gulf of Maine, 1997 Chl-a and SST 1D spectra

1/푘 1/푘

1/푘3 1/푘3

OCTS data, Gulf of Maine

Much steeper spectral fall-off (smoother spatial pattern) in some ocean regions • Sea surface temperature (SST) is another good passive scalar • The 1/푘3 power law is the predicted imprint of internal waves

P. B. Weichman and R. E. Glazman, “Spatial Variations of a Passive Tracer in a Random Wave Field”, JFM 453, 263 (2002) Internal Gravity Waves

Thermocline depth

Brundt-Väisälä frequency defines oscillation frequency of vertically displaced fluid parcels due to pressure-, temperature- and Internal waves live where density gradient is largest, salinity-induced above ~1 km depth density gradient • ~10 m wave amplitude, 1-100 km wavelength at these depths • But only ~5 cm signature at sea surface due to air-water 푁(푧) = −푔휕푧휌/휌 density contrast • Tiny compared to surface gravity waves, but much slower, hence visible via low frequency filtering (hours, days, weeks) • Internal wave speed ~2 m/s sets basic time scale 휌 푧 = 휌[푝 푧 , 푇 푧 , 푆 푧 ] SOFAR Channel

Aside: Same vertical structure produces a minimum at the thermocline depth in the acoustic sound speed (SOFAR waveguide channel), enabling basin-wide signal transmission (whale mating calls?) Overlapping Chl-a and SSH Spectra

“Slow” Eddy contribution “Fast” gravity wave contribution Insets: Topex/Poseidon satellite altimeter SSH spectra

P. B. Weichman and R. E. Glazman, “Turbulent Fluctuation and Transport −2.92 of Passive Scalars by 푘 Random Wave Fields”, PRL 83, 5011 (1999)

Chlorophyll-a spectra derived from OCTS multispectral Landsat Chlorophyll-a concentration spectrum imagery (Japanese NASDA ADEOS satellite) 60o N near Iceland (Gower et al., 1980) Long-term space-time coverage enables filtering of fast (hours, days) and slow (weeks, months, even years) components of SSH variability Data confirm that 1/푘3 Chl-a spectral behavior occurs in regions where wave motions dominate Passive scalar transport by random wave fields

휌1

휌2

In addition to the “mean flow” eddy velocity 퐯(퐫), internal waves generate (a spectrum of superimposed) smaller scale circulating patterns 퐮wave(퐫) • These create a pattern of horizontal compression and rarefaction regions on the surface that are visible in the passive scalar density • This horizontal motion effect is largest at the surface, even though vertical −2 motion is tiny due to large air-water contrast: 훿ℎ푠푢푟푓 ∼ 10 훿ℎ푡ℎ푒푟푚표푐푙𝑖푛푒

Unlike in eddy turbulence, for wave turbulence there is a small parameter −2 푢0/푐0 ∼ 10 that allows one to perform a systematic expansion for the passive scalar statistics 10 m • Fluid parcel speed 푢 ∼ ∼ 2 cm/s (for ~1 km wavelength) 0 10 min Δ𝜌 • Wave speed 푐 ∼ 푔ℎ ∼ (100 m/s) 10−3 ∼ 2 m/s 0 𝜌 Passive Scalar Dynamics Passive scalar transport 휓 by an externally imposed velocity field 퐯: 2 휕푡휓 + ∇ ⋅ 퐯휓 = 휅∇ 휓 Linearized (small fluctuations around a smooth mean 휓 :

휕푡훿휓 = −휓 ∇ ⋅ 퐯 ⇒ Concentration fluctuations are driven by fluid areal density fluctuations

′ ′ Formal solution to the passive scalar 휓 퐱, 푡 = 푑퐱 휓 퐱 , 푠 훿(퐱 − 퐙 ′ 푡 ) 퐱 푠 equation (neglecting diffusion 휅)

(Nonlinear) Lagrangian trajectory for a fluid 퐙퐱푠 푡 parcel (with entrained passive scalar) 휕푡퐙퐱푠 푡 = 퐯(퐙퐱푠 푡 , 푡) constrained to be at point 퐱 at time 푠

′ 푃 퐱, 푡; 퐱 , 푠 = 〈훿 퐱 − 퐙퐱′푠 푡 〉 Statistics computed from Markov-like transition probability

• Unlike for eddy turbulence, where statistics of 퐯 are very complicated, and poorly understood, very weakly interacting sinusoidal wave modes have near-Gaussian statistics • In addition, the small parameter 푢0/푐0, which does not exist for eddy motions, enables a systematic expansion for the Lagrangian trajectory Passive Scalar Spectra 푘2퐹 (푘) 퐿 Result for “renormalization” of passive scalar 푅PS 푘 = 2휓 2 휔 푘 spectrum by wave height spectrum 퐹퐿 푘 Wave dispersion relation; replaced e.g., by 휔 푘 = 푐0푘 • 휔 = 푔푘 for surface gravity waves

2 2 2 • 휔 = 푐0 푘 + 푓 for longer wavelength waves (larger than Rossby radius) that feel the Coriolis force (wave periods comparable to Earth rotation period) There is a remarkable “weak turbulence” theory of the wave spectrum (Zakharov et al.), based on slow exchange of energy via very weak nonlinear interactions between wave modes, and near-Gaussian statistics. • Again, unlike for Eddy turbulence, exact predictions for the Kolmogorov spectral exponents are then possible • Results depend on dispersion relation and exact form of nonlinear wave-wave interactions For internal waves, the theory produces: −4/3 • Larger scale inverse cascade region Scale set by energy injection 퐹 (푘) ∼ 푘 퐿 푘−3 • Smaller scale (typically below ~10 km) length scale (e.g., tidal flows direct cascade region over the continental shelf)

Predicted form spans a range that 푅 푘 ∼ 푘−4/3- 푘−3 푃푆 agrees with observations! End of Part 3