Team 3694 Page I of 37 Table of contents Table of Contents. Defining the problem. IL. Methods Mathematically Modeling Sea Level Rise, Temperature Data. The Ice Sheet. Mass Balance- Accumulation AN Mass balance ablation Mass balance and sea level rise Thermal Expansion. 13 Localization 14 l results A17 Output Sea Level Rise data Submersion simulation results 19 IV Discussion and conclusion v Recommendations Ref Appendix A Sea Level Rise Simulation Script Appendix B Topological Raster Matrix Creation Script. Appendix C Submersion Simulation Script. Appendix d Florida Cities Data Initialization
Team # 3694 Page 1 of 37 Table of Contents Table of Contents ................................................................................................................. 1 Defining the problem ....................................................................................................... 5 II. Methods ........................................................................................................................... 6 Mathematically Modeling Sea Level Rise ....................................................................... 6 Temperature Data ............................................................................................................. 7 The Ice Sheet .................................................................................................................... 8 Mass Balance – Accumulation ........................................................................................ 9 Mass Balance - Ablation ................................................................................................ 10 Mass Balance and Sea Level Rise ................................................................................. 13 Thermal Expansion ........................................................................................................ 13 Localization .................................................................................................................... 14 III. Results .......................................................................................................................... 17 Output Sea Level Rise Data ........................................................................................... 17 Submersion Simulation Results ..................................................................................... 19 IV. Discussion and Conclusion .......................................................................................... 23 V. Recommendations ......................................................................................................... 26 References .......................................................................................................................... 28 Appendix A Sea Level Rise Simulation Script .................................................................. 29 Appendix B Topological Raster Matrix Creation Script ................................................... 33 Appendix C Submersion Simulation Script ....................................................................... 35 Appendix D Florida Cities Data Initialization ................................................................... 37
Team 3694 Page 2 of 37 Introduction Strong evidence of a global warming trend exists, and powerful models have been created to estimate future climate. Temperatures have increased by about 0.5C 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 is 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 States A number of complex factors 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 observed increases. Mass balance of large ice sheets, 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 result 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 Av of a body of water with density Pocean due to melting of floating ice of weight w(assumed to be freshwater, with liquid density pater ) is given by Prater Pa ocean
Team # 3694 Page 2 of 37 I. Introduction Strong evidence of a global warming trend exists, and powerful models have been created to estimate future climate. Temperatures have increased 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 is 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 States alone. A number of complex factors 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 sheets, 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 result 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 ΔV of a body of water with density ρocean due to melting of floating ice of weight W (assumed to be freshwater, with liquid density ρwater) is given by ∆ = − water ocean V W ρ ρ 1 1 (1)
Team 3694 Page 3 of 37 The density of seawater is approximately 1024.8 kg/m; the mass of the Arctic sea ice approximately 2 x 10 kg. Thus, the volume change if all of the Arctic sea ice melted △V=2×10 1009-- 4.84×108m 1024.8 kg Approximating that 360 Gt of water causes a rise of 1 mm in sea level 484×10°m31000g.lGt m39072×10"kg360cn=0.0015mm This small change in sea level is inconsequential for our model, since the accuracy is well below one thousandth of a millimeter We also neglect the contribution 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 antara ice showed no statistically significant changes. Cavalieri et al projected minimal melting the Antarctic over the next 50 years. For this reason, only the Greenland Ice Sheet is considered 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 disagree with each other(see for example and). We wish to develop a model based 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. Furthermore, we develop a model that can be extended to several different temperature forcings, allowing us to compare firsthand the effect of carbon emissions on sea level rise
Team # 3694 Page 3 of 37 The density of seawater is approximately 1024.8 kg/m3 ; the mass of the Arctic sea ice is approximately 2 x 1013 kg . Thus, the volume change if all of the Arctic sea ice melted is given by: 8 3 3 3 13 4.84 10 1024.8 1 1000 1 2 10 m m kg m kg V kg = × ∆ = × − (2) Approximating that 360 Gt of water causes a rise of 1 mm in sea level , mm Gt mm kg Gt m kg m 0.0015 360 1 9.072 10 1000 1 4.84 10 3 11 8 3 ⋅ = × × ⋅ ⋅ (3) This small change in sea level is inconsequential for our model, since the accuracy is well below one thousandth of a millimeter. We also neglect the contribution 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 et al projected minimal melting in the Antarctic over the next 50 years . For this reason, only the Greenland Ice Sheet is considered 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 disagree 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. Furthermore, we develop a model that can be extended to several different temperature forcings, allowing us to compare firsthand the effect of carbon emissions on sea level rise
Team 3694 Page 4 of 37 Model overview a deeper understanding of ice sheet melting would provide valuable insight into sea level rise. By creating a framework that incorporates the contributions of ice sheet melting and hermal 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 data 2) Provide enough generality to predict sea level rise over a 50-year span 3)Compute sea level increases for Florida as a function of solely global temperature and time Ultimately, the model predicts consequences to human populations. In particular, we analyze the impact of sea level rise on the state of Florida, which many conside 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 result of sea level rise due to global warming Assumptions In order to streamline our model we have made several key assumptions 1) 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 are difficult to model accurately and have minimal effect on sea level rise 2) The air is the only heat source for melting the ice. Greenland's land is permafrost, and because 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 # 3694 Page 4 of 37 Model Overview A deeper understanding of ice sheet melting would provide valuable insight into sea level 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 data 2) Provide enough generality to predict sea level rise over a 50-year span 3) Compute sea level increases for Florida as a function of solely global temperature and time Ultimately, 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 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 result of sea level rise due to global warming. Assumptions In order to streamline our model we have made several key assumptions. 1) 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 are difficult to model accurately and have minimal effect on sea level rise . 2) The air is the only heat source for melting the ice. Greenland’s land is permafrost, and because 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 3694 Page 5 of 37 3)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 boundary onit 4)Sublimation and melting processes do not interfere with each other. This assumption drastically simplifies the computation needed for the model since sublimation and nelting can be considered separately. Additionally, the assumption is ver Sublimation primarily occurs at below freezing temperatures, a condition during which melting does not normally occur. Thus, the two processes are temporally isolated as in our 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 Greenland cannot be obtained in our framework. Additionally, we lack the computational resources and time to simulate such a variation, which would require the use of finite element methods and mesh generation for a complex topology Defining the problem Let m denote the mass balance of the greenland Ice Sheet 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) effect on Florida's 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 best prepare for and reduce SLR effects
Team # 3694 Page 5 of 37 3) 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 steadystate term from the heat equation, we can solve for the homogeneous boundary conditions. 4) Sublimation and melting processes 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 which melting does not normally occur. Thus, the two processes are temporally isolated as in our 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 Greenland cannot be obtained in our framework. Additionally, we lack the computational resources and time to simulate such a variation, which would require the use of finite element methods and mesh generation for a complex topology. Defining the problem Let M denote the mass balance of the Greenland Ice Sheet. 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) effect on Florida’s 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 best prepare for and reduce SLR effects
Team 3694 Page 6 of 37 l Methods Mathematically Modeling Sea Level rise Sea level rise results mostly 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. The logic of the simulation process is detailed in Figure 1 EdGCM Temp Figure 1: Simulation flow diagram
Team # 3694 Page 6 of 37 II. Methods Mathematically Modeling Sea Level Rise Sea level rise results mostly 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. The logic of the simulation process is detailed in Figure 1. Figure 1: Simulation flow diagram
Team 3694 Page 7 of 37 Temperature dat Temperature 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 one 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 limited 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 simulation)allowed us to analyze 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(intermediate), 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 global surface air temperature were obtained in this fashion Carben Dimeide al> Cabon Dioside l 1592 High> Carbon Donie trend Carbon Diusidexds Dioxide for 1s92aMed> Carbon Diode trend Carbon Diede ats>Carbon Dioride kr IS92ctow>Carbon( Figure 2: Carbon Dioxide Forcings for the EdGCM Models One 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 low
Team # 3694 Page 7 of 37 Temperature Data Temperature 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 one 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 limited 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 simulation) allowed us to analyze 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 (intermediate), 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 global surface air temperature were obtained in this fashion. Figure 2: Carbon Dioxide Forcings for the EdGCM Models One 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 low
Team 3694 Page 8 of 37 spatial accuracy. However, according to Chylek et al, the relationship between Greenland temperatures and global temperatures is well-approximated by △ GReenland=22×△T (4) This result is shown by Chylek et al for regions unaffected by the Nao and is predicted by climate model outputs The ice sheet The ice sheet is modeled as a 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 layer, has constant temperature Ti. a depiction of the ice sheet model is shown in Figure 3 T Figure 3: A profile view of the ice sheet model To compute heat flux and thus melting and sublimation through the ice sheet, we model as an infinite number of differential volumes shown in Figure 4
Team # 3694 Page 8 of 37 spatial accuracy. However, according to Chylek et al , the relationship between Greenland temperatures and global temperatures is well-approximated by ∆TGreenland = 2× ∆Tglobal 2. (4) This result is shown by Chylek et al for regions unaffected by the NAO and is predicted by climate model outputs. The Ice Sheet The ice sheet is modeled as a 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 layer, has constant temperature Tl. A depiction of the ice sheet model is shown in Figure 3. Figure 3: A profile view of the ice sheet model To compute heat flux and thus melting and sublimation through the ice sheet, we model it as an infinite number of differential volumes, shown in Figure 4. Ta T l
Team 3694 Page 9 of 37 h L Figure 4: Differential volumes of the ice sheet Initially, the height h is calculated using data provided by Williams et al Vol 26×10°km 1498/m brace 1736×106km alculated by subtracting the amount of ablation by the amount of accumulation ance is The primary mode of sea level rise in our model is through mass balance. Mass bal 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-Accumulation First we model accumulation huybrechts et al 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, the accumulation rate is 0.025 m/month. In terms of mass balance f=0.025LD where the product ld is the surface area of the ice sheet
Team # 3694 Page 9 of 37 Figure 4: Differential volumes of the ice sheet Initially, the height h is calculated using data provided by Williams et al . km km km Surface Vol h ice ice 1498 1.736 10 2.6 10 6 2 6 3 = × × = = The primary mode of sea level rise in our model is through mass balance. Mass balance is calculated by 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 – Accumulation First we model accumulation. Huybrechts et al 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, the accumulation rate is 0.025 m/month. In terms of mass balance, M ac = 0.025LD (5) where the product LD is the surface area of the ice sheet
Team 3694 Page 10 of 37 Mass balance ablate We then model the two parts of ablation, sublimation and melting Sublimation rate(mass flux )is given by M (T TRT where M is the molecular weight of water. This expression can be derived from the ideal gas law and the Maxwell-Boltzmann distribution. Substituting Buck's expression for e sat we obtain 18.678-/34s)r S=6.1121e 2nR(T+273.15) Buck's equation is applicable over a large 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 of ice Thus we can express rate of height change as follows 6.1121·d 257.14+T Sh 2mR(T+273.15 where d is the deposition factor, given by d=(l-deposition rate)=0.01. This term is needed because sublimation and deposition are in constant equilibrium. with the 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 via given S(1)=h-S
Team # 3694 Page 10 of 37 Mass Balance - Ablation We then model the two parts of ablation, sublimation and melting. Sublimation rate (mass flux) is given by: 2 1 0 2 ( ) = RT M S e T w sat π (6) where Mw is the molecular weight of water. This expression can be derived from the ideal gas law and the Maxwell-Boltzmann distribution . Substituting Buck’s expression for esat, we obtain: ( ) 2 1 257.14 234.5 18.678 0 2 ( 273.15) 6.1121 + = ⋅ + − R T M S e w T T T π (7) Buck’s equation is applicable over a large 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 of ice. Thus we can express rate of height change as follows: ( ) 2 1 257.14 234.5 18.678 2 ( 273.15) 6.1121 + ⋅ ⋅ = + − R T M e d S w T T T ice h ρ π (8) where 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 the 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 via sublimation is given by: S t h S t h ( ) = − ⋅ (9)
