Basic Concepts of
Numerical Weather Prediction
6 September 2012
Thematic Outline of Basic Concepts
• Representation of the spatial derivatives in the primitive
equations
• Time integration methods
• Representation of the initial atmospheric state
• Representation of the horizontal and vertical boundaries
• Physical process parameterizations
Note Before Proceeding
• The material in this lecture is not meant to be
comprehensive.
• Instead, it is meant to further introduce many of the
key topics that we will be covering in the remainder
of this course.
• Each of these topics will be discussed in greater
detail as we cover Chapters 3-6.
Spatial Derivatives
• Consider a simplified form of the u-momentum
equation:
u
u
u
u 1 p
 u
v w 
 fv
t
x
y
z  x
• How do you solve for the spatial derivatives in x and
y (and to a lesser extent, z)?
• There are two popular (and many other) means of
doing so…
Spatial Derivatives
• Method 1: Grid-Point / Finite Difference
• The atmosphere is represented as a threedimensional spatial grid defined on a specified map
projection.
• Grid points are generally evenly spaced (or nearly so).
• Exceptions: lat-lon grids; adaptive grids
• Spatial derivatives are solved using Taylor seriesderived finite difference approximations.
Spatial Derivatives
• The distance between grid points is selected such
that there are a sufficient number of grid points to
adequately represent the smallest feature of interest
• Related concept: truncation error (Section 3.4.1)
• ~6∆x (~10∆x) points needed to represent a feature (wave)
• Pitfalls of grid-point methods…
• Introduce non-physical properties to the model solution
• Have stability criteria, often necessitating a short time step
(one that requires more computations to get a forecast)
Spatial Derivatives
• Method 2: Spectral Methods
• Involves the forward transformation of the standard
dependent variables (u, v, T, etc.) into transform space
utilizing Fourier transforms.
• The resultant Fourier series and Fourier-Legendre functions
represent horizontal variability.
• Temporal and vertical derivatives are handled with conventional
methods.
• The dependent variables are subsequently transformed back
into physical space to get interpretable forecast fields.
Temporal Derivatives
• Again, consider a simplified form of the umomentum equation:
u
u
u
u 1 p
 u
v w 
 fv
t
x
y
z  x
• How do you solve for the temporal derivative in t and
thus integrate the model forward in time?
• We will discuss means of doing so in Section 3.3.1.
Stability Considerations
• For many temporal differencing schemes, the time
step (i.e., the time elapsed between individual
forecast times) is constrained.
• Constraining mechanism: the Courant number…
Ut
x
(U = speed of the fastest wave on the grid, ∆t = time step, ∆x = distance between horizontal grid
points chosen to adequately resolve the meteorological processes of interest)
Stability Considerations
• For grid-point methods, the Courant number must
be less than or equal to 1.
• This is known as the Courant-Friedrichs-Lewy (CFL)
criterion.
– This is the stability requirement for the advection terms in
an Eulerian (grid-point) framework.
• Typically, U can be determined and ∆x is specified.
We want the largest ∆t satisfying the CFL criterion.
Stability Considerations
• If we set ∆t too large, the CFL criterion will be
violated and non-meteorological features will grow
exponentially in the solution.
• If we set ∆t too small, the CFL criterion won’t be
violated but the forecast will take longer to
complete.
• In other words, the model will have to solve the primitive
equations more times for a specified forecast duration.
Stability Considerations
• Example: assume that the fastest wave propagates
along a 100 m s-1 upper-tropospheric jet stream…
– In other words, we have filtered out fast acoustic waves.
• What is the largest ∆t that we can have for
convective-scale, mesoscale, and synoptic-scale
simulations?
• Convective-scale (∆x = 4 km): 40 s
• Mesoscale (∆x = 20 km): 200 s
• Synoptic-scale (∆x = 50 km): 500 s
Stability Considerations
• If we want a 24-h forecast, we will need to solve our
equation set the following number of times…
• Convective-scale: 2160
• Mesoscale: 432
• Synoptic-scale: 173
• From the above, over an equally-sized domain, a
convective-scale simulation will take ~12.5 times as
long to complete as a synoptic-scale simulation!
Time Step Considerations
• The importance of a long time step becomes even
more apparent when the number of horizontal grid
points also enters the equation.
• Over a 1000 km x 1000 km domain, each simulation
has the following number of grid points:
• Convective-scale: 250 x 250 = 62500
• Mesoscale: 50 x 50 = 2500
• Synoptic-scale: 20 x 20 = 400
Time Step Considerations
• Thus, a convective-scale simulation over an equally-sized
domain will take 12.5 * (12.5 * 12.5) times as long as a
synoptic-scale simulation.
• This works out to ~1950 times longer!
• Similarly, a convective-scale simulation over an equallysized domain will take 5 * (5 * 5) times as long as a
mesoscale simulation.
• This works out to 125 times longer!
• Thus, using as long of a time step as possible is vital to
timely and efficient numerical weather prediction.
Boundary Conditions
• Numerical simulations are boundary-value problems.
• In the horizontal…
• Global simulations – domain is periodic; not a horizontal
boundary value problem
• Limited area simulations – domain is limited; need
information on the boundaries
• Boundary value data for limited area simulations
typically comes from larger-scale observational or
forecast data (e.g., from another simulation).
Boundary Conditions
• And in the vertical…
• Model atmosphere cannot extend upward to infinity
• Model atmosphere also constrained by the Earth’s surface
• The upper boundary is typically constrained to the
tropopause or lower stratosphere.
• The interaction between the atmosphere and the
lower boundary is typically parameterized.
Initial Conditions
• Numerical simulations are also initial value problems.
• The process of providing initial value data to a model
is known as initialization.
• Initial conditions are typically provided by a
numerical synthesis of available observations.
• Methods for obtaining initial conditions are covered
in Chapter 6.
Initial Conditions
• The quality of a numerical forecast is strongly
constrained by the quality of the initial conditions.
• But, our observing system is limited – the totality of
the true state of the atmosphere is unknowable!
• Thus, much effort is expended upon trying to obtain
as accurate of initial conditions as possible.
Initial Conditions
• There are two general types of initializations:
• Static initialization (cold start)
• Dynamic initialization (warm / hot start)
• Static initializations…
• Interpolate observations at t=0 to the model grid
• Ensure that initial conditions are appropriately balanced
• Begin model forward integration
• Static initializations require a “spin up” period
• No data are provided on scales smaller (e.g., convective or
topographic scale) than that of the available observations
Initial Conditions
• Dynamic initializations utilize some means of model
“spin up” to ensure that local circulations are
represented at the initial forecast time.
• Typically accomplished via forecast cycling.
• A short-term forecast from an earlier model simulation is
used as a “first guess” for the desired simulation.
• Available observations are assimilated, whether at
specified times (three-dimensional) or through time (fourdimensional), to improve the “first guess” analysis.
• The model is then integrated forward in time.
Forecast Cycling
Physical Parameterizations
• The “interesting” parts of equations (2.1)-(2.7) are
often difficult to incorporate into a numerical model!
• Relevant processes include…
•
•
•
•
•
Diabatic heating (such as with deep, moist convection)
Turbulence (friction, etc.)
Microphysical processes (phase changes)
Land-surface feedbacks
Solar (longwave) and atmospheric (shortwave) radiation
Physical Parameterizations
• Thus, the impacts of these physical processes upon
the atmosphere are parameterized.
• Why parameterize these processes?
• We may not know enough about them to explicitly resolve
them within the model.
• The process(es) may occur on unresolvable scales.
• The physical relationships may be sufficiently complex so
as to require excessive resources to represent explicitly.
Physical Parameterizations
• Underlying physical parameterizations: representing
a process based upon its known relationship to
resolved variables within the model.
• For example: we know (or think we know, at least)
that turbulence is related to vertical wind shear and
static stability, both of which a model can resolve.
• We will discuss the underlying formulations of
physical parameterizations in Chapters 4 and 5.