The trapped ion instability appears as a single mode in the Columbia Linear Machine and is expected to appear as such in tandem mirrors. Using three‐wave, nonresonant mode coupling of radially nonlocal eigenfunctions, two mode‐coupling equations for the unstable fundamental and stable first‐order radial harmonics are derived and solved. Mode coupling to the damped higher‐order radial harmonics is the principal physical saturation mechanism of the unstable fundamental mode.