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)