A New Numerical Approach to Solve 1D Viscous Plastic Sea Ice Momentum Equation

While there has been a colossal effort in ongoing decades, the ability to simulate oceanice has fallen behind various parts of the climate system and most Earth System Modelsare unable to capture the observed adversities of Arctic sea ice, which is, as it were,attributed to our frailty to determine sea ice dynamics. Viscous Plastic rheology is themost by and large recognized model for sea ice dynamics and it is expressed as a setof partial differential equations that are hard to tackle numerically. Using the 1D seaice momentum equation as a prototype, we use the method of lines based on Eulersbackward method. This results in a nonlinear PDE in space only. At that point, weapply the Damped Newtons method which has been introduced in Looper and Rapettiet al. [2] and used and generalized to 2D in Saumier et al. [1] to solve the Monge-Ampere equation. However, in our case, we need to solve 2nd order linear equationwith discontinuous coefficients during Newton iteration. To overcome this difficulty, weuse the Finite element method to solve the linear PDE at each Newton iteration. In thispaper, we will show that with the proper choice of a damping factor, convergence canbe guaranteed and the numerical solution indeed converges efficiently to the continuumsolution unlike other numerical approaches that typically solve an alternate set ofequations and avoid the difficulty of the Newton method for a large nonlinear algebraicsystem. The PDE approach is more stable.References[1] L.P. Saumier,B.Khouider. An Efficient Numerical Algorithm for the L2 OptimalTransport Problem with Applications to Image Processing, University of Victoria,2010.[2] G.Loeper and F.Rapetti. Numerical solution of the Monge-Ampere equation by aNewtons algorithm. C. R. Acad. Sci. Paris, I(340):319324, 2005.