Spectroscopy of Electric-Field Oscillations in the Solar Wind During the Passage of a Type III Radio Burst Using Observations Compared with Self-Similar Theory

Abstract

We present a spectrum of electric-field oscillations observed in situ in the solar wind by the WAVES experiment on the Wind spacecraft during the passage of a type III solar radio burst. 19 frequencies of this spectrum are compared with recent predictions of a self-similar nonlinear theory of two-dimensional electron oscillations (Osherovich and Fainberg, 2018).

Appendix:  Three Governing Equations for the Self-Similar Theory

For the two-dimensional case in cylindrical coordinates, the self-similar velocity $\vec{v}$ is the following:

$$ \vec{v} = \frac{y'(t)}{y(t)}r\vec{\mathrm{e}}_{r} + \frac{f'(t)}{f(t)}z\vec{\mathrm{e}}_{z}, $$

(12)

where $y(t)$ and $f(t)$ are the scalar functions in Equations 10 and 11.

The electron density is then

$$ n_{\mathrm{e}}(t) = n_{p}(y(t)/y_{eq})^{ - 2}(f(t)/f_{eq})^{ - 1} = n_{p}Y^{ - 2}F^{ - 1}, $$

(13)

where $y_{eq}$ and $f_{eq}$ are constants associated with equilibrium, and non-dimensional functions $Y$ and $F$ correspond to normalized $y$ and $f$, respectively. Using non-dimensional time from Equation 9 and self-similar variables $\eta$ and $\xi$ from Equations 10 and 11, we can present $\vec{v}$ and $\vec{E}$ in the following form:

$$ \vec{v} = \frac{dY}{d\tau} \eta \omega_{p0}\sqrt{2} y_{eq}\vec{\text{e}}_{r} + \frac{dF}{d\tau} \xi \omega_{p0}\sqrt{2} f_{eq}\vec{\text{e}}_{z} $$

(14)

$$ \vec{E} = 4\pi \text{e}n_{p}[E_{1}(\tau )\eta y(\tau )\vec{\text{e}}_{r} + E_{0}(\tau )\xi f(\tau )\vec{\text{e}}_{z}]. $$

(15)

Thus, the four functions $E_{0}(\tau)$, $E_{1}(\tau)$, $Y(\tau)$ and $F(\tau)$ are sufficient to find $n_{\mathrm{e}}$, $\vec{v}$ and $\vec{E}$. In fact, $E_{1}(\tau )$ is related to the other three functions algebraically (from Equation 17 in Osherovich and Fainberg (2018)):

$$ E_{1} = \frac{1}{2}(1 - Y^{ - 2}F^{ - 1} - E_{0}). $$

(16)

Thus, only three functions are needed to describe self-similar oscillations. For these functions there are three governing equations:

$$ \frac{d^{2}Y}{d\tau^{2}} - Y^{ - 1}F^{ - 1} + Y - YE_{0} = 0, $$

(17)

$$ \frac{d^{2}F}{d\tau^{2}} = - 2E_{0}F, $$

(18)

$$ \frac{dE_{0}}{d\tau} = \frac{dF}{d\tau} F^{ - 2}Y^{ - 2}. $$

(19)

These three equations derived in Osherovich and Fainberg (2018) have clear physical origins. Equation 17 is the radial component of the momentum Equation 2 after the separation of variables. Equation 18 corresponds to the $z$-component of the same momentum equation (force balance along axis of symmetry). Equation 19 comes from the z-component of Ampere’s law 4 with $\vec{H} = 0$. Equation 16, which is algebraic, comes from the Poisson’s equation 5 (again after separation of variables) and can be viewed as the definition of $E_{1}(\tau )$. The detailed derivation of the governing equations 17, 18, 19 is in Osherovich and Fainberg (2018). The spectrum of the electric field, $E_{0}$, along the axis of symmetry shown in Figure 2c is calculated from the numerical solution of the system of differential equations 17, 18, 19. The initial conditions used are the following: $F(0) = 1.151$, $\frac{dF}{d\tau} (0) = 0$, $Y(0) =1.29605$, $\frac{dY}{d\tau} (0) = - 0.007$ and $E_{0}(0) = 0.482773$. The theoretical frequencies in the fifth column of Table 1 have been found by solving the system of three equations with the above initial conditions.