Information on sources and transport of cosmic rays (CRs) are gained from three types of observations: spectra, composition and anisotropies. Among those three, anisotropies have a somewhat paradoxical status: on the one hand the observed arrival directions of CRs posses a low level of anisotropy – about 1 part in 1000 or 10 000 at TeV–PeV energies. Given the likely inhomogeneous distribution of sources in the Galaxy, this requires an efficient mechanism to randomise the directions of CRs and therefore constrains models of CR transport. On the other hand, a residual anisotropy in the arrival direction encodes information about the position and age of sources. This has been considered as a possibility for finding nearby sources, in particular in the context of the observed rise in the positron fraction. We note that this situation is somewhat reminiscent of the cosmic microwave background (CMB): while the angular power spectrum of anisotropies contains information on the parameters of the cosmological model, the high level of isotropy requires a mechanism that allows far away regions to be in causal contact at early times.

It has been known for a long time that pitch-angle scattering mediated by resonant interaction between CRs and turbulent magnetic fields can provide the necessary randomisation (as reviewed by, e.g., ). Pitch-angle scattering is also mediating spatial diffusion and if the distribution of sources is asymmetric with respect to the observer, e.g. there are more sources towards the Galactic centre than towards the anti-centre, a small degree of residual anisotropy is expected. The amplitude a of the dipole in the arrival directions, defined as the relative difference of observed maximum and minimum fluxes, ϕmax and ϕmin, is related to the gradient in the isotropic part f0 of the distribution function f(x, p, μ) = f0(x, p) (1 + aμ), a=ϕmax-ϕminϕmax+ϕmin≃3Dv|∇f0|f0, where D is the isotropic diffusion coefficient and v the particle speed.

Predictions for the dipole anisotropy start by assuming a spatial distribution of sources, e.g., from supernova remnant or pulsar surveys. From the propagated CR distributions, the dipole amplitude is then computed as a function of energy. This picture, however, does not compare well with measurements . As shown in Fig. , a prediction from the isotropic diffusion model (dotted line, cf. ) is almost two orders of magnitude higher than the measurements between 100 TeV and 1 PeV. This discrepancy has come to be known as the cosmic ray anisotropy problem . Nevertheless, using the observed distribution of arrival directions, in particular the dipole has been widely advertised as a means to infer the presence of young, nearby sources .

In the following we suggest two modifications to the theoretical picture that in combination can decrease the predicted anisotropy and bring it into agreement with observations. First, in the presence of a background magnetic field B0 and for low levels of turbulence – defined as the ratio of turbulent energy density and total energy density, η = δB2/(B02 + δB2) – diffusion is inherently anisotropic, with the diffusion along B0 much more efficient than perpendicular to it. Instead of Eq. (), the dipole amplitude should read (see, e.g. ) a=32∫-11dμμf(μ)f0=3v|∂f0/∂x|f0D∥. Note that now the amplitude depends on the parallel diffusion coefficient D∥ and the CR gradient along B0 (assumed to be in the x direction). This introduces a new degree of freedom, the angle between B0 and the CR gradient direction ∇f0/|∇f0|.

Second, the intermittency in the turbulent magnetic field can play a role. Analytical computations of the transport of CRs can only predict the average distribution function f for an ensemble of turbulent magnetic fields. Due to the stochastic nature of the turbulent fields, in one particular realisation of the turbulent magnetic field we expect deviations from the ensemble average and consequently also deviations of the dipole direction and amplitude from the ensemble averaged direction and amplitude. We note that such intermittency effects have also been argued to be the cause of the CR small scale anisotropy observed at TeV–PeV energies .

We explore both these effects, i.e., the anisotropic diffusion and turbulence intermittency effects, by numerically back-tracking particles through individual realisations of a turbulent magnetic field. Given a large number of trajectories, {xi(t), pi(t)}, we can compute the arrival directions seen by an observer at position xobs and time t0 from an assumed quasi-stationary CR distribution at an earlier time (t0 - Δt) and exploiting Liouville's theorem: f(xobs, pi(t0)) = f(xi(t0 - Δt), pi(t0 - Δt)). Note the intermittency effect is due to the local configuration of the turbulent magnetic field, i.e., the arrival directions can only reflect the turbulent field over the last few scattering lengths. This justifies truncating the expansion of the spatial distribution after the first derivative, i.e., the CR gradient ∇f0. Therefore, we adopt the quasi-stationary distribution f(x, t0 - Δt) = f0 + x∂f0/∂x. Note that in general there is an angle between the gradient direction ∇f0/|∇f0| and B0.

We solve the relativistic equations of motions with a 5th order adaptive Runge–Kutta algorithm at three energies, 10, 100 and 1000 TV. For the level of turbulence we consider η = 1 and 0.1 which should bracket its uncertainty. To bridge the large dynamical range between the particle gyroradius and a few times its scattering length, we set up the turbulent magnetic field on a set of nested grids , assuming a Kolmogorov spectrum, an outer scale L = 100 pc and a total RMS field strength of 4 µG.

In Fig.  we show the dipole directions in 50 realisations of the turbulent magnetic field in the absence of B0 for 1 PeV CRs. The position and direction of each circle shows the dipole direction and amplitude in one random realisation, respectively. Two things are noteworthy: It is clear that the individual dipole directions are in general not pointing in the CR gradient direction (denoted by the yellow star). Second, there is some scatter also in the amplitude of individual dipoles. The (vectorial) mean, however, shown by the red square, is in good agreement both with the expected direction and amplitude. For all energies considered, the ensemble averaged dipole amplitude reproduces the result of , cf. dotted line in Fig. . We further explore the scatter in the amplitude in Fig.  that shows the distribution of dipole amplitudes as a function of the longitude of the CR gradient direction. (As there is no other direction in this setup, we would not expect any systematic dependence on this direction which is in fact the case.) The dipole amplitudes scatter by a factor of a few around the expectation value |〈a〉| that is always in agreement with the prediction from the isotropic diffusion model, |aiso|. Still, the predictions are at least an order of magnitude above the upper limit from KASCADE.

Figure  shows the distribution of dipole directions and amplitudes in the presence of B0 with η = 0.1, i.e., a weak turbulent field, and assuming an angle of ∼ 90∘ between ∇f0 and B0. The amplitudes are significantly suppressed and the directions show again a large scatter, but now cluster around the B0 direction, indicated by the blue cross, which is also the direction of the ensemble average dipole. From Fig. , it can be seen how the distribution of dipole amplitudes depends on the angle between B0 and ∇f0: for small angles, the amplitudes closely track the prediction from the isotropic diffusion model, with relatively little scatter. At angles close to 90∘ though, the amplitude can be significantly suppressed, at the minimum limited only by the level of turbulence η. For the adopted η = 0.1, the mean amplitude is in agreement with KASCADE. We also note that the scatter is enhanced for angles around 90∘. About 20 % of the turbulent field realisations lead to a dipole amplitude that is compatible with the EAS-TOP measurement.

We now turn back to Fig.  which also shows the dipole amplitudes for 5 random realisations of the turbulent magnetic field (open circles connected by solid lines). It is interesting to see that not all configurations show the monotonous behaviour with energy of the ensemble average. One configuration in particular shows a falling amplitude between 10 and 100 TeV before rising again – very much like the data.

Observations of the dipole direction in the arrival distribution of CRs have been promoted as a means of source searches, yet the measured dipole amplitude is systematically overpredicted by isotropic diffusion models. We suggest two effects, that can resolve this so-called anisotropy problem: the anisotropic nature of diffusion in the presence of a background magnetic field and intermittency, that is fluctuations of the distribution function across individual realisations of the turbulent field. While the conjunction of both effects can sufficiently suppress the dipole amplitude (for small turbulence level η and if the background field and CR gradient are close to perpendicular), this is also casting doubt on the chances of finding sources in the dipole direction: in the high turbulence level case (η = 1) the directions scatter significantly around the CR gradient direction (see Fig. ). In the low turbulence case (η = 0.1), there is also scatter and in general the dipole directions cluster around the field direction, not the gradient direction. For the preferred case with a large angle between gradient and field, see Fig. , the scatter is so large that not even the field direction can be inferred.