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# Special Topics in Atmospheric and Oceanic Sciences ATOC 7500

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This 90 page Class Notes was uploaded by Jon Johns on Thursday October 29, 2015. The Class Notes belongs to ATOC 7500 at University of Colorado at Boulder taught by David Noone in Fall. Since its upload, it has received 11 views. For similar materials see /class/232049/atoc-7500-university-of-colorado-at-boulder in Marine Science at University of Colorado at Boulder.

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Date Created: 10/29/15

Climate models Reminder NCAR field trip 0 Meet at NCAR Mesa lab 230330 check out displays in lobby 0 Presentations by Phil Rasch and Bill Collins from 300400pm in Chapman meeting room Carpools David N 04 0 Dave K 134 0 Scott ONE WAY 103 L2 QG model for general circulation I How do you de ne the climatology In What Way does the simulation look like the real general circulation In What Ways does it not I How do you determine if the model is spun up How does the behavior change With tunable parameters Rossby radius diabatic heating mte drag coef cient I How can you include topography hoW does this change the simulation I What are some Ways to better choose the forcing How Would you expand this model to be a fully edged quotclimatequot model Zonal mean 1 not 11 Zonal mean zonal wind Vorlicvty any 10 x 10 vommy day 20 X m 2 2 gt u gt n 2 2 20 no A 20 do so A vonicily day an x 10 vnnic y day an X 10 2 2 gt o n 2 2 20 do an I 20 an an venich day 50 X 0 vomcny day an x m z z gt o gt o 2 2 o 20 m so 60 Lower layer Vomuily lime 50an Days Rossby radius Wmme W w M M r 1 WWW g Nlayers eg n3 Radiative equilibrium dTdt KCTeq 7T Teg radiative equilibrium pro le gives jet Which is baroclinically unstable Thus heating pushes atmosphere unstable While dynamics tries to return to isothermal case Circulation is competition and balance between the two Simple pro le give by Held and Suarez 1993 N3 means temperature at 2 layers Relaxation to radiative equilibrium 3 layer Some limitations of QG dynamics Can not predict stability recall lLZ f2cr5pZ can obtain spatial variation in temperature at each layer but not mean temperature This must be deduced from some other method or use n more complete equatio of a normally large tropical Wav I Model begins to fail in the tropics including development es Some of these aspects can be corrected by adding for instance strong dam 39 39 sion in the tropics Also note We are looki onl at the nondivergent part of the ow eld QG also gives the divergent part associated 39 stretching and vertical velocity ie the omega diagnostic equation This Would be needed to see the Hadley cells Also needed for trace gas transport Projects Option 1 start with QG model Option 2 build specific model Option 3 build some part for another model The project is science driven Modeling will be used to solve the problem but is only a tool Bad example I will build a model of clouds Good example I will show test if cloud properties change surface energy balance by building a model of clouds Option 1 will need coordination for group success Equot W 5 U 6 gt1 00 The effect of clouds on the regulation of the underlying SST over the tropic paci c Lin Cirrus clouds Chuck Effects of the ocean in the gem and the importance as a heat source to the atmosphere Xylina Role of cloud properties in the mass balance of seaice or role of vertical motions in the partitioning of phases in clouds with particular focus on mixedphase cloud conditions Matt S Accessing changes in climate from changes in sea ice in the Arctic Varability and mean state with ice loss Matt H By adding sea ice to an atmospheric model will increase atmospheric heat transport from the equator to the poles This will occur due to a larger thermal gradient from equator to pole Dave Add a simple sea salt aerosol component to our gem and do some RT calculations on the aerosol column and do a quick comparison to aeronet data Lansing Parametric study of teleconnection between a tropical wavemaking anomaly and high southern latitude climate especially the southern annular mode Scott QG mudel un a beta lane wnh mulurgnd rnethens Jasnn weather eventsannz ahzzarn and znnsHet speu staru 48 p emuus aseertar p n nee Whreh t rs hest reremsten why physres rs rrnpnrtantv Darren a Mars G CM tn Antar e rsn apes and rnruennrahseale SE surraee lempemlure mnauuns nunngthe 1ast glanal penud3 r7 kyr age Are rnnrsture snuree reguns m phase Dr nut er phase Dr naLhEr wnh Amarene tern erature ehange SE meand ENSO Anme Deesa lEI detrasemrelauvehumd ymtrease sml evapnratren ennughte where atrnnsphene temperature rs un avenge h her than s r1 ernperaturev ka e erthe Amazun Regunusmg the buundary eund mns rern NCEF data wnh a mudel er sur ee and atmusdphmc uxes errnersture and rsntep es usmg Raylegh theery an srrnple cundenEhunempumuun equauun D erek QG model 3r1 2 g VVqfKv q DltH Forerng H rs a strong eontrol on ehrnate r e govems equatorrpole energy 1mba1anee Model rs eonservatron of energy momentum and rnass Determmes how the ow can resolve the 1mba1anee r e make weather to transport heat 7 whreh we see as PV Additional components for GCM landsea mask topography possibly gravity waves hydrologic cycle convection clouds rainlatent heating radiative transfer atmosphere and surface GHGs and aerosols Land surface exchange drag latent and sensible heating Ocean slab EBM advection upwelling dynamics Sea ice Coordination Dynamic core adiabatic part Interface for physics diabatic forcing terms Number of levels How to couple with other components Interface subroutine Eg each person would work on a subroutine and adhere to the communication protocols Detailed time lines since all work depends on other components 4 April 11 April 18 April 25 April 2 May 8 May Suggested timeline Decide organization infrastructure etc Implementing code Finalizing code Performing experiments Finalizing experiments and write up Project due Vorticity on a sphere Spherical harmonics some useful properties w1 ZZWWWW szn n2 yn as imw2 a m lt1 2gtai n 391u391ltn 1gt6 uf1 2 2 gm n m 4112 1 Advection with spherical harmonics J a m amp VVq V39Vq at Non divergent altF G a m 1 EMF J 30 0FGn COSZ 81 cos a FZCOS uq Gzcosg vq 6 a A model Remember q the potential vorticity q f a a 1 V Vq KVZq Dq a 4g o g e 0 3 A e 3 a v a a a 65 so 9 2 got Lab assignment What will the weather be on Tuesday in Boulder Make a 5 day weather forecast from Thursday using the non divergent barotropic vorticity equation on a sphere v x7 f KV2 D Use a centered finite difference for time Use spherical harmonics to represent the spatial structure Include diffusion and friction extra part include mountain topography Implementing model code What is the state variable a series of complex coefficients Obtain vorticity field on a grid Derive spectral u and v from spectral vorticity Obtain u and v on a grid Calculate nonlinear uxes on a grid m and mi Assign drag momentum source Convert uxes to spectral form Compute ux divergence in spectral form alpha just the advectiorL as per last wee Time step the state variable in spectral form Spherical harmonic synthesis Spherical harmonic analysis Remember to output state every so often say 6 hours This is just the same as last week but now we also have u and v Initialization We start with known observed geopotential Can we use this to initialize our model Notice from momentum equation drag acts on relative vorticity not absolute vorticity How can we do this in our model hint think about the alpha operator See code dcnATOC7500week09 Fft99lF SphericalF90 Also vorg200603131800txt and topogtxt Also also advectionf90 Spherical harmonics subroutine sphiinitnlonnlatntrn1triang subroutine sphiana1nlonn1atn1eVgridntrnspec subroutine sphisynthntrnnleVspecnlonn1atgridnexin subroutine sphianalialphan1onn1atn1eVaunjtagbunitbgntrna1phas subroutine sphiuvcosntrn nleV vors divs ucos vcos subroutine sphilaplntrnn1eV drnn1aprnnpowiin Vorticity and streamfunction How does the result change if I aim to plant on the other side of the yard where the soil is clay rich How does the result change if we include a diurnal cycle Ad 5K What is the time mean soil temperature Is the result the same for all time steps Does the scheme conserve mass Does the scheme conserve variance Is the system positive de nite Is the phase and amplitude of the seasonal cycle the same at all depths What is the role of the surface heat ux and the magnitude of D in the result Do your predictions match the date given by the Farmers Almanac The 2d realm Advection in 2d Diffusion in 2d Can it be done with nite differences f r arms num v m a quot 39 39 Gr new 22 Tue 25 rsa mus 5mm Geapa enha HeigM am Vach e e vcEs quot A s 39 m Anoysis 122 Tue 23 m 2mm 5mm Genpoten a Haws dam vm m ty mim Vorticity advection absolute vorticity conserved 77Cf f 3 f2sgtsin6fo y d 10 iwgvgv dl at 6x 6y 5 a a 5 au f lv fl Vorticity and stream function 011 Stream function 6y ow along contours a v z i 6x For geostrophlc ow and QG we can thus deduce streamfunction from g geopotential height W 1 f0 Vorticity is the Laplacian of the C V2 1 streamfunction Thus streamfunction looks like a smooth or diffused version of vorticity Assignment What will the weather be next Tuesday Make a 5day weather forecast from Thursday using the non divergent barotropic vorticity equation on a beta plane Initialize with observed 500 hPa geopotential height Output predicted geopotential height vorticity and u and v wind components Choose a grid of nlon64 and nlat 16 with a 500 km spacing and a time step of 1 hour Domain is cyclic in longitude and bounded v 0 on the north and south boundaries Choose boundary condition quean at N and S edges bP N Coding challenges Calculate streamfunction from geopotential Calculate vorticity from for streamfunction Given vorticity obtain streamfunction Calculate u and v wind from streamfunction Predict evolution of vorticity 1 hour time step Loop for 5 days Code elements suggested subroutines Finite difference X and Y derivatives Finite difference Laplacian Finite difference Poisson solver invert Laplacian The art of climate and environmental modeling quotSeveral sciences are often necessary to form the groundwork of a single artquot Mills 1843 quot Science is knowledge which we understand so well that we can teach it to a computer and if we don t fully understand something it is an art to deal with itquot Knuth 1974 Climate system models Atmosphere momentum temperature mass humidity t L Ocean unomenmm temperature alinity m Evolution of Climate Models w um Elm mos mm mumm um um mm ms maul Wuwy 0 5 sun 11he davelapmvnl av climale may my me I35 25 years showing how mu camnonams are um develouen sevzrakaly and late cuuuled mm comurshen ans 5 F1 mnavem climaxe mod P0 p Q 012 Parts of models State variables Conservations laws mass momentum energy Tendency equations Analytic or numerical methods Discrete solution methods finite differences spectral methods others Codes to do it Hierarchy of models All do fundamentally the same thing but different levels of complexity Examples 1 2 3 e V39 Energy balance model Energy balance model With parameterized heat transport Energy balance With physics transport e g cyclone model Energy balance With explicit transport eg QG model We built 10 climate models 10 steps to making a climate model Onebox energy balance model EBM With a greenhouse effect Midlatit39ude cyclone Advection in onedimension Advection With Fourier coef cients Soil heat diffusion in one dimension Vorticity and strearnfunctions Tracer transport on a sphere Vorticity on a sphere 0 General circulation models dry 1 Climate models HHPWSQM eWF Note 10 steps plus an initial conditions means 11 models Energy balance models imple Conservation of energy Few p arameters Can have feedb acks Class ofmodels b ox models Solution reaches a steady state Stateoftheart until 1950 Important models by Sellers and Budyko on radiative feedb acks and albedo A model like this lead Arrheneous to realize that C02 due to coal burning will 5 39 earth s temperature 1886 I quotHMx arguing ever since Dynamical models Solutlon does not have a steady state but thae ls a quaslrequlllbrlum ortznt ayror e g penodlc an ow musthave3vana les lefamcebetween Weatha and cllmate Important work by Lorenz 1960571980 on predrctabrlrty Also Cllmate lowrdlmenslonal attractchY whlch has srnce lost favor as lt appears a c nsequence ofslmpllclty Slmllar low drmensronal clrmate models have lead to elegant but poorly tested theorres for rce sheet dynamrcs the role ocean crrculatron ln clrmate and the changes m the carbon cycle Nume cal methods Advectlon and dlffuslon are baslcbulldlngblocks of compler models Both descrlbe some klnd oftmnsport stabrlrty rssues Accura lssues Also elllptlc solvers common ln uld ow problems cm fallurewasthemaln reason thatRlchardson s r l famous flrst numerrcal weather predrctron by handl falled rn 1922 e e to predrct weather for 24 hours over orthem Euro e wlth 6 hour tlme st 5 Thls the most lmpresslve falled model expenment It was chamey 40 years later who worked out why Also Wrote Blg wharlx have latte whorls that feed on we velacl r and latte whartx have xmmlerwharls andxa an to lecoslry Weather predicti Initial value problem Limited predictability Nonlinear advection vorticity advects itself Main features well captured but many details were not Famous experienced in Chamey et al 1949 which we reproduced was one of the rst uses of computers Also demonstrated that weather prediction was possible Convinced US government to invest in numerica weather prediction 5 years ahead of Rossby s group in Swe n S Torriequ A W n General circulation Requirements Heating in tropics cooling in polar ion Mechanism for generating eddies Multilay ers for b aroc linic deve opment Conservation of mass momentum ant energy rick via QG and use of PV simpli es equations and assism book 39ng but retains 90 ofthe story Boundary value problem mmuu Important experiment by Phillips 1956 2layer QG on betaplane captured main aspects Essentially what we did but we used a sphere Also we used a spectral method that became popular in the mid1970s due to accuracy and e iciency Stateoftheart climate models As complete as possible Many individuals and fields of expertise inc u e Physics chemistry biology Lar e scale processes and small sca e process Enormous number of parameters Match observations exceedingly well Used as a centerpiece of climate change studies Especially important for understanding linkages Trusted by lots of informed non experts Not trusted by skeptics Not trusted by expert modelers re mass humidity m winwmrn Earthsystem Models of Intermediate Complexity EMICs Typically low resolution Possibly truncated dynamics Simple treatments of physics Sm all number of parameters Represent main features of climate well details are not so good Runs fast Have been used recently for paleoclimate experiments Becoming increasingly popular since full models are to costly to run Also they can be trusted since errors are easier to characterizemderstand Typically like stateoftheart climate models of the 19805 Very much like our group model Where is the art Where is the science What you should see on an exam if we had an exam Types of problems adyection diffusion wave equations elliptic equations Finite differences derivations from Taylor series basics of stability analysis existence of nonlinear instability Errors 7 amplitude phase dispersion Spectral methods all done with coefficients Spectraltransform methods linear parts spectral nonlinear on a grid Reasoning for parameter selection Fair use of parameters not too many not too few Modeling skills Fortran basics Structured programming breaking a complex code into smaller doable chunks possibly in subroutines Debugging strategies Strategies for testing and validation Is the strange result a bug bad physics or a Nobel Prize Visualization IDLMatlab time series line graphs contour plots map projections Basic analysis means variance differences More detailed analysis Fourier analysis EOFsPCA statistical tests Should have con dence in understanding uses of climate models and skills to build models of your own Reminder Project due 4 pm Monday paper copy Drop into my office Ekeley 8234 Leave in my mail box 3rd oor box 151 Also please return any books Remember also that your report should demonstrate application of what you have leamed this means you should refer to topics covered in the lectures stability accuracy methods conservation Diffusion and implicit solutions Spectral advection Does the scheme conserve mass Does the scheme conserve variance Is the system positive definite Are there errors in phase Are there errors in amplitude Are errors dependent on the shape you are advecting Is the computational mode an issue in this model If you truncate the solution does the model still behave well 1 m D T W yva r g r A 9 H s m m y m M I r r u T x m H 4 n r H x quot3 a l M 3 x a m Vvue nbp es m K b M W C 027 mm 02 w 3 6 J C053vm10 C053vnu02 Truncation Truncate Ak state Bk Will follow Assignment Remember to perform stability analysis of your models could present this as an appendix etc Lab assignment In Boulder the ground is frozen during winter and plants can not grow in frozen soil During spring the soil thaws 1 When should 1 plant tomatoes 2 When will the tomatoes die from the first frost Build a model of the top meter of soil Assume growing can start when the top 50 cm of soil thaws Heat flux through soil can be written proportional to the temperature gradient such that local temperature change convergence of flux is written 2 a K a T at 622 Solve for Ttz with appropriate value of the diffusivity K and a solution c eme The air temperature can be taken as mean of 280K of with an annual cycle of with an amplitude of 20 K Energy balance model Building models What is the actual problem How much complexity ie de ne scope What are the state variables ie de ne system Are there conservation laws Are there rules for changes in state variables ie what is the mathematical model How should this be implemented in a computer ie discrimination numerical methods What are useful diagnostics ie output that is needed to answer science objective How do you test the model is correct How do you double check the model is correct Global energy balance model ie radiative equilibrium dT S 01 1 06Z SUT4 Allows us to predict global mean temperature if we know the albedo emissivity and the solar constant Time stepping Can write change as change in state over some time That is a nite di erence TnOW W N Tam Tm T 7 future E N time difference 7 At As such we can make a prediction for one small time step At The advance in time future T becomes now by one time step and start again This is an integration of the di erential equation in time Today s objectives A How to log onto atoc 2 How to write and compile a Fortran program 3 How to write and compiler a useful Fortran program 4 Think about some experiments with your shiny new EB Remember use class web site and our wiki as a rst source of help Experiments discussion What is the radiative equilibrium temperature of Earth Does this depend on the choice of the time step Change the heat capacity so as to model the upper 70 meters of the ocean rather than the atmosphere Cp 295X108 JK Does the equilibrium temperature c ge Is the mean temperature the same if there is a diurnal cycle Annual cycle Glacial cycle If the sun should stop how long does it take for the temperature to become a factor of e391 its equilibrium value ie What is the efolding time for both the atmosphere and upper ocean How can the efolding time be found analytically from 1 What are some limitations of this model and What are their consequences How would you construct a coupled atmosphereocean model With this level of complexity For next Tuesday s class Onedimensional advection Advection physical interpretation 1 ua 1 at 6x udt 6q6x gt 0 6q6t lt 0 Advection describes the transport of a quantity into parcelregion of air xed in place ie a translation of the pattern surface no modi cation following the motion Total derivatives for Lagrangian View following the motion Partial derivatives give Eularian view xed in space Today s assignment Construct a nite difference model to test the dependence on the model accuracy on the details of the scheme Test sensitivity of the model results to the Courant number each of u AX and At strength of time lter zero is a good starting place different shaped initial distributions of q LANt I 51 ua 1 at 6x n1 nil 2u At qz1 qzril q qz 1 2M Consider difference between stability and accuracy Things to consider in building model 0 What type of finite difference scheme 0 What are the boundary conditions 0 What output do you need 0 How can you measure errors 0 What is truth What are 6 ways to double check the code Tasks Code tasks Write a subroutine to compute the tendency Write a subroutine to do the time stepping Output simulation results state only at some fraction of time steps and plot with IDL Output error diagnostics and plot with IDL Science tasks Does the scheme conserve mass Does the scheme conserve variance What is the critical Courant number stability try withwithout a time lter Is the forward time difference truly unstable as predicted Advection Discussion Is the initial condition balanced Is there an initial adjustment period What is a range of prediction in days Does this depend on the initial conditions What is the average time mean temperature difference How does this compare to the radiative equilibrium temperature difference Is the poleward heat transport always positive What is the mean windspeed Hint consider the temporal variance in windspeed Are your results sensitive to selection of free parameters Weather and climate In the model 0 how do you change the weather 0 how do you change the climate how do you DEFINE the climate Phase Puman 7 mm Days Devaun Pavamems Cyclone El 5 O n 5 Anticyclone 1 4 5 4n ran an Lorenz 1963 Low dimensional attractors Unclear if they are a good analogy to the true climate system but they have some appealing characteristics Nonlinear advection Linear models have relatively simple behavior consider an damped harmonic oscillator We may infer that nonlinearities make models interesting and often require numerical methods The cyclone model has interesting behavior because of the nonlinear advection product VT Advection is one key aspect of hydrodynamic ows Problems in time and space classes of 2nd order PDEs all z 02 all Wave equation hyperbolic 6t2 6x2 2 2 6T a T fxy Poisson equation elliptic 6x2 6y2 6T BZT Diffusion equation parabolic 6t 6x2 6T C 6T Ad f d t t t at 6x vec 10n equa 10n 1rs or er Plus suitable boundary and initial conditions These are the simplest canonical forms they get nasty quickly Advection Consider the conservation of a quantity q qXt could be COZ potential temperature ozone potential vorticity water 4 dt S is some source gig was 7 dz 61 dt x eg S 0 ldimensional advection equation one way wave equation 6g More generally a Z V 39 Vq S Finite difference schemes Upstream Flux in ux out u quotq 7 LOAx quot gt 0 at 7uq liq Ax ult0 More formal finite differences 39 Ward m 31 6q z qr a z u qr q centered in space at N 1 2m Cemered in time 6q m qr qlquot 391 u qr q centered in space at Mt ZAx One can of course construct many others Stability analysis 512 1 at 6x Von Neumann method Consider known solution q A eXpi 1 Check how amplitude changes with time 7 If A continues to grow with time then unstable 7 If bounded then stable but may not be accurate Certainly errors in amplitude but also consider phase errors Courant number C uAtAx lt 1 for stability This is the CFL condition CourantFredrichsLevy CFL condition Information can not move more than one grid box in one time step Al C lt 1 Ax Time it takes to move across one box is vAx must be greater than the time step Thus maximum speed of any stable motion is limited by the combination of Ax and At Another interpretation 611 1 Cy C6111 So advective translation is an interpolation in space If C gt 1 then it is an extrapolation and susceptible to exponentially growing errors amplitude grows each time step Lab exercise How accurate is the finite difference prediction Construct a 1d model of linear advection ie u constant to represent the zonal ow Develop metrics of error consider phase and amplitude Do results change with wave number time step Courant number shape of function time filter Note this assignment will be useful for the mid term assignment Spectral advection model Time step loop 6q u6q Or part 2 Model in Ak 6t 6x 905 2161141691 qk Ake k1 1 0A a iquk a itBkzkAk q q3391 2Ata q a Forward Fourier transform A Ar 2Ain a qn1 ZArHlexki It For output Inverse Fourier transform doo d91s emu Tips for implementation Begin with the advection model from last week Use 32 points from west to east and choose u to give one complete cycle every 20 days Set up a test to ensure that you can for the FFT for q Con rm the inverse FFT should give you back q For the rst step nd the derivative with Fourier coef cients be careful with real and complex variables It might be useful to output and plot the dqdX from both nite difference and spectral form to double check Increase the number of time steps Palt 2 Modify the time stepping to use Fourier coef cients rather than grid point form Do the FFT for output Fast Fourier Transform Takes grid point qX values and returns Fourier coef cients Ak Takes Fourier coef cients Ak an returns grid point values qX Subroutines in fft99F call 5etfft call fft99l call fft99l Initializes transform once Forward transform Inverse transform 1 1 nlon MUST be even and best if it has prime factors of only 2 ie 4 816 32 64128 The array of data going in and out of the subroutine must be nlon2 in length The input data is OVERWRITTEN with output FFT example The subroutine is in the file dcnATOC7500week05fft99F See the example in dcnATOC7500week05fftstubf90 compile with f90 fftstubf90 fft99F Also dcnATOC7500week05plotqpro can be used to plot the example output Discussion and experiments Does the scheme conserve mass Does the scheme conserve variance Is the system positive de nite Are there errors in phase Are there errors in amplitude Are errors dependent on the shape you are advecting Is the computational mode an issue in this model If you truncate the solution set coef cients of high wave number to zero does the model still behave well What have we gained from going to a spectral model Vorticity on a sphere Announcement Thursday April 6 we will meet at NCAR Talks by Phil Rasch and Bill Collins two of NCAR climate model experts 0 Interest in a tour of computing facilities 0 Transport car pool bike note hill NCAR shuttle I will be driving and can fit 4 people Discussion What fraction ofpollution emitted in Colorado in January is in the Southern Hemisphere by the end ofJune How did you choose the diffusion coef cient How does the diffusion change your answer How does the frequency at which you update the wind change your answer What do you expect happens to the trace r distribution after some long integration time Dav mm c Om Spherical harmonics some useful properties v41 Z EVILquot Pf A1 m nnl m Vivn Vn 6942quot m zm 01 Wquot 6 m m m m 12 a n8n1Vn1n 15n Vnil gm nz imz quot 4n271 Advection with spherical harmonics VWVVq aq39quot at a n 1 6F 00 F G m quot quot a quot cosz 02 COW w COS F uq G cos vq a a A model Remember q the potential vorticity q f a q VqqKV2q Dq 6t a g 0 9 0 9 6 o 609093 o 9393 Pa Lab assignment What will the weather be on Tuesday in Boulder Make a 5day weather forecast from Thursday using the nondivergent barotropic vorticity equation on a sphere 0 VVltfgtKV2 D Use a centered nite difference for time Use spherical harmonics to represent the spatial structure Include diffusion and friction extra part include mountain topography Implementing model code What is the state variable a series of complex coefficients Obtain vorticity field on a grid Spherical Derive spectral u and v from spectral vorticity harmonic Obtain u and v on a grid s nthes1s Calculate nonlinear uxes on a grid y uq and vq Ass1gn drag momentum source Spherical Convert uxes to spectral form harmonic Compute flux divergence in spectral form alpha just the advection as per last week analysis Time step the state variable in spectral form Remember to output state every so often say 6 hours This is just the same as last week but now we also have u and V Initialization We start with known observed geopotential Can we use this to initialize our model Notice from momentum equation drag acts on relative vorticity not absolute vorticity How can we do this in our model hint think about the alpha operator Weather prediction model a a a at u f V f zi 2 a 31 1 foqD Vl 14 3y v 3x n n as quot gal gij12Ata t iJ39 Piquot F if1j Gan Girlj71 at 1 2Ax 2M fzzgsin zfohgy Fu f GV f Assignment What will the weather be next Tuesday Make a 5 day weather forecast from Thursday using the non divergent barotropic vorticity equation on a beta plane Initialize with observed 500 hPa geopotential height Output predicted geopotential height vorticity and u and v wind components Choose a grid of nlon64 and nlat 16 with a 500 km spacing and a time step of 1 hour Domain is cyclic in longitude and bounded v 0 on the north and south boundaries Choose boundary condition Vzmean at N and S edges Model schematic Calculate streamfunction from geopotential Calculate vorticity from for streamfunction Given vorticity obtain streamfunction Calculate a and v wind from streamfunction Evaluate advection as voriticity ux diveregence Predict evolution of vorticity loop for 5 days bP Nt Code elements suggested subroutines Finite difference X and Y derivatives e g fddxc fddys Finite difference Laplacian eg fdlaps Finite difference Poisson solver invert Laplacian Suggested coding plan 39 Start With stub configure to run for 1 step 1 Read initial geopotental and get streamfunction 7 Compute vorticity 7 Compute u and v 7 Check the output is correct 2 Build Poisson solver subroutine most of the work today 7 Convince yourself that can recover the input streamfuction from your calculated vorticity 3 Try more steps With the Rossby wave test 4 Try the real initial conditions 39 Optionally add drag and horizontal di asion Possible starting point Code stub dcnATOC7500week07ndbvmstubf90 And a IDL plotting script dcnATOC7500week07bplotpro Also same directory a test initial conditions zSOOtest64x16txt and the real initial conditions z500inic64x l 6tXt EBM and beyond 0d EBM cpiZ l a 80T4 What is the temperature of the Earth choices for parameters Time step issues Able to predict meanstablesteacly state OK but what about change Can we predictmodel all we need to know i e given hindsight is the scope appropriate for our science question Discussion What is the radiative equilibrium temperature of Earth How does this depend on the choice of the time step Change the heat capacity so as to model the upper 70 meters of the ocean rather than the atmosphere Does the equilibrium temperature change Is the mean temperature the same if there is a diurnal cycle Annual cycle Glacial cycle If the sun should stop how long does it take for the temperature to become a factor of e391 its equilibrium value ie What is the e folding time for both the atmosphere and upper ocean How can the efolding time be found analytically from 1 How would you construct a coupled atmosphereocean model With this level of complexity What are some limitations of this model and What are their consequences What did didn t we gain Sarne balanced solution as analytical form Gain time dependent behavior Could not model greenhouse effect Can we answer the science question can we do hypothesis testing Global Heat Flows Reflected Solar incoming 235 Outgoing Fiaioiiation g 1 Solar Longwave iorw m392 Fiadiaiian Radiation 342Wm39 235Wm392 Reflected by Clouds and tmoaphere 7 i b quot 77 V 1 a 1 165 2 i 557 we a 4 T 555 g 39 lama Each ux arrow changes energy ehimd nameno 99 of the atmosphereclimate system A look at forward difference Formal almost origin Size of errors Better selection Energy balance and greenhouse effect S I E 0T4 Z E0T4 1 20 00 E039T E0T4 No atmosphere Greenhouse atmosphere What are the most important greenhouse gases How do we represent them Energy balance With latitude I1aS4 I f 113 cos I I TEP0T4 I I I ET0T4 I IHI I TI cos Tropics cl O to 30 Polar region 4 60 to 90 T gt ET ie energy surplus P lt EP ie energy de cit So there MUST be a poleward transport of energy Energy balance means T P ET EP So IT ET FIP EP FO Limitations of 0d EBM Lacks nonlinear or other interesting components Very simple response to external forcing oonsider some small perturbation Thursday s assignment Science question What is the global climate sensitivity Build a model that includes a greenhouse effect and estimate the climate sensitivity De ned as temperature change when CO2 is doubled Things to include in the model An atmosphere Some change in longwave due to CO2 Water vapor feedback Use centered time differences note trick for rst time step Seasons Albedo feedback Tropical and polar boxes does your model predict polar ampli cation Volunteer to present schemeflowchart for a model at the beginning of next class but discussion on the class wiki starts anytime Tracer transport model plume dispersion and the spectral method Discussion from last week Does your model correctly simulate Rossby waves Does your model simulate all wave number just as well How accurate is your prediction Does your solution remains stable for long simulations Why How does horizontal diffusion affect results How does the addition of surface drag effect results What else can we add to the model to make it more realistic 1 500mb eeupotenuax Heights dam Vm city e isec ms mums crs Anu ys xs 002 Tue 07 W 2005 Last Thursday 5 days today Issues Inaccurate advection ultimately screws things up need more accurate advection Advection is ultimately dispersive ie pattern smoothes out need more accurate advection Seems to be problems at boundaries consider a sphere For long term grid point models need more accurate way of representing advection but we know spectral models have perfect advection Spherical harmonics some useful properties v41 Z ZWZPXUWW Vzw nn1 Z a 0942quot m 21m 6 W71 6 m m m m m 1 Mai mam n 1gtsn mi Advection with spherical harmonics aq W 7VV iv V at q q Nondivergent 8g 70 FG 39 at n m 1 5Hquot 50quot was 7 coal M cos M cos F uq Gcos vq L1 a Assignment What fraction of pollution emitted in Colorado in January is in the Southern Hemisphere by the end of June On a grid nlon64 nlat 32 construct a 2d model in which wind is read in and a constituent is advected on a sphere using a spectral method and spherical harmonics Wind data is from NCEP reanalysis and available each 6 hours I will provide this Add a source as a known ux of tracer into the atmosphere from the grid point nearest Colorado The model 661 at V Vq S S is a source ux Use a centered nite difference for time Use spherical harmonics to represent the spatial structure As such the spatial derivatives are known analytically Implementing model code What is the state variable a series of complex coefficients S h 39 1 Read wind on a grid hp em armonic Obtain tracer eld on a grid synthes1s Calculate nonlinear uxes on a grid uq and vq Assign the surface uxes S Spherical Convert uxes to spectral form harmonic Compute ux divergence in spectral form analysis just the advection as per last week Time step the state variable in spectral form Remember to output state every so often say 6 hours I will provide subroutines to work with spherical harmonics Discussion questions How did you choose the diffusion coefficient How does the diffusion change your answer How does the frequency at which you update the wind change your answer What do you expect happens to the tracer distribution after some long integration time Note there will be no stub but there will be examples of how to use the spherical harmonic transform subroutines Diffusion and implicit solutions Lab assignment In Boulder the ground is frozen during winter and plants can not grow in frozen soil During spring the soil thaws 1 When should 1 plant tomatoes 2 When will the tomatoes die from the first frost Build a model of the top meter of soil Assume growing can start when the top 50 cm of soil thaws Heat flux through soil can be written proportional to the temperature gradient such that local temperature change convergence of flux is written 6T BZT K 2DTa T1 at Hz Solve for Ttz with appropriate value of the diffusivity K and a solution scheme The air temperature Ta can be taken as mean of 280K of with an annual cycle of with an amplitude of 20 k and the surface heat exchange coefficient can be taken as 1186400 seconds Implementation considerations 39 What are the state variables 39 How does time stepping work 39 How do you include the surface heat flux 39 What output do you need IDL task make a contour plot of T as a function of depth and time 39 If you re feeling daringfinished early what about an explicit scheme What about a spectral scheme Tridiagonal solver 39 Solves matrix system of equations T A Th1 soTn1 A391Th A is a tridiagonal matrix which has a well known efficiently calculated solution See dcnATOC7500week06tridiagf90 and dcnATOC7500week06tdstubf90 To compile f90 tdstubf90 tridiagf90 Remember the midterm assignment Models of the advectiondiffusion equation 2 a q ua qKa q at 8x 8x2 Greenhouse model The model dT S Cam 7 f17 1 25m 0T4 EmLup H t Solved with a centered in time nite difference This is a starting place You are the modeler so you can add change simplify etc etc as you see fit Objectives for today Model complex coupled set up Implementation and tricks of centered difference Parameter selection Debugging strategies Write output to a file to be read into IDL see example dcnATOC7500examples Importance of feedbacks when forced Tipssuggestions for implementation Start with the one parameter EBM from last week and change the time stepprng to be a centered d1fference leapfrog scheme Modify the code to compute the radiative equil39brium of the two reservoirs ocean and atmosphere at the same time Modify the solar ux partition absorption between atmosphere and ocean and add extra longwave radiation terms and the sensible and latent heat ux Add equation for changes in emissivity due to C02 Add equation for changes in emissivity due to water vapor Optionally add a second box to represent the polar regions and add a horizontal energy ux This model is substantially model complex than last week An astute modeler will know to expect bugs and plan ahead to minimize efforts finding em Finished early Time to update the wiki Build confidence and get a sense of the limitations of your model by trying many sensitivity experiments Also try a model with more boxes tropics poles maybe a stratosphere Model behavior predictability tempemtlre 3 on O 2 3 O Plots go well op the wiki 2000 000 6000 8000 time steosj CD See example Fottran and IDL plotting code in dcnATOC7500week2 on atoc Mode ouput temperature 0 if 20 30 40 50 See example Fortran and IDL plotting code in dcr ATOC7500week2 on atoc Discussion How can we validate the model How do we double check the model What are 3 more ways to double check the model On what criterion due you choose the drag coefficient k What fraction of the temperature change is due to the water vapor feedback Is the climate sensitivity the same for different epochs Today COZ365 ppmv preindustrial 280 ppmv Last Glacial Maximum C02 200 ppmv Is the water vapor feedback as important for all epochs Is the water vapor feedback linear Are you confident in your findings Why How does the model change if there is also an albedo feedback Is the water vapor feedback more or less important for snowball Earth Consider adding a secondbox so that there is both a tropics and a polar re ion With some meridional energy transport How does this change the climate senSitiVity Which aspects are most crucial Points from greenhouse model Climate sensitivity depends on strength of feedbacks Nonlinear feedbacks can amplify or damp variability or changes in the mean climate Feedbacks and gain AT ATforcing ATfeedback AT fATforcing Box models Quick very general Can capture fundamental behavior even when governing laws are not well known e g impact of national policies on climate Quantify interactions strength of feedbacks Assess time scales a related issue Examples swim1w maul x mm vrmsium 71 39P am mqu nuan quotu Fig 1 Simple model or the carbon cycle in mum Vennquot u r x quot39339 m WW i 133quot weik 5 1 a wan Global water balance ux model Atmosph re Gesteln Wisequot 1556 Sediment am mams1 Dynamical models start with a box model approach Consider a onelayer atmosphere with a tropical box and a polar box 20 m E z 3 10 3 k t 5 5 lt lt 0 0 0 D 20 30 40 50 ED 70 BO 90 O 10 20 30 4O 50 60 70 ED 90 LATlTUDE LATITUDE v mncl39on r kniiudv m Ilciphl in under minim rihcm w 1 William lmli7nntu llCdl runs lcmpumlurc gmdicni mm a much 39lnl MOSS Radiation pushes atmosphere into a baroclinically unstable regime which generates large scale turbulence cyclones that reduces the temperature gradient ie cyclones move heat Model of a midlatitude cyclone Relaxation to radiative equilibrium at north and south boundaries Lateral mixing to zonal mean recharge of air mass Warm air advection to the east cold air advection to the south Circulation change due to and frictional dissipation Model basis Temperature to north south east and west Circulation rotation rateWind speed Equation for change in circulation 4X equations for temperature change advection plus mixing E and W or healing N and S Model assignment Construct a model of a circulating midlatitude cyclone that transports heat poleward Science question 2 Given uncertainty in the initial conditions what is the useful range of a weather forecast Science question 1 What is the difference between the radiative equilibrium e uator to pole temperature difference and the dynamic equilibrium ifference Important points 1 Nonsimple behavior exists in even simple models 2 Difference between statistical properties say mean variance versus a deterministic state 3 How to plot results check web site for PDF

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