mhdDednerEosEqn

Defines the equations of ideal compressible magnetohydrodynamics with and arbitrary equation of state (EOS) and divergence cleaning:

\[\notag \begin{align} \frac{\partial \rho}{\partial t} + \nabla\cdot\left[ \rho\,\mathbf{u} \right] = 0 \\ \frac{\partial \rho\,\mathbf{u}}{\partial t} + \nabla\cdot\left[ \rho\,\mathbf{u}\,\mathbf{u}^{T} - \mathbf{b}\,\mathbf{b}^{T} + \mathbb{I} \left(P+ \tfrac{1}{2}\left|\mathbf{b}\right|^{2}\right) \right] = 0 \\ \frac{\partial E}{\partial t} + \nabla\cdot\left[ \left(E + P \right) \mathbf{u} + \mathbf{e} \times \mathbf{b} \right] = 0 \\ \frac{\partial \mathbf{b^{\mathrm{plasma}}}}{\partial t} + \nabla\times\mathbf{e} + \nabla \psi= 0 \\ \frac{\partial \psi}{\partial t} + \nabla\cdot\left[ c^{2}_{\mathrm{fast}} \mathbf{b} \right] = 0 \end{align}\]

Here, \(\mathbb{I}\) is the identity matrix and \(P\) is the pressure as specified by an external EOS. Updaters that compute all the data required from an EOS are found in vanDerWaalsComputeVariables, sesameComputeVariables and propaceosComputeVariables. The quantity \(c_{\mathrm{fast}}\) corresponds to the fastest wave speed over the entire simulation domain; divergence errors are advected out of the domain with this speed.

The electromagnetic fields are defined as:

\[\notag \begin{align} \mathbf{b} = \mathbf{b}^{\mathrm{plasma}}+\mathbf{b}^{\mathrm{external}} = \mu^{-1/2}_{0} \left(\mathbf{B}^{\mathrm{plasma}}+\mathbf{B}^{\mathrm{external}} \right) \\ \mathbf{e} = - \mathbf{u} \times \mathbf{b} + \mathbf{e}^{\mathrm{external}} = \mu^{-1/2}_{0} \left( - \mathbf{u} \times \mathbf{B} + \mathbf{E}^{\mathrm{external}} \right) \end{align}\]

Here, \(\mathbf{b}^{\mathrm{plasma}}\) is the magnetic field induced in the plasma by the inductive electric field, \(\mathbf{e}\), while \(\mathbf{e}^{\mathrm{external}}\) and \(\mathbf{b}^{\mathrm{external}}\) are electromagnetic fields computed “externally” to the ideal magnetohydrodynamic equations.

Parameters

basementPressure (float, optional)
The minimum pressure allowed. Pressures below this value will be replaced with this value for primitive state, eigensystem and flux computations. Defaults to zero.
basementDensity (float, optional)
The minimum density allowed. Densities below this value will be replaced with this value for primitive state, eigensystem and flux computations. Defaults to zero.
mu0 (float, optional)
Optional value for the constant \(\mu_0\). Defaults to \(4\pi\times 10^{-7}\).
externalEfield (string, optional)
Specifies the name of the data structure containing the externally computed electric field, \(\mathbf{e}^{\mathrm{external}}\).
externalBfield (string, optional)
Specifies the name of the data structure containing the externally computed magnetic field, \(\mathbf{b}^{\mathrm{external}}\).

Parent Updater Data

in (string vector, required)
Vector of Conserved Quantities (nodalArray, 9-components, required)

The vector of conserved quantities, \(\mathbf{q}\) has 9 entries:

  1. \(\rho\): mass density
  2. \(\rho\,u_{\hat{\mathbf{i}}} = \rho \mathbf{u} \cdot \hat{\mathbf{i}}\): momentum density in the \(\hat{\mathbf{i}}\) direction
  3. \(\rho\,u_{\hat{\mathbf{j}}} = \rho \mathbf{u} \cdot \hat{\mathbf{j}}\): momentum density in the \(\hat{\mathbf{j}}\) direction
  4. \(\rho\,u_{\hat{\mathbf{k}}} = \rho \mathbf{u} \cdot \hat{\mathbf{k}}\): momentum density in the \(\hat{\mathbf{k}}\) direction
  5. \(E = \rho\epsilon + \tfrac{1}{2}\rho|\mathbf{u}|^2 + \tfrac{1}{2}|\mathbf{b}|^2\): total energy density where \(\epsilon\) is the specific internal energy
  6. \(b_{\hat{\mathbf{i}}} = \mathbf{b} \cdot \hat{\mathbf{i}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{i}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{i}}\) direction
  7. \(b_{\hat{\mathbf{j}}} = \mathbf{b} \cdot \hat{\mathbf{j}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{j}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{j}}\) direction
  8. \(b_{\hat{\mathbf{k}}} = \mathbf{b} \cdot \hat{\mathbf{k}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{k}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{k}}\) direction
  9. \(\psi\): correction potential
Pressure (nodalArray, 1-component, required)
Value of the pressure as computed by the external EOS.
Sound speed squared (nodalArray, 1-component, required)
Value of the sound speed squared as computed by the external EOS.
internal energy (nodalArray, 1-component, required)
Value of the internal energy (\(\rho\epsilon\)) as computed by the external EOS.
Fastest Wave Speed (dynVector, 1-component, required)
The fastest wave speed across the entire simulation domain, \(c_{\mathrm{fast}}\). Can be computed using hyperbolic (1d, 2d, 3d) (see below).
Externally Computed Electric Field (nodalArray, 3-components, optional)

Additional terms in the generalized Ohm’s law, \(\mathbf{E}^{\mathrm{external}}\), computed “externally” to the ideal magnetohydrodynamic system. The data structure containing \(\mathbf{e}^{\mathrm{external}}\) is specified by the “externalEField” option described below.

  1. \({e}^{\mathrm{external}}_{\hat{\mathbf{i}}} = \mathbf{e}^{\mathrm{external}} \cdot \hat{\mathbf{i}} = \mu^{-1/2}_{0} \mathbf{E}^{\mathrm{external}} \cdot \hat{\mathbf{i}}\): “externally” computed electric field normalized by permeability of free-space in the \(\hat{\mathbf{i}}\) direction.
  2. \({e}^{\mathrm{external}}_{\hat{\mathbf{j}}} =\mathbf{e}^{\mathrm{external}} \cdot \hat{\mathbf{j}} = \mu^{-1/2}_{0} \mathbf{E}^{\mathrm{external}} \cdot \hat{\mathbf{j}}\):”externally” computed electric field normalized by permeability of free-space in the \(\hat{\mathbf{j}}\) direction
  3. \({e}^{\mathrm{external}}_{\hat{\mathbf{k}}} = \mathbf{e}^{\mathrm{external}} \cdot \hat{\mathbf{k}} = \mu^{-1/2}_{0} \mathbf{E}^{\mathrm{external}} \cdot \hat{\mathbf{k}}\): “externally” computed electric field normalized by permeability of free-space in the \(\hat{\mathbf{k}}\) direction
Externally Computed Magnetic Field (nodalArray, 3-components, optional)

Additional contribution to the magnetic field, \(\mathbf{b}^{\mathrm{external}}\), which is not evolved by the induction equation, but does contribute to the Lorentz force and the work done on the plasma. The data structure containing \(\mathbf{b}^{\mathrm{external}}\) is specified by the “externalBField” option described below.

  1. \({b}^{\mathrm{external}}_{\hat{\mathbf{i}}} = \mathbf{b}^{\mathrm{external}} \cdot \hat{\mathbf{i}} = \mu^{-1/2}_{0} \mathbf{B}^{\mathrm{external}} \cdot \hat{\mathbf{i}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{i}}\) direction
  2. \({b}^{\mathrm{external}}_{\hat{\mathbf{j}}} =\mathbf{b}^{\mathrm{external}} \cdot \hat{\mathbf{j}} = \mu^{-1/2}_{0} \mathbf{B}^{\mathrm{external}} \cdot \hat{\mathbf{j}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{j}}\) direction
  3. \({b}^{\mathrm{external}}_{\hat{\mathbf{k}}} = \mathbf{b}^{\mathrm{external}} \cdot \hat{\mathbf{k}} = \mu^{-1/2}_{0} \mathbf{B}^{\mathrm{external}} \cdot \hat{\mathbf{k}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{k}}\) direction
out (string vector, required)

For the mhdDednerEosEqn, one of four output variables are computed, depending on whether the equation is combined with an updater capable of computing fluxes (classicMusclUpdater (1d, 2d, 3d)), primitive variables (computePrimitiveState(1d, 2d, 3d)), the time step associated with the CFL condition (timeStepRestrictionUpdater (1d, 2d, 3d)) or the fastest wave speed in the grid (hyperbolic (1d, 2d, 3d)).

Vector of Fluxes (nodalArray, 9-components)

When combined with an updater that computes \(\nabla \cdot \mathcal{F}\left(\mathbf{w} \right)\) (e.g. classicMusclUpdater (1d, 2d, 3d)), the equation system returns:

  1. \(\nabla \cdot \mathcal{F}\left( \rho \right)\): mass flux
  2. \(\nabla \cdot \mathcal{F}\left( \rho\,u_{\hat{\mathbf{i}}} \right)\): \(\hat{\mathbf{i}}\) momentum flux
  3. \(\nabla \cdot \mathcal{F}\left( \rho\,u_{\hat{\mathbf{j}}} \right)\): \(\hat{\mathbf{j}}\) momentum flux
  4. \(\nabla \cdot \mathcal{F}\left( \rho\,u_{\hat{\mathbf{k}}} \right)\): \(\hat{\mathbf{k}}\) momentum flux
  5. \(\nabla \cdot \mathcal{F}\left( E \right)\): total energy flux
  6. \(\nabla \cdot \mathcal{F}\left( b_{\hat{\mathbf{i}}} \right)\): \(\hat{\mathbf{i}}\) magnetic field flux
  7. \(\nabla \cdot \mathcal{F}\left( b_{\hat{\mathbf{j}}} \right)\): \(\hat{\mathbf{j}}\) magnetic field flux
  8. \(\nabla \cdot \mathcal{F}\left( b_{\hat{\mathbf{k}}} \right)\): \(\hat{\mathbf{k}}\) magnetic field flux
  9. \(\nabla \cdot \mathcal{F}\left(\psi \right)\): correction potential flux
Vector of Primitive States (nodalArray, 9-components)

When combined with an updater that computes \(\mathbf{w} = \mathbf{w}(\mathbf{q})\) (e.g. computePrimitiveState(1d, 2d, 3d)), the equation systen returns:

  1. \(\rho\): mass density
  2. \(u_{\hat{\mathbf{i}}} = \mathbf{u} \cdot \hat{\mathbf{i}}\): velocity in the \(\hat{\mathbf{i}}\) direction
  3. \(u_{\hat{\mathbf{j}}} = \mathbf{u} \cdot \hat{\mathbf{j}}\): velocity in the \(\hat{\mathbf{j}}\) direction
  4. \(u_{\hat{\mathbf{k}}} = \mathbf{u} \cdot \hat{\mathbf{k}}\): velocity in the \(\hat{\mathbf{k}}\) direction
  5. \(P = \rho\epsilon(\gamma-1)\): ideal gas pressure
  6. \(b_{\hat{\mathbf{i}}} = \mathbf{b} \cdot \hat{\mathbf{i}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{i}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{i}}\) direction
  7. \(b_{\hat{\mathbf{j}}} = \mathbf{b} \cdot \hat{\mathbf{j}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{j}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{j}}\) direction
  8. \(b_{\hat{\mathbf{k}}} = \mathbf{b} \cdot \hat{\mathbf{k}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{k}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{k}}\) direction
  9. \(\psi\): correction potential
Time Step (dynVector, 1-component)
When combined with timeStepRestrictionUpdater (1d, 2d, 3d), the equation system returns the time step consisten with the CFL condition across the entire simulation domain.
Fastest Wave Speed (dynVector, 1-component)
When combined with hyperbolic (1d, 2d, 3d), the equation system returns the fastest wave speed across the entire simulation domain, \(c_{\mathrm{fast}}\).

Examples

The following block demonstrates the mhdDednerEosEqn used in combination with classicMusclUpdater (1d, 2d, 3d) to compute \(\nabla \cdot \mathcal{F}\left(\mathbf{w} \right)\):

<Updater hyper>
  kind = classicMuscl2d
  onGrid = domain

  # input nodal component arrays
  in=[q, pressure, soundSqr, intEnergy]

  # output nodal component arrays
  out = [qNew]

  # input dynVector containing fastest wave speed
  waveSpeeds = [waveSpeed]

  # the numerical flux to use
  numericalFlux = hlldFlux

  # CFL number to use
  cfl = 0.5

  # determines solve is conservative or primitive
  variableForm = conservative

  # Limiter; one per input nodal component array
  limiter=[muscl, muscl, muscl, muscl]

  # list of equations to solve
  equations = [mhd]

  <Equation mhd>
    kind=mhdDednerEosEqn
    mu0=1.0
  </Equation>

</Updater>

The following block demonstrates the mhdDednerEosEqn used in combination with timeStepRestrictionUpdater (1d, 2d, 3d) and hyperbolic (1d, 2d, 3d) to compute \(c_{\mathrm{fast}}\) with an externally supplied magnetic field:

<Updater getWaveSpeed>
  kind=timeStepRestrictionUpdater2d
  onGrid=domain

  # input nodal component arrays
  in=[q, pressure, soundSqr, intEnergy]

  # output dynVector containing fastest wave speed
  waveSpeeds=[waveSpeed]

  # list of equations to compute fastest wave speed for
  restrictions=[idealMhd]

  # courant condition to apply to the timestep
  courantCondition=0.5

  <TimeStepRestriction idealMhd>
    kind=hyperbolic1d
    model=mhdDednerEosEqn
    mu0=1.0
  </TimeStepRestriction>
</Updater>