Using particle-in-cell simulations of relativistic laser plasma wakes in the presence of an external magnetic field, we demonstrate that there exists a parameter window where the dynamics of the magnetized wake channel are largely independent of the laser wavelength $lambda_{rm las}$. One condition for this manifestation of limited similarity is that the electron density $n_{rm e}$ is highly subcritical, so that the plasma does not affect the laser. The freedom to choose a convenient laser wavelength can be useful in experiments and simulations. In simulations, an up-scaled wavelength (and, thus, a coarser mesh and larger time steps) reduces the computational effort, while limited similarity ensures that the overall structure and evolutionary phases of the wake channel are preserved. In our demonstrative example, we begin with a terrawatt$cdot$picosecond pulse from a ${rm CO}_2$ laser with $lambda_{rm las} = 10,mu{rm m}$, whose field reaches a relativistic amplitude at the center of a sub-millimeter-sized focal spot. The laser is shot into a sparse deuterium gas ($n_{rm e} sim 10^{13},{rm cm}^{-3}$) in the presence of a tesla-scale magnetic field. Limited similarity is demonstrated in 2D for $4,mu{rm m} leq lambda_{rm las} leq 40,mu{rm m}$ and is expected to extend to shorter wavelengths. Assuming that this limited similarity also holds in 3D, increasing the wavelength to $40,mu{rm m}$ enables us to simulate the after-glow dynamics of the wake channel all the way into the nanosecond regime.
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