Geophysical Fluid Dynamics: A Laboratory for Statistical Physics

Peter B. Weichman, BAE Systems

IGERT Summer Institute

Brandeis University

July 27-28, 2015

Jupiter Saturn

(S pole hexagon) Neptune Earth

(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

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

Outline (Part 1)

http://www.solarviews.com/cap/jup/vjupitr3.htm

Target Name: Jupiter

Spacecraft: Voyager

Produced by: NASA

Cross Reference: CMP 346

Date Released: 1990

http://nssdc.gsfc.nasa.gov/photo_gallery/photogallery-jupiter.html

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-neptune.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.

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, + and -

+

_

Final condition:

+ blob survives, appears stable

- blob disperses

t

Turbulent

cascade

O O

O O

Dynamical Stability Statistical equilibrium ? Vortex Hamiltonian?

YES!

Ergodicity? Sometimes!

Basic question:

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

2

2

( , )ˆ( , ) ( ) ( , ) ( , ) ( , )

( , )

( , )

( , )

1( , ) | ( , ) |

2

( , ) ( , )

( ) 2 sin( )

( , )

E L

D tp t f t t t

Dt

Dt

Dt t

t

p t

E d t t

t t

f

t

v rr r z v r v r f r

v r

v r

r

r r v r

ω r v r

r

F r

The Euler equation

- Convective derivative

- Velocity field

- Pressure field

- Kinetic energy

- Vorticity field (scalar in d=2)

- Viscosity

- Driving force, often stochastic

Basic

inviscid

Euler

Driving and

dissipation

Coriolis force

- Coriolis parameter (rotating coordinate system)

.]),([

0),(

constt

t

r

rv

),(),(

),(),(),(

2 tt

tt xy

rr

rrv

Constraints and Conservation Laws

(a) Incompressibility: Determines pressure field p(x,t) Implies existence of stream function:

(b) Angular momentum: (axially symmetric domains)

)termboundary(),(2

1),( 2 trdtd rrrvrrL

(c) Energy:

),'()',(),('2

1

|),(|2

1 2

tGtdd

tdE

rrrrrr

rvr

0|'|,|'|

ln2

1)',(

)condsbdy()'()',(

0

2

rrrr

rr

rrrr

RG

G

2D Coulomb

Green function

Analogy: Vorticity ↔ Charge density

dg

tdd

dGg

x

xxtdG

)(

)],([)(

)(

0,1

0,0)()],,([)(

rr

rr

(d) → () All powers of the vorticity!

More constraints and Conservation Laws

More generally:

)],([ tfdf rr

- Conserved for any

function f()

Convenient parametrization:

fractional area on which dt ),(r

Alternate form:

)()(

)(

gfd

gd

f

n

n

Vorticity is

freely self-

advecting

0),(

),(),(

0),(

0),(

),(),(

1

Dt

trDtrd

dt

dtrd

Dt

tD

t

tpDt

tD

nnn

n

rr

r

rv

rrv

All conserved integrals

may now be expressed

in terms of g():

All “charge

species” are

independently

conserved

)2.1,3.0(

)()1()()(

)()1()()(

q

qg

qG

Simple example: single charged

species (charge density q)

occupying fractional area . VtAqtVtA qq |)(|,),(:)( rr

)0( tAq

Infinitely folded fractal

structure: Statistics?

)( tAq

Dynamics fully specified by area

Relation to 2D turbulent cascade

Dynamical viewpoint on the formation of large-scale stucture:

The inverse energy cascade

22

22

2

2

|)(|||)2(

:Enstrophy

|)(|)2(2

1:Energy

kvkk

kvk

d

dE

Phase space: natural tendency for

“diffusion” to large k Conservation laws: constraints on

energy flow (absent in 3D due to

vortex line stretching bending, etc.)

Exists also in other systems,

e.g., ocean waves

(People)

($$$)

Dissipation, (Death and Taxes)

Driving, f (Birth and

Grants)

System scale L

Grid scale li

Final

steady

state

“Random”

(turbulent) initial

condition Energy

flux

Enstrophy

flux

Economic analogy: under “free” capitalistic

dynamics (total people & $$$ conserved),

people and money go in opposite

directions: an egalitarian/socialist initial

state is unstable towards one with a few

rich people and lots of poor people.

Statistical Mechanics

Low E:

“Kosterlitz-Thouless”

dipole gas phase

Raise E:

Momentumless “neutral

plasma” phase

Raise E further:

Macroscopic charge

segregation

aE

ji

ji

ji

||ln

2

1 rrN point

vortices

Entropy

picture

Macroscopic vortices

effectively require:

Standard Coulomb

energetics:

T > 0 i.e., E→-E,

or T < 0!

E

S

T

1

L. Onsager, “Statistical hydrodynamics”,

Nuovo Cimento Suppl. 6, 279 (1949).

Why are T < 0 states physical?

)1(/

|),(|2

1 2

OVE

tdE

rvrHydrodynamic flow energy

Expect energy density

Claim: All states with = O(1) must have E > E , i.e., T < 0,

in order to overcome screening

aaE

ji

ji

ji

||ln

2

1 4rr

Discrete version: a → 0

Well known fact: neutral Coulomb gas at T > 0 has

!0///:but

)sites#(/

242

4

aVNaVEaVN

NaE

Any T > 0 state has E/V = 0, hence all flows are microscopic: 0macro v

0requires0/ 24 TNE/aVE

Hydrodynamic states have “Super-extensive” lattice energy

REALITY intrudes:

Hydrodynamics is not in equilibrium with molecular

scales, which always have T > 0.

Communication between hydrodynamics and molecular

dynamics: T < 0 state must eventually decay away.

For << 1, there will exist a time scale tmolec << t << tvisc

over which equilibrium hydrodynamic description is

valid

T < 0

Viscosity

> 0

Pious

Hope

For now assume inviscid Euler equation to exact on all length scales.

Is the theory at least self-consistent?

YES!

Statistical Formalism Boltzmann/Gibbs

Free Energy

2 3

( 1 / )

1ln tr

1' ( ) ( , ') ( ')

2

( ) ( )

[ ( )]

1( )

2

( )

H

H

n

n

n

e T

F eV

H d d G

d h

d

h r r

r r r r r r

r r r

r r

r

Proper care and feeding of

conservation laws: Lagrange

multiplier/chemical potential for

each one.

Taylor coefficients correspond to

multipliers for vorticity powers n

Angular momentum multiplier

-plane/Coriolis potential term

Continuous spin Ising model! “Exchange” G(r,r’)

“Magnetic field” h(r)

“Spin weighting factor” ()

E.g., Energy/enstrophy theory (Kraichnan,…):

heoryGaussian t)( 2

2

Back to Jupiter for a moment:

Why is only one sign of vortex blob stable?

r0/L rmin/L

seeks minimum h(r)

seeks maximum h(r)

3/20,0

2

1)(

0

32

r

rrrh

Balance between angular momentum and Coriolis force produces an

effective potential minimum

Exact mean field theory

)'()',()('2

1rrrrrr GddE

This model can be solved exactly!

Hint from critical phenomena: Phase transitions in

models with long-ranged interactions are mean-field like.

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).

STSEF , Local entropy of mixing of noninteracting gas of vortices;

different species , different chemical potential ()

In terms of stream function :

2

[ ( )]

1| ( ) | , [ ( ) ( )]

2

( ) ln

E d S d W h

W d e

r r r r r

)()( oftransformLaplace~ eeW

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.

J. Miller, “Statistical mechanics of

Eulers equation in two dimensions”,

Phys. Rev. Lett. 65, 2137 (1990).

J. Miller, P. B. Weichman and M. C.

Cross, “Statistical mechanics, Euler’s

equation, and Jupiter’s Red Spot”,

Phys. Rev. A 45, 2328 (1992).

Details

Tomorrow!

)(

1

1)(

0)(),()1()()(

0

2

])([0

2

0

rr

r

r

r

V

dq

e

hqg

q

)]()([

)()]()([

0

00

2

0

0

0

),(

),()()(0)(

rr

rr

r

rrrr

hW

h

e

en

ndF

Mean field equations

Probability density for vortex of

charge density at r

)()(

)()(

0

0

rr

rr

“Order Parameter”

“Coarse-grained” stream function

To be solved with constraints:

Highly nonlinear

PDE

),()(

)( 0

r

rn

V

dFg Determines ()

for given g()

Example:

Hard-core → Fermi-like function

)(

)(

0

2 )(Q

1-

fixed,0,

r

r

rr

ed

e

qVQq

Point vortex limit:

An exact solution in this case predicts

collapse to a point at T = -1/8

Numerical solutions )LawsGauss'(,

,0)(

0

0

0

rrq

rrr

0T

T

q)(0 r

1

1

0,0

,)(

rr

rrqr

0T

0T

0T

1)/( 2

0 Lr

2

1 )/( Lr

10/Q

Point

vortices

More complex initial conditions,

with large number of vorticity

levels (e.g., for comparison with

numerical simulations): Discretize

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.

Verification of agreement between the

Monte Carlo result and the direct

solution for a case where the latter can

be obtained:

Experimental Realization: Guiding Center Plasmas

Nonneutral Plasma Group, Department of Physics, UC San Diego

http://sdphca.ucsd.edu/

Some beautiful experiments: Guiding center plasmas

Indivual electrons oscillate

rapidly up and down the

column, but the projected

charge density

),()(0

proj zndzn

L

rr

Obeys the 2D Euler

equation!

Euler dynamics arises

from the Lorentz force.

“Measurements of Symmetric Vortex Merger”, K.S.

Fine, C.F. Driscoll, J.H.Malmberg and T.B. Mitchell;

Phys. Rev. Lett. 67, 588 (1991).

There exists some theoretical work as well:

P. Chen and M. C. Cross: “Statistical two-

vortex equilibrium and vortex merger”, Phys.

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

Great Red Spot and White Ovals

Brown Barges (Jupiter northern

hemisphere)

Bouchet & Sommeria, JFM (2002) Phase diagram: energy vs. size in a confining weather band

(analogous to squeezed bubble surface tension effect)

Vortex-jet phase

transition line

Ocean Equilibria

Venaille & Bouchet, JPO (2011)

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

Chelton et. al, GRL (2007) Hallberg et. al, JPO (2006)

Rings

Jets

Westward

drift speed of

vortex rings

Equilibrium

prediction

Atmospheric Blocking Event: NE Pacific, Feb. 1-21, 1989

Ek & Swaters, J. Atmos. Sci. (1994)

𝑞 ≈ 𝐹(𝜓)

Signature of a near-

steady state:

End of Part 1

Part 2: Statistical mechanics of the Euler

equation (technical details & some

generalizations)

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)

Outline (Part 2)

Derivation of the Variational

Equations

Partition Function and Free Energy 𝐻 𝜔 = 𝐸 𝜔 − 𝐶𝜇 𝜔 − 𝑃[𝜔]

𝐸 𝜔 =1

2 𝑑2𝑟 𝑑2𝑟′𝜔 𝐫′ 𝐺 𝐫, 𝐫′ 𝜔(𝐫′)

𝐶𝜇 𝜔 = ∫ 𝑑2𝑟 𝜇[𝜔 𝐫 ]

𝑃 𝜔 = ∫ 𝑑2𝑟 ℎ 𝐫 𝜔(𝐫) ℎ 𝐫 =1

2𝛼𝑟2 + 𝛾𝑟3

Conservation of vorticity integrals

Conservation of angular

momentum, and Coriolis force

Fluid kinetic energy

∫ 𝐷 𝜔 = lim𝑎→0

𝑑𝜔𝑖

𝑞0

∞

−∞𝑖

Grand canonical partition function: Invariant phase space measure

(Liouville theorem): 𝑍(𝛽, 𝜇, ℎ) = ∫ 𝐷 𝜔 𝑒−𝛽𝐻[𝜔]

𝐹(𝛽, 𝜇, ℎ) = −1

𝛽ln(𝑍)

Free energy:

Hamiltonian functional

(expressed in terms of vorticity)

Independent integral over vorticity

level at each point in space

𝐺 𝐫, 𝐫′ ≈ −1

2𝜋ln

𝐫 − 𝐫′

𝑅0

Macro- vs. Micro-scale

𝑎-cell

𝑙-cell

𝐿

• Main barrier to

straightforward evaluation of

partition function 𝑍: Highly

nonlocal interaction 𝐺(𝐫, 𝐫′) • Solution (“asymptotic

freedom”): recognize that

interaction is dominated by

large scales, so integrate out

small scales first, where 𝐺 is

negligible (local ideal gas of

vortices), and then consider

large scales

• Variational principle

emerges here

• Mathematical approach:

consider scales 𝐿 ≫ 𝑙 ≫ 𝑎,

and take the limits

𝑎 → 0, 𝑙 → 0, but in such a

way that 𝑙/𝑎 → ∞

Neglecting interactions within an 𝑙-cell, partition function

contribution becomes an 𝑎-cell permutation count

Microscale vortex entropy Let 𝑛𝑙(𝜎𝑘) define the number of 𝑎-cells with vorticity

level 𝜎𝑘 in cell 𝑙

𝑁𝑙!

𝑛𝑙 𝜎1 ! 𝑛𝑙(𝜎2)!…𝑛𝑙(𝜎𝑀)!∼ 𝑒− 𝑛𝑙 𝜎𝑘 ln [𝑛𝑙 𝜎𝑘 /𝑁𝑙]

𝑀𝑘=1

Permutation factor: number of distinct ways of

rearranging vorticity within a given 𝑙-cell

(automatically preserves all conservation laws)

In the continuum limit, 𝑎 → 0, taking the limit of

continuous set of vorticity levels as well:

𝑛𝑙 𝜎𝑘 → 𝑛0(𝐫, 𝜎) Vorticity distribution at position 𝐫

𝐷[𝜔] = 𝐷 𝑛0 𝑒𝑆 𝑛0 /𝑎2 𝑆 𝑛0 = − 𝑑2𝑟 𝑑𝜎 𝑛0 𝐫, 𝜎 ln [𝑞0𝑛0(𝐫, 𝜎)]

Microscale configurational entropy density

Remaining integral over macroscale assignment of the microscale distribution function

• Depends only the intermediate scale 𝑙 • All fluctuations below this scale have been integrated out, accounted for in 𝑆[𝑛0]

Reformulation in terms of 𝑛0 𝐫, 𝜎

Constraints:

𝑑2𝑟 𝑛0 𝐫, 𝜎 = 𝑑2𝑟𝛿[𝜎 − 𝜔 𝐫 ] = 𝑔(𝜎)

𝑑𝜎 𝑛0 𝐫, 𝜎 =1 Normalization 𝜔0 𝐫 = 𝑑𝜎 𝜎 𝑛0(𝑟, 𝜎)

Equilibrium vorticity

𝑁𝜈 𝑛0 = 𝑑𝜎 𝑑2𝑟 𝜈 𝐫 𝑛0(𝐫, 𝜎)

Additional Lagrange multiplier for

normalization constraint

𝐶𝜇 𝑛0 = 𝑑2𝑟 𝑑𝜎 𝜇 𝜎 𝑛0(𝐫, 𝜎)

Global vorticity conservation

𝐸 𝑛0 =1

2 𝑑2𝑟 𝑑2𝑟′𝜔0 𝐫′ 𝐺 𝐫, 𝐫′ 𝜔0(𝐫

′)

𝑃 𝑛0 = 𝑑2𝑟 ℎ 𝐫 𝜔0(𝐫)

Can replace 𝜔 by 𝜔0 for any

smoothly varying interaction:

Express everything in terms of 𝑛0 𝐫, 𝜎 in order to complete the partition

function integral

Macroscale thermodynamics 𝑍(𝛽, 𝜇, 𝜈, 𝛼) = 𝐷 𝑛0 𝑒

−𝛽𝐺[𝑛0]

𝑇 =1

𝛽𝑎2=

𝑇

𝑎2

𝛽 =1

𝑇 𝑎2→ ∞

Key observation: Nontrivial balance between energy and

entropy requires the combination 𝛽 = 𝛽𝑎2 to remain finite in

the continuum limit

Since 𝛽 = 𝛽 /𝑎2 → ∞, the partition function integral is

dominated by the maximum of 𝐺[𝑛0]

G 𝑛0 = 𝐸 𝑛0 − 𝐶𝜇 𝑛0 − 𝑃[𝑛0] − 𝑁𝜈 𝑛0 − 𝑇 𝑆[𝑛0]

𝛿𝐺

𝛿𝑛0 𝐫, 𝜎= 0

Variational Equations

𝑛0 𝑟, 𝜎 = 𝑒𝑊[Ψ0(𝐫)−ℎ(𝐫)] 𝑒−𝛽 𝜎[Ψ0 𝐫 −ℎ 𝐫 −𝜇(𝜎)}

𝑊 𝜏 = −ln 𝑑𝜎

𝑞0𝑒𝛽 [𝜇 𝜎 −𝜎𝜏]

𝛿𝐺

𝛿𝑛0 𝐫, 𝜎= 0 ⇒

Ψ0 𝐫 = 𝑑2𝑟 𝐺 𝐫, 𝐫′ 𝜔0(𝐫′) Equilibrium stream function

From normalization condition

𝜔0 𝐫 = −∇2Ψ0 𝐫 = 𝑑𝜎 𝜎 𝑛0(𝐫, 𝜎) = 𝑇 𝑊′[Ψ0(𝐫) − ℎ(𝐫)]

Closed equation for the stream function

𝐹[Ψ0] = 𝑑2𝑟1

2∇Ψ0(𝐫)

2 − 𝑇 𝑊[Ψ0(𝐫) − ℎ(𝐫)]

Variational equation obtained by minimizing the free energy fucntional

Grand canonical entropy Kinetic energy

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

𝐷𝑄

𝐷𝑡= 0

Potential vorticity (PV) 𝑄(𝐫) = 𝜔(𝐫) + 𝑘𝑅2𝜓 𝐫 + 𝑓(𝐫)

𝑅0 = 1/𝑘𝑅 = 𝑐𝐾/𝑓 Rossby radius of deformation

Kelvin wave speed 𝑐𝐾 (speed of short wavelength inertia-gravity waves –

quantifies gravitational restoring force for surface height fluctuations)

𝜕𝑡 −∇2 + 𝑘𝑅

2 𝜓 + 𝐯 ⋅ ∇𝜔 + 𝛽𝜕𝑥𝜓 = 0

Coriolis parameter (Earth rotational force):

“Beta parameter”

Can be written in the form

𝑄 is advectively conserved in the same way that 𝜔 is

for the Euler equation

𝑓 = 2Ω𝐸 sin(𝜃𝐿)

𝛽 = 𝜕𝑦𝑓

𝜔 = −𝛽𝑘𝑥

𝑘2 + 𝑘𝑥2 Rossby wave dispersion relation (linearized dynamics)

QG Equilibria

𝐸 = 𝑑2𝑟 ∇𝜓(𝐫) 2 + 𝑘𝑅2𝜓(𝐫)2 =

1

2 𝑑2𝑟 𝑄 𝐫 − 𝑓 𝐫 𝐺𝑄 𝐫, 𝐫′ [𝑄 𝐫′ − 𝑓 𝐫′ ]

Energy function: Stream function follows surface height: 𝜓(𝐫) ∝ 𝛿ℎ(𝐫)

(−∇2+𝑘𝑅2)𝐺𝑄 𝐫, 𝐫′ = 𝛿(𝐫 − 𝐫′)

𝐺𝑄 𝐫, 𝐫′ = −1

2𝜋𝐾0( 𝐫 − 𝐫′ /𝑅0)

• Logarithmic singularity at the origin, but

exponential decay ∼ 𝑒−|𝐫−𝐫′|/𝑅0 at large

separation.

• 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

𝑆 𝑛0 = − 𝑑2𝑟 𝑑𝜎 𝑛0 𝐫, 𝜎 ln [𝑞0𝑛0(𝐫, 𝜎)] Here 𝜎 now denotes the values of 𝑄

𝐹[Ψ] = 𝑑2𝑟1

2𝛻Ψ 2 +

1

2𝑘𝑅2Ψ2 + 𝑓Ψ − 𝑇 𝑊 Ψ− ℎ

Equilibrium equations are derived by minimizing the functional:

QG Equilibirum Vortex

Two level system example: • Beautiful analogy with two

phase system, with phase

separation below a critical

temperature |𝑇 | < 𝑇𝑐

• Vortex may be thought of as

a droplet of one phase

inside the other

• Finite Rossby radius ⇒

Finite width interface

between phases, with PV

difference Δ𝜎(𝑇 ) and

surface tension Σ(𝑇 )

|𝑇 |/𝑇 𝑐

Σ(𝑇 )

Δ𝜎(𝑇 )

• Presence of Coriolis parameter 𝑓 𝑦 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 𝜕𝑡𝑄 + 𝐯 ⋅ ∇𝑄 = 0

Existence of a conserved energy functional (not necessarily quadratic)

• Assumed sufficiently smooth in space that 𝐸 𝑄 = 𝐸 𝑄 ≡ 𝐸[𝑄0]

Some vorticity-like field 𝑄(𝐫, 𝑡) that is advectively conserved

𝐸[𝑄]

𝜓 𝐫 =𝛿𝐸

𝛿𝑄 𝐫

Relation to stream function 𝜓, from

which velocity 𝐯 = ∇ × 𝜓 is derived

𝑛0 𝐫, 𝜎 = 𝑒𝑊[Ψ0(𝐫)−ℎ(𝐫)] 𝑒−𝛽 𝜎[Ψ0 𝐫 −ℎ 𝐫 −𝜇(𝜎)}

Integration over small scale fluctuations

produces the identical entropy

contribution, expressed in terms of the

𝑄-level distribution function 𝑛0 𝐫, 𝜎

𝑆 𝑛0 = − 𝑑2𝑟 𝑑𝜎 𝑛0 𝐫, 𝜎 ln [𝑞0𝑛0(𝐫, 𝜎)]

Exact variational condition for large scale structure produces the identical relation:

𝑊 𝜏 = −ln 𝑑𝜎

𝑞0𝑒𝛽 [𝜇 𝜎 −𝜎𝜏]

Equilibrium equations are then derived by minimizing the free energy functional:

𝐹 Ψ = 𝐿 Ψ − 𝑇 𝑑2𝑟𝑊[Ψ − ℎ]

𝐿 𝜓 = 𝑑2𝑟𝜓 𝐫 𝑄(𝐫) − 𝐸[𝑄] Convert to function of 𝜓

via Legendre transform

P. B. Weichman, Equilibrium theory

of coherent vortex and zonal jet

formation in a system of nonlinear

Rossby waves, Phys. Rev. E 73.

036313 (2006)

Higher Dimensional Example The collisionless Boltzmann equation: Flow equation for phase space

probability density 𝑓(𝐫, 𝐩) 𝜕𝑡𝑓 + 𝐫 ⋅ ∇𝑟𝑓 + 𝐩 ⋅ ∇𝑝𝑓 = 0

Newton’s laws provide 𝐫 , 𝐩 : 𝐫 = 𝐩/𝑚 𝐩 = 𝐅(𝐫)

D. Lynden-Bell & R. Wood, Mon. Not. R. Astron. Soc., 1968

• 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:

𝐹 𝐫 = −∇𝜙 𝐫 𝜙 𝐫 = 𝑑𝑑𝑟 𝑑𝑑𝑝𝑉 𝐫, 𝐫′ 𝑓(𝐫′, 𝐩)

𝐸 = 𝑑𝑑𝑟 𝑑𝑑𝑝𝐩 2

2𝑚𝑓(𝐫, 𝐩) +

1

2 𝑑𝑑𝑟 𝑑𝑑𝑝 𝑑𝑑𝑟′ 𝑑𝑑𝑝′𝑓 𝐫, 𝐩 𝑉 𝐫, 𝐫′ 𝑓(𝐫′, 𝐩′)

Energy functional:

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]

𝑛 𝐫 ≡ −∇2 Ψ 𝐫 = 𝑑𝑑𝑝 𝑓(𝐫, 𝐩)

𝐹 Ψ =1

2 𝑑𝑑𝑟 ∇Ψ 2 − 𝑇 𝑑𝑑𝑟 𝑑𝑑𝑝 𝑊[Ψ 𝐫 − |𝐩|2/2𝑚] 𝑇 = 𝑇/𝑎2𝑑

Debye-Hückel

theory of

electrolytes

provides another

example!

Near-Equilibrium Systems:

Weakly Driven & Dissipated

21Dp

Dt

vv f

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,…)

Generalizations to Weakly Driven Systems

𝛿𝐴 𝐫, 𝑡 = 𝑑𝐫′𝜒𝐴𝐵 𝐫, 𝐫′; 𝑡 − 𝑡′ ℎ𝐵(𝐫′, 𝑡) 𝜒𝐴𝐵 𝐫, 𝐫′, 𝑡 − 𝑡′ =

𝑖

2⟨ 𝐴 𝐫, 𝑡 , 𝐵 𝐫′, 𝑡 ⟩

Formalism possibly useful for treating evolution of ocean currents without

massive computational effort (predictability problem)

Thermodynamic response of density 𝐴 to field ℎ𝐵 conjugate to

density 𝐵, governed by dynamic response function 𝜒𝐴𝐵

See also recent kinetic equation approaches: • Nardini, Gupta, Ruffo, Dauxois, Bouchet, J. Stat. Mech. 2012

• Bouchet, Nardini, Tangarife, J. Stat. Phys. 2013

Weakly driven 2D Euler Equation

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

Bouchet, Simonnet, Phys. Rev. Lett. 2009

Some Investigations of

Ergodicity and Equilibration

Ergodicity Failure: Multiple solutions

Double

vortex

Symmetric

single vortex

Off-center

single vortex

• Entropy comparison for locally

stable states with the same total

vorticity 𝑄 = 0.2, angular

momentum 𝑀, and energy 𝐸(𝑀), • Largest entropy state is the global

free energy minimum

• Vortex separation decreases

with decreasing angular

momentum 𝑀

• Two vortex solution disappears

below a critical separation

• Generally consistent with

numerically observed

dynamical merger instability

𝑀 = 0.05 𝑀 = 0.0373

Chen & Cross, PRE 1996

Steady State Failure

Quadrupolar pattern time series 𝑊(𝑡)

𝑡 = 4, 40, 400, 4000 𝑡 = 0

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

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

P. B. Weichman and D. M. Petrich, “Statistical

equilibrium solutions of the shallow water

equations”, Phys. Rev. Lett. 86, 1761 (2001).

)/(

)/(

)(2

1||

2

1

)(

2

0

2

hfhdC

hhdC

hhdghdE

hgDt

D

t

hh

f

n

n

r

r

rvr

v

v

(also a model for compressible

flow: h →, g → )

There now exist gravity wave excitations

in addition to vortical excitations 0, ghcck

Conserved for

all n, f

potential + kinetic energy:

Coupled equations of

motion for height and

velocity fields

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

without viscosity. Basic question: Is there a nontrivial final

state? Or is all vortical energy eventually

“emitted” as waves?

Answer: YES! macroscopic vortices

survive.

Shallow Water Equations

𝜌1

𝜌2

Shallow Water Equilibria

𝐹 Ψ, ℎ = 𝑑2𝑟∇Ψ(𝐫) 2

2ℎ(𝐫)−1

2𝑔ℎ 𝐫 2 − 𝑇 ℎ(𝐫)𝑊[Ψ(𝐫)]

−∇ ⋅1

ℎ0 𝐫𝛻Ψ0 𝐫 = 𝑇 ℎ0(𝐫)𝑊

′[Ψ0(𝐫)]

∇Ψ0 𝐫 2

ℎ0 𝐫 2 = −𝑇 W Ψ0 𝐫 − 𝑔ℎ0(𝐫)

𝐯0 =1

ℎ0∇ × Ψ0

Additional hydrostatic balance requirement

∇ ⋅ ℎ0𝐯0 = 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

Free energy functional:

Equilibrium variational equations:

𝜔0 = −∇ ⋅1

ℎ0∇Ψ0

In equilibrium one must therefore have

Nontrivial equilibrium between interacting large scale negative temperature and

small scale positive temperature states is not possible

Magnetohydrodynamic

Equilibria

Ideal Magnetohydrodynamic Equations

Ideal MHD: 𝜕𝑡𝐯 + 𝐯 ⋅ ∇ 𝐯 + 𝐟 × 𝐯 = −∇𝑃 + 𝑱 × 𝑩

𝜕𝑡𝐁 = ∇ × (𝐯 × 𝐁)

𝐉 = ∇ × 𝐁

Lorentz force acting on electric current

passing through a fluid element

• Fluid is approximated as perfectly

conducting

• Electric fields are negligibly small

∇ ⋅ 𝐯 = 0

∇ ⋅ 𝐁 = 0

Advection of magnetic field by velocity field

• Magnetic field lines may be stretched and

tangled, but are otherwise attached to a

given fluid parcel

Quasistatic

Ampere law:

Incompressibility:

Closure equations:

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

• 𝐯, 𝐁 are horizontal ⇒ 𝐽, 𝜔 are normal to the plane,

and can be treated as scalars.

𝜕𝑡 𝜔 + 𝑓 + 𝐯 ⋅ ∇ 𝜔 + 𝑓 = 𝐁 ⋅ ∇𝐽

𝜕𝑡𝐴 + 𝐯 ⋅ ∇A = 0

𝐯 = ∇ × 𝜓

𝐁 = ∇ × 𝐴

𝐸 =1

2 𝑑2𝑟[ 𝐯(𝐫) 2 + 𝐁(𝐫) 2]

Conserved kinetic + EM energy

Resulting pair of scalar equations

Stream function &

vector potential

• Potential vorticity no longer

advectively conserved

• Replaced by advective conservation

of vector potential! Second derivative no longer controlled

• Microscopic fields much less regular!

• Leads to very different equilibria, with much stronger “subgrid” energetics

2D MHD Equilibrium Equations Two sets of conserved integrals:

𝑗 𝜎 = 𝑑2𝑟𝛿[𝜎 − 𝐴 𝐫 ] 𝑘 𝜎 = 𝑑2𝑟[𝜔 𝐫 + 𝑓 𝐫 ]𝛿[𝜎 − 𝐴 𝐫 ]

Controlled by Lagrange multipliers 𝜇 𝜎 , 𝜈(𝜎)

Equilibrium free energy functional:

𝐹 𝐴,Ψ = 𝑑2𝑟[1

2𝛻𝐴(𝐫) 2 +

1

2𝛻Ψ(𝐫) 2 − 𝜈′ 𝐴 𝛻A 𝐫 ⋅ 𝛻Ψ 𝐫 + ∇ℎ(𝐫) ⋅ ∇Ψ(𝐫)

−𝜇(𝐴 𝐫 ) − 𝑓(𝐫)𝜈 𝐴 𝐫 ] + 𝑊fluct[𝐴]

Microscopic fluctuation free energy

𝑊fluct[𝐴] is computed from a Gaussian

fluctuation Hamiltonian:

𝐻fluct 𝐴 =1

2 𝑑2𝑟 𝛻𝛿𝐴(𝐫) 2 + 𝛻𝛿Ψ(𝐫) 2 − 2𝜈′(𝐴 𝐫 )𝛻𝛿A(𝐫) ⋅ 𝛻𝛿Ψ(𝐫)

• 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 𝑊(𝜏) • Generates fluctuation corrections to the 𝐴-membrane surface tension

• Energy is no longer large scale: fluctuation contribution may dominate mean flow contribution

P. B. Weichman, “Long-Range Correlations and

Coherent Structures in Magnetohydrodynamic

Equilibria”, PRL 109, 235002 (2012)

Physics is that of two coupled elastic membranes!

• Generates long-range correlations

• External localizing potential provided by 𝜇, 𝜈

2D MHD Equilibria

• Jet and vortex-type equilibrium solutions continue to exist

• 2D Magnetic field lines follow contours of constant vector potential 𝐴0

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

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/𝑘 decay is characteristic prediction for the forward enstrophy cascade of 2D

eddy turbulence

Agulhas region

1/𝑘

1/𝑘3

Honshu region

𝜆 ≈ 60 km 𝜆 ≈ 6 km 𝜆 ≈ 600 km

1/𝑘

OCTS Chl-a

Gulf of Maine, 1997

Gulf Stream

Cape Cod

Nova Scotia

Chl-a and SST 1D spectra

1/𝑘

1/𝑘3

1/𝑘

1/𝑘3

Much steeper spectral fall-off (smoother spatial pattern) in some ocean regions

• Sea surface temperature (SST) is another good passive scalar

• The 1/𝑘3 power law is the predicted imprint of internal waves

OCTS data, Gulf of Maine

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

𝜌 𝑧 = 𝜌[𝑝 𝑧 , 𝑇 𝑧 , 𝑆 𝑧 ]

Internal waves live where density gradient is largest,

above ~1 km depth • ~10 m wave amplitude, 1-100 km wavelength at these depths

• But only ~5 cm signature at sea surface due to air-water

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

Brundt-Väisälä

frequency defines

oscillation frequency of

vertically displaced

fluid parcels due to

pressure-,

temperature- and

salinity-induced

density gradient

𝑁(𝑧) = −𝑔𝜕𝑧𝜌/𝜌

Thermocline

depth

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

𝑘−2.92

P. B. Weichman and R. E.

Glazman, “Turbulent

Fluctuation and Transport

of Passive Scalars by

Random Wave Fields”, PRL

83, 5011 (1999)

Landsat Chlorophyll-a concentration spectrum

60o N near Iceland (Gower et al., 1980)

“Slow” Eddy

contribution

“Fast” gravity

wave contribution

Insets:

Topex/Poseidon

satellite altimeter

SSH spectra

Chlorophyll-a spectra derived from OCTS multispectral

imagery (Japanese NASDA ADEOS satellite)

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/𝑘3 Chl-a spectral behavior occurs in

regions where wave motions dominate

Passive scalar transport by random wave fields

Unlike in eddy turbulence, for wave turbulence there is a small parameter

𝑢0/𝑐0 ∼ 10−2 that allows one to perform a systematic expansion for the

passive scalar statistics

• Fluid parcel speed 𝑢0 ∼10 m

10 min∼ 2 cm/s (for ~1 km wavelength)

• Wave speed 𝑐0 ∼ 𝑔ℎΔ𝜌

𝜌∼ (100 m/s) 10−3 ∼ 2 m/s

In addition to the “mean flow” eddy velocity 𝐯(𝐫), internal waves generate

(a spectrum of superimposed) smaller scale circulating patterns 𝐮wave(𝐫) • 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

motion is tiny due to large air-water contrast: 𝛿ℎ𝑠𝑢𝑟𝑓 ∼ 10−2𝛿ℎ𝑡ℎ𝑒𝑟𝑚𝑜𝑐𝑙𝑖𝑛𝑒

𝜌1

𝜌2

Passive Scalar Dynamics

𝜕𝑡𝐙𝐱𝑠 𝑡 = 𝐯(𝐙𝐱𝑠 𝑡 , 𝑡)

(Nonlinear) Lagrangian trajectory for a fluid

parcel (with entrained passive scalar)

constrained to be at point 𝐱 at time 𝑠

𝐙𝐱𝑠 𝑡

𝜓 𝐱, 𝑡 = 𝑑𝐱′𝜓 𝐱′, 𝑠 𝛿(𝐱 − 𝐙𝐱′𝑠 𝑡 ) Formal solution to the passive scalar

equation (neglecting diffusion 𝜅)

𝑃 𝐱, 𝑡; 𝐱′, 𝑠 = ⟨𝛿 𝐱 − 𝐙𝐱′𝑠 𝑡 ⟩ Statistics computed from Markov-like transition probability

• Unlike for eddy turbulence, where statistics of 𝐯 are very complicated, and poorly

understood, very weakly interacting sinusoidal wave modes have near-Gaussian

statistics

• In addition, the small parameter 𝑢0/𝑐0, which does not exist for eddy motions, enables a

systematic expansion for the Lagrangian trajectory

𝜕𝑡𝜓 + ∇ ⋅ 𝐯𝜓 = 𝜅∇2𝜓

𝜕𝑡𝛿𝜓 = −𝜓 ∇ ⋅ 𝐯

Linearized (small fluctuations around a smooth mean 𝜓 :

⇒ Concentration fluctuations are driven by fluid areal density fluctuations

Passive scalar transport 𝜓 by an externally imposed velocity field 𝐯:

Passive Scalar Spectra 𝑅 PS 𝑘 = 2𝜓

𝑘2𝐹 𝐿(𝑘)

𝜔 𝑘 2

𝜔 𝑘 = 𝑐0𝑘

𝐹 𝐿(𝑘) ∼ 𝑘−4/3

𝑘−3

• Larger scale inverse cascade region

• Smaller scale (typically below ~10 km)

direct cascade region

𝑅 𝑃𝑆 𝑘 ∼ 𝑘−4/3- 𝑘−3 Predicted form spans a range that

agrees with observations!

Scale set by energy injection

length scale (e.g., tidal flows

over the continental shelf)

Result for “renormalization” of passive scalar

spectrum by wave height spectrum 𝐹𝐿 𝑘

Wave dispersion relation; replaced e.g., by

• 𝜔 = 𝑔𝑘 for surface gravity waves

• 𝜔 = 𝑐02𝑘2 + 𝑓2 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:

End of Part 3