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How do Climate Models Work? What do they tell us?. Aaron Donohoe Cornish College Lecture 2/26/07. How do we model a physical system?. Simple Problem: What will the temperature of a pot of water on the stove be as a function of time?. What do we need to know?. Initial conditions

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How do climate models work what do they tell us l.jpg

How do Climate Models Work?What do they tell us?

Aaron Donohoe

Cornish College Lecture

2/26/07


How do we model a physical system l.jpg
How do we model a physical system?

  • Simple Problem: What will the temperature of a pot of water on the stove be as a function of time?


What do we need to know l.jpg
What do we need to know?

  • Initial conditions

    What is the water temperature at the start of the problem? [To]

  • Forcing

    How hot/large is stove top? [F]

  • Internal Physical Parameters

    How much water in the pot. [M]

    The properties of water (how much heat it takes to raise the temperature of water-heat capacity) [C]

    How much time has elapsed (we can choose this) [t]


How does this information relate to the temperature of the water l.jpg
How does this information relate to the temperature of the water?

Temperature =

Initial temperature +

(Rate of temperature change) X (time elapsed)

(Rate of temperature change)=

Heating Strengh / (Water Mass X Heat Capacity)


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Putting it together water?

Initial temperature. [To]

How hot/large is stove top? [F]

How much water in the pot. [M]

Heat capacity of water.[C]

How much time has elapsed [t]

Temperature [Tf]

Tf = To + (F/CM) X t


We can then repeat this time step l.jpg
We can then repeat this time step water?

Time stepping is more important in systems where the

change in the system depends on the state of the

system


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Back to our pot of water water?

  • What does our simple model say about the temperature of water after a long time has passed?

    Temperature increases indefinitely


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Temperature as a function of time water?

Tf = To + (F/CM) X t

Does this really happen?

We’ve missed something that happens as the pot

approaches a boil


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From an Energy Perspective water?

  • For any system, we can say:

    Change of Energy =

    Energy into the system –

    Energy out of the system


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Our simple model from an energy perspective water?

Tf = To + (F/CM) X t

  • We can rewrite this as

  • (Tf)CM – (To) CM = Ft

Energy put into the system

Initial Energy

Final Energy

Energy Change =

Energy put into the system


Slide11 l.jpg

Our Model water?

Alternative Simple Model

Neglects Energy out of the system (Evaporating water)

The energy of the system is constant- the system is

in equilibrium


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Back to our pot of water water?

We will reach a steady temperature when:

Energy into the system = Energy out of the system

Energy from stove = Energy lost by evaporating water

This occurs at the boiling point


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Two different types of models water?

  • Transient model

    The system is evolving

  • Equilibruim Model

    The system is in balance- is steady

    Equilibruim models are useful because they identify the state the system wants to be in


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The simpliest equilibrium model of the earth water?

The Equilibrium temperature of the Earth represents a

balance between incoming solar radiation and outgoing

long wave radiation


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Simple Energy Balance model of the Earth water?

  • Earth gains solar radiation from the sun

  • Some is reflected back to space (rest is absorbed)

  • Earth emits long wave radiation


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What do we need to know water?

  • Amount of solar radiation [So = 1368/4 W/m^2]

  • Amount (percent) of solar Insolation reflected by the Earths surface/clouds [a=.3]

  • Amount of long wave radiation emitted by the Earth as a function of temperature

    [bT^4, b= 5.67 X10^-8 W/(m^2 K^4)]

KEY CONCEPT: A Warm Earth Loses more Energy


Finding the earth temperature from an equilibrium energy model l.jpg

Energy from the sun water? =

(Solar radiation reflected) +

(long wave radiation emitted by the Earth)

In mathematical form:

So = a (So) + b T^4

Solving for T

T= [(So/4) (1-a)] ^.25 = 255 k = -18 C = 0 F

b

Finding the Earth Temperature from an Equilibrium Energy model

TOO COLD – What did we miss?


Greenhouse gases l.jpg
Greenhouse gases water?

The atmosphere absorbs RADIATION EMITTED BY THE EARTH, but lets solar

radiation pass through

Some of this radiation is remitted to space- some comes to Earth


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Add a single layer (Greenhouse) to the atmosphere in the model

  • Solar radiation passes through the atmosphere

  • Long Wave radiation is absorbed perfectly by the land surface and the atmosphere

  • The surface and the atmosphere emit long wave radiation according to there temperature


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Determining Earth’s Temperature in a single layer atmosphere model

Surface Energy Budget

So

+ b*(Ta^4)

= a*So

+ b*(Ts^4)

Atmospheric

Long Wave in

Emitted

Long Wave

Solar in

Reflected

Solar

Atmospheric Energy Budget

b* (Ts^4)

= 2*b*(Ta^4)

Absorbed long wave from surface

Long wave emitted to surface and space


Determining earth s temperature in a single layer atmosphere model21 l.jpg
Determining Earth’s Temperature in a single layer atmosphere model

  • The equations get complicated fairly quickly but

  • We have reduced the physical system to a mathematical equation

  • The resulting atmospheric temperature is 50 C

  • It’s too hot now – what can we do



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One-Dimensional atmosphere model

Climate Model


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What would be the next level of complexity to add? atmosphere model

  • The equator receives more solar radiation than the poles

South Pole Equator North Pole


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1 Dimensional Energy Balance Model atmosphere model

  • Require an energy balance at each latitude


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2 Dimensional Energy Balance Model atmosphere model

  • Resulting Temperature distribution

South Pole Equator North Pole


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What did we miss? atmosphere model

  • Energy can flow from equator to pole


Slide28 l.jpg

Two-Dimensional atmosphere model

Climate Model


Slide29 l.jpg

We must add atmospheric motion atmosphere model

  • % Code up of North and Coakley's seasonal EBM model

  • % Simplified to eliminate the ocean domain

  • % Designed to run without a seasonal cycle and hence

  • % to stop once an equilibrium solution is reached.

  • % The model uses an implicit trapezoidal method

  • % so the timestep can be long.

  • %size of domain.

  • jmx=151;

  • % Choose parameters.

  • % scaleQ

  • if (exist('scaleQ')==0); scaleQ=1.; end

  • % OLR constant.

  • if (exist('A')==0); A=203.3; end

  • % OLR coef.

  • if (exist('B')==0); B=2.09; end

  • %heat diffusion coefficient.

  • if (exist('Dmag')==0); Dmag = 0.44; end

  • %heat diffusion coefficient.

  • Toffset=0.;

  • if (exist('coldstartflag')==1);

  • if (coldstartflag==1), Toffset = -40; end

  • end

  • %Simulate Hadley Cell with Lindzen and Farrell plan

  • if (exist('hadleyflag')==0); hadleyflag = 0.; end

  • %Remove albedo feedback

  • if (exist('albedoflag')==0); albedoflag = 0.; end

  • %heat capacity over land.

  • Cl = 0.2; % something small to make it equilibriate quickly

  • %time step in fraction of year

  • delt=1./50;

  • NMAX=1000;

  • %set up x array.

  • delx = 2.0/jmx;

  • x = [-1.0+delx/2:delx:1.0-delx/2]';

  • phi = asin(x)*180/pi;

  • %obtain annual array of daily averaged-insolation.

  • %[insol] = sun(x);

  • %Legendre polynomial realizatin of mean annual insol.

  • if (exist('S')==1);

  • S=S;

  • else

  • Q = 338.5;

  • S = Q*(1-0.241*(3*x.^2-1));

  • S=scaleQ*S; S=S(:);

  • end

  • %set up inital T profile

  • T = 20*(1-2*x.^2);

  • T=T(:);

  • T=T+Toffset;

  • %load T_final

  • Tinit=T;

  • %setup D(x) if simulating the Hadley Cell

  • %and calculate the matrix Mh and invM.

  • if (hadleyflag)

  • xmp=[-1:delx:1];

  • D=Dmag*(1+9*exp(-(xmp/sin(25*pi/180)).^6));

  • D=D(:);

  • [invM,Mh]=setupfastM(delx,jmx,D,B,Cl,delt);

  • else

  • D=Dmag*ones(jmx+1,1);

  • [invM,Mh]=setupfastM(delx,jmx,D,B,Cl,delt);

  • end

  • %Boundary conditions

  • %Set up initial value for h.

  • alb=albedo(T,jmx,x,albedoflag);

  • src = (1-alb).*S/Cl-A/Cl; src=src(:);

  • h=Mh*T+src;

  • %Global mean temperature

  • Tglob=mean(T);

  • % Timestepping loop

  • for n=1:NMAX

  • Tglob_prev = Tglob;

  • % Calculate src for this loop.

  • alb=albedo(T,jmx,x,albedoflag);

  • src=((1-alb).*S-A)/Cl; src=src(:);

  • % Calculate new T.

  • T=-invM*(0.5*(h+src)+T/delt);

  • % Calculate h for next loop.

  • h=Mh*T+src;

  • % Check to see if global mean temperature has converged

  • Tglob=mean(T);

  • Tchange = Tglob-Tglob_prev;

  • if (abs(Tchange) < 1.0e-12), break; end

  • end

  • %save T_final.mat T

  • % compute meridional heat flux and its convergence

  • a=6.37e+6; % earth radius in meters

  • [invM,Mh]=setupfastM(delx,jmx,D,0.,1.0,delt);

  • Dmp=0.5*( D(2:jmx+1)+D(1:jmx) );

  • divF=Mh*T;

  • F=-2*pi*a^2*sqrt(1-x.^2).*Dmp.*gradient(T,delx);


Increasing levels of complexity l.jpg
Increasing levels of complexity atmosphere model

  • Add a seasonal cycle (model can be equilibrium or transient still)

  • Make the system cover the whole Earth which is a sphere

  • Add more vertical levels

  • Add an ocean, plants, ice, clouds,etc.


Slide31 l.jpg

Three-Dimensional atmosphere model

Climate Models

(GCM)


Slide32 l.jpg

The start: equations of motion: atmosphere model

  • Conservation of momentum

  • (F=ma)

x

y

Hydrostatic balance

(neglect vertical accelerations)

z

  • Conservation of mass

  • (no thermonuclear reactions)

 = dp/dt

= pressure velocity

  • Conservation of energy

  • (ditto)

  • Equation of state

  • (a gas)

All the trouble lies in Fx, Fy, J


Slide33 l.jpg

Solve equations on a sphere atmosphere model

Solve in the vertical

Solve in the horizontal

grid

Spherical harmonics


Slide34 l.jpg

Gets complicated fast… atmosphere model

Begin to think about clouds


Slide35 l.jpg

Gets complicated fast… atmosphere model

even this requires:


Slide36 l.jpg

Riesner et al. (1998) atmosphere model

MM5 model: microphysical scheme

  • Six categories:

  • water vapour

  • cloud water

  • cloud ice

  • snow

  • rain

  • graupel


Slide37 l.jpg

The Earth Simulator atmosphere model


Slide38 l.jpg

Not just the atmosphere… atmosphere model


Slide39 l.jpg

Hydrology… atmosphere model

Cartoon of processes

Schematic of CCSM

Hydrology model


Slide40 l.jpg

Dynamic vegetation… atmosphere model


Slide41 l.jpg

Prediction of vegetation atmosphere model

types as model simulation

Progresses…


Earth system model l.jpg

And ocean model, sea-ice model, land surface model, etc… atmosphere model

3D atmosphere

Atmospheric

CO2

3D ice sheets

2D sea ice

2D land surface

3D ocean

Land biogeochemistry

Ocean biogeochemistry

Ocean sediments

“Earth System Model”


How well do they work l.jpg
How well do they work? atmosphere model


Slide44 l.jpg

How well do they do? atmosphere model

Precipitation

Mean sea level

pressure






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Projections of Future Climate atmosphere model

Projected change in surface temperature over the next 100 years using two different GH gas emission scenarios (IPCC 2001).

  • The warming is projected to be greatest at higher latitudes and in winter.


Slide50 l.jpg

The Columbia Basin’s snow in a warming world atmosphere model

20th c.

2020s

2040s

April 1 snow water equivalent (mm)