Chapter 1 Introduction to Weather and Climate Models

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Chapter 1 Introduction to Weather and Climate Models

Christoph Schär Atmospheric and Climate Science ETH Zürich, Switzerland schaer@env.ethz.ch March 21, 2007

Handouts

Numerical Modelling of Weather and Climate

Christoph Schär and Ulrike Lohmann, Summer Term 2007

http://www.iac.ethz.ch/education/master/numerical_modelling_of_weather_and_climate

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Outline

Historical perspective

Governing equations Parameterizations

Initial conditions

Climate models

Regional models

Computational aspects

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Contents

Laboratory-Experiment of Kelvin-Helmholtz Instability

€

Ri = −g ρ

∂ρ

∂z



  

 

1 2 ∂v

∂z

−1

≤ 1 4 Richardson-Number Criteria:

Billow Couds (Gemsfairenstock, March 11, 2003)

Picture: Thomas Schumann

Sometimes the atmosphere exhibits physical principles in a textbook fashion,

….

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IR Satellite Picture MeteoSwiss, Zürich

…. more generally, however, the atmosphere looks very complicated, and a direct link to simple physical principles is not evident.

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Vilhelm Bjerknes (1862-1951)

In 1904, Bjerknes proposed that „weather forecasting should be considered as an initial value problem of mathematical physics.“

His proposal was motivated by his new circulation theorem, that overcame obvious limitations of Kelvin’s earlier theorem.

Circulation Theorem:

Circulation:

Kelvin‘s Theorem:

€

C=

∫

€

dC dt = 0

C

C

Bjerknes‘ Theorem:

€

dC

dt = −2ΩdA_e

dt −

∫ ρ

A

A_e Ω

Ω

Earth’s rotation

baroclinic effects

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Lewis F. Richardson (1881-1953)

In 1922, Richardson provided the first formulation of the atmospheric equations on a computational grid.

“If the coordinate chequer were 200 km square in plan, ... 64,000 computers would be needed to race the weather. In any case, the organisation indicated is a central forecast-factory.”

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Synchronization Communication Distributed Memory Parallelization

Extracts from Richardson’s Book

«Individual computers are specialising on the separate equations. Let us hope for their sake that they are moved on from time to time to new operations.»

«It took me the best part of six weeks to draw up the computing forms»

«The work of each region is coordinated by an official of higher rank. Numerous little "night signs" display the instantaneous values so their neighbouring computers can read them.»

«From the floor ... a tall pillar rises to half the height of the hall. It carries a large pulpit on its top. In this sits the man in charge of the whole theatre. ... One of his duties is to maintain an uniform speed of progress. ... He turns a beam of rosy light upon any region that is running ahead of the rest, and a beam of blue light upon those who are behindhand.»

«In a neighbouring building there is a research department. ...

Outside are playing fields, houses, mountains and lakes, for it was thought that those who compute the weather should breathe it freely.»

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Summary of gains of entropy and of water, both per mass of atmosphere during δt

“by stirring”

(sub-gridscale mixing)

“by precipitation” advection

Gains of water per mass

visible infrared

Gains of entropy Gains of energy

by radiation

“Omitted pending further theoretical investigation”

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Divergence of horizontal momentum-per-area.

Increase of pressure

∂p

∂t = 145.1 hPa/6h

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Outline

Historical perspective Governing equations

Parameterizations Initial conditions

Climate models

Regional models

Computational aspects

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Ingredients to Deterministic Weather Forecasting

1. Governing set of equations

•

Equations describing the motion and thermodynamics of the atmosphere (dynamics).

•

Equations describing the interaction of the atmospheric flow with a wide range of physical processes (parameterizations, “physics”). This includes: radiation, boundary layer processes, cloud microphysics, soil hydrology, etc.

2. Discretized form of equations on a computer (model)

3. Initial conditions to start integration

•

Sufficient measurements to initialize model (observing system)

•

Preparation of the observations on a computational grid (data

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“Euler Equations” in Cartesian Coordinates

Momentum equations

Equation of state

Thermodynamic equation Continuity equation

Du

Dt−fv=–1 ρ

∂p

∂x+F_x Dv

Dt+ fu=–1 ρ

∂p

∂y+F_y Dw

Dt =–1 ρ

∂p

∂z−g+Fz

D Dt= ∂

∂t+u ∂

∂x+v∂

∂y+w∂

∂z

p=ρR T DT

Dt– 1 c_pρ

Dp Dt =H

∂ρ

∂t +∂(uρ)

∂x +∂(vρ)

∂y +∂(wρ)

∂z =0 with

Why not the “Navier-Stokes Equations”?

Reynolds-number in atmospheric flows is far too large!

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requires Δz << Δx, Δy

Critical velocities

~300 m/s sound propagation

~100 m/s horizontal wind velocity

~20 m/s vertical gravity-wave (buoyancy-wave) propagation

Numerics: Courant-Friedrichs-Levy (CFL) stability criterion where U denotes largest velocity in system would require Δt ≤ 0.1 s

Anisotropy of Atmosphere

Diameter: 12'700 km Depth of

troposphere: 10 km

Diameter: ~5 cm Depth: 0.04 mm

U Δt Δz ≤1

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Vertical momentum equation:

dw dt =–1

ρ

∂p

∂z−g+F_z

Approach for large-scale models:

Hydrostatic Approximation

Balance between pressure-force and gravity, neglect vertical acceleration

Implications:

• Suppresses vertical sound propagation

• w must be diagnosed from continuity equation (diagnostic variable)

• much easier to maintain time-step criterion

• BUT: only valid for Δx > ~10 km

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Discretization on the Sphere

The simplest model grid on the sphere is a regular latitude / longitude grid.

It has some difficult side effects (e.g. pole problem).

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φ λ,( ϕ,t) = ψn m( )t

n=m N(m) m=−M

∑

M

∑

Global Spectral Models

Represent a two-dimensional field φ on the sphere as an expansion using spherical harmonics :Y_n^m(λ,ϕ)

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Example: ECMWF Model

(European Centre for Medium-Range Weather Forecasts, ECMWF)

T_L511 ~ 40 km

Resolution Upgrade February 2006 T_L799 ~ 25 km

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Vertical Discretization: Terrain-Following Coordinates

(European Centre for Medium-Range Weather Forecasts, ECMWF)

Resolution Upgrade February 2006 60 Levels 91 Levels

16 km 31 km 48 km 65 km 79 km

0 km

Typical vertical resolution in climate and numerical weather prediction models:

20-90 levels Hydrostatic models use a pressure-based coordinate system

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Outline

Historical perspective Governing equations

Parameterizations

Initial conditions Climate models Regional models Computational aspects

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The Parameterization Problem

Computational grid with Δx = 50 km Many small-scale processes

(e.g. convective clouds)

are too small to be

represented on the

computational grid

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Radar composite MeteoSchweiz

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Parameterized Processes

Typical atmospheric models have 10 km ≤ Δx ≤ 200 km Processes that are not explicitly represented at these resolutions are ”parameterized” instead, using physical understanding of the underlying processes, or semi- empirical relations.

Parameterized processes contribute substantially to

uncertainties in weather forecasting and climate models.

(ECMWF)

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Parameterized Processes

Du

Dt−fv=–1 ρ

∂p

∂x+F_x Dv

Dt+ fu=–1 ρ

∂p

∂y+F_y Dw

Dt =–1 ρ

∂p

∂z−g+Fz

D Dt= ∂

∂t+u ∂

∂x+v∂

∂y+w∂

∂z

p=ρR T DT

Dt– 1 cpρ

Dp Dt =H

∂ρ

∂t +∂(uρ)

∂x +∂(vρ)

∂y +∂(wρ)

∂z =0 with

Momentum equations

Equation of state

Thermodynamic equation Continuity equation

diabatic heating rate (F_x,F_y)/ρ

H/c_p

Parameterized terms non-conservative forces

Equations for specific water vapour and cloud water content Dq_vap

Dt =S_vap Dqcld

Dt =S_cld Additional equations

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Outline

Historical perspective Governing equations

Parameterizations Initial conditions

Climate models

Regional models

Computational aspects

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Initial conditions are obtained from observations

(ECMWF 2001)

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Example: Radiosonde Data

(ECMWF 2001)

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Global Data Assimilation

Global data assimilation systems ingest a wide range of data from various instruments and observation times. They run a general circulation model (GCM) in hindcast mode. The resulting analysis is a spatially and temporally coherent description of the actual state of the atmosphere. In data sparse regions, where few observations are available, these systems in essence provide a mixture between a short-range (e.g. 6 h) forecast and the available observations.

Observations: soundings, surface observations, satellite data, etc.

Analysis: spatially and temporally coherent dataset on a grid

Initial conditions: starting point of numerical model integration Global data assimilation system

06 UTC 12 UTC 18 UTC 00 UTC 06 UTC 12 UTC 18 UTC

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Outline

Historical perspective Governing equations

Parameterizations Initial conditions

Climate models

Regional models

Computational aspects

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The Coupled Climate System

Ocean

Feb 9, 2000 Sea Ice

Atmosphere

Land Surfaces

May 1989

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The Development

of Climate Models

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Phase space is spanned by all degrees of freedom of the system This example: Lorenz attractor 3 degrees of freedom

Global atmospheric model:

10⁸ degrees of freedom

Weather versus Climate Prediction

(Palmer, ECMWF)

Weather Prediction Climate Prediction

phase space

2^nd forecast

from perturbed initial condition

A weather forecast is a trajectory in phase space, starting from the initial conditions.

A climate forecast is a forecast of the attractor. Information on initial conditions is largely lost.

forecast initial condition

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The Global Energy Balance Energy Input = Energy Output

Sun ^visible

infrared visible

CO

Aerosols Aerosols

Aerosols

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Natural Climate Forcings

(Fröhlich and Lean 2004)

Solar Irradiance (W/m2)

2000 1995 1990 1985 1980

Solar Forcing

(Ammann et al. 2003)

Global Optical Depth

Volcanic Forcing

Agung El Chichon Pinatubo

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Internal variability

(Peixoto and Oort)

Observed climate records include

• response to forcings (natural and anthropogenic)

• internal variations (e.g. due to atmospheric variability)

Spectrum of

surface temperature variability

2500 y 100-400 y 10-30 y ~2 y 1 y 0.5 y 3-7 d 1 d 12 h

Some of the observed variations are due to:

• external forcings

• harmonics of external forcings

• internal variations

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Anthropogenic Climate Forcing

Greenhouse gases

CO2 [ppm]

CH4 [ppb]

(IPCC, TAR)

Aerosols (optical depth, year 2000)

(IPCC, TAR)

Land use (leaf area index)

Observed Potential

Vegetation

5.6 4.8 4.0 3.2 2.4 1.6 0.8 0 (Heck et al. 2001)

(IPCC, TAR)

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Carbon Dioxide (CO

)

Current CO₂ concentrations are higher than ever in the last 600,000 years Atmospheric CO₂-Concentration [ppm]

Years before present

400,000 300,000 200,000 100,000 present

200 240 280 320 360 400

2100: Scenarios up to 900 ppm 2007: Today, 380 ppm

1800: Preindustrial, 280 ppm

(Petit et al. 1999)

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Models with and without anthropogenic effects

(IPCC AR4)

Surface temperature

Observed

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Arctic Polar Ice

(Holland et al. 2006, GRL)

2000 2040

Minimal extension of arctic sea ice (September)

Beobachtung 1980-2006 Simulation 1900-2100

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Outline

Historical perspective Governing equations

Parameterizations Initial conditions

Climate models Regional models Computational aspects

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Atmospheric GCM

(ECHAM5, T106, ~120 km)

Coupled GCM

(HadCM3, ~300 km)

Model Chain for Climate Change Impact Study

Regional Model

(CHRM, 56 km)

Regional Model

(CHRM, 14 km)

Hydrological Model

(WaSiM, 1 km)

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Operational Model of MeteoSwiss (aLMo/COSMO)

(MeteoSwiss)

LM model of German Weather Service (COSMO Consortium) Horizontal Resolution:

7 km, 385x325 grid points Several daily runs extending to +72h

Data-assimilation using the nudging methodology

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Outline

Historical perspective Governing equations

Parameterizations Initial conditions

Climate models

Regional models

Computational aspects

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Development of CPU power

Cray XT3, CSCS 1,700 processors 1 TFLOP = 10¹²

10¹¹ 10¹⁰

10⁸ 10⁷ 1 MFLOP = 10⁶ 10⁵ 1 GFLOP = 10⁹

1 MFLOP =

One Million Floating Point Operations per Second

Blue Gene, DOE, USA 130,000 processors Earth Simulator, Japan 5,120 processors 10¹³

10¹⁴ 1 PFLOP = 10¹⁵

2010

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Horizontal Resolution

(MPI Hamburg)

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Development of Operational NWP Model Resolution

As a result of the increasing computer power, it was possible to refine the resolution of operational Numerical Weather Prediction (NWP) models:

Year Global Models Regional Models

1985 200 km 50 km

2006 40 km 7 km

200? ~20 km ~1 km

Potential of kilometer-scale modelling

• Improved representation of topography:

improved prediction of mountain-induced weather

• Explicit simulation of moist convection:

improved prediction of heavy precipitation

• Bridges resolution-gap between meteorological and hydrological models improved prediction of runoff and floods

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4500 4000 3500 3000 2500 2000 1500 1000 500 0

Swiss Topography at two Resolutions

Δx = 14 km

4500 4000 3500 3000 2500 2000 1500 1000 500 0

Δx = 1 km

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The simulation of weather and climate has come a long way towards realizing the visions of V. Bjerknes (1904) and L.F. Richardson (1922).

Some key issues:

• Small-scale processes are important for large scale features:

=> High spatial resolution desirable (computational challenge)

• Representation of sub-gridscale processes:

=> Improved parameterizations and understanding needed

• Nonlinearities imply intrinsic predictability limitations:

=> Probabilistic methods

• Validation of climate models on timescales of seasons to decades

=> Improved long-term homogeneous data sets needed

Summary