Mechanisms for multi-scale structures in dense degenerate astrophysical plasmas

Abstract

Two distinct routes lead to the creation of multi-scale equilibrium structures in dense degenerate plasmas, often met in astrophysical conditions. By analyzing an e-p-i plasma consisting of degenerate electrons and positrons with a small contamination of mobile classical ions, we show the creation of a new macro scale $L_{\mathrm{macro}}$ (controlled by ion concentration). The temperature and degeneracy enhancement effective inertia of bulk e-p components also makes the effective skin depths larger (much larger) than the standard skin depth. The emergence of these intermediate and macro scales lends immense richness to the process of structure formation, and vastly increases the channels for energy transformations. The possible role played by this mechanism in explaining the existence of large-scale structures in astrophysical objects with degenerate plasmas, is examined.

Appendices

Appendix A: Derivation of quadruple Beltrami equation

The Ampere’s law generally can be written in dimensionless variables as:

$$ \nabla\times\mathbf{B} = \frac{1}{2} \biggl[ \alpha\frac{N_{i}}{N_{0i}}\,{ \mathbf{V}}_{i} +(1-\alpha) \frac{N^{+}}{N_{0}^{+}}\,{\mathbf{V}}_{+} - \frac{N^{-}}{N_{0}^{-}}\,{\mathbf{V}}_{-} \biggr] $$

(A.1)

and

• (i) if $\alpha=1$ (e-i plasma, quasineutrality reads as $N_{i}=N^{-}=N$) we have:

  $$\mathbf{V}_{-}\equiv{\mathbf{V}}_{e}=\mathbf{V}_{i} - \frac{2}{N} \nabla\times\mathbf{B} $$

  leading to Double Beltrami (DB) states in e-i plasma with degenerate electrons (Berezhiani et al. 2015a);

• (ii) while when $\alpha=0$ (purely symmetric e-p plasma, quasineutrality reads as $N^{+}=N^{-}=N$) we have:

  $$\mathbf{V}_{-} - \mathbf{V}_{+} = - \frac{2}{N} \nabla\times\mathbf{B} $$

  that shall lead to higher Beltrami states when inertia effects in electron and positron fluids are taken into account [similar effect was discussed for relativistic non-degenerate plasmas in Yoshida and Mahajan 1999; Iqbal et al. 2008; Bhattacharyya et al. 2003; Mahajan and Lingam 2015].

Observations show that ion fluid fraction can be small ($\alpha \ll1$); also ion fluid velocity is much smaller than those for lighter elements—electron and positron fluids [$\mathbf{V}_{i} \ll\, {\mathbf{V}}_{-} , \mathbf{V}_{+}$] and, hence, one can imagine that the mobility of ions can be ignored in most of the cases (Oliveira and Tajima 1995) except when $\alpha= 1$ where, as it was shown in Mahajan et al. (2001, 2002, 2005, 2006), flow effects can be crucial in creating the structural richness in astrophysical environments as well as in the heating/cooling processes, Generalized Dynamo theory and flow acceleration phenomena. The case of $\alpha=1$—pure e-i plasma with degenerate electrons was already studied in Berezhiani et al. (2015a) and it was shown that when ignoring inertia effects in electron fluid the Double Beltrami states are accessible in the system. Hence, it is expected, that when ion fluid velocity is not neglected in e-p-i plasma with degenerate electrons and positrons number of relaxed states can be either 2 (when ignoring degenerate $e(p)$ fluids inertia effects although $G^{-}\neq G^{+}$) or 4 (when degenerate fluids inertia effects are taken into account); at the same time neglecting the ion flow effects ($\alpha\to0 , \mathbf{V}_{i}\to0$) we shall obtain the Single Beltrami state in former situation and the Triple Beltrami States in latter case—this problem is a scope of our study below [see Mahajan and Lingam 2015 and its results].

Let us now show how Beltrami states may acquire new structures due to degeneracy or/and the small fraction of mobile ions. We will study an incompressible e-p-i plasma with the simplifying assumption $\gamma_{+} \sim\gamma_{-}\equiv1$ that reduces the Generalized Bernoulli Conditions to $G^{+} + G^{-} = \mathit{const}$. The Ampere’s law (A.1), in dimensionless variables, is written as

$$ \nabla\times\mathbf{B} = \frac{1}{2} \bigl[\alpha\, \mathbf{V}_{i} + (1-\alpha)\,\mathbf{V}_{+} - \mathbf{V}_{-} \bigr]. $$

(A.2)

In terms of the e-p plasma bulk flow average velocity

$$ \mathbf{V} = \frac{1}{2} \bigl[(1-\alpha)\,\mathbf{V}_{+} + \mathbf{V}_{-} \bigr] $$

(A.3)

one can express the Generalized Momenta for positron and electron fluids as [$\mathbf{P}_{\pm}=G_{0}^{\pm}(n_{0}^{\pm })\,\mathbf{V}_{\pm}$]:

$$ \begin{aligned} &\mathbf{P}_{+} = \frac{G_{0}^{+}}{1-\alpha} \biggl( \mathbf{V} + \nabla\times\mathbf{B}-\frac{\alpha}{2}\, \mathbf{V}_{i} \biggr); \\ &\mathbf{P}_{-} = G_{0}^{-} \biggl(\mathbf{V} - \nabla \times\mathbf{B}+\frac{\alpha}{2}\,\mathbf{V}_{i} \biggr). \end{aligned} $$

(A.4)

Introducing $\beta\equiv\frac{G_{0}^{-}}{G_{0}^{+}}$ and using Eqs. (A.4) in Eq. (9), straightforward algebra leads to:

$$\begin{aligned} \mathbf{V} &= \eta \bigl( 2\beta G_{0}^{+} \nabla\times\nabla\times \mathbf{B} - \bigl[a_{+}(1-\alpha)\beta- a_{-} \bigr] \nabla\times\mathbf{B} \\ &\quad{}+ \bigl[1 + (1-\alpha) \beta \bigr]\,\mathbf{B} \bigr) - \alpha \beta \nabla\times\mathbf{V}_{i} \\ &\quad{} + \frac{\alpha}{2} \bigl[a_{+}(1-\alpha)\beta- a_{-} \bigr]\, \mathbf{V}_{i} \end{aligned}$$

(A.5)

with $\eta\equiv[a_{+}(1-\alpha)\beta+ a_{-}]^{-1}$.

We have to add the Ion flow Beltrami condition (13) written for incompressible case as

$$ \mathbf{B} + \zeta\nabla\times\mathbf{V}_{i} = \alpha a_{i}\, \mathbf{V}_{i} $$

(A.6)

to close the system of equations for incompressible e-p-i degenerate plasma.

Plugging Eq. (A.5) into Eqs. (A.4) and then using them in Eqs. (9) we get

$$\begin{aligned} &2\beta \bigl(G_{0}^{+} \bigr)^{2} \nabla\times\nabla \times \nabla\times\mathbf{B} - 2G_{0}^{+} \alpha_{1} \nabla \times \nabla\times\mathbf{B} \\ &\qquad{}+ \alpha_{2} \nabla\times\mathbf{B} - \alpha_{3}\, \mathbf{B} \\ &\quad= \alpha \bigl(G_{0}^{+} \bigr)^{2} \nabla\times\nabla\times\mathbf{V}_{i} - \alpha \bigl(G_{0}^{+} \bigr) \alpha_{1} \nabla\times \mathbf{V}_{i} \\ &\qquad{}- \alpha(1-\alpha) a_{+}a_{-} \mathbf{V}_{i}, \end{aligned}$$

(A.7)

where

$$\begin{aligned} \begin{aligned} &\alpha_{1} = \bigl[a_{+}(1- \alpha)\beta- a_{-} \bigr] , \\ &\alpha_{2} = \bigl[1 + (1-\alpha)\beta \bigr] G_{0}^{+} - 2 \beta(1-\alpha) a_{+} a_{-} , \\ &\alpha_{3}=(1-\alpha) (a_{+} + a_{-}) . \end{aligned} \end{aligned}$$

(A.8)

Equation (A.7) for immobile ions ($\mathbf{V}_{i} \equiv0$) will eventually give the so called “Triple Beltrami” equation for the magnetic field $\mathbf{B}$ (i.e. l.h.s. of Eq. (A.7) $\equiv0$).

Let us now simplify the equations when the ion density is just a small fraction of the density of the light species, i.e., $\alpha\ll1 \Longrightarrow(1-\alpha) \to1; [1+(1-\alpha)\beta] \to2$. After tedious but simple algebra, one obtains, in this limit, the quadruple Beltrami equation for $\mathbf{V}_{i}$:

$$ \begin{aligned}[b] & \bigl(G_{0}^{+} \bigr)^{2} \nabla\times\nabla\times\nabla\times\nabla\times \mathbf{V}_{i} - b_{1} \bigl(G_{0}^{+} \bigr)\nabla \times\nabla\times \nabla\times\mathbf{V}_{i} \\ &\quad{}+ b_{2} \bigl(G_{0}^{+} \bigr)\nabla\times\nabla \times\mathbf{V}_{i} - b_{3}\nabla\times \mathbf{V}_{i} + b_{4}\,\mathbf{V}_{i} = 0 , \end{aligned} $$

(A.9)

where

$$\begin{aligned} & \begin{aligned}[c] &b_{1} = (a_{+} - a_{-}) + \frac{1}{2} a_{i} \bigl(G_{0}^{+} \bigr); \\ &b_{2}= \bigl[ \bigl(G_{0}^{+} \bigr)-a_{+}a_{-} \bigr] + \frac{\alpha}{\zeta}a_{i} (a_{+} - a_{-}) + \frac{\alpha}{\zeta} \bigl(G_{0}^{+} \bigr) ; \end{aligned} \end{aligned}$$

(A.10)

$$\begin{aligned} & \begin{aligned} &b_{3} = (a_{+} - a_{-}) \biggl[1 - \frac{\alpha}{\zeta} \bigl(G_{0}^{+} \bigr) \biggr] + \frac{\alpha}{\zeta}a_{i} \bigl[ \bigl(G_{0}^{+} \bigr)-a_{+}a_{-} \bigr] ; \\ &b_{4} = \frac{\alpha}{\zeta} \bigl[a_{i} (a_{+} - a_{-}) -a_{+}a_{-} \bigr] . \end{aligned} \end{aligned}$$

(A.11)

Notice, that in above relations the terms with coefficient $(\alpha/\zeta)$ will vanish for either $\alpha\to0$ (when there is no fraction of ions at all!) or $m_{i}\to\infty$ (which means that ions are immobile!). In such case we arrive to Triple Beltrami Equations for $\mathbf{B}$ and $\mathbf{V}$ as mentioned above. Also, it is interesting to note that due to mobile ions (i.e. when $(\alpha/\zeta) \neq0$) the large scale is automatically there due to $b_{4}\neq0$ in Eq. (A.9). We will show this in detail in Appendix B.

Solving Eq. (A.9) for $\mathbf{V}_{i}$ and plugging it into (A.6) we will get the equation for $\mathbf{B}$; for the pure incompressible e-p plasma it is better to use Eq. (A.5) directly to find the magnetic field $\mathbf{B}$.

Appendix B: Scale separation in degenerate e-p-i plasma—quadruple Beltrami structures

Here we give some illustrative examples of Quadruple [Triple] Beltrami states for degenerate e-p-i [e-p] plasmas interesting for astrophysical context.

Introducing

$$ \begin{aligned} &b_{1}' = \frac{b_{1}}{G_{0}^{+}} ; \qquad b_{2}' = \frac{b_{2}}{G_{0}^{+}} ; \\ &b_{3}' = \frac{b_{3}}{(G_{0}^{+})^{2}} ; \qquad b'_{4} = \frac{b_{4}}{(G_{0}^{+})^{2}} \end{aligned} $$

(B.1)

Eq. (A.9) reads as:

$$\begin{aligned} &\nabla\times\nabla\times\nabla\times\nabla\times\mathbf{V}_{i} - b'_{1} \nabla\times\nabla\times\nabla\times \mathbf{V}_{i} \\ &\quad{}+ b'_{2} \nabla\times\nabla\times \mathbf{V}_{i} - b'_{3} \nabla\times \mathbf{V}_{i} + b'_{4}\, \mathbf{V}_{i} = 0 , \end{aligned}$$

(B.2)

Equation (B.2) can be written as

$$\begin{aligned} (\mathit{curl} - \mu_{1}) (\mathit{curl} - \mu_{2}) ( \mathit{curl} - \mu_{3}) (\mathit{curl} - \mu_{4})\, \mathbf{V}_{i} = 0, \end{aligned}$$

(B.3)

where $\mu_{k}$-s define the coefficients in Eq. (B.2) as

$$\begin{aligned} \begin{aligned} &b_{1}' = \mu_{1}+ \mu_{2}+\mu_{3}+\mu_{4} , \\ &b_{2}' =\mu_{1} \mu_{2} + \mu_{1}\mu_{3} + \mu_{1}\mu_{4} + \mu_{2}\mu_{3} + \mu_{2}\mu_{4} + \mu_{3} \mu_{4} , \\ &b_{3}' =\mu_{1} \mu_{2} \mu_{3} + \mu_{1} \mu_{2} \mu_{4} + \mu_{1} \mu_{3} \mu_{4} + \mu_{2} \mu_{3}\mu_{4} , \\ &b_{4}' = \mu_{1} \mu_{2} \mu_{3} \mu_{4}. \end{aligned} \end{aligned}$$

(B.4)

The general solution of Eq. (B.3) is a sum of four Beltrami fields $\mathbf{F}_{k}$ (solutions of Beltrami Equations $\nabla \times\mathbf{F}_{k} = \mu_{k}\, \mathbf{F}$) while eigenvalues ($\mu _{k}$) of the curl operator are the solutions of the fourth order equation

$$ \mu^{4}-b'_{1} \mu^{3} + b'_{2}\mu^{2} - b'_{3} \mu+ b'_{4}= 0. $$

(B.5)

Since $b'_{4} \to0$ for our case of study—e-p-i plasma with small fraction of mobile ions: $\alpha\ll1$; $m^{-} \ll m_{i} \Longrightarrow$ that Eq. (B.5) can be reduced to

$$ \mu(\mu- \mu_{2}) (\mu- \mu_{3}) (\mu- \mu_{4})= 0 . $$

(B.6)

solving of which gives $\mu_{1} \simeq0$—hence, the large scale structure existence is automatically guaranteed in such plasma due to the presence of small fraction of mobile ions (asymmetry due to small ion contamination); we stress here that this scale is always there.

Now we will have to solve the remaining equation:

$$ \mu^{3}-b'_{1} \mu^{2} + b'_{2}\mu- b'_{3}= 0 . $$

(B.7)

There are 3 possible simple, analytically tractable, scenarios:

(i) If

$$ (a_{+} - a_{-}) \biggl[1 - \frac{\alpha}{\zeta} \bigl(G_{0}^{+} \bigr) \biggr] = - \frac{\alpha}{\zeta}a_{i} \bigl[ \bigl(G_{0}^{+} \bigr)-a_{+}a_{-} \bigr] $$

(B.8)

then $b'_{3} \simeq0$, and Eq. (B.7) reduces to:

$$ \mu(\mu+ a_{1}) (\mu- a_{2}) = 0, $$

(B.9)

solving of which we find that $\mu_{2} \simeq0$ [another large scale now due to the asymmetry in degeneracy induced inertias of electrons and positrons] is also guaranteed in such plasma and two other short-scales are defined by Beltrami Parameters $a_{+}$ and $a_{-}$ as: $\mu_{3}=-a_{1}$; $\mu_{4}=a_{2}$, where

$$\begin{aligned} \begin{aligned} &a_{1} = \frac{1}{2}b'_{1} \mp\frac{1}{2}\sqrt{ \bigl(b'_{1} \bigr)^{2}-4b'_{2}} \quad\mbox{and} \\ &a_{2} = \frac{1}{2}b'_{1} \pm \frac{1}{2}\sqrt{ \bigl(b'_{1} \bigr)^{2}-4b'_{2}} . \end{aligned} \end{aligned}$$

(B.10)

In pure e-p plasma above conditions reduce to $a_{+}\sim a_{-}$ ($a_{1} = a_{+}$ while $a_{2} = a_{-}$) and eventually there are only 3 scales in total (defining equation is Triple Beltrami).

(ii) If $\alpha$ and $\beta$ and other parameters are such that both $b'_{1} \simeq0$ and $b'_{3} \simeq0$, then $b'_{2} = \mu_{3} \mu_{4} \neq0$. Here again one of the roots is zero (let it be $\mu_{2} = 0$), while other two satisfy the relations:

$$ \mu_{3} +\mu_{4} = 0 ; \qquad\mu_{3} \mu_{4} = b'_{2} . $$

(B.11)

Hence, also in this case length-scales are vastly separated (two scales are of similar range (short scales) and one is significantly large-scale): $|\mu_{2}|=|\mu_{3}|$; $\mu_{2}\simeq0$. For such scenario Beltrami coefficients $a_{\pm}$ must be such that $b'_{2}<0$ is maintained; for pure e-p plasma this is the case when $a_{+} = a_{-} = a > (G/n)^{1/2}$.

(iii) If $\alpha$ and $\beta$ and other parameters are such that all $b'_{1} \simeq0$, $b'_{2} \simeq0$ and $b'_{3} \simeq0$ can be established in addition to guaranteed $b'_{4} \simeq0$, then all the scale parameters $\mu_{1}$, $\mu_{2}$, $\mu_{3}$, $\mu_{4}$ become close to zero—no separation of length scales. For pure e-p plasma this is the case when $a_{+} = a_{-} = a = (G/n)^{1/2}$.

Thus, we can conclude, that for a rather big range of parameters there is a guaranteed scale separation in e-p-i plasma with degenerate lighter components.

Appendix C: Scale separation in compressible degenerate e-p plasma—triple Beltrami structures

Note that with no fraction of ions [$\alpha=0$, $\beta=1$] there is no charge separation and the scalar potential $\varphi \equiv0$ in Eqs. (3)–(4); hence, $n^{-} = n^{+} = n$, $\mathbf{B}=\nabla\times\mathbf{A}$; $\mathbf{E}=-\frac{1}{c}\frac {\partial}{\partial t}\,\mathbf{A}$ and the assumption of $T_{\pm}\to0$ leads to the initial temperature symmetry between species; we have consequently $G^{-}=G^{+}=G(n)$. The existence of soliton-like electromagnetic (EM) distributions in such a fully degenerate electron-positron plasma was studied in Berezhiani et al. (2015b).

Then, for pure compressible e-p plasma, if $\nabla [G^{\pm}(n^{\pm})]$ is at a much slower rate than the spatial derivatives of $\mathbf{B}$ and $\mathbf{V}_{\pm}$, we can write instead of (A.5) following relation [with corresponding $\eta= 1/(a_{+} + a_{-} )$]:

$$\begin{aligned} \mathbf{V} = \eta \biggl( 2\frac{G}{n} \nabla\times\frac{1}{n} \nabla\times\mathbf{B} - \frac{a_{+} - a_{-}}{n} \nabla\times\mathbf{B} \biggr) + \eta \frac{2{\mathbf{B}}}{n} \end{aligned}$$

(C.1)

and instead of (A.7) we obtain:

$$\begin{aligned} &\nabla\times \biggl(\frac{G}{n} \biggr)\nabla\times \biggl( \frac{1}{n} \biggr)\nabla\times\mathbf{B} -\kappa_{1}\nabla \times \biggl(\frac{1}{n} \biggr)\nabla\times\mathbf{B} \\ &\quad{}+ \kappa_{2} \nabla\times \biggl(\frac{G}{n} -a_{+} a_{-} \biggr)\, \mathbf{B} - \kappa_{3} \,\mathbf{B} = 0 , \end{aligned}$$

(C.2)

where

$$ \kappa_{1} = (a_{+} - a_{-}); \qquad\kappa_{2} = \frac{1}{G}; \qquad\kappa_{3} = \frac{a_{+} -a_{-}}{2G}. $$

(C.3)

After simple algebra we arrive to the defining equation for $\mathbf{V} = \frac{1}{2}(\mathbf{V}_{+} + \mathbf{V}_{-})$:

$$\begin{aligned} & \biggl(\frac{G}{n} \biggr)\nabla\times \biggl( \frac{1}{n} \biggr)\nabla\times\nabla\times\mathbf{V} - \kappa_{1} \biggl( \frac{1}{n} \biggr)\nabla\times\nabla\times\mathbf{V} \\ &\quad{}+ \kappa_{2} \nabla\times \biggl(\frac{G}{n} -a_{+} a_{-} \biggr) \biggl(\frac{n}{G} \biggr)\,\mathbf{V} - \kappa_{3} \, \mathbf{V} = 0 . \end{aligned}$$

(C.4)

Solution of Eqs. (C.2) and (C.4) is possible following the scenarios given in Appendix B, solutions will be similar to those given after Eq. (B.7), just density dependent. Estimation for the large scale $l_{\mathrm{meso}}$ in case of pure degenerate e-p plasma, derived from the dominant balance, gives:

$$ l_{\mathrm{meso}}= \frac{|\kappa_{2}|}{|\kappa_{3}|} |(G/n)-a_{+}a_{-}| = 2 \frac{|(G/n)-a_{+}a_{-}|}{|a_{+} - a_{-}|} \gg1 $$

(C.5)

if

$$ a_{+} = a_{-} = a \neq \biggl(\frac{G(n)}{n} \biggr)^{1/2} . $$

(C.6)

Hence, whenever the local density satisfies this condition there is a guaranteed scale separation in the degenerate e-p plasma with at least one large scale present.