Geophysical Fluid Dynamics - Brandeis UniversityGeophysical Flows (introduction & overview) 2....
Transcript of Geophysical Fluid Dynamics - Brandeis UniversityGeophysical Flows (introduction & overview) 2....
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. Eulerโs 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 Neptuneโs 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.
Jupiterโs Great Red Spot
A theoristโs/simulatorโs cartoon
-plane approximation:
โข Shear boundary conditions
โข Coriolis force
โข Weather bands
MODEL: (Marcus, Ingersol,โฆ)
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
( , )ห( , ) ( ) ( , ) ( , ) ( , )
( , )
( , )
( , )
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
ฯ 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 โ Charge density
dg
tdd
dGg
x
xxtdG
)(
)],([)(
)(
0,1
0,0)()],,([)(
rr
rr
(d) โ () 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 โcharge
speciesโ 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
โdiffusionโ 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)
($$$)
Dissipation, (Death and Taxes)
Driving, f (Birth and
Grants)
System scale L
Grid scale li
Final
steady
state
โRandomโ
(turbulent) initial
condition Energy
flux
Enstrophy
flux
Economic analogy: under โfreeโ capitalistic
dynamics (total people & $$$ 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:
โKosterlitz-Thoulessโ
dipole gas phase
Raise E:
Momentumless โneutral
plasmaโ 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., Eโ-E,
or T < 0!
E
S
T
1
L. Onsager, โStatistical hydrodynamicsโ,
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 โ 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 โSuper-extensiveโ 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! โExchangeโ G(r,rโ)
โMagnetic fieldโ h(r)
โSpin weighting factorโ ()
E.g., Energy/enstrophy theory (Kraichnan,โฆ):
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 rโ gives
negligible contribution to E.
Nearby vortices are essentially noninteracting (except for โhard coreโ 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, โStatistical mechanics of
Eulers equation in two dimensionsโ,
Phys. Rev. Lett. 65, 2137 (1990).
J. Miller, P. B. Weichman and M. C.
Cross, โStatistical mechanics, Eulerโs
equation, and Jupiterโs Red Spotโ,
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
โOrder Parameterโ
โCoarse-grainedโ stream function
To be solved with constraints:
Highly nonlinear
PDE
),()(
)( 0
r
rn
V
dFg Determines ()
for given g()
Example:
Hard-core โ 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.
โMeasurements of Symmetric Vortex Mergerโ, 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: โStatistical two-
vortex equilibrium and vortex mergerโ, 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, โRelaxation of 2D
turbulence to vortex crystals,โ 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
โข Appearance of meso-scale
coherent structures (rings and jets)
โข Westward drift speed of vortex rings
โข Poleward drift of cyclones
โข 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)
๐ โ ๐น(๐)
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 ๐ป ๐ = ๐ธ ๐ โ ๐ถ๐ ๐ โ ๐[๐]
๐ธ ๐ =1
2 ๐2๐ ๐2๐โฒ๐ ๐ซโฒ ๐บ ๐ซ, ๐ซโฒ ๐(๐ซโฒ)
๐ถ๐ ๐ = โซ ๐2๐ ๐[๐ ๐ซ ]
๐ ๐ = โซ ๐2๐ โ ๐ซ ๐(๐ซ) โ ๐ซ =1
2๐ผ๐2 + ๐พ๐3
Conservation of vorticity integrals
Conservation of angular
momentum, and Coriolis force
Fluid kinetic energy
โซ ๐ท ๐ = lim๐โ0
๐๐๐
๐0
โ
โโ๐
Grand canonical partition function: Invariant phase space measure
(Liouville theorem): ๐(๐ฝ, ๐, โ) = โซ ๐ท ๐ ๐โ๐ฝ๐ป[๐]
๐น(๐ฝ, ๐, โ) = โ1
๐ฝln(๐)
Free energy:
Hamiltonian functional
(expressed in terms of vorticity)
Independent integral over vorticity
level at each point in space
๐บ ๐ซ, ๐ซโฒ โ โ1
2๐ln
๐ซ โ ๐ซโฒ
๐ 0
Macro- vs. Micro-scale
๐-cell
๐-cell
๐ฟ
โข Main barrier to
straightforward evaluation of
partition function ๐: Highly
nonlocal interaction ๐บ(๐ซ, ๐ซโฒ) โข Solution (โasymptotic
freedomโ): recognize that
interaction is dominated by
large scales, so integrate out
small scales first, where ๐บ is
negligible (local ideal gas of
vortices), and then consider
large scales
โข Variational principle
emerges here
โข Mathematical approach:
consider scales ๐ฟ โซ ๐ โซ ๐,
and take the limits
๐ โ 0, ๐ โ 0, but in such a
way that ๐/๐ โ โ
Neglecting interactions within an ๐-cell, partition function
contribution becomes an ๐-cell permutation count
Microscale vortex entropy Let ๐๐(๐๐) define the number of ๐-cells with vorticity
level ๐๐ in cell ๐
๐๐!
๐๐ ๐1 ! ๐๐(๐2)!โฆ๐๐(๐๐)!โผ ๐โ ๐๐ ๐๐ ln [๐๐ ๐๐ /๐๐]
๐๐=1
Permutation factor: number of distinct ways of
rearranging vorticity within a given ๐-cell
(automatically preserves all conservation laws)
In the continuum limit, ๐ โ 0, taking the limit of
continuous set of vorticity levels as well:
๐๐ ๐๐ โ ๐0(๐ซ, ๐) Vorticity distribution at position ๐ซ
๐ท[๐] = ๐ท ๐0 ๐๐ ๐0 /๐2 ๐ ๐0 = โ ๐2๐ ๐๐ ๐0 ๐ซ, ๐ ln [๐0๐0(๐ซ, ๐)]
Microscale configurational entropy density
Remaining integral over macroscale assignment of the microscale distribution function
โข Depends only the intermediate scale ๐ โข All fluctuations below this scale have been integrated out, accounted for in ๐[๐0]
Reformulation in terms of ๐0 ๐ซ, ๐
Constraints:
๐2๐ ๐0 ๐ซ, ๐ = ๐2๐๐ฟ[๐ โ ๐ ๐ซ ] = ๐(๐)
๐๐ ๐0 ๐ซ, ๐ =1 Normalization ๐0 ๐ซ = ๐๐ ๐ ๐0(๐, ๐)
Equilibrium vorticity
๐๐ ๐0 = ๐๐ ๐2๐ ๐ ๐ซ ๐0(๐ซ, ๐)
Additional Lagrange multiplier for
normalization constraint
๐ถ๐ ๐0 = ๐2๐ ๐๐ ๐ ๐ ๐0(๐ซ, ๐)
Global vorticity conservation
๐ธ ๐0 =1
2 ๐2๐ ๐2๐โฒ๐0 ๐ซโฒ ๐บ ๐ซ, ๐ซโฒ ๐0(๐ซ
โฒ)
๐ ๐0 = ๐2๐ โ ๐ซ ๐0(๐ซ)
Can replace ๐ by ๐0 for any
smoothly varying interaction:
Express everything in terms of ๐0 ๐ซ, ๐ in order to complete the partition
function integral
Macroscale thermodynamics ๐(๐ฝ, ๐, ๐, ๐ผ) = ๐ท ๐0 ๐
โ๐ฝ๐บ[๐0]
๐ =1
๐ฝ๐2=
๐
๐2
๐ฝ =1
๐ ๐2โ โ
Key observation: Nontrivial balance between energy and
entropy requires the combination ๐ฝ = ๐ฝ๐2 to remain finite in
the continuum limit
Since ๐ฝ = ๐ฝ /๐2 โ โ, the partition function integral is
dominated by the maximum of ๐บ[๐0]
G ๐0 = ๐ธ ๐0 โ ๐ถ๐ ๐0 โ ๐[๐0] โ ๐๐ ๐0 โ ๐ ๐[๐0]
๐ฟ๐บ
๐ฟ๐0 ๐ซ, ๐= 0
Variational Equations
๐0 ๐, ๐ = ๐๐[ฮจ0(๐ซ)โโ(๐ซ)] ๐โ๐ฝ ๐[ฮจ0 ๐ซ โโ ๐ซ โ๐(๐)}
๐ ๐ = โln ๐๐
๐0๐๐ฝ [๐ ๐ โ๐๐]
๐ฟ๐บ
๐ฟ๐0 ๐ซ, ๐= 0 โ
ฮจ0 ๐ซ = ๐2๐ ๐บ ๐ซ, ๐ซโฒ ๐0(๐ซโฒ) Equilibrium stream function
From normalization condition
๐0 ๐ซ = โโ2ฮจ0 ๐ซ = ๐๐ ๐ ๐0(๐ซ, ๐) = ๐ ๐โฒ[ฮจ0(๐ซ) โ โ(๐ซ)]
Closed equation for the stream function
๐น[ฮจ0] = ๐2๐1
2โฮจ0(๐ซ)
2 โ ๐ ๐[ฮจ0(๐ซ) โ โ(๐ซ)]
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
๐ท๐
๐ท๐ก= 0
Potential vorticity (PV) ๐(๐ซ) = ๐(๐ซ) + ๐๐ 2๐ ๐ซ + ๐(๐ซ)
๐ 0 = 1/๐๐ = ๐๐พ/๐ Rossby radius of deformation
Kelvin wave speed ๐๐พ (speed of short wavelength inertia-gravity waves โ
quantifies gravitational restoring force for surface height fluctuations)
๐๐ก โโ2 + ๐๐
2 ๐ + ๐ฏ โ โ๐ + ๐ฝ๐๐ฅ๐ = 0
Coriolis parameter (Earth rotational force):
โBeta parameterโ
Can be written in the form
๐ is advectively conserved in the same way that ๐ is
for the Euler equation
๐ = 2ฮฉ๐ธ sin(๐๐ฟ)
๐ฝ = ๐๐ฆ๐
๐ = โ๐ฝ๐๐ฅ
๐2 + ๐๐ฅ2 Rossby wave dispersion relation (linearized dynamics)
QG Equilibria
๐ธ = ๐2๐ โ๐(๐ซ) 2 + ๐๐ 2๐(๐ซ)2 =
1
2 ๐2๐ ๐ ๐ซ โ ๐ ๐ซ ๐บ๐ ๐ซ, ๐ซโฒ [๐ ๐ซโฒ โ ๐ ๐ซโฒ ]
Energy function: Stream function follows surface height: ๐(๐ซ) โ ๐ฟโ(๐ซ)
(โโ2+๐๐ 2)๐บ๐ ๐ซ, ๐ซโฒ = ๐ฟ(๐ซ โ ๐ซโฒ)
๐บ๐ ๐ซ, ๐ซโฒ = โ1
2๐๐พ0( ๐ซ โ ๐ซโฒ /๐ 0)
โข Logarithmic singularity at the origin, but
exponential decay โผ ๐โ|๐ซโ๐ซโฒ|/๐ 0 at large
separation.
โข 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
๐ ๐0 = โ ๐2๐ ๐๐ ๐0 ๐ซ, ๐ ln [๐0๐0(๐ซ, ๐)] Here ๐ now denotes the values of ๐
๐น[ฮจ] = ๐2๐1
2๐ปฮจ 2 +
1
2๐๐ 2ฮจ2 + ๐ฮจ โ ๐ ๐ ฮจโ โ
Equilibrium equations are derived by minimizing the functional:
QG Equilibirum Vortex
Two level system example: โข Beautiful analogy with two
phase system, with phase
separation below a critical
temperature |๐ | < ๐๐
โข Vortex may be thought of as
a droplet of one phase
inside the other
โข Finite Rossby radius โ
Finite width interface
between phases, with PV
difference ฮ๐(๐ ) and
surface tension ฮฃ(๐ )
|๐ |/๐ ๐
ฮฃ(๐ )
ฮ๐(๐ )
โข Presence of Coriolis parameter ๐ ๐ฆ produces the equivalent of a gravitational field
โข Droplets are then unstable, and instead the denser phase coalesces below the
less dense phase, with a flat, narrow interface between โ โjetโ solution
โข Droplets in a more complex confining potential produce squeezed bubbles (Jupiter
โbargesโ)
Procedure for General Scalar Field Equilibria ๐๐ก๐ + ๐ฏ โ โ๐ = 0
Existence of a conserved energy functional (not necessarily quadratic)
โข Assumed sufficiently smooth in space that ๐ธ ๐ = ๐ธ ๐ โก ๐ธ[๐0]
Some vorticity-like field ๐(๐ซ, ๐ก) that is advectively conserved
๐ธ[๐]
๐ ๐ซ =๐ฟ๐ธ
๐ฟ๐ ๐ซ
Relation to stream function ๐, from
which velocity ๐ฏ = โ ร ๐ is derived
๐0 ๐ซ, ๐ = ๐๐[ฮจ0(๐ซ)โโ(๐ซ)] ๐โ๐ฝ ๐[ฮจ0 ๐ซ โโ ๐ซ โ๐(๐)}
Integration over small scale fluctuations
produces the identical entropy
contribution, expressed in terms of the
๐-level distribution function ๐0 ๐ซ, ๐
๐ ๐0 = โ ๐2๐ ๐๐ ๐0 ๐ซ, ๐ ln [๐0๐0(๐ซ, ๐)]
Exact variational condition for large scale structure produces the identical relation:
๐ ๐ = โln ๐๐
๐0๐๐ฝ [๐ ๐ โ๐๐]
Equilibrium equations are then derived by minimizing the free energy functional:
๐น ฮจ = ๐ฟ ฮจ โ ๐ ๐2๐๐[ฮจ โ โ]
๐ฟ ๐ = ๐2๐๐ ๐ซ ๐(๐ซ) โ ๐ธ[๐] Convert to function of ๐
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 ๐(๐ซ, ๐ฉ) ๐๐ก๐ + ๐ซ โ โ๐๐ + ๐ฉ โ โ๐๐ = 0
Newtonโs laws provide ๐ซ , ๐ฉ : ๐ซ = ๐ฉ/๐ ๐ฉ = ๐ (๐ซ)
D. Lynden-Bell & R. Wood, Mon. Not. R. Astron. Soc., 1968
โข For particles with long-ranged interactions, such as the Coulomb interaction, exact
integration of small-scale fluctuations is again permitted
โข Equilibrium equations are derived for the particle density:
๐น ๐ซ = โโ๐ ๐ซ ๐ ๐ซ = ๐๐๐ ๐๐๐๐ ๐ซ, ๐ซโฒ ๐(๐ซโฒ, ๐ฉ)
๐ธ = ๐๐๐ ๐๐๐๐ฉ 2
2๐๐(๐ซ, ๐ฉ) +
1
2 ๐๐๐ ๐๐๐ ๐๐๐โฒ ๐๐๐โฒ๐ ๐ซ, ๐ฉ ๐ ๐ซ, ๐ซโฒ ๐(๐ซโฒ, ๐ฉโฒ)
Energy functional:
These mean field equations for self gravitating systems, in the context of equilibration of star
clusters, were derived and studied in the 1960โs!
[But were found to produce unphysical solutions, likely due to absence of collisions]
๐ ๐ซ โก โโ2 ฮจ ๐ซ = ๐๐๐ ๐(๐ซ, ๐ฉ)
๐น ฮจ =1
2 ๐๐๐ โฮจ 2 โ ๐ ๐๐๐ ๐๐๐ ๐[ฮจ ๐ซ โ |๐ฉ|2/2๐] ๐ = ๐/๐2๐
Debye-Hรผckel
theory of
electrolytes
provides another
example!
Near-Equilibrium Systems:
Weakly Driven & Dissipated
21Dp
Dt
vv f
Near-equilibrium dynamics:
โข Can one derive a nonequilibrium statistical mechanics formalism for
steady states in the presence of small viscosity and weak driving?
โข Which equilibrium state is selected for given forcing pattern?
Possible tools from classic NESM:
โข Response functions, Kubo formulae, Kinetic equations,โฆ?
โข Required formal theoretical tools exist (Poisson bracket, invariant
phase space measure,โฆ)
Generalizations to Weakly Driven Systems
๐ฟ๐ด ๐ซ, ๐ก = ๐๐ซโฒ๐๐ด๐ต ๐ซ, ๐ซโฒ; ๐ก โ ๐กโฒ โ๐ต(๐ซโฒ, ๐ก) ๐๐ด๐ต ๐ซ, ๐ซโฒ, ๐ก โ ๐กโฒ =
๐
2โจ ๐ด ๐ซ, ๐ก , ๐ต ๐ซโฒ, ๐ก โฉ
Formalism possibly useful for treating evolution of ocean currents without
massive computational effort (predictability problem)
Thermodynamic response of density ๐ด to field โ๐ต conjugate to
density ๐ต, governed by dynamic response function ๐๐ด๐ต
See also recent kinetic equation approaches: โข Nardini, Gupta, Ruffo, Dauxois, Bouchet, J. Stat. Mech. 2012
โข Bouchet, Nardini, Tangarife, J. Stat. Phys. 2013
Weakly driven 2D Euler Equation
Simulations of stochastically driven transitions between near-equilibrium states
โข Close to an equilibrium phase transition between jet and vortex solutions
โข 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
โข Entropy comparison for locally
stable states with the same total
vorticity ๐ = 0.2, angular
momentum ๐, and energy ๐ธ(๐), โข Largest entropy state is the global
free energy minimum
โข Vortex separation decreases
with decreasing angular
momentum ๐
โข Two vortex solution disappears
below a critical separation
โข Generally consistent with
numerically observed
dynamical merger instability
๐ = 0.05 ๐ = 0.0373
Chen & Cross, PRE 1996
Steady State Failure
Quadrupolar pattern time series ๐(๐ก)
๐ก = 4, 40, 400, 4000 ๐ก = 0
High resolution numerical simulations: โข Spherical geometry blocks full equilibration, leaving an
oscillating pattern of four compact vortices, plus a
population of small-scale vortices
โข 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
โ Interaction between eddy and wave systems
2. Magnetohydrodynamic equilibria
โ Solar tachocline
โ Interaction between flow and electrodynamics
3. Ocean internal wave turbulence
โ 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, โStatistical
equilibrium solutions of the shallow water
equationsโ, 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 โ, g โ )
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
โemittedโ as waves?
Answer: YES! macroscopic vortices
survive.
Shallow Water Equations
๐1
๐2
Shallow Water Equilibria
๐น ฮจ, โ = ๐2๐โฮจ(๐ซ) 2
2โ(๐ซ)โ1
2๐โ ๐ซ 2 โ ๐ โ(๐ซ)๐[ฮจ(๐ซ)]
โโ โ 1
โ0 ๐ซ๐ปฮจ0 ๐ซ = ๐ โ0(๐ซ)๐
โฒ[ฮจ0(๐ซ)]
โฮจ0 ๐ซ 2
โ0 ๐ซ 2 = โ๐ W ฮจ0 ๐ซ โ ๐โ0(๐ซ)
๐ฏ0 =1
โ0โ ร ฮจ0
Additional hydrostatic balance requirement
โ โ โ0๐ฏ0 = 0
โข Existence of sensible equilibria requires the disappearance of compressive
(gravity wave) motions
โข E.g., forward cascade of wave energy to small scales, at which they are rapidly
dissipated, leaving only the large scale eddy dynamics
โข 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:
๐0 = โโ โ 1
โ0โฮจ0
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: ๐๐ก๐ฏ + ๐ฏ โ โ ๐ฏ + ๐ ร ๐ฏ = โโ๐ + ๐ฑ ร ๐ฉ
๐๐ก๐ = โ ร (๐ฏ ร ๐)
๐ = โ ร ๐
Lorentz force acting on electric current
passing through a fluid element
โข Fluid is approximated as perfectly
conducting
โข Electric fields are negligibly small
โ โ ๐ฏ = 0
โ โ ๐ = 0
Advection of magnetic field by velocity field
โข 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
โข E.g., solar tachocline โข Sharp boundary between rigidly rotating inner radiation
zone and differentially rotating outer convection zone
โข Large-scale organized structures here would have
strong implications for angular moment transport
between the two zones
โข ๐ฏ, ๐ are horizontal โ ๐ฝ, ๐ are normal to the plane,
and can be treated as scalars.
๐๐ก ๐ + ๐ + ๐ฏ โ โ ๐ + ๐ = ๐ โ โ๐ฝ
๐๐ก๐ด + ๐ฏ โ โA = 0
๐ฏ = โ ร ๐
๐ = โ ร ๐ด
๐ธ =1
2 ๐2๐[ ๐ฏ(๐ซ) 2 + ๐(๐ซ) 2]
Conserved kinetic + EM energy
Resulting pair of scalar equations
Stream function &
vector potential
โข Potential vorticity no longer
advectively conserved
โข Replaced by advective conservation
of vector potential! Second derivative no longer controlled
โข Microscopic fields much less regular!
โข Leads to very different equilibria, with much stronger โsubgridโ energetics
2D MHD Equilibrium Equations Two sets of conserved integrals:
๐ ๐ = ๐2๐๐ฟ[๐ โ ๐ด ๐ซ ] ๐ ๐ = ๐2๐[๐ ๐ซ + ๐ ๐ซ ]๐ฟ[๐ โ ๐ด ๐ซ ]
Controlled by Lagrange multipliers ๐ ๐ , ๐(๐)
Equilibrium free energy functional:
๐น ๐ด,ฮจ = ๐2๐[1
2๐ป๐ด(๐ซ) 2 +
1
2๐ปฮจ(๐ซ) 2 โ ๐โฒ ๐ด ๐ปA ๐ซ โ ๐ปฮจ ๐ซ + โโ(๐ซ) โ โฮจ(๐ซ)
โ๐(๐ด ๐ซ ) โ ๐(๐ซ)๐ ๐ด ๐ซ ] + ๐fluct[๐ด]
Microscopic fluctuation free energy
๐fluct[๐ด] is computed from a Gaussian
fluctuation Hamiltonian:
๐ปfluct ๐ด =1
2 ๐2๐ ๐ป๐ฟ๐ด(๐ซ) 2 + ๐ป๐ฟฮจ(๐ซ) 2 โ 2๐โฒ(๐ด ๐ซ )๐ป๐ฟA(๐ซ) โ ๐ป๐ฟฮจ(๐ซ)
โข Quantifies the effects of microscale magnetic and velocity fluctuations (no longer controlled by
the conservation laws)
โข Gaussian fluctuation entropy replaces Euler equation hard-core ideal gas entropy term ๐(๐) โข Generates fluctuation corrections to the ๐ด-membrane surface tension
โข Energy is no longer large scale: fluctuation contribution may dominate mean flow contribution
P. B. Weichman, โLong-Range Correlations and
Coherent Structures in Magnetohydrodynamic
Equilibriaโ, PRL 109, 235002 (2012)
Physics is that of two coupled elastic membranes!
โข Generates long-range correlations
โข External localizing potential provided by ๐, ๐
2D MHD Equilibria
โข Jet and vortex-type equilibrium solutions continue to exist
โข 2D Magnetic field lines follow contours of constant vector potential ๐ด0
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
โข Cholorphyll concentration field is freely advected by the fluid flow โ โpassive tracerโ
โข The flow leaves an imprint of the turbulence on the spatial pattern
โข Slow 1/๐ decay is characteristic prediction for the forward enstrophy cascade of 2D
eddy turbulence
Agulhas region
1/๐
1/๐3
Honshu region
๐ โ 60 km ๐ โ 6 km ๐ โ 600 km
1/๐
OCTS Chl-a
Gulf of Maine, 1997
Gulf Stream
Cape Cod
Nova Scotia
Chl-a and SST 1D spectra
1/๐
1/๐3
1/๐
1/๐3
Much steeper spectral fall-off (smoother spatial pattern) in some ocean regions
โข Sea surface temperature (SST) is another good passive scalar
โข The 1/๐3 power law is the predicted imprint of internal waves
OCTS data, Gulf of Maine
P. B. Weichman and R. E. Glazman, โSpatial Variations of a
Passive Tracer in a Random Wave Fieldโ, JFM 453, 263 (2002)
Internal Gravity Waves
๐ ๐ง = ๐[๐ ๐ง , ๐ ๐ง , ๐ ๐ง ]
Internal waves live where density gradient is largest,
above ~1 km depth โข ~10 m wave amplitude, 1-100 km wavelength at these depths
โข But only ~5 cm signature at sea surface due to air-water
density contrast
โข Tiny compared to surface gravity waves, but much slower,
hence visible via low frequency filtering (hours, days, weeks)
โข Internal wave speed ~2 m/s sets basic time scale
Brundt-Vรคisรคlรค
frequency defines
oscillation frequency of
vertically displaced
fluid parcels due to
pressure-,
temperature- and
salinity-induced
density gradient
๐(๐ง) = โ๐๐๐ง๐/๐
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
๐โ2.92
P. B. Weichman and R. E.
Glazman, โTurbulent
Fluctuation and Transport
of Passive Scalars by
Random Wave Fieldsโ, PRL
83, 5011 (1999)
Landsat Chlorophyll-a concentration spectrum
60o N near Iceland (Gower et al., 1980)
โSlowโ Eddy
contribution
โFastโ 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 1/๐3 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
๐ข0/๐0 โผ 10โ2 that allows one to perform a systematic expansion for the
passive scalar statistics
โข Fluid parcel speed ๐ข0 โผ10 m
10 minโผ 2 cm/s (for ~1 km wavelength)
โข Wave speed ๐0 โผ ๐โฮ๐
๐โผ (100 m/s) 10โ3 โผ 2 m/s
In addition to the โmean flowโ eddy velocity ๐ฏ(๐ซ), internal waves generate
(a spectrum of superimposed) smaller scale circulating patterns ๐ฎwave(๐ซ) โข These create a pattern of horizontal compression and rarefaction regions on
the surface that are visible in the passive scalar density
โข This horizontal motion effect is largest at the surface, even though vertical
motion is tiny due to large air-water contrast: ๐ฟโ๐ ๐ข๐๐ โผ 10โ2๐ฟโ๐กโ๐๐๐๐๐๐๐๐๐
๐1
๐2
Passive Scalar Dynamics
๐๐ก๐๐ฑ๐ ๐ก = ๐ฏ(๐๐ฑ๐ ๐ก , ๐ก)
(Nonlinear) Lagrangian trajectory for a fluid
parcel (with entrained passive scalar)
constrained to be at point ๐ฑ at time ๐
๐๐ฑ๐ ๐ก
๐ ๐ฑ, ๐ก = ๐๐ฑโฒ๐ ๐ฑโฒ, ๐ ๐ฟ(๐ฑ โ ๐๐ฑโฒ๐ ๐ก ) Formal solution to the passive scalar
equation (neglecting diffusion ๐ )
๐ ๐ฑ, ๐ก; ๐ฑโฒ, ๐ = โจ๐ฟ ๐ฑ โ ๐๐ฑโฒ๐ ๐ก โฉ Statistics computed from Markov-like transition probability
โข Unlike for eddy turbulence, where statistics of ๐ฏ are very complicated, and poorly
understood, very weakly interacting sinusoidal wave modes have near-Gaussian
statistics
โข In addition, the small parameter ๐ข0/๐0, which does not exist for eddy motions, enables a
systematic expansion for the Lagrangian trajectory
๐๐ก๐ + โ โ ๐ฏ๐ = ๐ โ2๐
๐๐ก๐ฟ๐ = โ๐ โ โ ๐ฏ
Linearized (small fluctuations around a smooth mean ๐ :
โ Concentration fluctuations are driven by fluid areal density fluctuations
Passive scalar transport ๐ by an externally imposed velocity field ๐ฏ:
Passive Scalar Spectra ๐ PS ๐ = 2๐
๐2๐น ๐ฟ(๐)
๐ ๐ 2
๐ ๐ = ๐0๐
๐น ๐ฟ(๐) โผ ๐โ4/3
๐โ3
โข Larger scale inverse cascade region
โข Smaller scale (typically below ~10 km)
direct cascade region
๐ ๐๐ ๐ โผ ๐โ4/3- ๐โ3 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 โrenormalizationโ of passive scalar
spectrum by wave height spectrum ๐น๐ฟ ๐
Wave dispersion relation; replaced e.g., by
โข ๐ = ๐๐ for surface gravity waves
โข ๐ = ๐02๐2 + ๐2 for longer wavelength waves (larger
than Rossby radius) that feel the Coriolis force (wave
periods comparable to Earth rotation period)
There is a remarkable โweak turbulenceโ 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.
โข Again, unlike for Eddy turbulence, exact predictions for the Kolmogorov spectral exponents
are then possible
โข Results depend on dispersion relation and exact form of nonlinear wave-wave interactions
For internal waves, the theory produces:
End of Part 3