Dissipative structures of diffuse molecular gas. II. The translucent environment of a dense core

International audienceAims. This paper belongs to a series of four, dedicated to the analysis of the dynamical, thermal and chemical properties of translucent molecular gas, with the perspective of characterizing the processes driving the dissipation of supersonic turbulence, an anticipated prerequisite of dense core formation. Methods: We analyze the small scale morphology and velocity structure of the parsec-scale environment of a low mass dense core (1 M_ȯ). Our work is based on large maps made with the IRAM-30 m telescope in the two lowest rotational transitions of 12CO and 13CO with high angular (20'' or 0.015 pc at 115 GHz) and spectral (0.055 kms s-1) resolutions. The field is translucent, hence providing strong constraints on the column density and physical conditions in the gas. Results: More than one third of the field mass (6.5 M_ȯ) lies in an elongated tail of dense and cold gas, possibly extending beyond the edge of the map and connected to the core in space and velocity. This core tail is highly turbulent and sub-structured into narrow filaments of aspect ratio up to 20. These are pure velocity structures with velocity shears in the range 2-10 kms s-1 pc-1. Another third of the mass, according to the weak extinction of the field, lies in more dilute molecular and atomic gas. Its molecular fraction, largely traced by optically thick 12CO lines, is even more turbulent than the dense core tail. The gas emitting in the broad wings of the 12CO lines is organized into a conspicuous network of narrow criss-crossed filaments, whose pattern at the parsec scale is seen for the first time. The gas there is optically thin in the 12CO(1-0) line (τ12