Mini-course on GFD What is GFD?

Mini-course on GFD
What is GFD?
The fluid dynamics of potential vorticity.
v( x,t ) = velocity of the fluid at location x and time t
ω ( x,t ) ≡ ∇ × v
€
homentropic case : p = p( ρ ) instead of p = p( ρ,S )
€
€
∂ ωi  ω j  ∂
∂v  ω i 
vi − v j
 = 
 
∂t  ρ   ρ  ∂x j
∂x j  ρ 
€
ω 
∂ ω 
  = −Lv  
∂t  ρ 
ρ
introduce : θ ( x,t )
Dθ
=0
Dt
€
€
The flow affects ∇θ in the opposite way from ω/ρ
∂
∇θ = −Lv ∇θ
∂t
€
€
ω 
∂ ω 
  = −Lv  
∂t  ρ 
ρ
∂
∇θ = −Lv ∇θ
∂t
€
⇒
€

ω

∂ ω
 ⋅ ∇θ  = −Lv  ⋅ ∇θ 
∂t  ρ

ρ

⇔
€

D ω
 ⋅ ∇θ  = 0
Dt  ρ

(Ertel’s theorem, homentropic case)
€
Ertel (1942), Cauchy (1827)
 D
D ω
Q=0
 ⋅ ∇θ  ≡
Dt  ρ
 Dt
Dθ1
Dθ 2
Dθ 3
= 0,
= 0,
=0
Dt €
Dt
Dt
€
(Lagrangian coordinates)
DQ1
DQ2
DQ3
= 0,
= 0,
=0
Dt
Dt
Dt
 ∂ ∂ ∂ 
Q ≡ (Q1,Q2 ,Q3 ) =  ,
,
 × ( A1, A2 , A3 )
 ∂θ1 ∂θ 2 ∂θ 3 
€
where
€
provided
€
v = A1∇θ1 + A2∇θ 2 + A3∇θ 3
dθ1 dθ 2 dθ 3 = d ( mass)
Summary of the homentropic case
( x, y,z,t ) coordinates
v = u∇x + v∇y + w∇z
€
€
(θ1,θ 2 ,θ3 , τ ) coordinates
v = A1∇θ1 + A2∇θ 2 + A3∇θ 3
∂ ∂ ∂ €
(ω1,ω 2,ω 3 ) =  , ,  × ( u,v,w)
 ∂x ∂ y ∂z 
€
ω
  ∂ ∂ ∂ 
ω
ω
(Q1,Q2,Q3 ) =  ⋅ ∇θ1, ⋅ ∇θ2, ⋅ ∇θ3  =  , ,  × ( A1, A2, A3 )
ρ
ρ
ρ
  ∂θ1 ∂θ 2 ∂θ 3 
∂ ω  ω 
ω
  =  ⋅ ∇ v − (v ⋅ ∇)
€∂t  ρ   ρ
ρ

∂
Q=0
∂τ
However, the simplicity of the Lagrangian description is offset
by the complexity of the transformation between coordinate systems.
€
General, nonhomentropic, case: p = p( ρ,S )
The presence of entropy gradients destroys 2/3 of
the conservation law,
€
DQ /Dt = 0
but endows the surviving 1/3 with a physical importance
that does€not decrease as the Lagrangian coordinates become
ever more convoluted.
This is the essence of GFD.
when entropy gradients are present…. pressure torque appears
 ω  ∇ρ × ∇p( ρ,S )
∂ ω 
  = −Lv   +
∂t  ρ 
ρ3
ρ
DQi ∇θ i ⋅ (∇ρ × ∇p( ρ,S ))
=
Dt
ρ3
€
Choose one of the
θi
to be
S
itself…
D  ω ⋅ ∇S 

=0
Dt  ρ 
€
The other two conservation laws are destroyed.
€
€
€ the surviving
However,
conservation law is more useful than
before because gyroscopic forces and gravitational restoring
forces resist the folding of entropy surfaces.
Gyroscopic or gravitational restoring forces resist the
irreversible folding of constant-entropy surfaces.
Motivated approach: Hamilton’s principle for a perfect fluid
∫ L dτ =
∫ dτ
∫∫∫
 1 ∂x ∂ x

da db dc
⋅
− E (α,S ( a,b,c )) − Φ( x, y,z) = 0
1
424
32 ∂τ ∂τ

dθ 1 dθ 2 dθ 3
∂ ( x, y,z)
α=
∂ ( a,b,c )
δx ( a,b,c, τ ), δy ( a,b,c, τ ), δz( a,b,c, τ ) ⇒
fluid equations
Eulerian version: δa( x, y,z,t ), δb( x, y,z,t ), δc ( x, y,z,t )
€
€
homentropic case:
δ
∂ ( a,b,c )
=0
∂ ( x, y,z)
€
⇒
∂ ∂ ∂
δ ( a,b,c ) =  , ,  × δT( a,b,c, τ )
 ∂a ∂ b ∂c 
⇒
δ ∫ L dτ =
∫ dτ ∫∫∫ da
∂
[∇ a × A ] ⋅ δ T = 0
∂τ
if entropy gradients are present:
∂ ( a,b,S )
δ
€ ∂ x, y,z = 0 and δS = 0
(
)
⇒
⇒
€
δa = −
∂
δψ,
∂b
δb =
∂
δψ
∂a
∂
[(∇ a × A ) ⋅ ∇ a S] = 0
∂τ
€
The particle-relabeling symmetry is the essence
of what it means to be a fluid.
∫ L dτ = ∫ dτ ∫∫∫
 1 ∂x ∂ x

 ∂ ( x, y,z)

da db dc 
⋅
− E
,S ( a,b,c ) − Φ( x, y,z)
2 ∂τ ∂τ

 ∂ ( a,b,c )

Potential vorticity is unique to fluid mechanics.
Approximations that do not respect potential vorticity conservation
must be viewed with suspicion.
Potential vorticity was first discovered (and has proved most useful)
in the study of rotating and/or stratified fluids (GFD) but it is
likely to be significant in the study of turbulence.
Within GFD: the “2+1 view” prevails
M. E. McIntyre & W. A. Norton (1988-2000):
“Potential vorticity inversion on a hemisphere”
Dynamical evolution using shallow-water dynamics:
potential vorticity
Reconstruction from the PV
at a fixed time:
“2+1 view of GFD”
Also, more recent work by Ali Mohebalhojeh & D. Dritschel
fluid depth
divergence
Shallow-water equations
Du
∂h
− fv = −g
Dt
∂x
Dv
∂h
+ fu = −g
Dt
∂y
∂h ∂
∂
+ ( hu) + ( hv ) = 0
∂t ∂x
∂y
D ∂
∂
∂
≡ +u +v
Dt ∂t
∂x
∂y
€
Dq
⇒
=0
Dt
u( x, y,t ), v ( x, y,t ), h ( x, y,t )
∂v ∂u
−
∂x ∂y
∂u ∂v
δ= +
∂x ∂y
ζ+ f
q=
h
ζ=
SWE are the prototype for the 3d primitive equations
= general fluid eqns + hydrostatic approx + Boussinesq approx
€
€
+traditional approx
D ζ + f 
in (x, y,S,t) coordinates :


Dt  ∂z /∂S 
“2+1 view” of the SWE: linearized dynamics
linear dynamics with f = f 0
waves ∝ exp(ikx + ily − iωt )
∂u
∂h
2
2
2
2
− f 0v = −g
ω
=
0
or
ω
=
f
+
gH
k
+
l
(
)
0
0
∂t
∂x
€
∂v
∂h
€
+ f 0 u = −g
Normal-mode variables:
∂t
∂y
€
 ∂u ∂v 
∂h
f
∂u ∂ v
g
+ H 0 +  = 0
q = ζ − 0 h, δ = + , ζ AG = ζ − ∇ 2 h
∂t
H0
∂x ∂ y
f0
 ∂x ∂y 
⇔
€
€
∂q
=0
∂t
€
∂δ
− f 0 ζ AG = 0
∂t
∂ζ AG
gH 0 2
+ f0 δ −
∇ δ=0
∂t
f0
Potential vorticity inversion at a fixed time :
take δ = ζ AG = 0
g 2
f
solve
∇ h − 0 h = q for h
f0
H0
g ∂h
g ∂h
then u = −
, v=
f 0 ∂y
f 0 ∂x
€
Nonlinear SWE with variable f and H
∂q
= {q,q} + {q, f ′} + {q, H ′} + {q,δ} + {q,ζ AG }
∂t
∂δ
− f 0 ζ AG = { , }
∂t
∂ζ AG
gH 0 2
+ f0 δ −
∇ δ ={ ,}
∂t
f0
Set δ = ζ AG = 0 in the first eqn, and throw away the others.
€
This is a Galerkin approximation.
€
the result:
It is also a (metric) projection.

∂g 2
f 0   g ∂h g ∂h   g 2
f0
f0
h  = − −
,
h+ f +
(H 0 − H )
 ∇ h−
 ⋅ ∇ ∇ h −
∂t  f 0
H 0   f 0 ∂y f 0 ∂x   f 0
H0
H0

∂q ∂ (ψ,q)
⇔
+
= 0,
∂t ∂€( x, y )
2
f
f
q = ∇ 2ψ − 0 ψ + f + 0 ( H 0 − H )
gH 0
H0
where ψ ≡
gh
f0
Single-layer QG dynamics
∂q ∂ (ψ,q)
+
= 0,
∂t ∂ ( x, y )
2
f0
f0
q=∇ ψ−
ψ+ f +
(H0 − H )
gH 0
H0
2
QG is the simplest useful balance model.
€

f0
∂ψ
∂ψ 
 h = ψ, u = − , v = + 
g
∂y
∂x 

Useful extensions: multi-layer flows, continuously stratified flow.
€ Early use in numerical weather prediction.
Biggest defect of QG: it is closely tied to a reference state.
Leading-order alternatives to QG avoid this at a cost in complexity.
Higher-order balance models.
QG is a metric projection onto the slow manifold
but phase space has no metric!
Hamiltonian fluid dynamics offers another kind of projection:
1. Hamilton’s principle + constraints
2. Restriction of a differential form
3. Dirac bracket
⇒
Semigeostrophic equations (SG)
SG, and an abstract view of QG
The state of the fluid corresponds to a point in phase space:
z = ( z1,z 2 ,z 3 K)
Exact dynamics in Hamiltonian form:
€
QG dynamics:
SG dynamics:
d
∂F ij ∂H
F ( z) = i J
dt
∂z
∂z i
d
∂F ij ∂H
∂F ij ∂µ( m)
F ( z) = i J
+ µ( m) i g
i
dt
∂z
∂z
∂z
∂z i
€
d
∂F ij ∂H
∂F ij ∂µ( m)
F ( z) = i J
+ µ( m) i J
i
dt
∂z
∂z
∂z
∂z i
€
SG corresponds to attaching constraints to Hamilton’s principle.
Unlike QG, SG is not tied to a particular reference state,
€
but it is much harder to solve than QG.
Single-layer QG
∂q ∂ (ψ,q)
+
= 0,
∂t ∂ ( x, y )
∂ 2ψ ∂ 2ψ f 0 2
f
q= 2 + 2 −
ψ + f + 0 (H0 − H )
∂x
∂y
gH 0
H0
3d QG
∂q ∂ (ψ,q)
+
= 0,
∂t ∂ ( x, y )
€
∂ 2ψ ∂ 2ψ ∂  f 0 2 ∂ψ 
 + f
q = 2 + 2 + 
2
∂x
∂y
∂ z  N ( z ) ∂z 
top/bottom boundary condition
€
€
 ∂ ∂ (ψ,•)   ∂ψ  N ( z) 2
w=0
 +
  +


f0
∂t ∂ ( x, y )  ∂z
Simplest case:
1. constant f
2. flat top & bottom boundaries
3. uniform N(z)
Simplest-case QG
Single layer:
 ∂ ∂ (ψ,•) 
 +

∂t ∂ ( x, y ) 
 ∂ ∂ (ψ,•) 
 +

∂t ∂ ( x, y ) 
3d:
€
 ∂ 2ψ ∂ 2ψ 
 2 + 2=0
∂y 
 ∂x
⇔ 2d Euler dynamics
 ∂ 2ψ ∂ 2ψ ∂ 2ψ 
 2 + 2 + 2 =0
∂y
∂z 
 ∂x
VERSUS:
€
3d Euler dynamics:
€
∂
∂
∂
ωi + v j
ωi − ω j
v i = 0,
∂t
∂x j
∂x j
∂v k
ω i = εijk
∂x j
Euler dynamics + viscosity = Navier-Stokes
dynamics
∂v
+ v ⋅ ∇v = −∇p + ν∇ 2 v,
∂t
2d Navier-Stokes
⇔
∇⋅v=0
Single-layer QG with no accessories
€
Essence of GFD is the huge difference between
2d and 3d Navier-Stokes
€
Conservation laws are the key.
In the limit ν → 0, 3d Euler conserves E =
€
and
1
v⋅ v
2
1
1
v ⋅ v = ∫∫ dx ∇ψ ⋅ ∇ψ
2
2
1
1 2 2
Z = ∫∫ dx ω ⋅ ω = ∫∫ dx (∇ ψ )
2
2
2d Euler conserves E =
€
∫∫∫ dx
∫∫ dx
Navier-Stokes dynamics
ε≡
dE
= −ν
dt
In 3d, Z is unbounded.
(ε ~ U
In 2d and in 3d,
€
∫∫∫ dx ω ⋅ ω ≡ −2ν Z
3
/L is independent of ν .)
Vortex stretching transfers energy to arbitrarily small scales
in a finite time.
€
In 2d, Z can only decrease. The impossibility of irreversible
vortex stretching traps the energy in large scales.
2d inviscid Euler
∂ζ
+ v ⋅ ∇ζ = 0
∂t
⇒
€
€
€
€
∂ (ψ,∇ 2ψ )
∂ 2
∇ψ+
=0
∂t
∂ ( x, y )
⇔
dE dZ
=
=0
dt
dt
1
E=
2
1
∫∫ dx dy v ⋅ v = 2
1
Z=
2
1
∫∫ dx dy ζ = 2
2
∞
∫∫ dx dy
∇ψ ⋅ ∇ψ =
∫ dk E (k )
0
∫∫ dx dy (∇ ψ )
2
2
∞
=
∫ dk
0
k 2 E (k )
3d inviscid QG
∂q ∂ (ψ,q)
+
= 0,
∂t ∂ ( x, y )
∂ 2ψ ∂ 2ψ f 0 2 ∂ 2ψ
q= 2 + 2 + 2 2
∂x
∂y
N 0 ∂z
dE dZ
=
=0
dt
dt
⇒
€
2
2
2



 ∂ψ 
1
∂ψ
f 0 2  ∂ψ  
E = ∫∫∫ dx dy dz   +   + 2    =
2
 ∂x  € ∂y  N 0  ∂z  
2
∂ 2ψ ∂ 2ψ f 2 ∂ 2ψ 
1
Z = ∫∫∫ dx dy dz  2 + 2 + 0 2 2  =
2
∂y
N 0 ∂z 
 ∂x
Energy transfer is to low
€
ktotal
∞
∞
∫ dk ∫ dk
H
0
E ( k H ,kV )
V
0
∞
∞
∫ dk H
∫ dkV ktotal E (k H ,kV )
0
0
2
f
2
2
2
≡ kx + ky + 0 2 kz
N0
2
3d inviscid QG: vertical modes
∂q ∂ (ψ,q)
+
= 0,
∂t ∂ ( x, y )
∂ 2ψ ∂ 2ψ f 0 2 ∂ 2ψ
q= 2 + 2 + 2 2
∂x
∂y
N 0 ∂z
€
Energy transfer is to low
kn =
2
2
ktotal ≡ k x + k y + k n
2
f 0 nπ
1
=
N 0 H 0 n - th deformation radius
€
Flow becomes depth-invariant at horizontal scales > 1/k1
€
Ocean currents observed south of the separated Gulf Stream
(Peter Rhines & Bill Schmitz)
3d inviscid QG: WKB form
∂q ∂ (ψ,q)
+
= 0,
∂t ∂ ( x, y )
2
∂ 2ψ ∂ 2ψ (βy ) ∂ 2ψ
q= 2 + 2 +
∂x
∂y
N 0 2 ∂z 2
Energy transfer is to low
€
kn ( y ) =
2
2
ktotal ≡ k x + k y + k n ( y )
2
βy nπ
1
=
N 0 H 0 n - th deformation radius
€
Energy transfer is toward the equator and into high vertical mode.
€
The energy in mode n shows an equatorial peak of width W,
determined by k y = k n i.e.
1
βW n nπ
=
Wn
N0 H0
€
⇒ Wn =
N 0H0
≡ equatorial deformation radius
β nπ
“Global characteristics of ocean variability estimated
from regional TOPEX/POSEIDON altimeter measurements”
D. Stammer, J. Phys. Oc. 1997
Eddy kinetic energy at the sea surface (cm/sec)**2
U. Send, C. Eden & F. Schott
“Atlantic equatorial deep jets…” J. Phys. Oc. 2002
Zonal flow (cm/sec) along the equator from six cruises.
€
Rectified flow
∂q ∂ (ψ,q)
+
= 0,
∂t ∂ ( x, y )
∂ 2ψ ∂ 2ψ
f
q = 2 + 2 + f + 0 (H0 − H )
∂x424
∂y3
H0
1
144
2443
h
ζ
Conserved quantities are
E€=
∫∫ dx
1
∇ψ ⋅ ∇ψ and
2
Z=
∫∫ dx
1 2
q =
2
∫∫ dx
1 2
ζ +
2
∫∫ dx ζ h + const
The system seeks the state of energy equipartition. If h = 0,
the conservation of ∫∫ dx ζ 2 prevents this.
However, if h ≠ 0 then
∫∫ €dx ζ h
2
dx
ζ
can increase if
∫∫
becomes negative.
Anticyclonic flow over seamounts
€
Numerical example: Seamount in a square ocean
Seamount 1 km high in an ocean 4 km deep and 4000 km wide
f = 2π /day
Navier-Stokes friction
Random initial conditions with rms velocity = 50 km/day
€
ocean depth
initial potential vorticity
Potential vorticity at subsequent times
6 days
19 days
31 days
Final velocity
Ocean basin with a protruding ridge
Ocean depth
Potential vorticity at 20 days
Mass transport after 60 days
No ridge
Ridge
How this might work…
Think of the deep ocean as a single layer with an external forcing.
Mean streamfunction at 1500-1750 m in NE Atlantic
based on 223 floats. Bower et al., Nature 2002
Westward intensification
At the largest spatial scales the lines of constant f/H do not close.
f/H lines on the North Atlantic
In the limit of a flat bottom, they are lines of constant latitude.
Single-layer, flat bottom QG in a basin
∂ζ
+ v ⋅ ∇(ζ + f ) = 0
∂t
For the flat-bottom case,
implies conservation of
€2
∫∫ dx (ζ + f ) =
Increasing
∫∫ dx ζ
2
∫∫ dx ζ
2
+
∫∫ dx ζ βy + L
implies decreasing ∫∫ dx ζ βy (westward mean flow)
€
€
Moreover:
d
dt
∫∫ dx ζ βy = − ∫∫ dx ∇ ⋅ (vζ ) βy = + ∫∫ dx vζ ⋅ ∇(βy )
€
 ∂v ∂ u 
= ∫∫ dx vζ β == ∫∫ dx v −  β = β
 ∂x ∂ y 
Westward intensification of eddy activity
€
[∫v
2
dy
]
east
west
An alternative approach to all this:
The Statistical Mechanics Viewpoint
Answers the question: What happens if you stir the ocean up
and then wait an infinitely long time?
Assumes:
1. Inviscid ocean
2. Truncated to finite degrees of freedom
For the present problem SM predicts:
1. Large-scale time-average flow
that obeys:
1

∇ ⋅  ∇ψ  + f
H

= aψ + b
H
2. Time fluctuations at the smallest scales
€
Best general reference: Carnevale & Frederiksen, JFM 1987
Recent application to this very problem:
Merryfield, Cummins & Holloway, JPO 2001
Maximum entropy flow states
1

∇ ⋅  ∇ψ  + f
H

= aψ + b
H
f/H lines
with shelf/slope
€
ψ
f = βy,
ψ
H = 4 km
€
‘Fofonoff flow’
€
€
Flat-bottom, beta-plane case:
Approach to ‘Fofonoff flow’
Wind-driven flow
East
wind
West
wind
Statistical-mechanical target state