Inner Structure of CME Shock Fronts Revealed by the Electromotive Force and Turbulent Transport Coefficients in Helios-2 Observations

In mean-field electrodynamics, the magnetic field and the velocity field are split into two parts: the background field and the perturbation on this background; see Steenbeck et al. (1966). The evolution of the background magnetic field ${{\boldsymbol{B}}}_{0}$ can be expressed by the mean induction equation:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {{\boldsymbol{B}}}_{0}}{\partial t}={\boldsymbol{\nabla }}\times ({{\boldsymbol{U}}}_{0}\times {{\boldsymbol{B}}}_{0})+{\boldsymbol{\nabla }}\times {\boldsymbol{M}}+\eta {{\boldsymbol{\nabla }}}^{2}{{\boldsymbol{B}}}_{0},\end{eqnarray} \tag{ 1 }$$

where ${{\boldsymbol{U}}}_{0}$ denotes the background flow velocity and η is the magnetic diffusivity; see Equation (22) in Yokoi (2013). The impact of small-scale turbulence on the large-scale magnetic field appears as an electromotive force ${\boldsymbol{M}}$.

The electromotive force can be derived from mean-field electrodynamics as the ensemble average of the cross product between the flow velocity fluctuations $\delta {\boldsymbol{U}}$ and the magnetic field fluctuations $\delta {\boldsymbol{B}}$:

$$\begin{eqnarray}&&{{\boldsymbol{M}}}_{1}=\langle \delta {\boldsymbol{U}}\times \delta {\boldsymbol{B}}\rangle .\end{eqnarray} \tag{ 2 }$$

However, Equation (2) does not give information on the actual magnetic field generation mechanisms.

To circumvent the shortcomings of this formulation, there are alternative models for ${\boldsymbol{M}}$ that are based on theoretical or numerical dynamo studies. Therefore, we also test a formulation based on the RFP model from Yoshizawa (1990) adjusted to SI units:

$$\begin{eqnarray}&&{{\boldsymbol{M}}}_{\mathrm{RFP}}=\alpha {{\boldsymbol{B}}}_{0}-\beta {\mu }_{0}{{\boldsymbol{J}}}_{0}+\gamma {{\boldsymbol{\Omega }}}_{0},\end{eqnarray} \tag{ 3 }$$

where μ₀ is the vacuum permeability. Yokoi (2013) also uses this separation for helically driven magnetic turbulent dynamos. The α term is proportional to the background magnetic field ${{\boldsymbol{B}}}_{0}$. It represents the kinetic helicity effect and refers to the α dynamo mechanism. Here, the large-scale magnetic field amplifies by turbulent helical twisting motions that convert poloidal into toroidal field, or vice versa. A visualization of the α effect is given in Figure 1; see Priest (1982). The β term comprises a curl of the background magnetic field: ${\mu }_{0}{{\boldsymbol{J}}}_{0}\,={\boldsymbol{\nabla }}\times {{\boldsymbol{B}}}_{0}$. It represents the turbulent diffusion effect (β term) that causes diffusion of the large-scale magnetic field by small-scale turbulent disturbances. Finally, the γ term is a different kind of magnetic field amplification mechanism driven by the cross-helicity effect. It was first proposed by Yoshizawa (1990) and is also used by Sur & Brandenburg (2009), Yokoi (2013). This effect operates if cross-helicity is present in the magnetohydrodynamic description of turbulence. The cross-helicity describes the linkeage between magnetic field and vortex flows. Like the β term, the γ also contains a curl of a background field, here of the flow velocity: ${{\boldsymbol{\Omega }}}_{0}\,={\boldsymbol{\nabla }}\times {{\boldsymbol{U}}}_{0}$.

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Various models of the transport coefficients α and β are possible. An early formulation of α and β was derived from the mean-field electrodynamics by Krause & Raedler (1980):

$$\begin{eqnarray}\alpha & = & \tfrac{1}{3}\tau \langle -\delta {\boldsymbol{U}}\cdot \delta {\boldsymbol{\Omega }}\rangle \\ \beta & = & \tfrac{1}{3}\tau \langle \delta {\boldsymbol{U}}\cdot \delta {\boldsymbol{U}}\rangle \end{eqnarray} \tag{ 4 }$$

τ denotes the characteristic timescale of the fluctuations and describes the half-time of the decay of the turbulent energy and the helical strucures. Analogous to α and β we add the γ transport coefficient to our model as follows:

$$\begin{eqnarray}&&\gamma =\tfrac{1}{3}\tau \langle \delta {\boldsymbol{U}}\cdot \delta {\boldsymbol{B}}\rangle \end{eqnarray} \tag{ 5 }$$

A second form gives a more general modeling for the transport coefficients; see Equation (23)–(25) in Yoshizawa (1990) and Equation 36(a)–(c) in Yokoi (2013). In particular, for turbulent flows with high magnetic Reynolds or Lundquist numbers, a dependency on the current helicity appears in the α coefficient (Pouquet et al. 1976):

$$\begin{eqnarray}&&\alpha ={C}_{\alpha }\tau \,\langle -\delta {\boldsymbol{u}}\cdot \delta {\boldsymbol{\Omega }}+\delta {\boldsymbol{b}}\cdot \delta {\boldsymbol{J}}\rangle ,\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\beta ={C}_{\beta }\tau \,\tfrac{1}{2}\langle | \delta {\boldsymbol{u}}{| }^{2}+| \delta {\boldsymbol{b}}{| }^{2}\rangle ,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\gamma ={C}_{\gamma }\tau \,\langle \delta {\boldsymbol{u}}\cdot \delta {\boldsymbol{b}}\rangle .\end{eqnarray} \tag{ 8 }$$

The magnetic field is normalized here to Alfvén velocity units: ${\boldsymbol{b}}={\boldsymbol{B}}/\sqrt{{\mu }_{0}{\rho }_{0}}$ and the plasma flow velocity as ${\boldsymbol{u}}={\boldsymbol{U}}$. The coefficients C_α, C_β, and C_γ are model parameters. They were estimated by Hamba (1992) as C_α = O(10⁻²), C_β = O(10⁻¹), and C_γ = O(10⁻¹) for the turbulent electromotive force.

The structure of this work is as following: in Section 2 we use Taylor's frozen-in flow hypothesis (Taylor 1938) to reduce the curl to one-dimensional derivatives and calculate the transport coefficients from Equation (4). Then we construct a shock-front model of a helical magnetic field and a vortex in the plasma flow velocity, which corresponds to a field configuration as expected from kinetic helicity. With an initial estimate of free model parameters we find specific signatures in the α and β terms. In Section 3 we employ our method to the spacecraft data and compare the resulting signatures to the model prediction.

The difficulty in evaluating the transport coefficients lies in the task of determining the spatial derivatives from spacecraft data in order to calculate the vorticity ${\boldsymbol{\Omega }}={\boldsymbol{\nabla }}\times {\boldsymbol{U}}$ in the plasma bulk flow, where ${\boldsymbol{U}}$ is a function of space and time. Also the electric current density is defined by spatial derivatives of the magnetic field data through Ampère's law: ${\mu }_{0}{\boldsymbol{J}}={\boldsymbol{\nabla }}\times {\boldsymbol{B}}$. The spatial derivatives can be obtained from multi-spacecraft mission data using four spacecraft in a tetrahedral formation, like Cluster (Balogh et al. 1997) or MMS (Torbert et al. 2016). Such data allow us to separate time derivatives from spatial derivatives within the three-dimensional (3D) vectors of the magnetic field and plasma flow velocity. Unfortunately, the above-mentioned spacecraft orbit around the Earth and do not provide data from the heliosphere below Earth orbit. Therefore, we use data from the Helios spacecraft that provide information on the proton velocity, density, and the magnetic field down to a distance of 0.3 au from the Sun. However, the two Helios spacecraft do not orbit in formation, so we can only use single-spacecraft data and the field vectors depend only on time and have no spatial derivative. To overcome this issue, we use a streamwise coordinate system, with the coordinate z in the direction to the mean flow.

We obtain a spatial derivative from the time series using Taylor's frozen-in flow hypothesis (Taylor 1938). If we assume that $\tfrac{\delta U}{{U}_{0}}\ll 1$, the turbulent eddies evolve on a timescale larger than the observed event and thus the proper change in their structure is negligible. We can then replace the spatial derivative along z with the time derivative: $\partial z={U}_{0}\partial t$. This leads to a reduced form of the nabla operator that only consists of the z derivative:

$$\begin{eqnarray}&&{\boldsymbol{\nabla }}=\left(\begin{array}{c}\partial /\partial x\\ \partial /\partial y\\ \partial /\partial z\end{array}\right)\to \displaystyle \frac{1}{{U}_{0}}\left(\begin{array}{c}0\\ 0\\ \partial /\partial t\end{array}\right).\end{eqnarray} \tag{ 9 }$$

The curl of a function h then becomes:

$$\begin{eqnarray}&&{\boldsymbol{\nabla }}\times \left(\begin{array}{c}{h}_{x}\\ {h}_{y}\\ {h}_{z}\end{array}\right)\to \displaystyle \frac{1}{{U}_{0}}\left(\begin{array}{c}-\partial {h}_{y}/\partial t\\ +\partial {h}_{x}/\partial t\\ 0\end{array}\right).\end{eqnarray} \tag{ 10 }$$

In large-eddy simulations (LESs) any quantity can be separated into a resolved grid scale (GS) and an unresolved sub-grid scale (SGS) part. This is possible, e.g., with a spatial averaging, a time averaging, or with a frequency-domain filtering method. The GS then represents the homogenous mean field while the SGS contains the inhomogenous turbulent part. For in situ measurements of the solar wind, the SGS quantity cannot be obtained for a single event because the fluctuations are of spatial scales larger than the resolved GS. If we applied here the same method as used for LESs, the curl of our mean fields would either vanish or become negligible—and consequently only the α term would give significant non-zero contributions. Therefore, we require an alternative formulation of the β and γ terms that contain such a curl.

For our CME model, we consider the Parker spiral as the background field, which is relatively constant while the magnetic transient event is the fluctuation that passes by the spacecraft. From the observational data, we determine the mean field through a moving Gaussian convolution over large spatial scales and obtain the small-scale fluctuations as the residual after subtracting the obtained mean field. The Gaussian-convolution filtered signal, as well as its residual, contains contributions from scales larger and smaller than the characteristic length scale used for the filtering, which is set via σ₀ in Section 3.2. By this method, the obtained fluctuations $\delta {\boldsymbol{B}}$ and $\delta {\boldsymbol{U}}$ both still contain parts of the background (or mean) fields.

In contrast to the Gaussian-convolution separation described above, a frequency-based filtering method would split those contributions strictly at the characteristic scale, as often applied in turbulence research and LESs, see Figure 2. This leads to different fluctuations $\delta {\boldsymbol{B}}$ and $\delta {\boldsymbol{U}}$ than obtained through the Gaussian convolution. Also the exact choice of this characteristic separation scale has an influence on the result. It is worth investigating, in future work, how the separation method for background field and fluctuation could be improved for in situ observations. One possibility may be to split the Gaussian-convolution residuals further with a frequency-based filter into their resolved GS and unresolved SGS components. This would allow us to formulate the electromotive force more consistently with previous works in turbulent-transport theory.

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In the following, instead of the curl of the background quantities ${\boldsymbol{\nabla }}\times {{\boldsymbol{B}}}_{0}$ and ${\boldsymbol{\nabla }}\times {{\boldsymbol{U}}}_{0}$ that would both vanish here, we now use the curl of the Gaussian-convolution residuals; see Section 2.2. The contributions from larger and smaller scales contained in these residuals still have sufficiently large derivatives to get a non-zero current density and vorticity with the curls of $\delta {\boldsymbol{B}}$ and $\delta {\boldsymbol{U}}$. This leads us from Equation (12) to an alternative formulation of the electromotive force ${{\boldsymbol{M}}}_{2}$ as:

$$\begin{eqnarray}&&{{\boldsymbol{M}}}_{2}=\alpha {{\boldsymbol{B}}}_{0}-\beta ({\boldsymbol{\nabla }}\times \delta {\boldsymbol{B}})+\gamma ({\boldsymbol{\nabla }}\times \delta {\boldsymbol{U}})\end{eqnarray} \tag{ 11 }$$

We can now insert the transport coefficients from Equations (4) and (5) into (11) and get:

$$\begin{eqnarray}{{\boldsymbol{M}}}_{2} & = & \tfrac{1}{3}\tau \langle -\delta {\boldsymbol{U}}\cdot ({\boldsymbol{\nabla }}\times \delta {\boldsymbol{U}})\rangle {{\boldsymbol{B}}}_{0}\\ & & -\tfrac{1}{3}\tau \langle \delta {\boldsymbol{U}}\cdot \delta {\boldsymbol{U}}\rangle ({\boldsymbol{\nabla }}\times \delta {\boldsymbol{B}})\\ & & +\tfrac{1}{3}\tau \langle \delta {\boldsymbol{U}}\cdot \delta {\boldsymbol{B}}\rangle ({\boldsymbol{\nabla }}\times \delta {\boldsymbol{U}})\end{eqnarray} \tag{ 12 }$$

where τ is the timescale used for the averaging in order to determine the observed background fields, see Section 3.2.

Realistic turbulence models usually need to contain the production and dissipation mechanisms on large and small scales. However, in turbulent systems the large scales contain the energy, which determines the turbulent transport properties. In the region of the heliosphere that we analyze, from 0.4 au, the solar wind is no longer accelerated and starts to decay. We expect that the injection of energy on the largest scales starts to fade out and that the dissipation begins to dominate. This allows us to construct a simplified model, where the turbulent transport coefficients may be described also while neglecting the production mechanism.

We use a streamwise coordinate system with the third component aligned with the mean flow ${{\boldsymbol{U}}}_{0}$. Due to the proximity to the Sun (about 0.4 au) the Parker spiral is still rather radial. Therefore, we assume the mean magnetic field ${{\boldsymbol{B}}}_{0}$ to be parallel with ${{\boldsymbol{U}}}_{0}$. Alltogether, we define the large-scale solar wind background, representing a quiet solar wind, as

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{0}=\left(\begin{array}{c}0\\ 0\\ {B}_{0}\end{array}\right),{{\boldsymbol{U}}}_{0}=\left(\begin{array}{c}0\\ 0\\ {U}_{0}\end{array}\right).\end{eqnarray} \tag{ 13 }$$

We define a second set of fields that represent the perturbation caused by a magnetic transient event. This region contains a helically twisted magnetic field ${{\boldsymbol{B}}}_{1}$ and a vortex in the plasma flow velocity ${{\boldsymbol{U}}}_{1}$. The construction is made so that the large-scale fields (${{\boldsymbol{B}}}_{0}$, ${{\boldsymbol{U}}}_{0}$) and small-scale fields (${{\boldsymbol{B}}}_{1}$, ${{\boldsymbol{U}}}_{1}$) are of the same order of magnitude.

The model contains several free parameters: B₀ and U₀ are the strength of the mean magnetic field and mean plasma flow velocity.

From these background values, we expect the mean fields to increase to a plateau value before and after a magnetic transient event, which we express through _x, _y, _z and ι_x, ι_y, ι_z. Additionally, we define a peak value that sets the amplitude of the fields for the center of the vortex structure through ψ_x, ψ_y, ψ_z and ϕ_x, ϕ_y, ϕ_z. The direction of the field vectors depends on Θ(y, z) = atan(y/z). g_B describes the shape of the peak in the magnetic field, while we use f_p for the non-diverging center of the velocity vortex, see Equations (18) and (19). We define the disturbed solar-wind field vectors as:

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{1}={B}_{0}\left[\left(\begin{array}{l}{\epsilon }_{x}\\ -{\epsilon }_{y}\sin {\rm{\Theta }}\\ {\epsilon }_{z}\cos {\rm{\Theta }}\end{array}\right)+\left(\begin{array}{l}{\psi }_{x}\\ -{\psi }_{y}\sin {\rm{\Theta }}\\ {\psi }_{z}\cos {\rm{\Theta }}\end{array}\right){g}_{B}\right],\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{{\boldsymbol{U}}}_{1}={U}_{0}\left[\left(\begin{array}{c}{\iota }_{x}\\ {\iota }_{y}\\ {\iota }_{z}\end{array}\right)+\left(\begin{array}{l}{\phi }_{x}\\ {\phi }_{y}(\cos {\rm{\Theta }}+\sin {\rm{\Theta }})\\ {\phi }_{z}(\cos {\rm{\Theta }}-\sin {\rm{\Theta }})\end{array}\right)| {f}_{p}| \right].\end{eqnarray} \tag{ 15 }$$

We now focus on a one-dimensional cut through the center of the vortex. This means, when a 3D magnetic shock front passes by a spacecraft, one always observes the innermost structure of that shock front. Therefore, we may choose cosΘ = 0 for all times t and replace sinΘ with

$$\begin{eqnarray}\sin {\rm{\Theta }}=\left\{\begin{array}{rcl}-1 & : & t\lt 0\\ 1 & : & t\gt 0\\ 0 & : & t=0\end{array}\right.\end{eqnarray} \tag{ 16 }$$

where t = 0 specifies the center of the shock front in the time series. However, to avoid that ${{\boldsymbol{B}}}_{1}$ contains a non-steady step function with diverging derivatives at the vortex center, we use a smooth transition instead, here a fifth-order polynomial with zero first and second derivatives at the boundaries:

$$\begin{eqnarray}&&{s}_{t}\left(\xi =\displaystyle \frac{t}{{c}_{g}}\right)=\displaystyle \frac{1}{2}+\displaystyle \frac{15}{16}\xi -\displaystyle \frac{5}{8}{\xi }^{3}+\displaystyle \frac{3}{16}{\xi }^{5}\end{eqnarray} \tag{ 17 }$$

with $| t| \leqslant {c}_{g}$ and c_g as the width of the transition. The coefficients of the terms in s_t follow from the boundary conditions s_t(−1) = 0, s_t(0) = 1/2, and s_t(1) = 1.

With the assumption of incompressibility the radial profile of a vortex asympthotically decreases as 1/R, with $R(t)=| t|$. This would lead to a diverging velocity at the vortex center R → 0. But, for a realistic plasma vortex motion, the velocity at the center should vanish. Therefore, we replace 1/R with the radial profile

$$\begin{eqnarray}&&{f}_{p}(t,{c}_{f})=\displaystyle \frac{t}{\cosh (t/{c}_{f})},\end{eqnarray} \tag{ 18 }$$

where c_f is another free parameter that limits the derivatives at the vortex center.

We amplify the helical magnetic field with a Gaussian profile to mimic a peak in the magnetic field around the shock front:

$$\begin{eqnarray}&&{g}_{B}(t,{\sigma }_{B})=\displaystyle \frac{1}{\sqrt{2\pi {{\sigma }_{B}}^{2}}}\exp \left(-\displaystyle \frac{{t}^{2}}{2{{\sigma }_{B}}^{2}}\right),\end{eqnarray} \tag{ 19 }$$

where σ_B sets the width of the peak and completes our set of free parameters.

The fluctuating fields can now be written as:

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{1}={B}_{0}\left(\begin{array}{l}{\epsilon }_{x}+{\psi }_{x}{g}_{B}\\ -({\epsilon }_{y}+{\psi }_{y}{g}_{B})\left(2{s}_{t}\left(\displaystyle \frac{t}{{c}_{g}}\right)-1\right)\\ \quad 0\end{array}\right),\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{{\boldsymbol{U}}}_{1}={U}_{0}\left(\begin{array}{l}{\iota }_{x}+{\phi }_{x}| {f}_{p}| \\ {\iota }_{y}+{\phi }_{y}{f}_{p}\\ {\iota }_{z}-{\phi }_{z}{f}_{p}\end{array}\right).\end{eqnarray} \tag{ 21 }$$

As a final step, we use a weight function w_H to define a specific interval during which the helical fields dominate. This allows us to switch smoothly between quiet and helical solar wind with the smooth transition s_t defined in Equation (17):

$$\begin{eqnarray}{w}_{H}=\left\{\begin{array}{rc}\qquad 0: & \qquad t\lt -{\rm{\Delta }}-{c}_{h}\\ \quad {s}_{t}\left(\displaystyle \frac{t}{{c}_{h}}\right): & -{\rm{\Delta }}-{c}_{h}\lt t\lt -{\rm{\Delta }}\\ \qquad 1: & \qquad -{\rm{\Delta }}\lt t\lt {\rm{\Delta }}\\ 1-{s}_{t}\left(\displaystyle \frac{t}{{c}_{h}}\right): & \qquad {\rm{\Delta }}\lt t\lt {\rm{\Delta }}+{c}_{h}\\ \quad 0: & \quad {\rm{\Delta }}+{c}_{h}\lt t\end{array}\right.\end{eqnarray} \tag{ 22 }$$

with c_h = 1.8 hr and Δ = 3.6 hr. In Figure 3 we show w_H and the radial profile f_p.

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The total vector fields are now given by the sums of the smoothly tapered background and helical fields:

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{tot}}=(1-{\kappa }_{B}{w}_{H}){{\boldsymbol{B}}}_{0}+{w}_{H}{{\boldsymbol{B}}}_{1},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{{\boldsymbol{U}}}_{\mathrm{tot}}=(1-{\kappa }_{U}{w}_{H}){{\boldsymbol{U}}}_{0}+{w}_{H}{{\boldsymbol{U}}}_{1}.\end{eqnarray} \tag{ 24 }$$

For this study we choose κ_B = κ_U = 0, meaning that the background fields remain constantly present throughout the whole event, while ${{\boldsymbol{B}}}_{1}$ and ${{\boldsymbol{U}}}_{1}$ describe the distrubances. This allows us to take ${{\boldsymbol{B}}}_{0}$ and ${{\boldsymbol{U}}}_{0}$ as the mean fields, as well as ${{\boldsymbol{B}}}_{1}$ and ${{\boldsymbol{U}}}_{1}$ as the fluctuations. The transport coefficients and the electromotive force can then be expressed with our model parameters as:

$$\begin{eqnarray}&&\alpha =-\displaystyle \frac{\tau }{3}{U}_{0}{\iota }_{x}{\phi }_{y}{{w}_{H}}^{2}\displaystyle \frac{\partial {f}_{p}}{\partial t},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}\beta & = & \displaystyle \frac{\tau }{3}{{U}_{0}}^{2}{{w}_{H}}^{2}({{\iota }_{x}}^{2}+{{\iota }_{y}}^{2}+{{\iota }_{z}}^{2}+2{\iota }_{y}{\phi }_{y}{f}_{p}\\ & & -2{\iota }_{z}{\phi }_{z}{f}_{p}+{{\phi }_{y}}^{2}{{f}_{p}}^{2}+{{\phi }_{z}}^{2}{{f}_{p}}^{2}),\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}\gamma & = & \displaystyle \frac{\tau }{3}{U}_{0}{B}_{0}{{w}_{H}}^{2}[{\iota }_{x}({\epsilon }_{x}+{\psi }_{x}{g}_{B})\\ & & -({\iota }_{y}+{\phi }_{y}{f}_{p})({\epsilon }_{y}{s}_{t}+{\psi }_{y}{s}_{t}{g}_{B})],\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}{M}_{1} & = & {U}_{0}{B}_{0}{{w}_{H}}^{2}\{{({\iota }_{z}-{\phi }_{z}{f}_{p})}^{2}{({\epsilon }_{y}{s}_{t}+{\psi }_{y}{s}_{t}{g}_{B})}^{2}\\ & & +[{\iota }_{x}({\epsilon }_{y}{s}_{t}+{\psi }_{y}{s}_{t}{g}_{B})\\ & & {+({\iota }_{y}+{\phi }_{y}{f}_{p})({\epsilon }_{x}+{\psi }_{x}{g}_{B})]}^{2}\\ & & {+{({\iota }_{z}-{\phi }_{z}{f}_{p})}^{2}{({\epsilon }_{x}+{\psi }_{x}{g}_{B})}^{2}\}}^{\tfrac{1}{2}},\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}{M}_{2} & = & \{{(\alpha {B}_{0})}^{2}\\ & & \,+\left[-\beta \displaystyle \frac{{B}_{0}}{{U}_{0}}\left({\epsilon }_{y}\displaystyle \frac{\partial {w}_{H}{s}_{t}}{\partial t}+{\psi }_{y}\displaystyle \frac{\partial {w}_{H}{s}_{t}{g}_{B}}{\partial t}\right)\right.\\ & & \,{\left.+\gamma \left({\iota }_{y}\displaystyle \frac{\partial {w}_{H}}{\partial t}+{\phi }_{y}\displaystyle \frac{\partial {w}_{H}{f}_{p}}{\partial t}\right)\right]}^{2}\\ & & \,+\left[-\beta \displaystyle \frac{{B}_{0}}{{U}_{0}}\left({\epsilon }_{x}\displaystyle \frac{\partial {w}_{H}}{\partial t}+{\psi }_{x}\displaystyle \frac{\partial {w}_{H}{g}_{B}}{\partial t}\right)\right.\\ & & {\left.{\left.+\gamma \left({\iota }_{x}\displaystyle \frac{\partial {w}_{H}}{\partial t}\right)\right]}^{2}\right\}}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 29 }$$

As the z component of the helical magnetic field is always zero, see Equation (20), the two parameters _z and ψ_z can be removed from our model. Without loss of generality we assume a vortex in the y–z plane and hence set ϕ_x = 0. The other free parameters are chosen by an educated guess: we estimate the strength of the large-scale fields in the solar wind as B_SW = 25 nT and U_SW = 450 km s⁻¹. With _x = −1, we assume a helical magnetic field of the same magnitude as the solar wind background field in the x-direction and a smaller amplitude in the y-direction, _y = −0.15. We set the peak values of the helical magnetic field by the parameters ψ_x = −1.25 and ψ_y = −3.5. For the plasma flow velocity, we assume that there is a small component normal to the vortex ι_x = 0.15 and a three times smaller component along y and z: ι_y = − 0,05, ι_z = 0.05. We choose the amplitude of the plasma flow vortex as ϕ_y = −0.25 and ϕ_z = −0.15. The broadness parameters of g_B, f_p, s_t, and w_H we estimate to be σ_B = 0.45 hr, c_f = 0.36 hr, c_g = 0.36 hr, and c_h = 1.8 hr and the center of the vortex (t = 0) we set at 1978 April 18, 18:25:38 UT. The time resolution Δt = 648 s is chosen to be identical to the time resolution of the Helios data (Section 3.1). The resulting magnetic and velocity fields are plotted in Figures 4 and 5.

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Our initial choice for the characteristic timescale of the fluctuations is τ = 7.5 Δt = 1.35 hr. This timescale is chosen with respect to the evaluation of the background fields from the Helios spacecraft data; see Section 3.2.

We can now compute the transport coefficients from Equations (25)–(27). In the upper panel of Figure 12 we find a single peak in the kinetic helicity (α) surrounded by two smaller peaks with opposite sign. However, the smaller peaks come from the derivative of the radial profile and do not necessarily mean that there is instantaneous dynamo action in the opposite sense.

The turbulent diffusion (β) features a double peak in the center of the vortex with the left peak being smaller than the right peak; see the lower panel of Figure 12. The double peak in β is formed by the radial profile of the vortex f_p; the different amplitudes of the peaks are caused by the ratio between the plateaus and peak values in the plasma flow velocity (ι_y, ι_z, ϕ_y, ϕ_z). Unlike the α term, the turbulent diffusion is not strongest at the vortex center, because the magnetic field fluctuation disappears there. The plateau in β is mainly formed by the x component of the plasma flow normal to the vortex plane. This normal component needs to be different from zero, so that the α transport coefficient may exist in our model.

The cross-helicity (γ) term resembles the β term, but with an opposite sign. There is a plateau and a double peak at the vortex center with the left peak being smaller than the right peak. However, the asymmetry between both peaks in the γ term is less strong than for the β transport coefficient of the model. When the sign is opposite to the β term, this reflects the magnetic polarity and, in particular, that the magnetic field vector is antiparallel to the plasma flow velocity. If both vectors are parallel, one expects the same sign for β and γ.

The magnitude of the electromotive force M₁ (Figure 13) calculated from mean-field electrodynamics (Equation (28)) features a plateau and a double peak around the vortex center similar to the turbulent diffusion term with the left peak being smaller than the right peak. Here, the sign of the plateau parameters in the plasma flow velocity (ι_y, ι_z) determines which peak is higher.

On the other hand, if we use Equation (29) to calculate the magnitude of the electromotive force from the turbulent transport coefficients (M₂), we find a superposition of the three peaks with an enhanced amplitude on the right side. The plateaus in the β and γ coefficients do not carry over to the model electromotive force M₂ because there are no plateaus in the current density and vorticity of our model. This is similar to the observational data, where the plateau phase in M₂ is less significant than in M₁; see the logarithmic plot in Figure 14.

When we compare both formulations of the electromotive force (M₁ and M₂), the double peaks visible in M₁ seem to be merged in M₂; see the red dashed line in the lower panel of Figure 13. The reason for this less significant double-peak structure in M₂ is not solely that the α coefficient shows a single peak in the vortex center. Indeed, it is the contribution to M₂ of the whole γ term that features a strong single peak at the vortex center, which overlays the still significant double-peak structure obtained from the sum of the α and β terms. The γ term itself is dominated by the vorticity that has a strong single peak in the vortex center due to the large derivatives in the velocity field.

We use magnetic field and proton velocities obtained by the Helios-2 spacecraft (Musmann et al. 1975; Schwenn et al. 1975) from 1978 April 18–19. At that point in time, Helios-2 was located at a distance of 0.41 au from the Sun. The Helios data are in the Spacecraft Solar Ecliptic coordinate system: the XY-plane is the Earth mean ecliptic of date and the +X-axis is the projection of the vector spacecraft–Sun on the XY-plane. The +Z-axis is the direction of the ecliptic south pole (Fränz & Harper 2002). According to the header of the data files, the magnetic field and plasma flow velocity are given in radial–tangential–normal (RTN) coordinates with R = −X, T = −Y, and N = +Z. There was no correction made regarding the RTN system being defined relative to the helioequatorial plane instead of the ecliptic plane, which results in errors up to 1%. The magnetic field and plasma flow velocity, however, are consistent with each other and can be compared directly; see the Acknowledgments for the link to the Helios data. These data have a sampling rate of 40.5 s, which is the highest rate that is accessible for both the magnetic field and the proton flow velocity. The magnetic field data are obtainable at a higher sampling rate, but here they are reduced to the same resolution as the proton flow velocity. However, the sampling rate is not constant over the entire interval, because of changes between different observing modes, plus there are data gaps. These irregularities in the data we reduce by binning the data to an equidistant time discretization. As we observe a time interval with a magnetic cloud passing by, we choose the sampling rate at the shock front (Δt = 648 s) as the resolution for the whole interval. We replace single missing data points by a quadratic interpolation.

We perform a coordinate transformation to streamwise coordinates using the median values of the three components of the plasma flow velocity. This turns our coordinate system into the direction of the mean flow. In this case, however, as the radial component before the transformation is in the direction of the Sun, the resulting coordinate system is very close to the initial one, therefore we persist with the RTN notation. To determine the mean magnetic field and the mean plasma flow velocity, we use a smoothing function with a Gaussian kernel with a standard deviation σ₀ of 11 data points. This σ₀ mimics a box-car smoothing with a moving average over 30 neighboring points, but avoids the unwanted artifacts of this smoothing, such as those due to outliers in the data. The fluctuation we calculate as the difference of the original and the mean field. For the half-time of the fluctuations τ, we assume that 4τ is the minimum time for the decay of turbulent structures to a negligible value, here 1/2⁴. With that, we obtain a characteristic timescale of one fourth of the smoothing value: τ = 30/4Δt = 1.35 hr.

Figures 6–14 show the results of the Helios data analysis. The center of a structure in the magnetic transient that we interpret as a vortical structure is marked by a vertical dotted line. Our model is in the coordinate system x = N, y = T, and z = R. When comparing the magnetic field and plasma flow velocity components to our model, we have to keep in mind that we assume the case of a vortical structure that lies entirely in the yz-plane (or TR-plane in case of the observation). Therefore, our model does not feature a radial profile in the x(N) component. Also, by assuming a one-dimensional cut through the vortex center, the radial magnetic field component does not show any helical features and is constant.

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Figures 6 and 7 show the observed RTN components and the magnitude of the magnetic field and plasma flow velocity respectively. In comparison to our model (Figures 4 and 5), there is an initial increase and subsequent drop in all three magnetic field components, resulting in a change of sign around the center of the possible vortical structure, which could be explained by the structure not being restricted to the TR-plane. The plasma flow velocity components show a similar behavior with a decrease in front of the structure followed by an increase after the structure. However, the most prominent resemblance to the model can be found in the magnitude values with the magnitude of the plasma flow velocity resembling a shock front with a sudden increase followed by a slow relaxation over several hours. Simultanously, the magnitude of the magnetic field features a temporary increase around the region of interest which could be caused by induction work on ${\boldsymbol{B}}$, indicating the possibility of vortical plasma motions.

Figure 8 shows the mean magnetic field components. We observe that the normal and tangential components are not zero as they are in our model. As in the model, the radial component dominates with a magnitude of about 30 nT. While the observed mean fields are not constant, they do not show any sudden changes either, which shows this assumption is reasonable on the timescale of the fluctuations.

When using a streamwise coordinate system, the normal and tangential mean plasma flow velocity components are supposed to be zero and indeed that is the picture we get in the upper two panels of Figure 9. As for the mean magnetic field, the mean plasma flow velocity is not a constant but the changes are slow enough for the mean vorticity ${{\boldsymbol{\Omega }}}_{0}$ to be assumed constant on the fluctuation timescale.

We are especially interested in the components of the fluctuations of the fields: as they are centered around zero, they are useful for identifying changes of sign that are indicators of vortical structures. The fluctuating magnetic field components are shown in Figure 10, where we see interesting behavior of all three components around the time of the presumed vortical structure. The tangential and radial components feature a change of sign from positive to negative while the normal component δB_N shows an isolated peak from about +20 to −80 nT shortly after the center of the transition layer, at the same time the magnitude of the magnetic field reaches its maximum. This can be interpreted as a helical structure that would explain the increase in the magnetic field magnitude as a result of induction work.

The fluctuations of the plasma flow velocity components (Figure 11) interestingly change sign in all components before the δB_N peak, and again after the event has passed the spacecraft. After the event, the velocity components are more alike fluctuations around zero, similar to the radial profile we used in our model. While this does not directly prove the existence of a vortex in the solar wind, such behavior would be expected from vortical plasma motions. Additionally, we find that the cross-helicity effect is relevant for the turbulent transport within the solar wind.

The transport coefficients are calculated with Equation (4) using numerical derivatives. Just as in the model (see Figure 12), we find a single peak in the kinetic helicity (α), as well as an asymmetric double peak in the turbulent diffusion (β) and the cross-helicity (γ) coefficient; see Figure 12. However, the peak in α is higher than in the model and there are no minor peaks with opposite sign around it, which means that if the observed plasma flow velocity describes a vortex it must follow a different radial profile than assumed in our model. In both observation and model, we find a double-peak structure in β, even though it is more symmetric in the observation. As in the model, we see a plateau phase in β and γ around their peaks in the vortex center. The γ coefficient from the observation fits well in its asymmetry and amplitude to our model prediction.

In the upper panel of Figure 13 we show the magnitude of the turbulent electromotive force M₁ (solid line) that shows a basic agreement with our model prediction (red dotted line). Apparently, our model of helical magnetic field and vortical plasma motion fits reasonably well to the observed electromotive force formulation derived from mean-field electrodynamics (M₁). We find a double peak at the vortex center with similar asymmetry and a spatially extended plateau phase.

For the absolute value of M₂ we find a peak next to the vortex center, similar to our model prediction; see the lower panel of Figure 13. As in the model, there is no prominent plateau phase around the peak in M₂. However, the overall agreement between model and observation is less good than for M₁: while the peak in our model prediction has a similar asymmetry and amplitude to the observed one, there is a slight spatial shift inbetween them. This might stem from a too steep velocity gradient at our model's vortex center because the central peak is dominated by the cross-helicity term's contribution.

The increase to and the decrease from the plateau value can be better identified in a logarithmic plot; see Figure 14.