Introduction to
Numerical Weather Prediction
4 September 2012
Introduction
“You can create any solution that you want to with a
numerical model.”
Scary thought, eh?
How do we go about getting the “right” solution – or at
least a reasonable facsimile thereof?
Components of an NWP System
Key Thematic Elements
• Underlying framework for NWP (Chs. 2-3)
• Physical process parameterizations (Chs. 4-5)
• Model initialization methods (Ch. 6)
• Applications of numerical models (Chs. 7+)
Course Structure
We will make heavy use of the assigned text for this
course. I anticipate lecturing for approximately 60 min
each class, leaving 15 min for discussion and questions.
Thus, please do read the section(s) to be covered in
class ahead of time!
Basic Equation Set
Horizontal Momentum Equations
u
u
u
u uv tan uw 1 p
 u
v w 


 2( w cos  v sin  )  Fx
t
x
y
z
a
a  x
time derivs. advection terms
curvature terms pres. grad. Coriolis terms
v
v
v
v u 2 tan uw 1 p
 u  v  w 


 2u sin   Fy
t
x
y
z
a
a  y
φ = latitude, a = radius of the Earth, Ω = rotational frequency of Earth, F = friction
friction
Basic Equation Set
Vertical Momentum Equation
w
w
w
w u 2  v 2 1 p
 u
v
w


 2u cos  g  Fz
t
x
y
z
a
 z
time deriv.
advection terms
curvature
term
pres. grad. Coriolis term gravity friction
φ = latitude, a = radius of the Earth, Ω = rotational frequency of Earth, F = friction
Basic Equation Set
Thermodynamic Equation
T
T
T
1 dH
 u
v
 w(   d ) 
t
x
y
c p dt
or, alternately…
T
T
T
T
Q
 u
v
w
 w d 
t
x
y
z
cp
time deriv.
advection terms
dry diabatic
adiabatic heating
term
γ = lapse rate of temperature, γd = dry adiabatic lapse rate, Q = diabatic heating rate
Basic Equation Set
Continuity Equations
(i.e., mass – top – and water vapor – bottom – are neither created nor destroyed)




u v w
 u
v
w
 (   )
t
x
y
z
x y z
time derivs.
advection terms
divergence
term
qv
q
q
q
 u v  v v  w v  Qv
t
x
y
z
qv = water vapor mixing ratio, Qv = source/sink of qv due to phase changes
Basic Equation Set
Ideal Gas Law
p  RT
If you can solve the set of equations (often referred to
as the primitive equations) given in the past few slides,
you can do NWP!
…but…
…how do we actually solve these equations???
What Do We Need?
•
•
•
•
•
•
•
How do we represent these equations on a map?
How do we integrate them in time?
How do we integrate them in space?
How do we handle resolvable versus unresolvable processes?
How do we handle diabatic processes?
How do we handle friction?
How do we handle microphysical phase changes (water vapor
as well as other species – cloud, ice, graupel, rain, etc.)?
• How do we obtain our initial atmospheric state?
…all among many relevant questions we will address!
Another scary thought:
Nearly everything we describe from here on out
involves some sort of approximation.
This is true for the equations themselves, the methods
used to solve them, the initial and boundary data used
to drive the model, and so on.
Before proceeding, a couple of notes…
Prognostic vs. Diagnostic
• Prognostic: any equation with a time-derivative is a
prognostic equation; it can be integrated in time to
produce a prediction
• Diagnostic: any equation without a time-derivative is
a diagnostic equation; it can only be used to
diagnose what is happening at a given time
Vertical Coordinate
• As given before, our vertical coordinate is height (z)
– Note: the comment on pg. 7 is wrong (should be z, not p)
• Other vertical coordinates may be used if
appropriate substitutions are made…
– Pressure (p)
– Potential temperature (θ)
– Terrain-following (σ)
• We’ll discuss these further in a later chapter.
Reynolds Averaging
• The equations we outlined earlier are valid on all
scales of motion
• But, in a numerical model, we have a grid with finite
horizontal and vertical resolution…
• This leads to there being two scales of physical
processes: those that we can resolve (larger) and
those that we cannot resolve (smaller)
Reynolds Averaging
• The goal of Reynolds averaging is to separate out the
resolvable and unresolvable scales of motion.
• We do so by splitting our dependent variables (u, T,
p, etc.) into mean (resolved) and turbulent
(perturbation/unresolved) components, e.g.,
u  u  u'
T  T T'
p  p  p'
Reynolds Averaging
• General idea:
– Substitute such expressions into the primitive equations
– Take the mean of each of the primitive equations
– Simplify the result using Reynolds’ postulates
• Reynolds’ postulates…
a'  0
mean of all perturbations is zero
aa
mean of a mean is equal to the mean
ab  ab  ab
ab'  ab'  ab'  0
mean operator is commutative unless both variables
are turbulent/perturbation components
Reynolds Averaging: Example
Equation (2.11), dropping frictional parameterization…
u
u
u
u 1 p
 u
v w 
 fv
t
x
y
z  x
We desire to substitute for u, v, w, p, and ρ, average the
entire equation, and ultimately simplify the result using
Reynolds’ postulates.
Reynolds Averaging: Example
u
u
u
u 1  p
u'
u'
u'
 u
v w 
 f v  u'
 v'
 w'
t
x
y
z  x
x
y
z
resolvable scales
aggregate (mean) effects on
unresolvable scales
Reynolds Averaging: Example
• Typically, the unresolvable scale terms are written in
terms of turbulent stresses.
– See also: discussion related to equations (2.14)-(2.16).
– These stresses are typically approximated utilizing physical
parameterization packages, to be covered in later lectures.
• We commonly drop the overbars from the primitive
equations, making the Reynolds average(s) implicit.
Reynolds Averaging: An Aside
• The discussion of Reynolds averaging in the text
discusses two things that we have neglected:
– Frictional parameterization
– Tensor notation
• We will also visit these topics later in the course.
Squelching Acoustic Waves
• The biggest thorn in the side of modelers is acoustic
waves…
– Acoustic waves aren’t meteorologically relevant.
– But, their very high frequency means that a very short
time step (and a LOT of computer power) is needed to
solve the primitive equations…
– …unless we do something to filter them out exclusively.
• Recall: the restoring mechanism for acoustic waves
is their compressibility.
Squelching Acoustic Waves
• In other words, the propagation of acoustic waves is
reliant upon the density adjusting to horizontal
compression and expansion within the waves.
• To filter out acoustic waves, we need to eliminate the
possibility of compressibility.
• Specifying conditions under which density can (and
cannot) vary enables us to filter out acoustic waves.
Squelching Acoustic Waves
Squelching Acoustic Waves
• Method 2: Boussinesq Approximation
– Assumes that density is constant except where it is tied to
buoyancy (or gravity).
– Equivalently obtained (in part) by replacing (2.5) with:
u v w
 
0
x y z
– This relates density perturbations to temperature, rather
than pressure, perturbations.
• Note: Boussinesq continuity equation conserves
volume (not mass), not necessarily a good trade-off!
Squelching Acoustic Waves
• Method 3: Anelastic Approximation
• The Boussinesq approximation can be viewed as a
simplified subset of the anelastic approximation.
• Obtained (in part) by replacing (2.5) with:
 
 
 



u 
v 
w  0
x
y
z
Summary
• We have introduced the basic equation set used for
numerical weather prediction.
• We have briefly described what is needed in order to
solve this equation set.
• We have outlined the need for separating the
resolvable from the unresolvable scales of motion.
• We have described the need for filtering out acoustic
waves from the primitive equations.