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Numerical methods for Numerical methods for atmospheric dynamicsatmospheric dynamics
Maria Francesca CarforaMaria Francesca Carfora
IAC CNRIAC CNR
22
Plan of the Plan of the talktalk
• Operational models for NWP: an overviewOperational models for NWP: an overview
• The Primitive EquationsThe Primitive Equations
• Numerical methods: Numerical methods:
– a semi Lagrangian approach a semi Lagrangian approach
– on traditional grids on traditional grids
– on geodesic gridson geodesic grids
Grenoble – Oct 2004 3
What is NWP?
The technique used to obtain an objective forecast of the future weather (up to possibly two weeks) by solving a set of governing equations that describe the evolution of variables that define the present state of the atmosphere.
Feasible only using computers.
Grenoble – Oct 2004 4
NWP system
NWP entails not just the design and development of atmospheric models, but includes all the different components of an NWP system
It is an integrated, end-to-end forecast process system.
As an example…
Grenoble – Oct 2004 5
NWP model (ECMWF)Model details:
–Spectral resolution T511–Reduced gaussian grid N256
(resolution 40 km) –60 hybrid vertical levels
from the ground level to a height of 65 km
–Time step 15’ –2-time level semi-
Lagrangian dynamics–Physical parameterization
Grenoble – Oct 2004 6
Model geometry (ECMWF)
Horizontal resolution T511 ~ 40 km
Vertical resolution60 levels
Grenoble – Oct 2004 7
Modules in a NWP model
• Dynamics
• Physics parameterization
• Data assimilation
• Predictability - validation
Grenoble – Oct 2004 8
PhysicsSub-scale processes to be parameterized
Grenoble – Oct 2004 9
Physics
Grid-scale precipitation (large scale condensation)Deep and shallow convectionMicrophysics (increasingly becoming important)Evaporation PBL processes, including turbulenceRadiationCloud-radiation interactionDiffusionGravity wave dragChemistry (e.g., ozone, aeorosols)
Grenoble – Oct 2004 10
Data assimilation
Data sources for the ECMWF
Meteorological Operational System
(EMOS).
Numbers refer to amount of received data in 24 hours
Grenoble – Oct 2004 11
Observations feed prediction models. But…
Observations are (or may be):
-unevenly distributed (in space and/or in time)
-incomplete
-of poor quality
Data assimilation
Need for an assimilation procedure
Grenoble – Oct 2004 12
There are errors in the model and in the observations, so we can never be sure which one to trust. However we can look for a strategy that minimizes on average the difference between the analysis and the truth.
Data assimilation is an analysis technique in which the observed information is accumulated into the model state by taking advantage of consistency constraints with laws of time evolution and physical properties.
Data assimilation
Grenoble – Oct 2004 13
Variational data assimilation:• Running the NWP model we obtain an estimate for the observed
quantities (analysis)• A cost function (J0) measures the distance between the analysis and the
observations
• Minimizing J0 we determine a corrected forecast which is closer to the observations.
• This forecast gives the values for the observed variables to be introduced in the model
Data assimilation
Grenoble – Oct 2004 14
Predictability – forecast error
Sources of error in NWP:–Errors in the initial conditions
–Errors in the model
–Intrinsic predictability limitations
Grenoble – Oct 2004 15
Sources of Errors - continued
Initial Condition Errors
1 Observational Data Coverage
a Spatial Density
b Temporal Frequency
2 Errors in the Data
a Instrument Errors
b Representativeness Errors
3 Errors in Quality Control
4 Errors in Objective Analysis
5 Errors in Data Assimilation
6 Missing Variables
Model Errors
1 Equations of Motion Incomplete
2 Errors in Numerical Approximations
a Horizontal Resolution
b Vertical Resolution
c Time Integration Procedure
3 Boundary Conditions
a Horizontal
b Vertical
4 Terrain
5 Physical Processes
Grenoble – Oct 2004 16
Predictability – forecast error
Predictability limitations:
The deterministic approach to numerical weather prediction provides one single forecast for the "true" time evolution of the system. The ensemble approach to numerical weather prediction tries to estimate the probability density function of forecast states. Ideally, the ensemble probability density function estimate includes the true state of the system as a possible solution.
Grenoble – Oct 2004 17
Predictability – forecast error
Grenoble – Oct 2004 18
Dynamics
It was recognized early in the history of NWP that primitive equations were best suited for NWP
Governing equations can be derived from the conservation principles and approximations.
It is important to understand the resulting wave solutions and their relationship to the chosen approximations.
Grenoble – Oct 2004 19
• Mass conservation
• Momentum conservation
• Energy conservation
• (water, gaseous and aerosol components conservation)
Key Conservation Principles
Grenoble – Oct 2004 20
Primitive equations
dtdr
zdtd
rvdtd
ru ;;cos
Conservation laws in spherical geometry:
(,,r)Spherical coordinates
Velocity components
Grenoble – Oct 2004 21
Prognostic variables
Horizontal and vertical wind components
Pressure, height or potential temperature
Surface pressure
Specific humidity/mixing ratio
Mixing ratios of cloud water, cloud ice, rain, snow
Mixing ratio of chemical species
Grenoble – Oct 2004 22
Primitive equations
QTtd
dC
TRpdtd
gzp
Fa
uufp
atdvd
Fa
vuvfp
atdud
p
V
tan1
tancos1
2
Longitudinal velocity(along the parallels)
Latitudinal velocity(along the meridians)
Hydrostatic approssim.
Mass conservation
State equation
Energy conservation
Grenoble – Oct 2004 23
Numerics
1) Space discretization:
A. Horizontal discretization
B. Vertical discretization
2) Time discretization
Grenoble – Oct 2004 24
A) Horizontal discretization
• Finite differences• Finite elements• Spectral methods:
variables are represented by truncated spherical armonics
Numerics
where is longitude, is sin(latitude) and Pnm are Legendre polynomials
Grenoble – Oct 2004 25
In the case of finite differences:• uniform grids (in longitude and latitude)
• reduced (or stretched) grids
• geodesic grids
• Spatial staggering (velocity and pressure)
Numerics
Grenoble – Oct 2004 26
Horizontal discretization:
A uniform longitude-latitude
grid (i.e. variable space resolution)
Adjustments:
• reduced grids
• stretched grids
Grenoble – Oct 2004 27
Horizontal discretization:
A geodesic grid (quasi uniform space resolution)
Grenoble – Oct 2004 28
Horizontal discretization:
Variables collocation
u,v,h
A (unstaggered) B
D
C
E
u,v
h h
u
v
u
v
h
hu,v
grid length
Grenoble – Oct 2004 29
B) Vertical discretization:
Finite differences, with several vertical coordinates:• Height on the mean sea level (z )• Pressure ( p )
• Normalized pressure ( =p / p* )
• Hybrid coordinates (k= Ak p + Bk p* )
• Potential temperature ( )
Numerics
Grenoble – Oct 2004 30
Vertical discretization:
Height coordinate Pressure coordinate
Normalized pressure coordinate
Grenoble – Oct 2004 31
Time discretization schemes
• Two-level (e.g., Forward or backward)• Three-level (e.g., Leapfrog)• Multistage (e.g., Forward-backward)• Higher-order schemes (e.g., Runge-Kutta)• Time splitting (split explicit)• Semi-implicit• Semi-Lagrangian
Numerics
Grenoble – Oct 2004 32
Time discretization schemes
for semi-Lagrangian
…with two interpolations
…with one interpolation
…without interpolations
Numerics
Grenoble – Oct 2004 33
Lagrangian viewpoint:
the total derivative is seen as the time evolution along a trajectory, which is called the characteristic line
A regular grid at time tn evolvs in an irregular one
at time tn+1
semi-Lagrangiantechnique:moving backward along the characteristic line one can determine its starting point
A regular grid at time tn+1 originates from an
irregular one at time tn
dtd
yv
xu
t
Grenoble – Oct 2004 34
semi-Lagrangian technique:moving backward along the characteristic line one can determine its starting point
The trajectory from A to B is approximated by the straight
line A'B
Grenoble – Oct 2004 35
Toy model: shallow water eqs.
0))(( sh
h
dt
d
fdt
dVk
V
,cos
1,
Ryxh
V = (u,v) wind field
f = Coriolis parameter
= geopotential
s = orography
h = horizontal
Grenoble – Oct 2004 36
Main features of the method
• Vectorial discretization for the momentum equation;
• semi-Lagrangian procedure with sub-stepping for an accurate reconstruction of the characteristic lines;
• semi-implicit treatment of some terms to obtain unconditional stability;
• Finite volumes for the continuity equation to obtain exact mass conservation;
• Full or splitted scheme
Grenoble – Oct 2004 37
Semi-Lagrangian advection in spherical geometry
Grenoble – Oct 2004 38
Momentum equation
• Semi-Lagrangian• Semi-implicit
“Lagrangian” terms
Grenoble – Oct 2004 39
Evaluation of Lagrangian terms
Characteristic system
Eulero with substepping (Casulli, 1990)
Runge Kutta 2 (Heun)
Grenoble – Oct 2004 40
Continuity equation
• Finite volumes• Semi-implicit
Grenoble – Oct 2004 41
Geopotential equation
(9 diagonals, unsymmetric)
Grenoble – Oct 2004 42
Splitting
• First step:
• Second step:
(to be coupled with the continuity equation)
(to be solved separately)
Grenoble – Oct 2004 43
Geopotential equation
(5 diagonals, symmetric, positive definite)
Grenoble – Oct 2004 44
Linear stability analysis – full scheme
1)2/sin(2)2/sin(2
)2/sin(21)1(
)2/sin(2)1(1
C
ytKIxtKI
ytKItf
xtKItf
j
j
2/1
22
1
1211RCB
AA
kk ww RCB 1
1)2/sin(2)2/sin(2
)2/sin(21
)2/sin(21
B
ytKIxtKI
ytKItf
xtKItf
j
j
Th.
where 2/sin2/sin4 22222 yKxKtfA j
Grenoble – Oct 2004 45
Linear stability analysis – splitted scheme
kk ww CB 1
1)2/sin(2)2/sin(2
)2/sin(210
)2/sin(201
B
ytKIxtKI
ytKI
xtKI
j
j
1)2/sin(2)2/sin(2
)2/sin(2
)2/sin(2
C 43
21
ytKIxtKI
ytKIcc
xtKIcc
j
j
tyx
Htf
tf
111
21
11,1maxCB1 22
122
12
221
2
2
1Th.
Grenoble – Oct 2004 46
Linear stability results
• Unconditional stability of the full scheme for
≥ 0.5
• Unconditional stability of the splitted scheme for
= 1 pressure and divergence terms1 ≥ 0.5 Coriolis terms
Grenoble – Oct 2004 47
The method on a
geodesic grid
• Advection
• Shallow water
Grenoble – Oct 2004 48
Logical structure:10 diamonds (0,nk) x (1, nk+1)
Parameters:k = refinement level
nk = 2k
nodes = 10*nk2+2
cells = 20*nk2
k nk nodes cells Lmax
0 1 12 20 7040
1 2 42 80 3520
2 4 162 320 1760
3 8 642 1280 880
4 16 2562 5120 440
5 32 10242 20480 220
6 64 40962 80960 110
7 128
163842
327680
55
Grenoble – Oct 2004 49
Geodesic grid:
PRO
• Quasi - uniform resolution
• “Natural” solution to the pole problem
• Only normal velocities
CON
• More complicated logical structure
• Triangular cells
• Only normal velocities
Grenoble – Oct 2004 50
Interpolation:
CwBwAwP CBA
Geodesic grid:
Baricentric coordinates :
)(PBCareawA
Grenoble – Oct 2004 51
Tests: • Solid-body rotation• Deformation flow
Grenoble – Oct 2004 53
Higher order interpolation procedures
Variable resolution
Extension to the Shallow water on a
geodesic grid
Current perspectives:
Grenoble – Oct 2004 54
References
• Amato, U. and Carfora, M.F., (2000), “Semi-Lagrangian Treatment of Advection on the Sphere with Accurate Spatial and Temporal Approximations”, Mathematical and Computer Modelling, 32, 981-995.
• Carfora, M.F., (2000), “An Unconditionally Stable Semi-Lagrangian Method for the Spherical Atmospherical Shallow Water Equations”, Int. J. for Numer. Meth. in Fluids 34, 6, 527-558.
• Carfora, M.F., (2001), “Effectiveness of the operator splitting for solving the atmospherical shallow water equations”, Int. J. Numer. Meth. Heat and Fluid Flow, 11, 3, 213-226 .
• Abrugia, G. and Carfora, M.F., (2003), “Semi-Lagrangian Advection on a Spherical Geodesic Grid”, Tech. Rep. IAC-NA n.274/03 (submitted).
• Carfora, M.F. and Noviello, G., (2004), “Shallow water equations on a spherical geodesic grid”, IAC Tech. Rep. n. 291/04
5555
Thank you!Thank you!