Gravitoelectromagnetic sheath

Abstract

In this paper we propose a gravito-electro-magnetic sheath (GEMS) model to explore the equilibrium properties of the solar plasma system. It describes the solar interior plasma (SIP) on the bounded scale and the solar wind plasma (SWP) on the unbounded scale from the viewpoint of plasma-based theory. This differs from the previously reported gravito-electrostatic sheath (GES) model employed to precisely define the solar surface boundary (SSB) on the fact that the present investigation incorporates variable temperature, magnetic field and collisional processes on the solar plasma flow dynamics. We show that the included parameters play important roles in the solar plasma dynamics. We demonstrate that the SSB location shifts outward as a result of the magnetic field by 14 % in comparison with that predicted by the GES model. As a consequence of the interaction of the plasma with magnetic field, the width of the sheath broadens by 25 % in comparison with the GES model predicted value. This physically means that the magnetic field decreases the distribution of the tiny (inertialess) electrons relative to the massive (inertial) ions, which in turn increases the confining wall potential value resulting in the increased width. Besides, the sonic point moves inward by 8 % as a result of collisions in the SIP that leads to rapid acceleration. Here, collisional dynamics plays an important role in the conversion process of the electron thermal energy into the bulk plasma flow energy. An interesting feature of continuous and smooth transition of the electric current density from the SIP to the SWP (with finite positive divergence on both the scales) through the SSB under inhomogeneous temperature distribution is also reported. Finally, the analyses may be applied to understand the realistic equilibrium dynamics of stellar plasmas never addressed before within the earlier GES framework like establishment of current-field correlation, properties of the slow solar wind and its effects on the interaction mechanism with the magnetic field, heliospheric current sheet, etc.

Appendices

Appendix A: Derivation of the SIP current density

The SIP current density contains currents from the four sources. These are the current due to free fall under gravity (j _g ), the current due to the electric field or potential (j _θ ) and two currents due to thermal motion of the electrons (j _e ) and ions (j _i ). The presumed uniform magnetic field in the SIP is not taken into account as a source, which is well-justified by the Maxwell equation (j _B =∇×B/μ ₀=0). The total current density is the sum of the four current densities which can be written in the following way,

$$\begin{aligned} j = - j_{g} + j_{\theta} + j_{e} - j_{i}. \end{aligned}$$

(A.1)

Now, we derive the expression for each component of the current density separately. The equation describing the evolution of the current due to free fall under gravity can be written as follows,

$$\begin{aligned} j_{g} = n_{0}ev, \end{aligned}$$

(A.2)

where, n ₀ is the equilibrium population density of plasma species having charge e and flowing with the velocity v in the SIP scale. The flow velocity of a particle undergoing gravitational free-fall is described by the kinematic equation as given below,

$$\begin{aligned} v^{2} = u^{2} + 2ra, \end{aligned}$$

where, u is the initial velocity of the object, a the acceleration due to gravity and r the radial distance from the initial position (heliocenter). In the present case, the initial velocity of the particle is assumed to be zero (u=0). The normalized form of the reduced equation with required simplification looks as follows,

$$\begin{aligned} M = \sqrt{2g_{S}\xi}, \end{aligned}$$

where, as described in the main text, M is the ion Mach number, g _S the normalized solar self-gravity and ξ the normalized radial position with respect to the heliocenter. Substituting the expression of M in Eq. (A.2), the current density due to gravitational free-fall takes the form as given below,

$$\begin{aligned} j_{g} = n_{0}ec_{S}\sqrt{2g_{S} \xi}, \end{aligned}$$

(A.3)

where, c _S is the phase speed of the sound plasma mode. The generalized equation to describe the evolution of the current density due to the electric field or potential can be written as follows,

$$\begin{aligned} j_{\theta} = n_{0}ev. \end{aligned}$$

(A.4)

The kinetic energy of a charged particle in a potential field is given by,

$$\begin{aligned} \frac{1}{2}m_{i}v^{2} = - e\phi, \end{aligned}$$

where, ϕ is the electrostatic plasma potential and m _i the ion mass. The normalized form of the above equation after simplifications can be presented as follows,

$$\begin{aligned} M = \sqrt{2( - \theta )}, \end{aligned}$$

where, θ is the normalized electrostatic potential. Substituting this in Eq. (A.4), the normalized expression of the current density takes the form as given below,

$$\begin{aligned} j_{\theta} = n_{0}ec_{S}\sqrt{2( - \theta )}. \end{aligned}$$

(A.5)

The generalized current density due to the electron thermal motion can be expressed as follows,

$$\begin{aligned} j_{e} = n_{e}ev, \end{aligned}$$

(A.6)

where, n _e is the electron population density. The kinetic energy of the thermal electron is given by,

$$\begin{aligned} \frac{1}{2}m_{e}v^{2} = \frac{1}{2}k_{B}T, \end{aligned}$$

where, m _e is the electron mass, k _B the Boltzmann constant and T the equilibrium plasma temperature. The simplified form of the above equation after normalization becomes as shown below,

$$\begin{aligned} M = \sqrt{\frac{m_{i}}{m_{e}}\tau}, \end{aligned}$$

where, τ is the normalized equilibrium temperature and m _i the ion mass. Using this in Eq. (A.6), the normalized current density due to the electron thermal motion takes the form as described by the following equation,

$$\begin{aligned} j_{e} = n_{0}ec_{S}\sqrt{\frac{m_{i}}{m_{e}} \tau} e^{\theta}. \end{aligned}$$

(A.7)

Here, n _e =n ₀ N _e where, N _e =e ^θ is the normalized electron population density. The current density evolution due to the ion thermal motion can be expressed as given below,

$$\begin{aligned} j_{i} = n_{i}ev, \end{aligned}$$

(A.8)

where, n _i is the ion population density. The kinetic energy of the thermal ion is expressed by the following equation,

$$\begin{aligned} \frac{1}{2}m_{i}v^{2} = \frac{1}{2}k_{B}T. \end{aligned}$$

The simplified form of the above equation after normalization can be written as follows,

$$\begin{aligned} M = \sqrt{\tau}. \end{aligned}$$

Using this in Eq. (A.8), the normalized current density due to the ion thermal motion takes the form as given below,

$$\begin{aligned} j_{i} = n_{0}ec_{S}\sqrt{\tau} e^{\theta}. \end{aligned}$$

(A.9)

Now, substituting all the individual current densities in Eq. (A.1), we get the total SIP current density as follows,

$$\begin{aligned} &j_{\mathit{SIP}} = n_{0}ec_{S} \biggl\{ - \sqrt{2g_{S}\xi} + \sqrt{2( - \theta )} - \biggl( \sqrt{\tau} - \sqrt{ \frac{m_{i}}{m_{e}}\tau} \biggr)e^{\theta} \biggr\} . \end{aligned}$$

(A.10)

We normalize the above current density by the Bohm current density given by j _B =n ₀ ec _S . The normalized current density in the SIP scale now can be written as follows,

$$\begin{aligned} J_{\mathit{SIP}} = - \sqrt{2g_{S}\xi} + \sqrt{2( - \theta )} - \biggl( \sqrt{\tau} - \sqrt{\frac{m_{i}}{m_{e}}\tau} \biggr)e^{\theta}. \end{aligned}$$

(A.11)

Appendix B: Derivation of the Lorentz force

Let us consider that the SWP is steady-state with variation only in radial and azimuthal directions. The velocity and magnetic field of the SWP in the given configuration can be written as follows,

$$\begin{aligned} v = v_{r}\hat{r} + v_{\phi} \hat{\phi}\quad \mbox{and}\quad B = B_{r}\hat{r} + B_{\phi} \hat{\phi}. \end{aligned}$$

Here, v _r and B _r are the respective radial components of the velocity and magnetic field; and, v _ϕ and B _ϕ are the corresponding azimuthal components. The radial SWP ion flux conservation in steady-state yields,

$$\begin{aligned} \frac{1}{r^{2}}\frac{d}{dr} \bigl( r^{2}\rho v_{r} \bigr) = 0, \end{aligned}$$

where, ρ is the mass density in kg m⁻³ and r is the radial distance. Integrating along the radial direction, we get the following simplified algebraic equation,

$$\begin{aligned} r^{2}\rho v_{r} = \mathit{const}. \end{aligned}$$

(B.1)

Now, Maxwell equation to describe the evolution of the electric field due to change in the magnetic field is given as follows,

$$\begin{aligned} \nabla \times E = - \frac{dB}{dt}, \end{aligned}$$

(B.2)

where, E and B denote the electric and magnetic fields, respectively. The equation can be simplified by invoking the concept of convective derivative in time-stationary form (∂B/∂t∼0), and using the identity of vector triple product, [A×(B×C)=B(A.C)−C(A.B)], as given below,

$$\begin{aligned} \nabla \times E = \nabla \times ( v \times B ) = \frac{1}{r} \frac{d}{dr} \bigl[ r ( v_{r}B_{\phi} - v_{\phi} B_{r} ) \bigr]. \end{aligned}$$

(B.3)

Since, the SWP is a conducting fluid and velocity is parallel to the magnetic field, hence v×B=0. Integration of above equation and subsequent simplification gives the azimuthal component (B _ϕ ) of B as given below,

$$\begin{aligned} B_{\phi} = \frac{v_{\phi} B_{r} - \varOmega rB_{r}}{v_{r}}, \end{aligned}$$

(B.4)

where, Ω is the angular velocity of the roots of the lines of force in the Sun. Now, another Maxwell equation, which describes the non-existence of magnetic monopole, ∇.B=0 in spherical geometry gives,

$$\begin{aligned} \frac{1}{r^{2}}\frac{d}{dr} \bigl( r^{2}B_{r} \bigr) = 0. \end{aligned}$$

Integrating it along radial direction, we get the radial component (B _r ) of B as given below,

$$\begin{aligned} B_{r} = \biggl( \frac{r_{0}}{r} \biggr)^{2}b_{r}, \end{aligned}$$

(B.5)

where, b _r is an arbitrary constant magnetic field at a reference point r=r ₀. The azimuthal momentum conservation of the SWP is described by the ϕ-component of momentum equation as given below,

$$\begin{aligned} \rho v_{r}\frac{1}{r}\frac{d}{dr} ( rv_{\phi} ) =& ( j \times B )_{\phi} \\ =& \frac{1}{\mu_{0}} \bigl[ ( \nabla \times B ) \times B \bigr]_{\phi} = \frac{B_{r}}{\mu_{0}r}\frac{d}{dr} ( rB_{\phi} ), \end{aligned}$$

where, μ ₀ is the permeability of free space. We rewrite the above equation in simplified form as given below,

$$\begin{aligned} \frac{d}{dr} ( rv_{\phi} ) = \frac{B_{r}}{\mu_{0}\rho v_{r}}\frac{d}{dr} ( rB_{\phi} ). \end{aligned}$$

(B.6)

Using Eqs. (B.1) and (B.5) we get the following condition,

$$\begin{aligned} \frac{B_{r}}{\mu_{0}\rho v_{r}} = \frac{B_{r}r^{2}}{\mu_{0}\rho v_{r}r^{2}} = \mathit{const}. \end{aligned}$$

This allows us to integrate Eq. (B.6), which gives the following algebraic equation,

$$\begin{aligned} rv_{\phi} - \frac{B_{r}}{\mu_{0}\rho v_{r}}rB_{\phi} = \mathit{const} = L. \end{aligned}$$

(B.7)

Using the expression of B _ϕ from Eq. (B.4) in Eq. (B.7), we get the azimuthal component (v _ϕ ) of v as presented below,

$$\begin{aligned} v_{\phi} = \frac{B_{r}^{2}}{\mu_{0}\rho v_{r}^{2}} ( v_{\phi} - \varOmega r ) + \frac{L}{r}. \end{aligned}$$

(B.8)

In the above equation, the term B _r /(μ ₀ ρ)^1/2 represents the Alfven speed (c _A ) in radial direction. The plasma flow speed is sub-Alfvenic in the solar surface while it becomes super-Alfvenic in the outer solar atmosphere. So, there exist a radial point where it becomes equal to the Alfven speed. Let this point be r _a . At the radial point r=r _a , Eq. (B.8) reduces to the following expression,

$$\begin{aligned} L = \varOmega r_{a}^{2}. \end{aligned}$$

(B.9)

Substituting Eqs. (B.8) and (B.9) in Eq. (B.4), and after some manipulation using Eq. (B.5), we get the expression for B _ϕ as follows,

$$\begin{aligned} B_{\phi} = - b_{r}r_{0}^{2}\varOmega \mu_{0}\rho v_{r}r\frac{r_{a}^{2} - r^{2}}{ ( b_{r}^{2}r_{0}^{4} - \mu_{0}\rho v_{r}^{2}r^{4} )}. \end{aligned}$$

(B.10)

The Lorentz force due to the above magnetic field can be expressed by the following equation,

$$\begin{aligned} F = ( j \times B )_{r} = - \frac{1}{\mu_{0}r}B_{\phi} \frac{d}{dr} ( rB_{\phi} ). \end{aligned}$$

(B.11)

Appendix C: Derivation of the SWP current density

The SWP current density consists of the currents from five sources, fifth source being the magnetic field (j _B ). Other four sources are same as in the SIP, i.e., free fall under gravity (j _g ), the electric field or potential (j _θ ), the electron thermal motion (j _e ) and the ion thermal motion (j _i ). Thus, the total current density for the SWP scale (j _SWP ) can be written as,

$$\begin{aligned} j_{\mathit{SWP}} = - j_{g} + j_{\theta} - j_{i} + j_{e} + j_{B}. \end{aligned}$$

(C.1)

Now, we derive the expression for the individual current densities separately in the similar manner as in the SIP. The generalized current density due to free fall under gravity can be written as follows,

$$\begin{aligned} j_{g} = n_{0}ev. \end{aligned}$$

(C.2)

All the notations throughout the derivation holds similar meaning as used in the SIP analysis (Appendix A). The velocity of a particle undergoing gravitational free-fall can be set out as,

$$\begin{aligned} v^{2} = u^{2} + 2ra. \end{aligned}$$

In the present case, the acceleration due to gravity is presumed to be constant in the outer solar atmosphere. In normalized form, the simplified equation for the velocity can be written as,

$$\begin{aligned} M = \sqrt{\frac{2a_{0}}{\xi}}. \end{aligned}$$

Here, a ₀ is a constant defined by, $a_{0} = GM_{ \odot} / c_{S}^{2}\lambda_{J}$, where, G is the universal gravitational constant, M _⊙ the solar mass and λ _J the critical Jeans scale length. Substituting this in Eq. (C.2), the normalized current density due to gravitational free-fall dynamics takes the following form,

$$\begin{aligned} j_{g} = n_{0}ec_{S}\sqrt{\frac{2a_{0}}{\xi}}. \end{aligned}$$

(C.3)

Other three current densities, i.e., current due to the electric field or potential (j _θ ), the electron thermal motion (j _e ) and the ion thermal motion (j _i ), have same expressions and derivation as in the SIP scale, although they are different in magnitude. We rewrite the respective final expression of each of the parameters below,

$$\begin{aligned} j_{\theta} = n_{0}ec_{S}\sqrt{2( - \theta )}, \end{aligned}$$

(C.4)

$$\begin{aligned} j_{e} = n_{0}ec_{S}\sqrt{\frac{m_{i}}{m_{e}} \tau} e^{\theta},\quad \mbox{and} \end{aligned}$$

(C.5)

$$\begin{aligned} j_{i} = n_{0}ec_{S}\sqrt{\tau} e^{\theta}. \end{aligned}$$

(C.6)

The fifth contributor is the SWP inhomogeneous magnetic field. The general expression of the current density due to the magnetic field is given by the Maxwell equation as described below,

$$\begin{aligned} j_{B} = \frac{1}{\mu_{0}}\nabla \times B. \end{aligned}$$

(C.7)

Here, we consider only the azimuthal component (B _ϕ ) of the magnetic field (B). Therefore, the curl of B reduces to dB _ϕ /dr. The current density in normalized form thus can be written as below,

$$\begin{aligned} j_{B} = j_{0}\frac{dB_{N\phi}}{d\xi}. \end{aligned}$$

(C.8)

where, j ₀ is a current-scaling factor defined by j ₀=B ₀/μ ₀ λ _J . Here, the magnetic field and position are written in normalized form. Substituting the expression of each components in Eq. (C.1), we get the equation to describe the evolution of the SWP current density as follows,

$$\begin{aligned} j_{\mathit{SWP}} =& n_{0}ec_{S} \biggl\{ - \sqrt{ \frac{2a_{0}}{\xi}} + \sqrt{2( - \theta )} - \biggl(\sqrt{\tau} + \sqrt{ \frac{m_{i}}{m_{e}}\tau}\,\biggr) e^{\theta} \biggr\} \\ &{} + \frac{B_{0}}{\mu_{0}\lambda_{J}}\frac{dB_{N\phi}}{d\xi}. \end{aligned}$$

(C.9)

We normalize the above current density by the Bohm current density, given by j _B =n ₀ ec _S . The normalized current density is described by the following equation,

$$\begin{aligned} J_{\mathit{SWP}} =& - \sqrt{\frac{2a_{0}}{\xi}} + \sqrt{2( - \theta )} - \biggl( \sqrt{\tau} - \sqrt{\frac{m_{i}}{m_{e}}\tau} \biggr)e^{\theta} \\ &{} + \rho_{J}\frac{dB_{N\phi}}{d\xi}, \end{aligned}$$

(C.10)

where, ρ _J is a dimensionless constant defined by ρ _J =B ₀/μ ₀ λ _J n ₀ ec _S .