Some fundamentals of
numerical weather prediction
(for ATM 562)
Robert Fovell
Atmospheric and Environmental Sciences
University at Albany, SUNY
rfovell@albany.edu
Equations
• Navier-Stokes equations
– Newton’s 2nd law: real & apparent forces
– Turbulence and mixing
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1st law of thermodynamics
Ideal gas law
Continuity equation
Clausius-Clapeyron equation
Domain
• Local, regional or global
in scale
Domain
• Local, regional or global
in scale
• Discretize into grid
volumes
– virtual internal walls
– boundary conditions
• Initialize each volume
• Make some forecasts…
Extrapolation
• Equations predict tendencies
– Initial forecasts start with observations but
subsequent forecasts based on forecasts
– Forecasts used to recalculate tendencies
– Success depends on quality of initialization and
accuracy of tendencies
A simplistic example:
temperature in your backyard
The model forecast will consider an enormous number
of factors to estimate present tendency to project future
value
In the absence of such information, you simply guess…
and wait to see how good your guess was.
The model does not wait. It uses each forecast to
recalculate the tendencies to make the next forecast
Verification of your forecast:
OK since you didn’t project out too far
Forecast time steps cannot be too long.
Tendencies have the tendency to CHANGE.
Model forecasts will reflect…
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Radiative processes
Surface and sub-surface processes
Cloud development and microphysics
Advection and mixing
Convergence and divergence
Ascent and subsidence
Sea-breezes, cold and warm fronts
Cyclones, anticyclones, troughs, ridges
Model resolution
• Things to try to resolve
– Clouds, mountains, lakes & rivers,
hurricane eyes and rainbands, fronts &
drylines, tornadoes, much more
• What isn’t resolved is subgrid
• At least 2 grid boxes across a feature for
model to even “see” it… and at least 6 to
render it properly
Wave-like features are ubiquitous,
in space and time
Models sample a wave only at grid points.
Example: 4 points across each wave.
Models “connect the dots”.
Suppose we have only 3 points across the wave…
or just 2 points…
These do not look much like the actual wave at all.
With only 2 points, wave may be invisible.
High resolution Hurricane Katrina satellite picture
= composed of 1 km pixels.
High resolution model = possibly accurate, definitely expensive,
and extremely time-consuming
As model resolution increases, we see more, but have to DO more.
Plus, the time step has to decrease, to maintain linear stability
Compromising on the resolution – now it’s 10 km
30 km resolution
At what point would we not be able to tell that’s a
hurricane if we had not known it from the start?
This is the world as seen by global weather models not
so long ago, and many climate models today.
“My favorite story is about the model that perfectly predicts
tomorrow’s weather, but it takes two days to run the program.”
Outlook is good…
• Models are improving
– Faster computers
– More and better input data (satellites)
– Better ways of using those data
– Progressively better numerical techniques
– High resolution models… so less is missed
(subgrid)
– … but some things we can never capture
Parameterization
• Parameterization = an attempt to represent what we
cannot see, based largely on what we can
• Parameterizations in a typical NWP model include and are NOT limited to…
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Boundary layer processes and subgrid mixing
Cloud microphysics [resolve cloud, can’t resolve drops]
Convective parameterizations [can’t resolve clouds]
Surface processes [heat, moisture fluxes]
Subsurface processes [soil model, ocean layers]
Radiative transfer, including how radiation interacts with
clouds
• Here is an example…
Another view of Katrina... But focusing on roll clouds.
Roll clouds are revealing of boundary layer mixing.
How roll clouds form
• Consider the sun
warming the land
during the day
How roll clouds form
• With uneven
heating, wind and
vertical wind shear,
roll-like circulations
can start
• You can simulate
roll formation in a
high resolution
model (∆x ≤ 500 m)
How roll clouds form
• As the land warms
up, the rolls get
deeper (also wider)
• The rolls mix heat
vertically
– air is a lousy conductor
• Also mixing
momentum,
moisture
How roll clouds form
• If conditions are
favorable, clouds
will form above the
roll updrafts
• These make the
rolls visible
• We see these roll
clouds on satellite
pictures
• In a sense, being able
to simulate the roll
clouds means we’ve
done many things well - radiation, winds,
mixing, surface/soil and
saturation processes
Parameterizing mixing…
• In many models, however, the roll clouds and the
mixing that created them are subgrid (and if they DO
appear, they are almost certainly mishandled)
• But that mixing is important. It influences structure
and stability of the atmosphere, including CAPE, CIN
and convective initiation
• If we cannot resolve it, we must parameterize it
• NWP models are full of parameterizations, each a
potential model shortcoming, each a possible source
of problems, of error, of uncertainty regarding the
future
Birth of NWP
Prof. Cleveland Abbe
“There is a physical basis for
all meteorological phenomena.
There are laws of mechanics
and heat that apply to the
atmosphere, and as fast as we
acquire the ability to discover
and reason out their
consequences, we shall
perceive that LAW and
ORDER prevail in all the
complex phenomena of the
weather and the climate.”
(1901)
First head of US Weather Bureau
Prof. Vilhelm Bjerknes
• 1904 vision on weather
prediction
• Lamented the
“unscientific” basis of
meteorology
• Goal: to make
meteorology a more
exact science
• … by making
predictions
Bjerknes’ vision
• Two key ingredients:
– Sufficiently accurate knowledge of the state of the
atmosphere at the initial time
– Sufficiently accurate knowledge of the physical laws that
govern how the atmospheric state evolves
• Identified 7 fundamental variables -- T, p, density,
humidity, and three wind components -- and the
equations that calculated their tendencies
• These are nasty equations, without simple solutions.
Only numerical methods could be brought to bear on
them.
Max Margules
• Austrian meteorologist
• Tried to predict surface
pressure using continuity
equation… and found it
could produce very poor
results
• In 1904, he declared this
would be impossible and that
weather forecasting was
“immoral and damaging to
the character of a
meteorologist.”
Lewis Fry Richardson
• Among first to try to
solve the
weather/climate
equations numerically
• Invented many
concepts still used
today
• Made the first numerical
weather forecast
• Prediction was horribly
wrong
Richardson’s technique
• Richardson laid out a grid, collected his
observations, created novel numerical
approximations for his equations and
crunched his numbers, by hand and slide rule
• Actually, it was a hindcast that took laborious
computations using old, tabulated data
dreamstime.com
Aside
• “In 1967, Keuffel & Esser, commissioned a
study of the future. The report predicted that,
by the year 2067, Americans would live in
domed cities… and watch three-dimensional
television. Unfortunately for the company, the
report failed to predict that slide rules would
be obsolete in less than ten years.”
Richardson’s grid
Richardson used 5 vertical levels, including surface
Richardson’s forecast
• He predicted a surface pressure rise of
145 mb (about 14%) in only 6 h
• In reality the pressure hardly changed at
all
• First forecast = first forecast failure
• What went wrong?
Richardson EXTRAPOLATED too far.
His technique made monsters out of meaningless oscillations
that happen as air wiggles up and down in a stable atmosphere.
Richardson’s book
• Richardson revealed the details of his blown
forecast in “Weather Prediction by Numerical
Process”, published in 1922.
• Undaunted, he imagined his pencil and paper
technique applied to the entire global
atmosphere…
– In a huge circular ampitheatre in which human
calculators would do arithmetic and, guided by a
conductor at the center, pass results around to
neighbors
Richardson’s forecasting
theater
After Richardson
• Richardson’s confidence was not unfounded.
• Mathematicians and meteorologists realized the flaws
of his techniques
• The digital computer was created
• The meteorological observation network was
expanding rapidly
• It seemed only a matter of time until the vision of
Abbe, the goal of Bjerknes, the dream of Richardson,
became a reality…
• Then in 1962, Ed Lorenz did his little experiment…
Prof. Ed Lorenz
• MIT professor
• Landmark 1962
paper “Deterministic
Nonperiodic Flow”
• Led to “chaos
theory” and
dynamical systems
• Associated with
“butterfly effect”
The Lorenz Experiment
• Lorenz’ model wasn’t
a weather model, and
didn’t even have grid
points
• 3 simple equations,
which can describe fluid
flow in a cylinder with
heated bottom and
cooled top
• He called his variables
X, Y and Z
The Lorenz Experiment
• X indicated the
magnitude and
direction of the
overturning motion
• As X changed sign,
the fluid circulation
reversed
The Lorenz Experiment
• Y was proportional
to the horizontal T
gradient
The Lorenz Experiment
• And Z revealed the
fluid’s stability
The Lorenz Model
• Three simple equations
• But in important ways
they were like the
equations we use in
weather forecasting
– They are coupled
– They are nonlinear
Simulation similar to Lorenz’ original experiment.
X = circulation strength & magnitude.
The model was started with unbalanced initial values for
X, Y and Z, creating a shock. The model was seeking a
suitable balance.
Next, a spin-up period with swings that grow until…
The fluid chaotically shifts from CW to CCW circulations
in an nonperiodic fashion
This was Lorenz’ discovery… sensitive dependence
on initial conditions
Dependence on initial
conditions
• Caused by nonlinear terms
• Even if model is perfect, any error in initial
conditions means forecast skill decreases
with time
• Reality: models are far from perfect
• Long range weather prediction is
impossible
• Lorenz: “We certainly hadn’t been successful
at doing that anyway and now we had an
excuse.”
From Lorenz (1963)
• “Two states differing by imperceptible amounts may
eventually evolve into two considerably different
states. If, then, there is any error whatever in
observing the present state — and in any real system
such errors seem inevitable — an acceptable
prediction of an instantaneous state in the distant
future may well be impossible.”
From Lorenz (1963)
• “In view of the inevitable inaccuracy and
incompleteness of weather observations, precise
very-long-range forecasting would seem to be nonexistent…. There remains the very important
question as to how long is ‘very-long-range’. Our
results do not give the answer for the atmosphere;
conceivably it could be a few days or a few
centuries.”
Lorenz attractor
Wikipedia
• 3D space with coordinates
being the Lorenz variables,
X Y and Z
• Predicted values can be
plotted as a point in this
space
• Start with an initial conditions
that are slightly different…,
• … they diverge in this space
• Birth of chaos theory
• Poorly named: chaos ≠
random
The butterfly metaphor
• ‘Butterfly effect’ did not originate w/ Lorenz. In a
1963 paper, he referred to seagulls.
• In 1969, Joe Smagorinsky wrote “would the flutter of
a butterfly’s wings ultimately amplify to the point
where numerical simulation departs from reality?”
(BAMS article)
• In 1993, Lorenz gave a talk entitled, “Does the flap of
a butterfly’s wings in Brazil stir up a tornado in
Texas?”. But, Lorenz did not write the title.
• ‘Butterfly effect’ was coined by James Gleick in his
1987 book, “Chaos: Making a New Science”
Summary
• NWP models = great tools for understanding our
present and predicting our future
• Models initialized with data to compute tendencies,
via approximation combined with extrapolation
• Models have limitations
– Resolution
– Unresolvable features and processes
– Incomplete or erroneous initial conditions
• New forecasts based on previous ones, so error
grows
• Limit to predictability (Lorenz)
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