Modeling of the coupled land-atmosphere.

Schär, ETH Zürich

Sonia I. Seneviratne and Christoph Schär Land-Atmosphere-Climate Interactions Winter term 2006/07

Modeling of the coupled land-atmosphere.

System. Part (a)

Christoph Schär

Institute for Atmospheric and Climate Science ETH Zürich

schaer@env.ethz.ch

Schär, ETH Zürich

3

Outline

Climate models

• basics

• governing equations

• discretization on the sphere

• parameterization

• computational resolution

• components of the climate system

Model evaluation and applications: a few examples Boundary layer parameterization

Schär, ETH Zürich

Discretization in Cartesian Coordinates

Continuous Discretized Schematics

T(x) T(x_i) with x_i=i!x

!x

!x

!y T(x,y) T(x_i, y_j) with x_i=i!x, y_i=j!y

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6

Example: linear one-dimensional transport equation for quantity and velocity u.

partial differential equation

Discretization in space; grid spacing !x:

coupled system of ordinary differential equations

Discretization in time; time step !t:

coupled system of finite difference equations

Solve for

time-stepping algorithm

How to convert equations onto the computational grid?

Finite Difference Method

!"

!t +u!"

!x =0 (u=const)

!(x,t)

!"_j

!t + u "_j+1 # "_j#1

2 $x =0

!_jⁿ := !(j"x, n"t)

!_j(t) := !(j"x,t)

!_jⁿ⁺¹ " !ⁿ_j^"1

2 #t + u !_jⁿ₊₁ " !ⁿ_j"1

2 #x =0

!ⁿ⁺¹

!_jⁿ⁺¹ = !ⁿ_j^"1 " u #t

#x

[

]

Schär, ETH Zürich (see course “Numerical Methods in Environmental Sciences”) Time-stepping algorithm

The Courant number is a dimensionless parameter that depends upon the (somewhat arbitrary) choice of !x and !t. For most numerical time-stepping algorithm, it determines the numerical stability of the algorithm.

Courant-Friedrichs Levy (CFL) stability criterion

Thus, the information must travel not further than one grid-point per time-step.

Changing the grid-spacing requires changing the time-step!

!_jⁿ⁺¹ = !ⁿ_j^"1 " u #t

#x

[

]

" = Courant Number

!

u "t

"x #1

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9

Governing equations in cartesian coordinates

Du

Dt ! fv= –1

"

#p

#x +F_x Dv

Dt+ fu= –1

!

"p

"y +F_y Dw

Dt = –1

!

"p

"z #g+F_z D

Dt = !

!t+u !

!x +v !

!y+w !

!z

p =!R T

DT Dt – 1

c_p! Dp Dt =H

!"

!t +!(u")

!x +!(v")

!y +!(w")

!z = 0

with Momentum equations

Equation of state

Thermodynamic equation Continuity equation

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requires !z << !x, !y

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 !t!0.1 s

Anisotropy of Atmosphere

Diameter: 12'700 km Depth of

troposphere: 10 km

Diameter: ~5 cm Depth: 0.04 mm

U !t

!z "1

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11

Vertical momentum equation:

dw dt =–1

!

"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 !x > ~10 km

Schär, ETH Zürich

Governing equations in cartesian coordinates

Du

Dt ! fv= –1

"

#p

#x +F_x Dv

Dt+ fu= –1

!

"p

"y +F_y Dw

Dt = –1

!

"p

"z #g+F_z D

Dt = !

!t+u !

!x +v !

!y+w !

!z

p =!R T

DT Dt – 1

c_p! Dp Dt =H

!"

!t +!(u")

!x +!(v")

!y +!(w")

!z = 0

with Momentum equations

Equation of state

Thermodynamic equation Continuity equation

Hydrostatic approximation

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13

Governing equations in pressure coordinates

p =!R T with Hor. momentum equations

Equation of state

Thermodynamic equation Continuity equation

D Dt= !

!t+u !

!x

"

#

$

% _p+v ! dy

"

#

& $

% '

p

+( !

!p ! = Dp

and Dt

!"

!p=–1

# with !=gz

Hydrostatic equation

Du

D t– f v=– !"

!x

#

$

% &

' (

p

+F^x D v

Dt +f u=– !"

!y

#

$

% &

' (

p

+F^y

!

DT D t " #

$c_p=H

!u

!x

"

# $ %

&

'

p

+ !v

!y

"

# $ %

&

'

p

+!(

!p=0

The formulation in pressure coordinates is superior, as several equations (horizontal momentum

equation, continuity equation) are dramatically simplified.

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diabatic heating rate

!

"(F_x,F_y) H/c_p

Parameterized terms non-conservative forces

Governing equations in pressure coordinates

p =!R T with Hor. momentum equations

Equation of state

Thermodynamic equation Continuity equation

D Dt= !

!t+u !

!x

"

#

$

% _p+v ! dy

"

# & $

% '

p

+( !

!p ! = Dp

and Dt

!"

!p=–1

# with !=gz

Hydrostatic equation

Du

D t– f v=– !"

!x

#

$ % &

' (

p

+F^x D v

Dt +f u=– !"

!y

#

$ % &

' (

p

+F^y

!

DT D t " #

$c_p=H

!u

!x

"

#

$ %

&

'

p

+ !v

!y

"

#

$ %

&

'

p

+!(

!p=0

Equations for specific water vapour and cloud water content Dq_vap

Dt =S_vap Dq_cld Dt = S_cld Additional equations

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16

Parameterized Processes

Typical atmospheric models have 10 km ! !x ! 200 km processes which are not explicitly represented are ”parameterized”

using physical understanding of the underlying processes

Parameterized processes contribute substantially to uncertainties in weather

forecasting and climate models.

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Temperature tendencies due to turbulence scheme

[K/day]

(Köhler, ECMWF) Jan. 1999

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Temperature tendencies due to convection scheme

[K/day]

Jan. 1999

(Köhler, ECMWF)

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Discretization on the Sphere

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

It suffers from the pole problem:

As !x–> 0, the CFL-criterion

implies that the time step requires special care.

!

u "t

"x #1

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Global Gridpoint Models: Icosahedral Global Mesh 21

(German Weather Service, DWD)

This grid is constructed from a projection of an icosahedron on the sphere, and subsequent refinement of the 20 triangles.

Used by the German Weather Service.

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! ",#,t( ) = $nm

( )t

n=m N(m) m=&M %

%M ^Ynm(",#)

Global Spectral Models

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

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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)

60 Levels

Resolution Upgrade February 2006 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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Vertical Discretization over Complex Terrain

Sigma-type coordinate

5 10 [km]

5 10

[km] Smooth coordinate

(Schär et al. 2002)

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Horizontal Resolution

(MPI Hamburg)

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Resolution versus Grid-Spacing

The terms “resolution” and “grid-spacing” are often used interchangeably.

Careful!

At least 4 grid points are needed to represent some structure:

!x=100 km <=> resolved scales ! 400 km

!x

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Horizontal Resolution

Issue: Resolution of models (IPCC AR4 GCMs)

(compiled by M. Litschi) Zurich-Stockholm

Zurich-Florence

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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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The Coupled Climate System

Ocean

Feb 9, 2000 Sea Ice

Atmosphere

Land Surfaces

May 1989

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32

Coupling of oceanic and atmospheric models

Information exchange between ocean and atmosphere:

Ocean => Atmosphere:

- sea surface temperature (SST) - sea ice extent

Atmosphere => Ocean:

- momentum fluxes (windstress) - net heat flux

- net freshwater flux

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surface windstress:

!_x =" C_m u ur ₁

!_y =" C_m u vr ₁ Freshwaterflux:

F_f = P # E

P: Niederschlag über Ozeanen E: Verdunstung aus Ozeanen Net ocean heat flux:

H_O =

[

]

Coupling of oceanic and atmospheric models

Atmosphere => Ocean

P: precipitation

E: evaporation from ocean surface

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34

The Development of Climate Models

(IPCC TAR)

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Outline

Climate models

Model evaluation and applications: a few examples

• model validation

• climate change scenarios

Boundary layer parameterization

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37

Zonal mean temperature and precipiation distribution of 15 climate models.

Observations (black)

mm/Tag K

Latitude Latitude

Temperature Precipitation

Validation of global climate models

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Validation of regional climate models

DJF precipitation [mm/d]

Regional climate models (RCM, 56 km resolution):

RCM-Simulations (CHRM / ETH) driven by

• ERA-15 reanalysis (bottom right)

• HadAM3 GCM CTL run (top right) Validation against

• CRU (Climate Research Unit)

• ERA-15 (ECMWF Reanalysis)

(Vidale et al. 2003)

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39

Natural forcing Natural and anthropogenic forcing

Global mean surface temperature 1860-2000

(IPCC TAR, 2001)

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Climate change around 2100 (IPCC SRES A2)

Change in yearly mean surface temperature [K]

(Martin Wild, ETH Zürich)

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Climate change around 2100 (IPCC SRES A2) 43

Change in yearly mean precipitation [mm/d]

(Martin Wild, ETH Zürich)

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Arctic sea ice

(Holland et al. 2006, GRL)

2000 2040

Minimal extent of arctic sea ice (September)

Observation 1980-2006 Simulation 1900-2100

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45

European climate-change signal

Temperature [^oC]

Precipitation [mm/d]

Scenario–Control, CHRM/ETH

warm and dry

summers in the south.

mild and wet winters in the north.

(Vidale et al. 2007)

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Summer Temperatures and Heatwaves (2070-2100)

(Schär et al. 2004, Nature, 427, 332-336) [ºC]

Change in Temperature !T

[%]

Change in Variability !"/"

(StdDev of seasonal T)

1961-1990 2071-2100 Zurich Temperature Series

Year Year

• Not only changes in mean, but also changes in variability •

• Together these combine to increase the frequency of heatwaves •

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47

PRUDENCE: Summer precipitation

Mean 99% percentile for n = 5 d

• Substantial reduction in mean precipitation, increases in strong events

• Appears to be main summer signature in most PRUDENCE models

• Uncertainties due to important role of convection and land-surface processes

Change in precipitation [%], July-Aug-Sept

(Christensen and Christensen 2003, Nature)

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Outline

Climate models

Model evaluation and applications: a few examples Boundary layer parameterization

• Basics

• Reynolds averaging

• Simple PBL parameterization

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Laboratory observations: transition to turbulence

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Space and time scales

• Subgrid-scale transport in the atmosphere is dominated by turbulence.

• Time scale of turbulence varies from seconds to half an hour.

• Length scale varies from mm (dissipative eddies) to 100 m (transporting eddies).

• Largest eddies are the most efficient in terms of transport.

1 hour

100 hours 0.01 hour

microscale turbulence

spectral gap diurnal

cycle cyclones

(Köhler, 2006, ECMWF)

Spectral analysis of turbulent kinetic energy

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52

Structure of atmospheric boundary layers

surface layer entrainment layer

boundary layer

~ 50 m

~ 1 km

(see course “Boundary layer meteorology and pollutant transport”)

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Boundary layer processes

+

Mittlere Strömung Resultierende Turbulenz

– + –

u (z), w (z), T (z), q (z) u ,! w ,! T ,! q !

Mean flow Resulting turbulence

! = ! (z) + !' (x,y,z,t) Decompose variables T, u, w, q :

$ = Air density (u,v,w) = Wind

q = Specific humidity

= Average over time

´ = Perturbations

Turbulence in boundary layers is

driven by shear and/or surface heating.

Turbulence implies vertical transport of momentum, heat, moisture and other atmospheric constituents.

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54

Reynolds averaging (1/3)

!

"u

"t +u"u

"x +w"u

"z =#1

$

"p

"x

Momentum equation (f=0, 2D)

!

"(u +u')

"t +(u +u')"(u +u')

"x +(w +w')"(u +u')

"z =#1

$

"(p + p')

"x

Plug into momentum equation and collect terms Introduce perturbation variables

!

u=u +u'

!

v= v +v'

!

p= p + p'

!

"'

neglect

Term due to turbulence

!

"u

"t +u "u

"x +w "u

"z =#1

$

"p

"x #u'"u'

"x #w'"u'

"z

Momentum equation for mean flow

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Reynolds averaging (2/3)

!

F^x ="v'#$u'

Momentum equation with turbulence term

momentum equation for mean flow Simplify

!

"'#0

!

" #(v'$)^%0

implies

!

"u

"t +u "u

"x +w "u

"z =#1

$

"p

"x #u'"u'

"x #w'"u'

"z

!

F^x ="1

#$ %

(

)

!

"v'u'="

(

)

Determined by correlations between different Velocity components

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56

Reynolds averaging (3/3)

In the atmosphere, the vertical flux of momentum often dominates

!

F^x ="1

#$ %

(

)

#

'

(

)

! 'z

"u

"t +u "u

"x +w "u

"z =#1

$

"p

"x +F^x

Vertical flux of horizontal momentum (turbulent stress)

!

" =#w'u' [N/m²]

Similar: Vertical flux of moisture and heat/energy (see section 3b of lecture)

LH = L ET = L ! q " w "

SH = c_p ! T " w "

[kg/(m²s)]

[W/m²]

approximately constant in surface layer

approximately constant in surface layer

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The neutral boundary layer: shear-driven turbulence

(Oke 1978)

z_o # 1-10cm z_o # 50cm z_o # 1m

!

u (z)= u*

"

#

$ % &

' ( ln z z_o

#

$ % &

' (

Analytical solution:

u* friction velocity

% von Karman constant (% ^=0.4) z_o roughness length

!

u*= "/#

unperturbed geostrophic flow

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58

dH_s/dt

H₂O, CO₂

SW_net LW_net LH=$E SH

G

Surface energy balance

Role of surface energy balance

How does the boundary layer react to the partitioning of the net energy balance into sensible and latent heat fluxes?

Schär, ETH Zürich

Diurnal boundary layer: thermally driven turbulence

Potential temperature &

Mixing ratio q

Relative humidity RH

well-mixed boundary layer 12 LT

06 LT

z z z

Adiabatically conserved quantities (i.e. D%/Dt=0, Dq/Dt=0) are homogenized in the well-mixed boundary layer.

The net heat input determines the depth of the well-mixed layer.

The implied increase in RH can lead to condensation (cloud- topped boundary layers) and initiate convection

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60

DRY

WET

1000 hPa 500 hPa

lower sensible heat flux leads to

shallower boundary layer

Development of diurnal PBL over dry and wet surfaces

(Schär et al. 1999)

1000 hPa 500 hPa

top of PBL

higher latent heat flux is concentrated in

shallower PBL

more unstable profile, stronger convection,

more precipitation

Schär, ETH Zürich

Simple turbulence parameterization

K-diffusion method: analogy to molecular diffusion

!

w'u' " –K#u

#z

!

F^x ="1

#

$

(

)

$z % $

$z K $u

$z

&

' ( )

* + % K$²u

$z²

For a parameterization, the subgrid-scale terms, i.e.

must be expressed in terms of the resolved variables. This is referred to as closure.

!

w'u'

!

w'q'

!

w'T'

The value of the diffusion coefficient K depends upon a number of factors, most importantly on atmospheric stability.