Generation of axisymmetric magnetosonic shock waves: Applications to static equilibria and accretion disks

Núñez, Manuel

doi:10.1016/j.ijnonlinmec.2016.07.013

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…The first coefficients to find is

\begin{matrix} true \\ right q_{0} & = & left \frac{1}{c} boldL \cdot (\nabla_{u} A_{n} \cdot R) \cdot boldR \\ right q & = & left \frac{q_{0}}{boldL \cdot boldR} . \end{matrix}

Since we know formula , it is easy to show

\begin{matrix} true \\ right false(\nabla_{u} A_{n} \cdot boldR false) \cdot boldR & = & left (0, 0, 2 ρ c_{s}), q_{0} = \frac{2 ρ c_{s}^{2}}{c} \\ right boldL \cdot boldR & = & left 2 ρ c_{s}, q = \frac{c_{s}}{c} . \end{matrix}

Now we consider the equilibrium state, which with our assumptions is (in the basis

scriptC

)

{boldu}_{0} = false(v_{x}, 0, ρ false)

. We define

\begin{matrix} true \\ right p_{0} & left = boldL \cdot (() A_{x} \cdot \frac{\partial R}{\partial x} + A_{y} \cdot \frac{\partial R}{\partial y} + false(\nabla_{u} A_{x} \cdot boldR false) \cdot \frac{\partial {boldu}_{0}}{\partial x} + false(\nabla_{u} A_{y} \cdot boldR false) \cdot \frac{\partial {boldu}_{0}}{\partial y}) \\ right p & left = \frac{p_{0}}{boldR \cdot boldL} . \end{matrix}

The coefficient p 0 may also be set in terms of the basis

scriptB

, [], but then we need to have some i...…”

Section: Localized Perturbations and Shocksmentioning

confidence: 99%

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Wavefronts and shocks in the vicinity of sonic lines

Núñez

2018

Z Angew Math Mech

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Sonic points in an equilibrium fluid are those for which the speed of sound equals the flow velocity. We study how high frequency waves evolve in the presence of sonic points. It is shown that their behavior depends in an essential way on the original direction of the wave normal, and the effects of sonic points extend far beyond their immediate vicinity. The transport equation along rays of perturbations of the original equilibrium is also detailed, obtaining conditions for their evolution into shocks.

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“…The first coefficients to find is

\begin{matrix} true \\ right q_{0} & = & left \frac{1}{c} boldL \cdot (\nabla_{u} A_{n} \cdot R) \cdot boldR \\ right q & = & left \frac{q_{0}}{boldL \cdot boldR} . \end{matrix}

Since we know formula , it is easy to show

\begin{matrix} true \\ right false(\nabla_{u} A_{n} \cdot boldR false) \cdot boldR & = & left (0, 0, 2 ρ c_{s}), q_{0} = \frac{2 ρ c_{s}^{2}}{c} \\ right boldL \cdot boldR & = & left 2 ρ c_{s}, q = \frac{c_{s}}{c} . \end{matrix}

Now we consider the equilibrium state, which with our assumptions is (in the basis

scriptC

)

{boldu}_{0} = false(v_{x}, 0, ρ false)

. We define

\begin{matrix} true \\ right p_{0} & left = boldL \cdot (() A_{x} \cdot \frac{\partial R}{\partial x} + A_{y} \cdot \frac{\partial R}{\partial y} + false(\nabla_{u} A_{x} \cdot boldR false) \cdot \frac{\partial {boldu}_{0}}{\partial x} + false(\nabla_{u} A_{y} \cdot boldR false) \cdot \frac{\partial {boldu}_{0}}{\partial y}) \\ right p & left = \frac{p_{0}}{boldR \cdot boldL} . \end{matrix}

The coefficient p 0 may also be set in terms of the basis

scriptB

, [], but then we need to have some i...…”

Section: Localized Perturbations and Shocksmentioning

confidence: 99%

“…What is remarkable is that both approaches yield the same result for the appearance of shocks: for concrete examples see e.g. []. We will consider the first case and recall briefly the main theoretical results, as applied to system .…”

Section: Localized Perturbations and Shocksmentioning

confidence: 99%