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

1. How do Climate Models Work?What do they tell us? Aaron Donohoe Cornish College Lecture 2/26/07

2. 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?

3. 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]

4. 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)

5. Putting it together 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

6. We can then repeat this time step Time stepping is more important in systems where the change in the system depends on the state of the system

7. Back to our pot of water • What does our simple model say about the temperature of water after a long time has passed? Temperature increases indefinitely

8. Temperature as a function of time Tf = To + (F/CM) X t Does this really happen? We’ve missed something that happens as the pot approaches a boil

9. From an Energy Perspective • For any system, we can say: Change of Energy = Energy into the system – Energy out of the system

10. Our simple model from an energy perspective 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

11. Our Model Alternative Simple Model Neglects Energy out of the system (Evaporating water) The energy of the system is constant- the system is in equilibrium

12. Back to our pot of 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

13. Two different types of models • 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

14. The simpliest equilibrium model of the earth The Equilibrium temperature of the Earth represents a balance between incoming solar radiation and outgoing long wave radiation

15. Simple Energy Balance model of the Earth • Earth gains solar radiation from the sun • Some is reflected back to space (rest is absorbed) • Earth emits long wave radiation

16. What do we need to know • 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

17. Energy from the sun = (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?

18. Greenhouse gases 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

19. 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

20. 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

21. 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

22. More complicated energy budget

23. One-Dimensional Climate Model

24. What would be the next level of complexity to add? • The equator receives more solar radiation than the poles South Pole Equator North Pole

25. 1 Dimensional Energy Balance Model • Require an energy balance at each latitude

26. 2 Dimensional Energy Balance Model • Resulting Temperature distribution South Pole Equator North Pole

27. What did we miss? • Energy can flow from equator to pole

28. Two-Dimensional Climate Model

29. We must add atmospheric motion • % 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);

30. Increasing levels of complexity • 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.

31. Three-Dimensional Climate Models (GCM)

32. The start: equations of motion: • 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

33. Solve equations on a sphere Solve in the vertical Solve in the horizontal grid Spherical harmonics

34. Gets complicated fast… Begin to think about clouds

35. Gets complicated fast… even this requires:

36. Riesner et al. (1998) MM5 model: microphysical scheme • Six categories: • water vapour • cloud water • cloud ice • snow • rain • graupel

37. The Earth Simulator

38. Not just the atmosphere…

39. Hydrology… Cartoon of processes Schematic of CCSM Hydrology model

40. Dynamic vegetation…

41. Prediction of vegetation types as model simulation Progresses…

42. And ocean model, sea-ice model, land surface model, etc… 3D atmosphere Atmospheric CO2 3D ice sheets 2D sea ice 2D land surface 3D ocean Land biogeochemistry Ocean biogeochemistry Ocean sediments “Earth System Model”

43. How well do they work?

44. How well do they do? Precipitation Mean sea level pressure

45. How well do the models do clouds?

46. How well do the models do sea ice?

47. How well do models do snow cover?

48. Modeled Future Climate Change

49. Projections of Future Climate 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.

50. The Columbia Basin’s snow in a warming world 20th c. 2020s 2040s April 1 snow water equivalent (mm)