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, magnetohydrodynamics, 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.
2.
3.
4.
5.
6.
The Great Red Spot and geophysical simulations
Euler’s equation and conservation laws
Relation to 2D turbulence: inverse energy cascade
Thermodynamics and statistical mechanics
Equilibrium solutions
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 highresolution 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:
_
O O
Two blobs of opposite vorticity, + and
Turbulent
cascade
+
t 
Final condition:
+ blob survives, appears stable
O
- blob disperses
O
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
inviscid
Euler
Driving and
Dv ( r , t )
2
 p(r, t )  f (r ) zˆ  v(r, t )   v(r, t )  f (r, t ) dissipation
Dt
D 
  v (r , t )  
- Convective derivative
Dt t
v (r , t )
- Velocity field
- Pressure field
p (r , t )
1
E   dr (r, t ) | v(r, t ) |2
- Kinetic energy
2
ω(r, t )    v(r, t )
- Vorticity field (scalar in d=2)
f (r )  2 E sin( L ) - Coriolis parameter (rotating coordinate system)

- Viscosity
F( r , t )
- Driving force, often stochastic
Constraints and Conservation Laws
Determines pressure field p(x,t)
Implies existence of stream function:
(a) Incompressibility:
  v(r, t )  0
 v(r, t )    (r, t )  ( y , x )
[  (r, t )  const.]
  (r, t )  2 (r, t )
(b) Angular momentum:
L   dr r  v(r, t )  
(axially symmetric domains)
1
2
d
r
r
 (r, t )  (boundary term)

2
(c) Energy:
1
E   dr | v(r, t ) |2
2
1
  dr  dr ' (r, t )G (r, r ' ) (r ' , t )
2
Analogy: Vorticity ↔ Charge density
2D Coulomb
Green function
  2G (r, r ' )   (r  r ' ) ( bdy conds )
G (r, r ' )  
1  | r  r'| 
, | r  r ' | 0
ln 
2  R0 
More constraints and Conservation Laws
(d) → () All powers of the vorticity!
Dv(r, t )
 p(r, t )
Dt
  v(r, t )  0
Vorticity is

D

(
r
,
t
)

 0 freely self 
Dt
advecting

 n   dr  (r , t ) n 
d n
D (r , t )
  dr  (r , t ) n 1
0
dt
Dt
Convenient parametrization:
0, x  0
G ( )   dr  [   (r, t )],  ( x)  
1, x  0
Alternate form:
More generally:
 f   dr f [ (r, t )]
- Conserved for any
function f()
All conserved integrals
may now be expressed
in terms of g():
 n   d  n g ( )
All “charge
dG ( )
g ( ) 
  dr  [   (r, t )]
species” are  f  d f ( ) g ( )
d
independently
g ( )d  fractional area on which conserved
   (r, t )    d

Simple example: single charged
species (charge density q)
occupying fractional area .
Dynamics fully specified by area
Aq (t )  r V :  (r, t )  q, | Aq (t ) | V
Aq (t  0)
G ( )   (  q)  (1   ) ( )
g ( )   (  q)  (1   ) ( )
(  0.3, q  1.2)
Aq (t  )
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
| v(k ) |2
2

2 (2 )
dk
2
2
Enstrophy:  2  
|
k
|
|
v
(
k
)
|
(2 ) 2
Energy :
(People)
E
($$$)
Phase space: natural tendency for
“diffusion” to large k
Conservation laws: constraints on
energy flow (absent in 3D due to
vortex line stretching bending, etc.)
Exists also in other systems,
e.g., ocean waves
Economic analogy: under “free” capitalistic
dynamics (total people & $$$ 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.
“Random”
(turbulent) initial
condition
Energy
flux
Final
steady
state
System scale L
Grid scale li
Driving, f
(Birth and
Grants)
Enstrophy
flux
Dissipation, 
(Death and Taxes)
Statistical Mechanics
L. Onsager, “Statistical hydrodynamics”,
Nuovo Cimento Suppl. 6, 279 (1949).
Low E:
“Kosterlitz-Thouless”
dipole gas phase
Standard Coulomb
energetics:
T>0
Raise E:
Momentumless “neutral
plasma” phase
Entropy
picture
 | ri  r j | 
1

E   i j ln 
2 i j
 a 
N point
vortices
Raise E further:
Macroscopic charge
segregation
Macroscopic vortices
effectively require:
1 S

T E
i.e., E→-E,
or T < 0!
Why are T < 0 states physical?
1
dr | v(r, t ) |2

2
  E / V  O(1)
E
Hydrodynamic flow energy
Expect energy density
Claim: All states with  = O(1) must have E > E  , i.e., T < 0,
in order to overcome screening
Discrete version: a → 0
 | ri  r j | 
1 4

E  a  i j ln 
2 i j
 a 
Well known fact: neutral Coulomb gas at T > 0 has
E / a 4  N (# sites)
but : N  V / a 2

E / V  Na 4 / V  a 2  0 !
Any T > 0 state has E/V = 0, hence all flows are microscopic: v macro  0
  E / V  0 requires E/a 4  N 2  T  0
Hydrodynamic states have “Super-extensive” lattice energy
REALITY intrudes:
T<0

Hydrodynamics is not in equilibrium with molecular
scales, which always have T > 0.
Viscosity

Communication between hydrodynamics and molecular
dynamics: T < 0 state must eventually decay away.
>0
Pious
Hope

For  << 1, there will exist a time scale tmolec << t << tvisc
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   e   H (   1 / T )
Free Energy
F 
H
1
ln tr  e   H 
V
1
dr  dr '  (r )G (r, r') (r ')

2
  dr h(r ) (r )
  dr [ (r )]
Proper care and feeding of
conservation laws: Lagrange
multiplier/chemical potential for
each one.
1 2
Angular momentum multiplier 
h(r )   r   r 3 -plane/Coriolis potential term
2
 ( )   n n Taylor coefficients correspond to
n
Continuous spin Ising model!
“Exchange”
G(r,r’)
“Magnetic field”
h(r)
“Spin weighting factor” ()
multipliers for vorticity powers n
E.g., Energy/enstrophy theory (Kraichnan,…):
 ( )   2 2  Gaussian theory
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
2
  0,   0  r0  2 / 3
seeks minimum h(r)
seeks maximum h(r)
Balance between angular momentum and Coriolis force produces an
effective potential minimum
Exact mean field theory
This model can be solved exactly!
Hint from critical phenomena: Phase transitions in
models with long-ranged interactions are mean-field like.
1
E   dr  dr' (r)G(r, r' ) (r' )
2
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).
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 :
1
E   dr |  (r ) |2 , S   dr W [ (r )  h(r )]
2
W ( )   ln



d e
  [   ( )]

e
W ( )
Details
Tomorrow!
~ Laplace transformof 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
 0   0 (r )    0 (r )   d n0 (r,  )

 (r )
e    [ 0 (r ) h (r )]  ( )
n0 (r,  ) 
eW [ 0 (r ) h (r )]
 0 (r )   (r )
 0 (r )   (r )
Highly nonlinear
PDE
Probability density for vortex of
charge density  at r
“Order Parameter”
“Coarse-grained” stream function
To be solved with constraints:
g ( )  
F
dr
  n0 (r,  )
 ( )
V
Determines ()
for given g()
Example:
g ( )   (  q)  (1   ) ( ), h(r )  0
1
  2 0 (r ) 
1  e  [ q 0 (r )   ]
Hard-core → Fermi-like function
dr
q     2 0 (r )
V
Point vortex limit:
q  ,   0, Q  qV fixed
1 2
e   (r )
-   0 (r ) 
  ( r )
Q
d
r
e

An exact solution in this case predicts
collapse to a point at T = -1/8
0, r  r0
(Gauss's Law )
q, r  r0
 0 (r )  
Numerical solutions
Q   / 10
(r0 / L) 2  1  
T 0
T 0

Point
vortices
T  
0 (r)  q
T  0
T 0
(r1 / L) 2  
q, r  r1
0, r  r1
 0 (r )  
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
Some beautiful experiments: Guiding center plasmas
Indivual electrons oscillate
rapidly up and down the
column, but the projected
charge density
L
nproj(r )   dz n(r, z )
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.
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 twovortex 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
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 semiquantitatively 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
prediction
Jets
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 nearsteady state:
𝑞 ≈ 𝐹(𝜓)
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
(expressed in terms of vorticity)
𝐻 𝜔 = 𝐸 𝜔 − 𝐶𝜇 𝜔 − 𝑃[𝜔]
1
𝐸𝜔 =
2
2
𝑑 𝑟
2 ′
′
𝑃𝜔 =
′
𝑑 𝑟 𝜔 𝐫 𝐺 𝐫, 𝐫 𝜔(𝐫 )
𝐶𝜇 𝜔 = ∫ 𝑑 2 𝑟 𝜇[𝜔 𝐫 ]
∫ 𝑑2𝑟
′
ℎ 𝐫 𝜔(𝐫)
Conservation of vorticity integrals
1 2
ℎ 𝐫 = 𝛼𝑟 + 𝛾𝑟 3
2
Grand canonical partition function:
𝑍(𝛽, 𝜇, ℎ) = ∫ 𝐷 𝜔 𝑒 −𝛽𝐻[𝜔]
Free energy:
1
𝐹(𝛽, 𝜇, ℎ) = − ln(𝑍)
𝛽
Fluid kinetic energy
1
𝐫 − 𝐫′
′
𝐺 𝐫, 𝐫 ≈ −
ln
2𝜋
𝑅0
Conservation of angular
momentum, and Coriolis force
Invariant phase space measure
(Liouville theorem):
∞
∫ 𝐷 𝜔 = lim
𝑎→0
𝑖
𝑑𝜔𝑖
−∞ 𝑞0
Independent integral over vorticity
level at each point in space
Macro- vs. Micro-scale
𝑙-cell
𝑎-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
Permutation factor: number of distinct ways of
rearranging vorticity within a given 𝑙-cell
(automatically preserves all conservation laws)
𝑁𝑙 !
∼ 𝑒−
𝑛𝑙 𝜎1 ! 𝑛𝑙 (𝜎2 )! … 𝑛𝑙 (𝜎𝑀 )!
𝑀
𝑘=1 𝑛𝑙
𝜎𝑘 ln[𝑛𝑙 𝜎𝑘 /𝑁𝑙 ]
In the continuum limit, 𝑎 → 0, taking the limit of
continuous set of vorticity levels as well:
𝑛𝑙 𝜎𝑘 → 𝑛0 (𝐫, 𝜎)
𝐷[𝜔] =
𝐷 𝑛0 𝑒 𝑆 𝑛0 /𝑎
2
𝑆 𝑛0 = −
Vorticity distribution at position 𝐫
𝑑2𝑟
𝑑𝜎 𝑛0 𝐫, 𝜎 ln[𝑞0 𝑛0 (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
Express everything in terms of 𝑛0 𝐫, 𝜎 in order to complete the partition
function integral
Constraints:
Equilibrium vorticity
𝑑𝜎 𝑛0 𝐫, 𝜎 = 1
𝑑 2 𝑟 𝑛0 𝐫, 𝜎 =
Normalization
𝜔0 𝐫 =
𝑑𝜎 𝜎 𝑛0 (𝑟, 𝜎)
𝑑 2 𝑟𝛿[𝜎 − 𝜔 𝐫 ] = 𝑔(𝜎)
Global vorticity conservation
𝐶𝜇 𝑛0 =
𝑑2𝑟
𝑑𝜎 𝜇 𝜎 𝑛0 (𝐫, 𝜎)
Additional Lagrange multiplier for
normalization constraint
𝑁𝜈 𝑛0 =
𝑑𝜎
𝑑 2 𝑟 𝜈 𝐫 𝑛0 (𝐫, 𝜎)
Can replace 𝜔 by 𝜔0 for any
smoothly varying interaction:
𝐸 𝑛0
1
=
2
𝑃 𝑛0 =
𝑑2𝑟
𝑑 2 𝑟 ′ 𝜔0 𝐫 ′ 𝐺 𝐫, 𝐫 ′ 𝜔0 (𝐫 ′ )
𝑑 2 𝑟 ℎ 𝐫 𝜔0 (𝐫)
Macroscale thermodynamics
𝑍(𝛽, 𝜇, 𝜈, 𝛼) =
𝐷 𝑛0 𝑒 −𝛽𝐺[𝑛0]
G 𝑛0 = 𝐸 𝑛0 − 𝐶𝜇 𝑛0 − 𝑃[𝑛0 ] − 𝑁𝜈 𝑛0 − 𝑇𝑆[𝑛0 ]
𝑇=
1
𝑇
=
𝛽𝑎2 𝑎2
1
𝛽= 2→∞
𝑇𝑎
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 ]
𝛿𝐺
=0
𝛿𝑛0 𝐫, 𝜎
Variational Equations
𝛿𝐺
=0
𝛿𝑛0 𝐫, 𝜎
𝑊 𝜏 = −ln
Ψ0 𝐫 =
⇒
𝑑𝜎 𝛽[𝜇
𝑒
𝑞0
𝑛0 𝑟, 𝜎 = 𝑒 𝑊[Ψ0(𝐫)−ℎ(𝐫)] 𝑒 −𝛽
𝜎 −𝜎𝜏]
𝑑 2 𝑟 𝐺 𝐫, 𝐫 ′ 𝜔0 (𝐫′)
𝜎[Ψ0 𝐫 −ℎ 𝐫 −𝜇(𝜎)}
From normalization condition
Equilibrium stream function
Closed equation for the stream function
𝜔0 𝐫 = −∇2 Ψ0 𝐫 =
𝑑𝜎 𝜎 𝑛0 (𝐫, 𝜎) = 𝑇 𝑊′[Ψ0 (𝐫) − ℎ(𝐫)]
Variational equation obtained by minimizing the free energy fucntional
𝐹[Ψ0 ] =
𝑑2𝑟
1
∇Ψ0 (𝐫)
2
2
− 𝑇𝑊[Ψ0 (𝐫) − ℎ(𝐫)]
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
Potential vorticity (PV)
𝑄(𝐫) = 𝜔(𝐫) + 𝑘𝑅2 𝜓 𝐫 + 𝑓(𝐫)
Rossby radius of deformation
𝑅0 = 1/𝑘𝑅 = 𝑐𝐾 /𝑓
Kelvin wave speed 𝑐𝐾 (speed of short wavelength inertia-gravity waves –
quantifies gravitational restoring force for surface height fluctuations)
Coriolis parameter (Earth rotational force):
𝜕𝑡 −∇2 + 𝑘𝑅2 𝜓 + 𝐯 ⋅ ∇𝜔 + 𝛽𝜕𝑥 𝜓 = 0
𝑓 = 2Ω𝐸 sin(𝜃𝐿 )
“Beta parameter”
𝛽 = 𝜕𝑦 𝑓
Can be written in the form
𝐷𝑄
=0
𝐷𝑡
𝑄 is advectively conserved in the same way that 𝜔 is
for the Euler equation
𝛽𝑘𝑥
𝜔=− 2
𝑘 + 𝑘𝑥2
Rossby wave dispersion relation (linearized dynamics)
QG Equilibria
Stream function follows surface height: 𝜓(𝐫) ∝ 𝛿ℎ(𝐫)
Energy function:
𝐸=
2
𝑑 𝑟 ∇𝜓(𝐫)
2
+
𝑘𝑅2 𝜓(𝐫)2
(−∇2 +𝑘𝑅2 )𝐺𝑄 𝐫, 𝐫 ′ = 𝛿(𝐫 − 𝐫 ′ )
𝐺𝑄
𝐫, 𝐫 ′
1
=−
𝐾0 ( 𝐫 − 𝐫 ′ /𝑅0 )
2𝜋
1
=
2
𝑑 2 𝑟 𝑄 𝐫 − 𝑓 𝐫 𝐺𝑄 𝐫, 𝐫 ′ [𝑄 𝐫 ′ − 𝑓 𝐫 ′ ]
• 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[𝑞0 𝑛0 (𝐫, 𝜎)]
Here 𝜎 now denotes the values of 𝑄
Equilibrium equations are derived by minimizing the functional:
𝐹[Ψ] =
𝑑2𝑟
1
𝛻Ψ
2
2
1 2 2
+ 𝑘𝑅 Ψ + 𝑓Ψ − 𝑇𝑊 Ψ − ℎ
2
QG Equilibirum Vortex
Two level system example:
Δ𝜎(𝑇)
Σ(𝑇)
• 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|/𝑇𝑐
• 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
𝜕𝑡 𝑄 + 𝐯 ⋅ ∇𝑄 = 0
𝐸[𝑄]
𝜓 𝐫 =
𝐿𝜓 =
Some vorticity-like field 𝑄(𝐫, 𝑡) that is advectively conserved
Existence of a conserved energy functional (not necessarily quadratic)
• Assumed sufficiently smooth in space that 𝐸 𝑄 = 𝐸 𝑄 ≡ 𝐸[𝑄0 ]
𝛿𝐸
𝛿𝑄 𝐫
Relation to stream function 𝜓, from
which velocity 𝐯 = ∇ × 𝜓 is derived
𝑑 2 𝑟𝜓 𝐫 𝑄(𝐫) − 𝐸[𝑄]
𝑆 𝑛0 = −
𝑑2𝑟
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)
Convert to function of 𝜓
via Legendre transform
𝑑𝜎 𝑛0 𝐫, 𝜎 ln[𝑞0 𝑛0 (𝐫, 𝜎)]
Integration over small scale fluctuations
produces the identical entropy
contribution, expressed in terms of the
𝑄-level distribution function 𝑛0 𝐫, 𝜎
Exact variational condition for large scale structure produces the identical relation:
𝑛0 𝐫, 𝜎 = 𝑒 𝑊[Ψ0(𝐫)−ℎ(𝐫)] 𝑒 −𝛽
𝜎[Ψ0 𝐫 −ℎ 𝐫 −𝜇(𝜎)}
Equilibrium equations are then derived by minimizing the free energy functional:
𝐹 Ψ =𝐿 Ψ −𝑇
𝑑 2 𝑟𝑊[Ψ − ℎ]
𝑊 𝜏 = −ln
𝑑𝜎 𝛽[𝜇
𝑒
𝑞0
𝜎 −𝜎𝜏]
Higher Dimensional Example
The collisionless Boltzmann equation: Flow equation for phase space
probability density 𝑓(𝐫, 𝐩)
𝜕𝑡 𝑓 + 𝐫 ⋅ ∇𝑟 𝑓 + 𝐩 ⋅ ∇𝑝 𝑓 = 0
Newton’s laws provide 𝐫, 𝐩:
𝐹 𝐫 = −∇𝜙 𝐫
𝐫 = 𝐩/𝑚
𝜙 𝐫 =
𝑑𝑑 𝑟
𝐩 = 𝐅(𝐫)
𝑑 𝑑 𝑝 𝑉 𝐫, 𝐫 ′ 𝑓(𝐫 ′ , 𝐩)
Debye-Hückel
theory of
electrolytes
provides another
example!
Energy functional:
𝐸=
•
•
𝑑𝑑 𝑟
𝑑𝑑 𝑝
𝐩2
1
𝑓(𝐫, 𝐩) +
2𝑚
2
𝑑𝑑 𝑟
𝑑𝑑 𝑝
𝑑 𝑑 𝑟′
𝑑 𝑑 𝑝′ 𝑓 𝐫, 𝐩 𝑉 𝐫, 𝐫 ′ 𝑓(𝐫′, 𝐩′ )
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: 𝑛 𝐫 ≡ −∇2 Ψ 𝐫 = 𝑑𝑑 𝑝 𝑓(𝐫, 𝐩)
𝐹Ψ =
1
2
𝑑 𝑑 𝑟 ∇Ψ
2
−𝑇
𝑑𝑑 𝑟
𝑑 𝑑 𝑝 𝑊[Ψ 𝐫 − |𝐩|2 /2𝑚]
𝑇 = 𝑇/𝑎2𝑑
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    2 v  f
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,…)
𝛿𝐴 𝐫, 𝑡 =
𝑑𝐫 ′ 𝜒𝐴𝐵
𝐫, 𝐫 ′ ; 𝑡
−
𝑡′
ℎ𝐵
(𝐫 ′ , 𝑡)
𝑖
𝜒𝐴𝐵 𝐫, 𝐫 ′ , 𝑡 − 𝑡 ′ = ⟨ 𝐴 𝐫, 𝑡 , 𝐵 𝐫 ′ , 𝑡 ⟩
2
Thermodynamic response of density 𝐴 to field ℎ𝐵 conjugate to
density 𝐵, governed by dynamic response function 𝜒𝐴𝐵
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
𝑀 = 0.05
Off-center
single vortex
𝑀 = 0.0373
Symmetric
single vortex
Double
vortex
• Vortex separation decreases
with decreasing angular
momentum 𝑀
• Two vortex solution disappears
below a critical separation
• Generally consistent with
numerically observed
dynamical merger instability
• 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
Chen & Cross, PRE 1996
Steady State Failure
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
Quadrupolar pattern time series 𝑊(𝑡)
Dritschell, Qi, Marston, JFM (2015)
𝑡 = 4, 40, 400, 4000
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
𝜌1
𝜌2
(also a model for compressible
flow: h →, g → )
h 
Coupled equations of


t motion for height and

Dv
  gh  velocity fields

Dt
  ( hv )  
potential + kinetic energy:
1
1
2
2
d
r
h
|
v
|

g
d
r
(
h

h
)
0
2
2 
Cn   dr h ( / h) n 
Conserved for

C f   dr h f ( / h) all n, f

E
There now exist gravity wave excitations
  ck , c  gh0
in addition to vortical excitations
Basic question: Is there a nontrivial final
state? Or is all vortical energy eventually
“emitted” as waves?
Answer: YES! macroscopic vortices
survive.
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.
P. B. Weichman and D. M. Petrich, “Statistical
equilibrium solutions of the shallow water
equations”, Phys. Rev. Lett. 86, 1761 (2001).
Shallow Water Equilibria
Free energy functional:
𝐹 Ψ, ℎ =
∇Ψ(𝐫) 2 1
𝑑 𝑟
− 𝑔ℎ 𝐫
2ℎ(𝐫)
2
2
2
− 𝑇ℎ(𝐫)𝑊[Ψ(𝐫)]
Equilibrium variational equations:
1
−∇ ⋅
𝛻Ψ0 𝐫 = 𝑇ℎ0 (𝐫)𝑊 ′ [Ψ0 (𝐫)]
ℎ0 𝐫
∇Ψ0 𝐫 2
= −𝑇W Ψ0 𝐫 − 𝑔ℎ0 (𝐫)
ℎ0 𝐫 2
1
𝐯0 = ∇ × Ψ0
ℎ0
1
𝜔0 = −∇ ⋅
∇Ψ0
ℎ0
Additional hydrostatic balance requirement
In equilibrium one must therefore have ∇ ⋅ ℎ0 𝐯0 = 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
𝜕𝑡 𝐯 + 𝐯 ⋅ ∇ 𝐯 + 𝐟 × 𝐯 = −∇𝑃 + 𝑱 × 𝑩
Ideal MHD:
𝜕𝑡 𝐁 = ∇ × (𝐯 × 𝐁)
Closure equations:
Quasistatic
Ampere law:
Incompressibility:
𝐉= ∇×𝐁
∇⋅𝐯= 0
∇⋅𝐁=0
Lorentz force acting on electric current
passing through a fluid element
• Fluid is approximated as perfectly
conducting
• Electric fields are negligibly small
Advection of magnetic field by velocity field
• Magnetic field lines may be stretched and
tangled, but are otherwise attached to a
given fluid parcel
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.
Resulting pair of scalar equations
𝜕𝑡 𝜔 + 𝑓 + 𝐯 ⋅ ∇ 𝜔 + 𝑓 = 𝐁 ⋅ ∇𝐽
𝜕𝑡 𝐴 + 𝐯 ⋅ ∇A = 0
• 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
1
𝐸=
2
𝑑 2 𝑟[
𝐯(𝐫)
2
+ 𝐁(𝐫)
Conserved kinetic + EM energy
2]
𝐯= ∇×𝜓
𝐁=∇×𝐴
Stream function &
vector potential
2D MHD Equilibrium Equations
Two sets of conserved integrals:
𝑗 𝜎 =
𝑑 2 𝑟𝛿[𝜎 − 𝐴 𝐫 ]
𝑑 2 𝑟[𝜔 𝐫 + 𝑓 𝐫 ]𝛿[𝜎 − 𝐴 𝐫 ]
𝑘 𝜎 =
Controlled by Lagrange multipliers 𝜇 𝜎 , 𝜈(𝜎)
P. B. Weichman, “Long-Range Correlations and
Coherent Structures in Magnetohydrodynamic
Equilibria”, PRL 109, 235002 (2012)
Equilibrium free energy functional:
𝐹 𝐴, Ψ =
𝑑 2 𝑟[
1
𝛻𝐴(𝐫)
2
2
+
1
𝛻Ψ(𝐫)
2
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
•
•
•
•
1
𝐴 =
2
𝑑 2 𝑟 𝛻𝛿𝐴(𝐫)
2
+ 𝛻𝛿Ψ(𝐫)
Physics is that of two coupled elastic membranes!
• Generates long-range correlations
• External localizing potential provided by 𝜇, 𝜈
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
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
Agulhas region
Honshu region
1/𝑘
1/𝑘 3
1/𝑘
Peak features may be due
to tidal period resonances
𝜆 ≈ 600 km
𝜆 ≈ 60 km
𝜆 ≈ 6 km
• 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
OCTS Chl-a
Nova Scotia
Cape Cod
Gulf Stream
Gulf of Maine, 1997
Chl-a and SST 1D spectra
1/𝑘
1/𝑘
1/𝑘 3
1/𝑘 3
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/𝑘 3 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
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𝜕𝑧 𝜌/𝜌
𝜌 𝑧 = 𝜌[𝑝 𝑧 , 𝑇 𝑧 , 𝑆 𝑧 ]
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
of Passive Scalars by
Random Wave Fields”, PRL
83, 5011 (1999)
Chlorophyll-a spectra derived from OCTS multispectral
imagery (Japanese NASDA ADEOS satellite)
𝑘 −2.92
Landsat Chlorophyll-a concentration spectrum
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/𝑘 3 Chl-a spectral behavior occurs in
regions where wave motions dominate
Passive scalar transport by random wave fields
𝜌1
𝜌2
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 ∼ 10−2 𝛿ℎ𝑡ℎ𝑒𝑟𝑚𝑜𝑐𝑙𝑖𝑛𝑒
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
10 m
• Fluid parcel speed 𝑢0 ∼
∼ 2 cm/s (for ~1 km wavelength)
• Wave speed 𝑐0 ∼
10 min
Δ𝜌
𝑔ℎ 𝜌 ∼ (100
m/s) 10−3 ∼ 2 m/s
Passive Scalar Dynamics
Passive scalar transport 𝜓 by an externally imposed velocity field 𝐯:
𝜕𝑡 𝜓 + ∇ ⋅ 𝐯𝜓 = 𝜅∇2 𝜓
Linearized (small fluctuations around a smooth mean 𝜓:
𝜕𝑡 𝛿𝜓 = −𝜓∇ ⋅ 𝐯
⇒ Concentration fluctuations are driven by fluid areal density fluctuations
𝜓 𝐱, 𝑡 =
𝑑𝐱 ′ 𝜓 𝐱 ′ , 𝑠 𝛿(𝐱 − 𝐙𝐱′ 𝑠 𝑡 )
𝐙𝐱𝑠 𝑡
𝜕𝑡 𝐙𝐱𝑠 𝑡 = 𝐯(𝐙𝐱𝑠 𝑡 , 𝑡)
Formal solution to the passive scalar
equation (neglecting diffusion 𝜅)
(Nonlinear) Lagrangian trajectory for a fluid
parcel (with entrained passive scalar)
constrained to be at point 𝐱 at time 𝑠
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
Passive Scalar Spectra
𝑅PS
𝑘 2 𝐹𝐿 (𝑘)
𝑘 = 2𝜓
𝜔 𝑘 2
𝜔 𝑘 = 𝑐0 𝑘
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=
𝑐02 𝑘 2 + 𝑓 2 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
𝐹𝐿 (𝑘) ∼ 𝑘 −3
• Smaller scale (typically below ~10 km)
𝑘
direct cascade region
𝑅𝑃𝑆 𝑘 ∼ 𝑘 −4/3 - 𝑘 −3
Scale set by energy injection
length scale (e.g., tidal flows
over the continental shelf)
Predicted form spans a range that
agrees with observations!
End of Part 3