Parker-spiral theory predicts that the heliospheric magnetic field (HMF) will have components of opposite polarity radially toward the Sun and tangentially antiparallel to the solar rotation direction (i.e., in Geocentric Solar Ecliptic (GSE) coordinates, with B _{X}/B _{Y} < 0). This theory explains the average orientation of the HMF very well indeed but does not predict the so-called “ortho-gardenhose” (hereafter OGH) flux with B _{X}/B _{Y} > 0 which is frequently observed. We here study the occurrence and structure of OGH flux, as seen in near-Earth space (heliocentric distance r = 1~mbox{AU}) by the Wind and Advanced Composition Explorer (ACE) spacecraft (for 1995 – 2017, inclusive) and by the Helios-1 and -2 spacecraft at 0.29~mbox{AU} < r leq 1~mbox{AU} (for December 1974 to August 1981), in order to evaluate the contributions to OGH flux generation of the various mechanisms and factors that are not accounted for by Parker-spiral theory. We study the loss of OGH flux with increasing averaging timescale [tau ] between 16 seconds and 100 hours and so determine its spectrum of spatial/temporal scale sizes. OGH flux at Earth at sunspot minimum is shown to be more common than at sunspot maximum and caused by smaller-scale structure in the HMF (with a mode temporal scale at a fixed point of tau_{text{mp}} approx 10 hours compared to tau_{text{mp}} approx 40 hours for sunspot maximum, corresponding to about 5.5∘ and 22∘ (respectively) of heliocentric angular width for corotational motion or 21 mathrm{R}_{odot} and 84 mathrm{R}_{odot} for radial solar-wind flow (where mathrm{R}_{odot} is the mean solar radius). OGH field generated by rotating the HMF through the radial direction is also shown to differ in its spectrum of scale sizes from that generated by rotating the HMF through the tangential direction – the former does not contribute to the “excess” open heliospheric flux at a given r but the latter does. We show that roughly half of the HMF deflection from the ideal Parker-spiral needed to give the observed occurrence of OGH flux at Earth occurs at r below 0.3 AU. By comparing the Helios and near-Earth data we highlight some questions that can be addressed by the Parker Solar Probe mission, which will study the HMF down to r = 0.046~mbox{AU}. We suggest that with decreasing heliocentric distance, Probe will detect decreased OGH field due to draping around transient ejecta, such as blobs and coronal mass ejections, but increasing structure in the radial field within traditional HMF sectors that are remnant Alfvénic disturbances in outflow regions from coronal reconnection sites.
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