Difference between revisions of "Blatter-Pattyn Boundary Conditions"

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   & \hat{x}:\quad 4\frac{\partial u}{\partial x}\frac{\partial b}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial b}{\partial y}-\frac{\partial u}{\partial z}=-2\frac{\partial v}{\partial y}\frac{\partial b}{\partial x}-\frac{\partial v}{\partial x}\frac{\partial b}{\partial y}-\frac{\tau _{bx}}{\eta }, \\  
 
   & \hat{x}:\quad 4\frac{\partial u}{\partial x}\frac{\partial b}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial b}{\partial y}-\frac{\partial u}{\partial z}=-2\frac{\partial v}{\partial y}\frac{\partial b}{\partial x}-\frac{\partial v}{\partial x}\frac{\partial b}{\partial y}-\frac{\tau _{bx}}{\eta }, \\  
 
  & \hat{y}:\quad 4\frac{\partial v}{\partial y}\frac{\partial b}{\partial y}+\frac{\partial v}{\partial x}\frac{\partial b}{\partial x}-\frac{\partial v}{\partial z}=-2\frac{\partial u}{\partial x}\frac{\partial b}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial b}{\partial x}-\frac{\tau _{by}}{\eta }. \\\end{align}</math>
 
  & \hat{y}:\quad 4\frac{\partial v}{\partial y}\frac{\partial b}{\partial y}+\frac{\partial v}{\partial x}\frac{\partial b}{\partial x}-\frac{\partial v}{\partial z}=-2\frac{\partial u}{\partial x}\frac{\partial b}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial b}{\partial x}-\frac{\tau _{by}}{\eta }. \\\end{align}</math>
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We assume that basal traction is related to basal sliding velocity through a traction paramter, ''<big>&beta;</big>'', such that
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<math>\tau _{bx}=\left. \beta u \right|_{z=b},\quad \tau _{by}=\left. \beta v \right|_{z=b}</math>.
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Substituting this expression into the above gives our final horizontal boundary conditions at the ice sheet base

Revision as of 20:50, 3 August 2009

We will go through an approximate derivation of the boundary conditions that are implemented with Glimmer/CISM's higher-order scheme. By "approximate" we mean that some of the derivation is guided by physical intuition and what appear to be "reasonable" arguments, rather than through the application of rigorous mathematics. We take comfort in the fact that, in the end, we wind up with the same sets of equations that one ends up with from the more rigorous approach. We will look at the derivation in three parts, (1) the free surface boundary condition, (2) the specified basal traction boundary condition, and (3) lateral boundary conditions.

Stress Free Surface

At the ice surface, a stress-free boundary condition is applied. The traction vector, T, must be continuous at the ice sheet surface and, assuming that atmospheric pressure and surface tension are small, we have


\begin{align}
  & T_{i}=-T_{i(boundary)}\approx 0 \\ 
 & T_{i}=\sigma _{ij}n_{j}=\sigma _{i1}n_{1}+\sigma _{i2}n_{2}+\sigma _{i3}n_{3}=0\\\end{align}


where the ni are the components of the outward facing, unit normal vector in Cartesisan coordinates.

For a function F(x,y,z) = f(x,y) - z = 0, where z = f(x,y) defines the surface, the gradient of F(x,y,z) gives the components of the surface normal vector:


a=\sqrt{\left( \frac{\partial f}{\partial x} \right)^{2}+\left( \frac{\partial f}{\partial y} \right)^{2}+1^{2}}


For the case of the ice sheet surface, s = f(x,y) and the surface normal is given by


n_{i}=\left( \frac{\partial s}{\partial x},\frac{\partial s}{\partial y},-1 \right)\frac{1}{a}


and


a=\sqrt{\left( \frac{\partial s}{\partial x} \right)^{2}+\left( \frac{\partial s}{\partial x} \right)^{2}+1^{2}}\approx \sqrt{1^{2}}=1


This simplification comes about because, in general, the slopes on glaciers and ice sheets are small, in which case the slope squared is very small. Thus, to first order, the surface normal vector components are simply given by


n_{i}=\left( \frac{\partial s}{\partial x},\frac{\partial s}{\partial y},-1 \right)


The expression above for Ti=0 gives three equations that must be satisfied for a free surface boundary condition:


\begin{align}
  & i=x:\quad T_{x}=\sigma _{xx}n_{x}+\sigma _{xy}n_{y}+\sigma _{xz}n_{z}=0, \\ 
 & i=y:\quad T_{y}=\sigma _{yx}n_{x}+\sigma _{yy}n_{y}+\sigma _{yz}n_{z}=0, \\ 
 & i=z:\quad T_{z}=\sigma _{zx}n_{x}+\sigma _{zy}n_{y}+\sigma _{zz}n_{z}=0. \\ 
\end{align}


Expanding the last one and expressing stresses in terms of strain rates and pressures, where η is the effective viscosity, gives


\left( 2\eta \dot{\varepsilon }_{zx} \right)n_{x}+\left( 2\eta \dot{\varepsilon }_{zy} \right)n_{y}+\left( 2\eta \dot{\varepsilon }_{zz}-P \right)n_{z}=0,


which, when solved for the pressure gives


Pn_{z}=\left( 2\eta \dot{\varepsilon }_{zz} \right)n_{z}+\left( 2\eta \dot{\varepsilon }_{zx} \right)n_{x}+\left( 2\eta \dot{\varepsilon }_{zy} \right)n_{y}.


Expanding the above expression in terms of velocity gradients and normal vector components we have


P=2\eta \frac{\partial w}{\partial z}-\left( \eta \frac{\partial u}{\partial z} \right)\frac{\partial s}{\partial x}-\left( \eta \frac{\partial v}{\partial z} \right)\frac{\partial s}{\partial y}


where we have made the usual 1st-order approximation


\frac{\partial w}{\partial x}=\frac{\partial w}{\partial y}\approx 0 .


Now we use this expression for the pressure and expand the two horizontal boundary condition expressions


\begin{align}
  & i=x:\quad T_{x}=\sigma _{xx}n_{x}+\sigma _{xy}n_{y}+\sigma _{xz}n_{z}=0, \\ 
 & i=y:\quad T_{y}=\sigma _{yx}n_{x}+\sigma _{yy}n_{y}+\sigma _{yz}n_{z}=0, \\\end{align}


in terms of velocity gradients and the effective viscosity to obtain


\begin{align}
  & \hat{x}:\quad 2\eta \frac{\partial u}{\partial x}\frac{\partial s}{\partial x}+\eta \left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)\frac{\partial s}{\partial y}-\eta \frac{\partial u}{\partial z}= \\ 
 & \quad \quad \quad \quad \quad 2\eta \frac{\partial w}{\partial z}\frac{\partial s}{\partial x}-\eta \frac{\partial u}{\partial z}\left[ \frac{\partial s}{\partial x}\frac{\partial s}{\partial x} \right]-\eta \frac{\partial v}{\partial z}\left[ \frac{\partial s}{\partial y}\frac{\partial s}{\partial x} \right], \\ 
\end{align}


\begin{align}
  & \hat{y}:\quad 2\eta \frac{\partial v}{\partial y}\frac{\partial s}{\partial y}+\eta \left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)\frac{\partial s}{\partial x}-\eta \frac{\partial v}{\partial z}= \\ 
 & \quad \quad \quad \quad \quad 2\eta \frac{\partial w}{\partial z}\frac{\partial s}{\partial y}-\eta \frac{\partial u}{\partial z}\left[ \frac{\partial s}{\partial x}\frac{\partial s}{\partial y} \right]-\eta \frac{\partial v}{\partial z}\left[ \frac{\partial s}{\partial y}\frac{\partial s}{\partial y} \right]. \\ 
\end{align}


In both of these expression, the terms in square brackets are ~0 because, as noted above slopes on ice sheets are small (and the slope squared is exceedingly small). From continuity, we also have


\frac{\partial w}{\partial z}=-\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}.


Using this expression for the normal vertical velocity gradient and removing the terms in square brackets our two horizontal boundary condition expressions become


\begin{align}
  & \hat{x}:\quad 2\eta \frac{\partial u}{\partial x}\frac{\partial s}{\partial x}+\eta \left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)\frac{\partial s}{\partial y}-\eta \frac{\partial u}{\partial z}=-2\eta \left( \frac{\partial u}{\partial x}\frac{\partial s}{\partial x}+\frac{\partial v}{\partial y}\frac{\partial s}{\partial x} \right), \\ 
 & \hat{y}:\quad 2\eta \frac{\partial v}{\partial y}\frac{\partial s}{\partial y}+\eta \left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)\frac{\partial s}{\partial x}-\eta \frac{\partial v}{\partial z}=-2\eta \left( \frac{\partial u}{\partial x}\frac{\partial s}{\partial y}+\frac{\partial v}{\partial y}\frac{\partial s}{\partial y} \right). \\ 
\end{align}


As in the derivation of the Blatter-Pattyn model, collect terms of a particular velocity gradient (that is, u or v) and move them to one side of the equation and, lastly, divide through by the effective viscosity η to get the final form of the 1st-order free surface boundary conditions


\begin{align}
  & \hat{x}:\quad 4\frac{\partial u}{\partial x}\frac{\partial s}{\partial x}+ \frac{\partial u}{\partial y}\frac{\partial s}{\partial y}-\frac{\partial u}{\partial z}=-2 \frac{\partial v}{\partial y}\frac{\partial s}{\partial x}-\frac{\partial v}{\partial x}\frac{\partial s}{\partial y}, \\ 
 & \hat{y}:\quad 4\frac{\partial v}{\partial y}\frac{\partial s}{\partial y}+\frac{\partial v}{\partial x}\frac{\partial s}{\partial x}-\frac{\partial v}{\partial z}=-2\frac{\partial u}{\partial x}\frac{\partial s}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial s}{\partial x}. \\ 
\end{align}


Specified Basal Traction

Derivation of the basal boundary condition follows that above for the free surface except that (1) the right-hand side of the equation is not zero, rather it consists of an assumed basal traction vector with components \left( \tau _{bx},\tau _{by} \right), (2) the normal vector components now consist of horizontal gradients in the basal surface, and (3) the effective viscosity does not disappear from the equations when we divide through. Making these substitutions we obtain the 1st-order basal boundary conditions for a specified basal traction


\begin{align}
  & \hat{x}:\quad 4\frac{\partial u}{\partial x}\frac{\partial b}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial b}{\partial y}-\frac{\partial u}{\partial z}=-2\frac{\partial v}{\partial y}\frac{\partial b}{\partial x}-\frac{\partial v}{\partial x}\frac{\partial b}{\partial y}-\frac{\tau _{bx}}{\eta }, \\ 
 & \hat{y}:\quad 4\frac{\partial v}{\partial y}\frac{\partial b}{\partial y}+\frac{\partial v}{\partial x}\frac{\partial b}{\partial x}-\frac{\partial v}{\partial z}=-2\frac{\partial u}{\partial x}\frac{\partial b}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial b}{\partial x}-\frac{\tau _{by}}{\eta }. \\\end{align}


We assume that basal traction is related to basal sliding velocity through a traction paramter, β, such that


\tau _{bx}=\left. \beta u \right|_{z=b},\quad \tau _{by}=\left. \beta v \right|_{z=b}.


Substituting this expression into the above gives our final horizontal boundary conditions at the ice sheet base