  The cross sections of solar coronal loops are suggested to be rarely
circular. We examine linear kink oscillations in straight, density-enhanced,
magnetic cylinders with elliptical cross-sections by solving the
three-dimensional magnetohydrodynamic equations from an initial-value-problem
perspective. Motivated by relevant eigen-mode analyses, we distinguish between
two independent polarizations, one along the major axis (the M-modes) and the
other along the minor one (the m-modes). We find that, as happens for coronal
loops with circular cross-sections, the apparent damping of the transverse
displacement of the loop axis is accompanied by the accumulation of transverse
Alfv\'enic motions and the consequent development of small-scales therein,
suggesting the robustness of the concepts of resonant absorption and
phase-mixing. In addition, two stages can in general be told apart in the
temporal evolution of the loop displacement; a Gaussian time dependence
precedes an exponential one. For the two examined density ratios between loops
and their surroundings, the periods of the M-modes (m-modes) tend to increase
(decrease) with the major-to-minor-half-axis ratio, and the damping times in
the exponential stage for the M-modes tend to exceed their m-mode counterparts.
This is true for the two transverse profiles we examine. However, the relative
magnitudes of the damping times in the exponential stage for different
polarizations depend on the specification of the transverse profile and/or the
density contrast. The applications of our numerical findings are discussed in
the context of coronal seismology.
