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2008 A a convenient truth a model for sea level rise fore

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    2008 A a convenient truth a model for sea level rise fore

    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 0.5
    o
    C 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
    WV
    ρρ
    11
    (1)数学中国
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    madio.net
    本内容由645617861l同学与工作人员共同整理,恭喜同学成功完成数学中国第一期威客项目
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    Team # 3694 Page 3 of 37
    The density of seawater is approximately 1024.8 kg/m
    3
    ; the mass of the Arctic sea ice is
    approximately 2 x 10
    13
    kg . Thus, the volume change if all of the Arctic sea ice melted is
    given by:
    38
    3 3
    13
    1084.4
    8.1024
    1
    1000
    1102 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
    kgm 0015.0
    360
    1
    10072.9
    1000 11084.4 3 11
    38
    =⋅
    ×
    ⋅⋅× (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.数学中国
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    madio.net
    本内容由645617861l同学与工作人员共同整理,恭喜同学成功完成数学中国第一期威客项目
    数学中国:www.madio.net,最专业的数学建模平台
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    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.数学中国
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    madio.net
    本内容由645617861l同学与工作人员共同整理,恭喜同学成功完成数学中国第一期威客项目
    数学中国:www.madio.net,最专业的数学建模平台
    备战美赛,美赛资料下载:http://www.madio.net/forum-108-1.html
    美赛讨论加入群:2017美赛官方群287336869
    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
    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.


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