Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 / 3 / a / Solution

The shearing sheet is a local Cartesian expansion about a reference circular orbit in a differentially rotating disk. One works in a frame rotating at $\Omega$, identifies $x,y,z$ with radial, azimuthal, and vertical directions, retains the linear background shear and radial tidal acceleration, and neglects curvature and background variation across a patch much smaller than the orbital radius. The incompressible version also takes constant reference density and imposes $\nabla \cdot \mathbf{u}=0$, filtering sound waves.

It represents local three-dimensional vortical motion, epicyclic motion, shear dynamics, inertial waves, and, after adding buoyancy, internal gravity waves and convection. It cannot represent global geometry or boundaries, order-radius structures, density-changing compressible motion, or acoustic waves.