  Observations of magnetic clouds, within interplanetary coronal mass ejections
(ICMEs), are often well described by flux rope models. Most of these assume
either a cylindrical or toroidal geometry. In some cases, these models are also
capable of accounting for non-axisymmetric crosssections but they generally all
assume axial invariance. It can be expected that any ICME, and its flux rope,
will be deformed along its axis due to influences such as the solar wind. In
this work, we aim to develop a writhed analytical magnetic flux rope model
which would allow us to analytically describe a flux rope structure with
varying curvature and torsion so that we are no longer constrained to a
cylindrical or toroidal geometry. In this first iteration of our model we will
solely focus on a circular cross-section of constant size. We describe our flux
rope geometry in terms of a parametrized flux rope axis and a parallel
transport frame. We derive expressions for the axial and poloidal magnetic
field components under the assumption that the total axial magnetic flux is
conserved. We find an entire class of possible solutions, which differ by the
choice of integration constants, and present the results for a specific
example. In general, we find that the twist of the magnetic field locally
changes when the geometry deviates from a cylinder or torus. This new approach
also allows us to generate completely new types of in situ magnetic field
profiles which strongly deviate from those generated by cylindrical or toroidal
models.
