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We present a study of the relative orientation between the magnetic field andelongated cloud structures for the $\rho$ Oph A and $\rho$ Oph E regions inL1688 in the Ophiuchus molecular cloud. Combining inferred magnetic fieldorientation from HAWC+ 154 $\mu$m observations of polarized thermal emissionwith column density maps created using Herschel submillimeter observations, wefind consistent perpendicular relative alignment at scales of $0.02$ pc($33.6"$ at $d \approx 137$ pc) using the histogram of relative orientations(HRO) technique. This supports the conclusions of previous work using Planckpolarimetry and extends the results to higher column densities. Combining thisHAWC+ HRO analysis with a new Planck HRO analysis of L1688, the transition fromparallel to perpendicular alignment in L1688 is observed to occur at amolecular hydrogen column density of approximately $10^{21.7}$ cm$^{-2}$. Thisvalue for the alignment transition column density agrees well with values foundfor nearby clouds via previous studies using only Planck observations. Usingexisting turbulent, magnetohydrodynamic simulations of molecular clouds formedby colliding flows as a model for L1688, we conclude that the molecularhydrogen volume density associated with this transition is approximately$\sim10^{4}$ cm$^{-3}$. We discuss the limitations of our analysis, includingincomplete sampling of the dense regions in L1688 by HAWC+.

HAWC+/SOFIA Polarimetry in L1688: Relative Orientation of Magnetic Field and Elongated Cloud Structure