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1、Team # 3694Page 1 of 37更多免費學習資料，請關(guān)注微博：學神資料站更多學習資料，請關(guān)注淘寶店鋪：學神資料站/Table of ContentsTable of Contents1Defining the problem5II. Methods6Mathematically Modeling Sea Level Rise6Temperature Data7The Ice Sheet8Mass Balance Accumulation9Mass Balance - Ablation10Mass Balance and

2、 Sea Level Rise13Thermal Expansion13Localization14III. Results17Output Sea Level Rise Data17Submersion Simulation Results19IV. Discussion and Conclusion23V. Recommendations26References28Appendix A Sea Level Rise Simulation Script29Appendix B Topological Raster Matrix Creation Script33Appendix C Subm

3、ersion Simulation Script35Appendix D Florida Cities Data Initialization37更多數(shù)學建模資料請關(guān)注微店店鋪“數(shù)學建模學習交流”/RHO6PSpATeam # 3694Page 2 of 37I. IntroductionStrong evidence of a global warming trend exists, and powerful models have been created to estimate future climate. Temperatures have i

4、ncreased by about 0.5oC over the last 15 years, and global temperature is at its highest level in the past millennium . Although the warming trend is quite evident, the consequences of such wide scale climate change are still poorly understood. One of the most-feared consequences of global warming i

5、s sea level rise, and for good reason. TOPEX/Poseidon satellite altimeter indicates that sea levels rose 3.2 0.2 mm annually during 1993 -1998 . Indeed, Titus et al estimate that a 1 meter rise in sea levels could cause $270-475 billion in damages in the United Statesalone.A number of complex factor

6、s underlie sea level rise. Thermal expansion of water due to temperature changes has long been implicated as the major component of sea level rise; however, recent studies have shown that thermal expansion alone cannot account for a majority of the observed increases . Mass balance of large ice shee

7、ts, in particular the Greenland Ice Sheet, is now believed to play a major role in sea level. The mass balance is controlled by two major processes, accumulation (influx of ice to the sheet) and ablation (loss of ice from the sheet) . Accumulation is primarily the result of snowfall;ablation is a re

8、sult of sublimation and melting.Contrary to popular belief, however, floating ice does not play a significant role in sea level rise. By Archimedes Principle, the volume increase V of a body of water with density ocean due to melting of floating ice of weight W (assumed to be freshwater, withliquid

9、density water) is given by11DV = W r- r(1)waterocean Team # 3694Page 3 of 37The density of seawater is approximately 1024.8 kg/m3 ; the mass of the Arctic sea ice isapproximately 2 x 1013 kg . Thus, the volume change if all of the Arctic sea ice melted is given by: 1 1DV = 2 10 kg13-= 4.84 10 m83(2)

10、kgkg10001024.8m3m3Approximating that 360 Gt of water causes a rise of 1 mm in sea level ,4.84 108 m3 1000kg 1Gt1mm= 0.0015mm(3)9.072 1011 kg 360Gtm3This small change in sea level is inconsequential for our model, since the accuracy is wellbelow one thousandth of a millimeter.We also neglect the cont

11、ribution of Antarctic Ice Sheet because its overall effect on sea level rise is minimal and difficult to quantify. Between 1978 to1987, satellite-borne microwave radiometer data indicated that Arctic ice decreased by 3.5%, while Antarctic ice showed no statistically significant changes . Cavalieri e

12、t al projected minimal melting in the Antarctic over the next 50 years . For this reason, only the Greenland Ice Sheet isconsidered in the model.Several models already exist for mass balance and for thermal expansion. However, these models are very complex with respect to many variables, and often d

13、isagree with each other (see for example and ). We wish to develop a model based on simple physical processes, as solely a function of temperature and time. In this way the analysis of the effects of the warming is simplified, and the dependence of sea level rise on temperature becomes evident. Furt

14、hermore, we develop a model that can be extended to several different temperature forcings, allowing us to compare firsthand the effect of carbonemissions on sea level rise.Team # 3694Page 4 of 37Model OverviewA deeper understanding of ice sheet melting would provide valuable insight into sea level

15、rise. By creating a framework that incorporates the contributions of ice sheet melting and thermal expansion, we can estimate global mean sea level over a 50-year time period.The model achieves several important objectives :1) Accurately fits past sea level rise data2) Provide enough generality to p

16、redict sea level rise over a 50-year span3) Compute sea level increases for Florida as a function of solely global temperature and timeUltimately, the model predicts consequences to human populations. In particular, we analyze the impact of sea level rise on the state of Florida, which many consider

17、 particularly vulnerable due to its generally low elevation and proximity to the Atlantic Ocean. From this analysis, we assess possible strategies to minimize damage as a resultof sea level rise due to global warming.AssumptionsIn order to streamline our model we have made several key assumptions.1)

18、 The sea level rise is primarily due to two factors, the balance of accumulation/ablation of the Greenland Ice Sheet and the thermal expansion of the ocean. This ignores the contribution of processes such as calving and direct human intervention, which aredifficult to model accurately and have minim

19、al effect on sea level rise .2) The air is the only heat source for melting the ice. Greenlands land is permafrost, andbecause of large amounts of ice on its surface it is assumed at a relatively constant temperature. This allows us to use convection as a mode of heat transfer.Team # 3694Page 5 of 3

20、73) The temperature within the ice changes linearly at the steady-state. This assumption allows us to solve the heat equation for Neumann conditions. By subtracting the steady- state term from the heat equation, we can solve for the homogeneous boundaryconditions.4) Sublimation and melting processes

21、 do not interfere with each other. This assumption drastically simplifies the computation needed for the model since sublimation and melting can be considered separately. Additionally, the assumption is very reasonable. Sublimation primarily occurs at below freezing temperatures, a condition during

22、which melting does not normally occur. Thus, the two processes are temporally isolated as inour model.5) The surface of the ice sheet is homogeneous with regards to temperature, pressure, and chemical composition. This assumption is necessary because high-resolution spatial temperature data for Gree

23、nland cannot be obtained in our framework. Additionally, we lack the computational resources and time to simulate such a variation, which wouldrequire the use of finite element methods and mesh generation for a complex topology.Defining the problemLet M denote the mass balance of the Greenland Ice S

24、heet. Given a temperature forcing function, we must quantitatively estimate the sea level increases SLR that occur as a result. These increases are a sum of M and thermal expansion TE effects, corrected for local trends. Further, we must quantitatively and qualitatively the long-term (50 years) effe

25、ct on Floridas major cities and metropolitan areas from global warming, as a result of high SLR. This analysis can be used to make recommendations as to how to bestprepare for and reduce SLR effects.Team # 3694Page 6 of 37II. MethodsMathematically Modeling Sea Level RiseSea level rise results mostly

26、 from mass balance of the Greenland Ice Sheet and thermal expansion due to warming. In order to model sea level increases, a mass balance model and thermal expansion model are used, as well as other post-computation effects. Thelogic of the simulation process is detailed in Figure 1.Figure 1: Simula

27、tion flow diagramTeam # 3694Page 7 of 37Temperature DataTemperature data is the sole forcing in our model and thus shall be considered carefully. Because we needed to model several different scenarios, our temperature data must include several scenarios that are very controlled and only differ in on

28、e variable. Further, the temperature data must be of very good quality and provide the correct temporal resolution for our simulation. For these reasons, we decided to use a Global Climate Model (GCM) to create our own temperature data, using input forcings that we could easily control. Because of l

29、imited computational power and time restrictions, we chose the EdGCM . EdGCM is a fast model for educational purposes. The program is based on the NASA GISS model for climate change. The program fit all of our needs; in particular, the rapid simulation (about 10 hours for a 50 year climate simulatio

30、n) allowed us toanalyze several different temperature scenarios.The temperature scenarios we analyzed incorporate the three estimates of carbon emissions resulting from the IPCC Third Assessment Report (TAR) the low, high, and medium projections in the IS92 series . The IS92e (high), IS92a (intermed

31、iate), and the IS92c (low) scenarios were all closely approximated using the tools in EdGCM. These approximated carbon forcings are shown in graphical form in Figure 2. All other forcings were kept at default according to the NASA GISS model. Three time series for globalsurface air temperature were

32、obtained in this fashion.Figure 2: Carbon Dioxide Forcings for the EdGCM ModelsOne downside to the EdGCM is that it can only output global temperature changes .Regional temperature changes are calculated, but are difficult to access and have lowTeam # 3694Page 8 of 37spatial accuracy. However, accor

33、ding to Chylek et al , the relationship betweenGreenland temperatures and global temperatures is well-approximated byDTGreenland= 2.2 DTglobal(4)This result is shown by Chylek et al for regions unaffected by the NAO and is predictedby climate model outputs.The Ice SheetThe ice sheet is modeled as a

34、simplified rectangular box. Each point on the upper surface of the ice sheet is assumed at constant temperature, Ta. This is because our climate model does not have accurate spatial resolution for areas in Greenland, so the small temperature differences are ignored. The lower surface, the permafrost

35、 layer, has constant temperatureTl. A depiction of the ice sheet model is shown in Figure 3.TaTlFigure 3: A profile view of the ice sheet modelTo compute heat flux and thus melting and sublimation through the ice sheet, we model itas an infinite number of differential volumes, shown in Figure 4.Team

36、 # 3694Page 9 of 37Figure 4: Differential volumes of the ice sheetInitially, the height h is calculated using data provided by Williams et al .2.6 106 km3 Vol= 1498kmh =Surfac=e ice1.736 106 km2iceThe primary mode of sea level rise in our model is through mass balance. Mass balance is calculated by

37、subtracting the amount of ablation by the amount of accumulation.Accumulation, the addition of ice to the ice sheet, is primarily in the form of snowfall.Ablation is primarily the result of two processes, sublimation and melting.Mass Balance AccumulationFirst we model accumulation. Huybrechts et al

38、showed that the temperature of Greenland is not high enough to melt significant amounts of snow. Furthermore, Knight showed empirically that rate of accumulation is well-approximated by a linear relationship with time, and that accumulation over Greenland continental ice is 0.30 m/year. Thus, theacc

39、umulation rate is 0.025 m/month. In terms of mass balance,= 0.025LDM ac(5)where the product LD is the surface area of the ice sheet.Team # 3694Page 10 of 37Mass Balance - AblationWe then model the two parts of ablation, sublimation and melting.Sublimation rate (mass flux) is given by:1S = e(T )M w2(

40、6)0sat 2pRT where Mw is the molecular weight of water. This expression can be derived from the idealgas law and the Maxwell-Boltzmann distribution . Substituting Bucksesat, we obtain:expression for (18.678-T)T 1 2 234.5 M w257.14+T(7)2pR(T + 273.15)S0 = 6.1121 eBucks equation is applicable over a la

41、rge range of temperatures and pressures, including the environment of Greenland. The approximation fails at extreme temperatures and pressures but is computationally simple (relatively). To convert mass flux into rate of thickness change of the ice, we divide the mass flux expression by the density

42、of ice.Thus we can express rate of height change as follows:(18.678-)T T1 234.5 6.1121 d e M w 2257.14+T(8)Sh = r 2pR(T + 273.15) icewhere d is the deposition factor, given by d = (1-deposition rate) = 0.01 . This term is needed because sublimation and deposition are in constant equilibrium. With th

43、e sublimation rate expression, it is now trivial to find the thickness of the ice sheet after one timestep of the computational model. Indeed, the new thickness due to ablation viasublimation is given by:S (t) = h - Sh t(9)Team # 3694Page 11 of 37where h is the current thickness of the ice sheet and

44、 t is the elapsed time after onetimestep. Substituting for Sh with the expression we derived and substituting for the known value of the molecular weight of water yields (18.678-T)T 1 234.5 6.112110 tr ice-20.0003448 2257.14+T(10)e S (t) = h - (T + 273.15) This equation governs the sublimation of th

45、e ice.To model melting, the second component of ablation, we apply the heat equation. Theheat equation governs the relationshipUt (x, t) = kU xx (x, t)(11)where k=0.0104 is the thermal diffusivity of the ice . In order to solve the heat equation for the Neumann conditions, we assume a steady-state U

46、s with the same boundary conditions as U and that is independent of time. The residual temperature V has homogeneous boundary conditions and initial conditions found by U-Us. Thus we canrewrite the heat equation as:U (x, t) = V (x, t) + Us (x, t)(12)The steady-state solution of the heat equation is

47、given by:+ Ta - TlU = Tx(13)slS (t)subject to the constraints 0 x S(t) and 0 t k, U(x, t) 0 for fixed t, the ice will melt for k x h. Thus, we seek the solution to U(k, t)=0 for k to determine ablation. Computationally, we solve this expression using the first 100 terms of the Fourier series expansi

48、on and the MATLAB function fzero. The solution of this equation for k is the primary computational step for the MATLAB simulation (see Appendix A). The new value of k is used to renew h as the new thickness of the ice sheet, and a consequent timestep can begin calculation.Team # 3694Page 13 of 37Wit

49、h these two components we can now finalize an expression for ablation and apply it to a computational model. The sum of the infinitesimal changes in ice sheet thickness for each differential volume gives the total change in thickness. To find these changes, wefirst note thatMass Balance Loss Due to

50、Sublimation = (h-S)LDMass Balance Loss Due to Melting = (S-k)*LD(20)(21)where the product LD is the surface area of the ice sheet. Note that in these equations, the“mass balance” refers to net volume change. Thus, ablation is given by= (h - S )LD + (S - k )LD = (h - k)LDM ab(22)Mass Balance and Sea

51、Level RiseCombining accumulation and ablation into an expression for mass balance, we haveM = M ac - M ab = 0.025LD - (h - k)LD(23)Relating this to sea level rise, we use the approximation 360 Gt water = 1mm sea levelrise. Thus,1mmSLR= M r(24)mbice360Gtwhich quantifies the sea level rise due to mass

52、 balance.Thermal ExpansionA second mode of sea level rise is also considered: thermal expansion due to warming.According to various literature , thermal expansion of the oceans due to increase in globalTeam # 3694Page 14 of 37temperature will contribute a significant portion of the rise in future se

53、a level, at least asmuch as melting of polar ice for the current century , . Therefore, we incorporated this component into our model for further accuracy and a more comprehensive understanding.Thermal expansion operates depending on various factors. Temperature plays the primary role, but the diffusion of radiated heat, mixing of the ocean, and various other complexities concerning