**6 - Applications**

These figures show advection on the sphere using a latitude-longitude grid. Refinement is only not allowed near the poles. The finest level grid cells near the equator are roughly the same size as the coarsest level grid cells at the poles.

The test shown here is advection over the poles (solid body rotation), with a return to the initial position at time 1. Target pattern initial data is advected. On the latitude-longitude grid the pattern should move down in the center, spread out to the sides, move up along the sides and come back together at the top.

**6.2 Advection in saturated
groundwater flow **

A pressure drop is specified between the left and right boundaries with no flow at top and bottom. Embedded elliptical regions have permeabilities varying between 0.01 and 50 against background permeability 1. The stream function and velocities are computed based on Darcy's law, using the "Immersed Interface Method" for solving elliptic equations with discontinuous coefficients on uniform Cartesian grids.

Transport of tracer moving in from left boundary (injected for 0 < t < 3.0) is computed using AMRCLAW. Adaptive mesh refinement is partially visible. The regions near the front where no mesh is shown have refinement by an additional factor 2 over the adjacent regions, for an effective 160 x 160 grid.

Results shown below

**6.3 Acoustics with discontinuous
sound speed **

Plane wave pressure pulse in a region with discontinuous wave speeds.

The upper ellipse has speed 2, the lower ellipse speed .001 with the background speed 1.

Results are shown below

**6.4 Gravity-driven unstable
interface **

Gas dynamics with gravity.

Initial conditions: greater density to left of sinusoidal vertical interface than to right.

These results are *very* preliminary... just a first pass at getting a sensible problem.

Results are shown below