We have observed five sulphur-bearing molecules in foreground diffuse molecular clouds lying along the sight-lines to five bright continuum sources. We have used the GREAT instrument on SOFIA to observe the 1383 GHz $^2Pi_{3/2} J=5/2-3/2$ transitions of SH towards the star-forming regions W31C, G29.96-0.02, G34.3+0.1, W49N and W51, detecting foreground absorption towards all five sources; and the EMIR receivers on the IRAM 30m telescope at Pico Veleta to detect the H$_2$S 1(10)-1(01), CS J=2-1 and SO 3(2)-2(1) transitions. In nine foreground absorption components detected towards these sources, the inferred column densities of the four detected molecules showed relatively constant ratios, with N(SH)/N(H$_2$S) in the range 1.1 - 3.0, N(CS)/N(H$_2$S) in the range 0.32 - 0.61, and N(SO)/N(H$_2$S) in the range 0.08 - 0.30. The observed SH/H$_2$ ratios - in the range (0.5-2.6) $times 10^{-8}$ - indicate that SH (and other sulphur-bearing molecules) account for << 1% of the gas-phase sulphur nuclei. The observed abundances of sulphur-bearing molecules, however, greatly exceed those predicted by standard models of cold diffuse molecular clouds, providing further evidence for the enhancement of endothermic reaction rates by elevated temperatures or ion-neutral drift. We have considered the observed abundance ratios in the context of shock and turbulent dissipation region (TDR) models. Using the TDR model, we find that the turbulent energy available at large scale in the diffuse ISM is sufficient to explain the observed column densities of SH and CS. Standard shock and TDR models, however, fail to reproduce the column densities of H$_2$S and SO by a factor of about 10; more elaborate shock models - in which account is taken of the velocity drift, relative to H$_2$, of SH molecules produced by the dissociative recombination of H$_3$S$^+$ - reduce this discrepancy to a factor ~ 3.
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