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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