2. Pressure as a vertical coordinate
a. Explain how the equations are simplified when using pressure as a vertical coordinate. What other
assumption must be made to simplify the equations?
b. Suppose the size of a volume in a three-dimensional grid with a pressure coordinate has the
following dimensions: $\Delta x = \Delta y = 5$ km and $\Delta p = 10$ hPa. Suppose the scalar velocities at the
west, east, north, south, and lower boundaries are $u_{west} = -3 \text{ m s}^{-1}$, $u_{east} = -1 \text{ m s}^{-1}$,
$u_{north} = -2 \text{ m s}^{-1}$, $u_{south} = 2 \text{ m s}^{-1}$, and $\omega_{inf} = 0.02 \text{ hPa s}^{-1}$, respectively. Use the
continuity equation to estimate the vertical velocity at the top of the cell $\omega_{sup}$.
c. Mathematically and conceptually describe what the velocity $\omega_{sup}$ is.