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S. Lovejoy McGill, Montreal
The weather and climate as problems
in physics: scale invariance and
multifractals
McGillApril 13, 2012
The Weather and ClimateEmergent Laws and Multifractal Cascades
SHAUN LOVEJOY and DANIEL SCHERTZER
(in press, Oct. 2012)
Required reading for this course
The Weather
3
Meteorologists
The Emergence of physical laws
Low level (fundamental)
high level(simpler if applicable)
Quantum mechanics Classical MechanicsLarge scales(usually)
Statistical mechanicsContinuum mechanics, Fluid mechanics thermodynamics
Large numbers of particles
General Relativity Special RelativityLow energy mass density
Special Relativity
Velocities << speed of light
Classical (Galilean) Relativity
Example: The emergence of Thermodynamics from
Newton’s laws
F=ma
F=ma
F=ma
First law: conservation of energySecond law: increase in entropy
Newton’s laws: Thermodynamics:
ex.: Boyle’s law: (pressure) x (volume) = constant
Low level, (difficult to handle for many
particles)
High level (Valid when many particles
are present)
Large number ofparticles
Pioneers of turbulence
7
Richardson1881 - 1953
Kolmogorov1903 – 1987
Corrsin1920 – 1986
8
Obukhov1918 – 1989
Ralph Bolgiano, Jr. 1922 — 2002
The emergence of turbulence dynamics (Classical)
Laws of turbulenceClassical:
Richardson, Kolmogorov, Corrsin, Obukhov, Bolgiano
Fluid mechanics
Strong stirring (nonlinearity)
Low level (fundamental)
High level
Vortices in strongly turbulent fluid(M. Wiczek, numerical simulation, 2010)“Spaghetti”
Emergent laws reduce seeming complexity to
simplicity at another level
Mandelbrot1924-2010
Complex?
The Mandelbrot set
(“self-similar”, scale invariant, fractal)
Blowing up gives the same type of shapes
Or simple?
Generating the Mandelbrot set
-Take a number.
-Multiply it by itself.
-Add a constant.
-Repeat.
(I forgot to mention: take a COMPLEX number)
Complex?
20 steps: 1:1 20 steps: 5:1
20,000 steps: 1:1 20,000 steps: 5:1
200 steps: 1:1 200 steps: 5:1
Drunkard’s walk
Δx
Or simple?
(distance) x (distance) = number of bars visited
(Brownian motion)
From initial barAverage number of bars visited (or displacements made)
Complex?… or simple?
20481536102451201100
1200
1300
1400
1500
1600
1700
t
v(t)
1 second of wind data (roof of Rutherford
building, McGill)
Infra Red satellite effective temperatures, January 16, 2008
The Atmosphere
mm/s
Brute force…
Atmosphere: Laws of Fluid mechanics(low level)
windEarth angular velocity
pressure
Gravitational potential
Specific heat Heating ratetemperature
Specific volume=1/ρ
Gas constant
density
Governing atmospheric laws
Friction
Brute force numerical solution of the equations (2)…
Discretization of the equations
Brute force numerical solution of the equations (3)…
Earth system modelling
Or simplicity?
Atmosphere: Emergent laws(high level)
Fluctuations ≈ (turbulent flux) x (scale)H
Size:Anisotropic Space-time Scale function
Fluctuation/conservation exponent
Cascading Turbulent flux
Differences, tendencies, wavelet coefficients
Fluctuation = change in time and/or spaceScale = sizeTurbulent flux = strength of stirring
Power law
These laws are scale invariant
Which Richardson?The father of
Numerical Weather
Prediction…
The father of numerical weather prediction
1922
The weather prediction factory
(artist: Francois Schuiten)
Richardson’s numerical
grid for integrating
Each column was divided into 5 vertical cells and defined 7 quantities: pressure, temperature, density, water content, 3 velocity components
“It took me the best part of six weeks to draw up the computing forms and to work out the new distribution in two vertical columns for the first time. My office was a heap of hay in cold rest billet. With practice the work of an average computer might go perhaps ten times faster. If the time-step were 3 hours, then 32 individuals could just compute two-points so as to keep up with the weather.”
-Richardson 1922
... or the grandfather of cascades?
Weather prediction by Numerical Process 1922, p.66
Scale by scale simplicity: cascades
CASCADE LEVELS
0 --
1 --
2 -- . . .
n --
xy
ε
0l
l0 / λ1
2
n
l0 / λ
l0 / λ
multiplication by 4independent random(multiplicative)increments
multiplication by 16independent random(multiplicative)increments
“Does the wind have a velocity?”
0.2 0.4 0.6 0.8 1
-2
-1
1
2
3
4
W(t)
t
“Although at first sight strange, the question grows upon acquaintance…” - Richardson 1926
Richardson suggested that the trajectory of a particle could be like a Wierstrass function (1872)
20481536102451201100
1200
1300
1400
1500
1600
1700
t
v(t)
1 second of wind data (roof of Rutherford
building, McGill)
mm/s
Scale invarianceand fractals
0.2 0.4 0.6 0.8 1
-2-1
1234
0.2 0.4 0.6 0.8
-2
-1
1
2
3
4 λΗ
1
λ
1
Wierstrass function showing scale invariance under anisotropic “blowup” (H=1/3 in this example, λ=3)
blowup
λ=3
Cantor set
• Let us start with:
and let us iterate:
x3A small part is same as the whole if “blown up” by a factor 3 (“scale invariance”, “self-similarity”)
Sierpinski Triangle
1 2
Koch snowflakeLet us start with:
and let us iterate:
Sierpinski Pyramid
• First iteration:
10 th iteration:
Menger Sponge
• motif:
iterations:
Hom
ogen
eous Interm
ittent
Parent eddy
Daughter eddies
Grand-daughter eddies
CASCADES
β-modelFractal set
“active”
“calm”
Cascades and Multifractals
Aircraft temperature transect (12km altitude)
500 1000 1500 2000 x (km)
- 53
- 52
- 51
- 50
- 49Temperature (oC)
Turbulent flux (gradient of the above)
1 σ
500 1000 1500 2000 x5
101520 ε
500 1000 1500 2000 x48
12 ε
500 1000 1500 2000 x2468 ε
500 1000 1500 2000 x12345 ε
500 1000 1500 2000
500 1000 1500 2000 x0.51.5
3.0 ε
500 1000 1500 2000 x0.51.01.5 ε
x0.40.81.2 ε
Temperature turbulent flux ε
at 280m resolution
High to low Resolution:degrading by factors of 4
km
ε0 ε1
Cascades and Multifractals
(“α model”)
Simulations: adding small scale details (low resolution to high)
Cascades
DRESSEDDRESSEDENERGYENERGY
FLUX FLUX DENSITYDENSITYFIELDSFIELDS
BAREBAREENERGYENERGY
FLUX FLUX DENSITYDENSITYFIELDSFIELDS
Dressed densityaveraged over
16x16
y εx y Π
x
1.1.
1.1.
1.1.
1.1.
1.
Cascade • level 1
bare density
1.
Cascade • level 3
bare density
Cascade • level 4
bare density
Cascade • level 5
bare density
Cascade • level 6
bare density
Cascade • level 7
bare density
Dressed densityaveraged over
8x8
Dressed densityaveraged over
4x4
Dressed densityaveraged over
2x2
Multifractal
0.5 1.0 1.5 2.0
Log10λ0.1
0.2
0.3
0.4 Log10
Mq
0.5 1.0 1.5 2.0
0.1
0.2
0.3
0.4
0.5 1.0 1.5 2.0
0.1
0.2
0.3
0.4
0.5 1.0 1.5 2.0
0.1
0.2
0.3
0.4
EW wind NS wind
Temphs humidity
Log10
Mq
Log10λ
Log10
Mq Log10
Mq
Mq ≈ λK q( )
q=2
q=1.5
q=2
q=1.5
q=2
q=1.5
q=2
q=1.5
20000km20000km
20000km 20000km
200km
200km
200km
200km
Cascades work!!
Cloud liquid water (top)
Cloud liquid water (side)
Cloud top, infra red
Cloud top visible
Cloud bottom visible
Cascade modeling: clouds and radiative transfer
Cascade Simulations
The Climate
48
The produc+on of maple syrup is affected by global warming...
Instead... have you tried my delicious 100% Canadian syrup on your pancakes?
Tar sands syrup
What is the climate?
“Climate is conventionally defined as the long-term statistics of the weather…”. -Committee on Radiative Forcing Effects on Climate, 2005 US National Academy of Science
"Climate is what you expect, weather is what you get.”
-Farmers Almanac
3 Three regimes:
three types of variability: not two!
51
100 200 300 400 500 600 700
- 2
2
4
6
8
10
1 Century,Vostok, 20-92kyr BP
20 days,75oN,100 oW, 1967 - 2008
1 hour,Lander, 10 Feb.-12 March, 2005
0
�T/
t
�
Climate(10-30 yrs to 50,000 yrs)
Weather(up to 10 days)
Low frequencyweather(10 days to 10-30 yrs)
TemperatureFluctuations Growing
Fluctuations Growing
Fluctuations Decreasing
100 200 300 400
-8
-6
-4
-2
2ΔT (K)
Age (kyr)
Vostok core (Antarctica ≈75oS) ≈ 130 yr resolution (average)
Holocene
≈40yr resolution
≈600yr resolutionLast four Glacials
Today
53
20 40 60 80
-
-
-
-
-
-
44
42
40
38
36
34
Age (kyr BP)
Holo
cene
Last glacial
“Dansgaard events”
� 8K
�18O
Paleotemperatures: GRIP (GReenland Ice core Project), summit location (≈75oN), High (5.2 yr ) resolution section
NOW
Moburg
Ljundquist
Huang
Esper
Crowley
1600 1700 1800 1900
-
-
-
-
0.8
0.6
0.4
0.2
0.2
BriffaMann
Jones
T(K)Date
1980
Eight “Reconstructions” of the Northern hemisphere
temperatures(30 year average applied)
Multiproxy Temperatures
post 2003
1900 1920 1940 1960 1980 2000
-
-
0.4
0.2
0.2
0.4
0.6T (K)
Date
Global temperature anomaly mean and “spread” from
three different instrumental estimates
The climate?
Do Global Climate models predict...
56
...or low frequency weather?
1 year 100 years10 years
- 0.5 0.0 0.5 1.0 1.5 2.0 2.5
-
-
-
-
0.9
0.7
0.5
0.3
Atmosphere (3km)
Mean surface
IPSL
Log10
Δt (yrs)
0.2
0.5
1
0.1
- 0.1
0.1
0.34
GCM
Data
averagetemperaturechange
OC
OC
OC
OC
OC
2
-0.5
0.5
Global Surface
>2003 multiproxies, 1500-1980
Vostok(Antarctic)
105yrs
10-2
Log10Δt (yrs)
±2K
±3K5oC
0.1oC
20CR global scale
0.4
weather Low frequency weather climate
Low frequency climate
1oC
GlobalTemperature
averagetemperaturechange
1 year10000 years100 years10 days
GCM’s
Implications for global warming
• By comparing model and natural variability, we found that GCM’s seem to be missing a long-time mechanism of internal variability such as land-ice.
• Anthropogenic contributions to 20th warming and 21st C warming scenarios may thus be either over - or under estimated.
59
2. Emergent Atmospheric laws are power lawsFluctuations are scaling, their exponents are scale invariant
Conclusions1. Low level laws: complex (Fluid mechanics)High level laws simplicity (emergent turbulent laws)
3. There are three different regimes:Weather to ≈ 10 days, Low frequency weather to ≈ 10-30 yrs,Climate to ≈ 50- 100kyrs.
4. Without special forcing GCM’s produce low frequency weathernot climate type variability