Abstract.A force-free magnetic field with constant alpha for a circular cylindrical flux rope (Lundquist solution) is widely used to describe the magnetic field configuration in interplanetary flux ropes. Observations as well as MHD simulations indicate that interplanetary flux ropes are not circular but have an oblate shape. Here we present an analytical solution for a force-free magnetic field with constant alpha in an elliptic flux rope which may be regarded as a direct generalization of the Lundquist solution. An alternative simpler solution for a force-free magnetic field with constant alpha in an oblate flux rope is discussed.

Context. Detailed studies of magnetic cloud observations in the solar wind in recent years indicate that magnetic clouds are interplanetary flux ropes with a low twist. Commonly, their magnetic fields are fit by the axially symmetric linear force-free field in a cylinder (Lundquist field), which in contrast has a strong and increasing twist toward the boundary of the flux rope. Therefore another field, the axially symmetric uniform-twist force-free field in a cylinder (Gold-Hoyle field) has become employed to analyze magnetic clouds. Aims. Magnetic clouds are bent, and for some observations, a toroidal rather than a cylindrical flux rope is needed for a local approximation of the cloud fields. We therefore try to derive an axially symmetric uniform-twist force-free field in a toroid, either exactly, or approximately, and to compare it with observations. Methods. Equations following from the conditions of solenoidality and force-freeness in toroidally curved cylindrical coordinates were solved analytically. The magnetic field and velocity observations of a magnetic cloud were compared with solutions obtained using a nonlinear least-squares method. Results. Three solutions of (nearly) uniform-twist magnetic fields in a toroid were obtained. All are exactly solenoidal, and in the limit of high aspect ratios, they tend to the Gold-Hoyle field. The first solution has an exactly uniform twist, the other two solutions have a nearly uniform twist and approximate force-free fields. The analysis of a magnetic cloud observation showed that these fields may fit the observed field equally well as the already known approximately linear force-free (Miller-Turner) field, but it also revealed that the geometric parameters of the toroid might not be reliably determined from fits, when (nearly) uniform-twist model fields are used. Sets of parameters largely differing in the size of the toroid and its aspect ratio yield fits of a comparable quality.

[1] An analytical solution of force-free magnetic fields inside a toroid with an arbitrary aspect ratio was found. A scalar equation was derived which when solved gives the heart of the solution of the vector field in toroidal coordinates, in closed form in terms of hypergeometric functions. The solution can be used for interpretation of magnetic cloud measurements in the interplanetary space. If it is assumed that a magnetic cloud is a large loop with roots at the Sun, then its part can be locally treated as a part of a toroid. The former solution by Miller and Turner [1981] was limited only to larger aspect ratios. The presented solution coincides with the Lundquist [1950] one for cylindrical clouds in case of a very large aspect ratio torus.

No abstract