Appendix B: Derivation of the steady state diffusive particle transport equation in cylindrical coordinates

Consider for the present purpose a cylindrical jet model where the plasma moves along the z-axis at constant (relativistic) v_z while the perpendicular velocity component is purely azimuthal, i.e. characterized by the angular frequency $\Omega$, in which case it proves useful to apply cylindrical coordinates $x^{\alpha}=(ct,r,\phi,z)$. One may then choose a set of (non-normalized) holonomic basis vectors $\{{\bf {\vec{e}_{\alpha}}}, \alpha=0,1,2,3\}$ with ${\bf {\vec{e}_{\alpha}}}=\partial{\bf\vec{x}}/\partial x^{\alpha}$, which determine a 1-form basis $\{{\bf {\vec{e}^{\alpha}}}\}$, known as its dual basis. For cylindrical coordinates $x^{\alpha}$ the metric tensor $g_{\alpha \beta}$ becomes coordinate-dependent, i.e. for the covariant metric tensor we have

$$(g_{\alpha \beta})={\rm diag}\{-1,1,r^2,1\}~,$$ (B.1)

while for the contravariant counterpart is consequently given by $(g^{\alpha \beta})={\rm diag}\{-1,1,1/r^2,1\}$. In particular, all partial derivatives of the metric coefficients vanish, except for the g₂₂ coefficient, and therefore all connection coefficients or Christoffel symbols of second order vanish, except for

$$\Gamma^1_{22}=-r~, \quad \Gamma^2_{21}=\Gamma^2_{12}=\frac{1}{r}\cdot$$ (B.2)

Using the chosen holonomic basis we consider a simple plasma flow where the four velocity could be written in coordinate form as $u^{\alpha}{\bf {\vec{e}_{\alpha}}} =\gamma~{\bf\vec{e}_0}+(\gamma~\Omega/c)~{\bf\vec{e}_2}+(\gamma~v_z/c)~ {\bf\vec{e}_3}$ and $u_{\alpha}{\bf\vec{e}^{\alpha}}=-\gamma~{\bf {\vec{e}^0}}+ (\gamma~r^2~\Omega/c)~{\bf\vec{e}^2}+(\gamma~v_z/c)~{\bf\vec{e}^3}$, respectively, or in shortened notation as

 

$$u^{\alpha}= \gamma~(1,0,\Omega/c,v_z/c)~,$$ (B.3)

$$u_{\alpha}= \gamma~(-1,0,\Omega~r^2/c,v_z/c)~,$$ (B.4)

where the normalization

$$\gamma=\frac{1}{\sqrt{1-\Omega^2 r^2/c^2-v_z^2/c^2}}$$ (B.5)

denotes the Lorentz factor of the flow and where the angular frequency may be selected to be a function of the radial coordinate, i.e. $\Omega=\Omega(r)$.

Generally, for a contravariant four vector $A^{\alpha}$ the covariant derivative is given by

$$A^{\alpha}_{\;\;\vert\vert\beta}=\frac{\partial A^{\alpha}}{\partial x^{\beta}}+ \Gamma^{\alpha}_{\beta \mu}~A^{\mu}~,$$ (B.6)

while for the covariant derivative of a covariant four vector $A_{\alpha}$ one has

$$A_{\alpha\vert\vert\beta}=\frac{\partial A_{\alpha}}{\partial x^{\beta}}- \Gamma^{\mu}_{\alpha \beta}~A_{\mu}~.$$ (B.7)

Hence, for the assumed four velocity Eq. (B.3), the fluid four divergence becomes zero, i.e. $\nabla_{\beta}~ u^{\beta} =0$, while the fluid four acceleration Eq. (A.4) reduces to

$$\dot{u}_{\alpha}~{\bf {\vec{e}^{\alpha}}}=-u^2~(\Gamma^2_{2... ...)~ {\bf\vec{e}^1}=-(\gamma^2~\Omega^2~r/c^2)~{\bf\vec{e}^1}.$$ (B.8)

For the components of the shear tensor Eq. (A.6) we may then derive the following relations

 

$$\sigma_{01}=\sigma_{10} =-(\gamma^3~r^2/c^2)~\Omega~ \frac{\rm d\Omega}{{\rm d}r}~$$

$$\sigma_{00}=\sigma_{11} =\sigma_{22}=\sigma_{33}=\sigma_{02} =\sigma_{20}=\sigma_{23}=\sigma_{32}=0~$$

$$\sigma_{12}=\sigma_{21} =(\gamma^3~r^2/c)~\frac{\rm d\Omega}{{\rm d}r}~ (1-v_z^2/c^2)~$$

$$\sigma_{13}=\sigma_{31} =(\gamma^3~r^2~v_z/c^3)~\Omega~ \frac{\rm d\Omega}{{\rm d}r}\cdot$$ (B.9)

The viscous energization coefficient Eq. (A.5) then becomes

$$\Gamma=\frac{1}{15}~\gamma^4~r^2~\left(\frac{\rm d\Omega}{{\rm d}r}\right)^2 (1-v_z^2/c^2),$$ (B.10)

noting that $\sigma^{01}=-\sigma_{01}$, $\sigma^{12}= \sigma_{12}/r^2$ and $\sigma^{13}=\sigma_{13}$.

As the fluid four divergence vanishes, the first term in brackets of Eq. (A.1) becomes zero, while for the second term in brackets one finds

$$\dot{u}_{\alpha}~q^{\alpha}= \dot{u}_1~q^1~,$$ (B.11)

with the radial particle current $q^1 \equiv q^r$ (cf. Eq. (A.3))

$$q^1 \equiv q^r=-\kappa~\left(\frac{\partial f}{\partial r}+... ...rac{(p'^0)^2}{p'} \frac{\partial f}{\partial p'}\right)\cdot$$ (B.12)

In the steady-state the fifth term in brackets of Eq. (A.1) becomes

$$\nabla_{\alpha}~ q^{\alpha} =\frac{1}{r}\frac{\partial}{\part... ...ppa~(1+\gamma^2 v_z^2/c^2)\frac{\partial^2 f}{\partial z^2}~,$$ (B.13)

noting that the third component of the heat flux is given by

$$q^3=-\kappa~(1+\gamma^2 v_z^2/c^2) \frac{\partial f}{\partial z}\cdot$$ (B.14)

Finally, for the fourth term in brackets of Eq. (A.1) we have

$$u^{\alpha} \nabla_{\alpha}f =(\gamma v_z/c) \frac{\partial f}{\partial z}\cdot$$ (B.15)

Now, by collecting together all the relevant terms of Eq. (A.1) and by introducing a source term Q (which may depend on r, z and p'), we finally arrive at the relevant relativistic steady-state transport equation in cylindrical coordinates being appropriate for the considered rotating and shearing jet flows:

  

For purely azimuthal, special relativistic flows with v_z=0, the transport Eq. (B.16) reduces to Eq. (5.2) derived in WJM 94.

As suggested by WJM 94, the derived transport equation may be cast in a more suitable form by introducing the variable

$$\Phi=\ln(H)$$ (B.17)

replacing the comoving particle momentum variable p'. Following WJM 94, we may define H such that $(\partial f/\partial r)_H= -q^r/\kappa$, with q^r given by Eq. (B.12), the index H denoting a derivative at constant H, i.e.

$$H=p'^0~c~\exp\left(-\int^r \mbox{d}r'~\frac{\gamma^2~\Omega^2~r'}{c^2} \right)\cdot$$ (B.18)

With respect to a physical interpretation of H we would like to note, that in the case of rigid rotation (i.e. $\Omega=\rm const.$) H could be related to the Hamiltonian for a bead on a rigidly rotating wire (cf. WJM 94; see also Eq. (C.12)). Hence, by writing $f(r,z,p')\rightarrow f(r,z,\Phi)$, the relevant derivatives transform like

 

$$\left(\frac{\partial f(r,z,\Phi)}{\partial r}\right)_{\Phi} \frac{\partial f(r,z,p')}{\partial r} + \left(\frac{\partial p'}{\partial r}\right)_{\Phi} \frac{\partial f(r,z,p')}{\partial p'}$$ =

$$-\frac{q^r}{\kappa},$$ = (B.19)

using $\partial p'/\partial p'^0=p'^0/p'$ and noting that $p'^0~c=\exp[\int \mbox{d}r'~\gamma^2~\Omega^2~r'/c^2]~\exp\Phi$. As usual, the index $\Phi$ in Eq. (B.19) denotes a derivative at constant $\Phi$. Similarly, for the momentum-derivatives we have

 

$$\frac{\partial f(r,z,p')}{\partial p'}=\frac{\partial\Phi}{\parti... ...(r,z,\Phi)}{\partial\Phi}=\frac{p'}{(p'^0)^2} \frac{\partial f(r,z,\Phi)}{\Phi}$$ (B.20)

and consequently

 

$$\frac{\partial^2 f(r,z,p')}{\partial p'^2} =\frac{(p'^0)^2-2~p'^2... ...i} +~ \frac{p'^2}{(p'^0)^4} ~\frac{\partial^2 f(r,z,\Phi)}{\partial\Phi^2}\cdot$$ (B.21)

Now, using Eq. (B.19) and collecting all terms together in the transport Eq. (B.16) which depend on $\partial f/\partial r$ and additionally recalling that $\kappa= (v'^2~\tau_{\rm c})/3$ we may arrive at

 

$$\left[\frac{1}{r}+\frac{\partial\kappa/\partial r}{\kappa}\right]... ...rac{\gamma^2~\Omega^2~r}{c^2} \left(\frac{\partial f}{\partial r}\right)_{\Phi}$$ (B.22)

where the position and momentum dependence of the collision time $\tau_{\rm c}$ (and thus of the diffusion coefficient) has been caught into the definition of the variables $\alpha$ and $\beta$, i.e. $\alpha$ and $\beta$ are given by

$$\alpha =\frac{\partial\ln \tau_c}{\partial\ln p'}\ \qquad ... ...nd} \qquad \beta = \frac{\partial\ln \tau_c}{\partial\ln r},$$ (B.23)

respectively. Equation (B.22) has been obtained noting that v'² = c² p'²/(p'⁰)² and $[\partial\tau_{\rm c}/\partial p']/\tau_{\rm c} =\alpha/p'$ and using

$$\frac{\partial\kappa}{\partial p'}=\frac{1}{3}\left(\frac{\pa... ...~\tau_c + v'^2 \frac{\partial\tau_c}{\partial p'}\right)\cdot$$ (B.24)

In a similar manner, the terms depending on $\partial f/\partial p'$ in Eq. (B.16) may be rewritten using Eq. (B.20) as

$$\frac{1}{p'^2}\left[\frac{\partial}{\partial p'}\left(\frac{\gamm... ...x{d}\Omega}{\mbox{d}r}\right)^2~ \frac{\partial f(r,z,\Phi)}{\partial\Phi}\cdot$$ (B.25)

Likewise, for the terms in Eq. (B.16) which depend on $\partial^2 f/\partial p'^2$ one finds

$$\frac{1}{p'^2}\left(\frac{\gamma^4~r^2}{15~c^2}~ (1-\frac{v_z^2}{... ... + \frac{p'^2}{(p'^0)^2}~\frac{\partial^2 f(r,z,\Phi)}{\partial\Phi^2} \right)~$$ (B.26)

where Eq. (B.21) has been used.

Now, collecting all relevant terms together the steady state transport Eq. (B.16) may finally be rewritten as

  

with $f \equiv f(r,z,\Phi)$ and where the r-derivatives should be understood as derivatives keeping $\Phi$ constant, i.e. we may treat $r,z,\Phi$ as independent variables. Again, for v_z=0 the partial differential equation (B.27) reduces to the steady state transport equation given in WJM 94 (e.g. see their Eq. (5.5)).

For the present application where we are interested in highly relativistic particles with $p'^0 \simeq p'$, the steady-state transport Eq. (B.27) simplifies to