# Notes about the flow-line SSA¶

Using the same notation as in the rest of the manual, the flow-line case the shallow shelf approximation reads

Here \(u\) is the ice velocity, \(\nu = \nu(u')\) is the ice viscosity, \(H\) is the ice thickness, and \(h\) is the ice surface elevation.

Let \(\alpha = (1 - \rhoi / \rhow)\) and note that \(h = \alpha\, H\) whenever ice is floating, so

We can easily integrate this, getting

Note

This is a non-linear *first-order* ODE for the ice velocity \(u\).

## An observation¶

Consider a boundary-value problem for the velocity \(u\) of a floating ice shelf on an interval \([0, L]\):

Now consider a similar BVP on \([0, a]\) for some positive \(a < L\):

Because (58) is a first-order ODE, the solutions \(u\) of
(58) and \(v\) of (59) *coincide* on \([0, a]\),
i.e. \(u(x) = v(x)\) for all \(x \in [0, a]\).

Note

This implies that the velocity \(u(x)\) of a floating flow-line ice shelf modeled by (58) is not sensitive to ice geometry perturbations at locations downstream from \(x\).

## Discrete analog of this property¶

Let \(N > 2\) be an integer and define the \(N\)-point grid \(x_{j} = 0 + j\dx\), \(\dx = L/(N-1)\).

Discretizing (58) on this grid results in a non-linear system with \(N-1\) unknowns (call this system \(S_{L}\)).

Let \(a = L - \dx\) and write down a system of equations by discretizing (59) on the grid consisting of the first \(N-1\) of \(x_{j}\). This system (call it \(S_{a}\)) has \(N-2\) unknowns.

The property we would like our discretization to have is this:

If \(u_{1},\dots,u_{N-1}\) is the solution of \(S_{L}\) and \(v_{1},\dots,v_{N-2}\) solves \(S_{a}\), then \(u_{k} = v_{k}\) for all \(k = 1,\dots,N-2\).

Note that the first \(N-3\) equations in \(S_{L}\) and \(S_{a}\) are the same.

## Discretization¶

Let \([X]_{j}\) be an approximation of \(X\) at a grid location \(j\).

The standard approach *within* the domain is to use centered finite differences and linear
interpolation to approximate staggered-grid values, i.e. averaging values at immediate
regular grid point neighbors:

### Interior¶

### Ice front¶

Let \(n\) be the last grid point with non-zero ice thickness, i.e. assume that \(H_{k} = 0\) for \(k > n\).

The implementation of the stress boundary condition at the ice front amounts to adding one more equation (see (54)):

We can then combine (62) with (61) (with \(i\) replaced by \(n\)) to get the discretization at the ice front:

### Choosing FD approximations¶

Assuming that the ice front is at \(n\)

If we assume that the ice front is at \(n\), the last equation in the system looks like (63):

Assuming that the ice front is at \(n+1\)

If we assume that the ice front is at \(n+1\), the \(n\)-th equation looks like a “generic” interior equation (61) and we have one more equation ((64)) shifted by \(1\):

Note that the second equation in (65) is the same as
(64), but with the index shifted by \(1\). Both correspond to
locations *at the ice front*.

The goal

We want to choose FD approximations \([\tau_{\text{stat}}]_{*}\) and \([H\, H']_{*}\) in a way that would make it possible to obtain (64) by transforming equations (65).

We propose using *constant extrapolation* to approximate \(H_{n+\frac12}\)

This gives us the following approximation of derivatives:

After substituting (60) this becomes

Note

The ice front case in (67) is the **one half**
of the standard one-sided finite-difference approximation of \(H'\).

Checking if (66) is the right choice

Consider the first equation in (65) and note that it
corresponds to the case in which \(n\) is **not** at the ice front.

Multiplying by \(\dx\) and moving one of the terms to the right hand side, we get

Now consider the second equation in (65). Note that here
\(n+1\) **is** at the ice front.

Put together, (68) and (69) read as follows:

Substituting the second equation into the first produces

Compare (71) to
(69). Note that *they are the same*, except for the
index shift. In other words, (71) is the same as
(64), as desired.

This confirms that finite difference approximations (66) and (67) result in a discretization with the property we seek:

modeled ice velocity at a given location \(x\) along a flow-line ice shelf is not sensitive to geometry perturbations downstream from \(x\).

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