numerical methods for atmospheric dynamics

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.


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…


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


Model geometry (ECMWF)

Horizontal resolution T511 ~ 40 km

Vertical resolution60 levels


Modules in a NWP model

• Dynamics

• Physics parameterization

• Data assimilation

• Predictability - validation


PhysicsSub-scale processes to be parameterized


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)


Data assimilation

Data sources for the ECMWF

Meteorological Operational System

(EMOS).

Numbers refer to amount of received data in 24 hours


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


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


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


Predictability – forecast error

Sources of error in NWP:–Errors in the initial conditions

–Errors in the model

–Intrinsic predictability limitations


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



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.


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.


• Mass conservation

• Momentum conservation

• Energy conservation

• (water, gaseous and aerosol components conservation)

Key Conservation Principles


Primitive equations

dtdr

zdtd

rvdtd

ru ;;cos

Conservation laws in spherical geometry:

(,,r)Spherical coordinates

Velocity components


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


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


Numerics

1) Space discretization:

A. Horizontal discretization

B. Vertical discretization

2) Time discretization


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


In the case of finite differences:• uniform grids (in longitude and latitude)

• reduced (or stretched) grids

• geodesic grids

• Spatial staggering (velocity and pressure)

Numerics


Horizontal discretization:

A uniform longitude-latitude

grid (i.e. variable space resolution)

Adjustments:

• reduced grids

• stretched grids



A geodesic grid (quasi uniform space resolution)



Variables collocation

u,v,h

A (unstaggered) B

D

C

E

u,v

h h

u

v

u

v

h

hu,v

grid length


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


Vertical discretization:

Height coordinate Pressure coordinate

Normalized pressure coordinate


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


Time discretization schemes

for semi-Lagrangian

…with two interpolations

…with one interpolation

…without interpolations

Numerics


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


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


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


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


Semi-Lagrangian advection in spherical geometry


Momentum equation

• Semi-Lagrangian• Semi-implicit

“Lagrangian” terms


Evaluation of Lagrangian terms

Characteristic system

Eulero with substepping (Casulli, 1990)

Runge Kutta 2 (Heun)


Continuity equation

• Finite volumes• Semi-implicit


Geopotential equation

(9 diagonals, unsymmetric)


Splitting

• First step:

• Second step:

(to be coupled with the continuity equation)

(to be solved separately)


Geopotential equation

(5 diagonals, symmetric, positive definite)


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


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.


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


The method on a

geodesic grid

• Advection

• Shallow water


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


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


Interpolation:

CwBwAwP CBA

Geodesic grid:

Baricentric coordinates :

)(PBCareawA


Tests: • Solid-body rotation• Deformation flow


Higher order interpolation procedures

Variable resolution

Extension to the Shallow water on a

geodesic grid

Current perspectives:


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!

numerical methods for atmospheric dynamics

Documents

nwp systemit

objective forecast

model state

corrected forecast

scale processes

time step

analysis technique

observed variables