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Equations

Fluids, radiation, magnetic fields, and dust

Assuming the speed of light is not reduced (\(\hat{c} = c\)), Quokka solves the following conservation laws for the cell-centered variables, as written below in the Heaviside–Lorentz system of units:

\[\frac{\partial \vec{U}}{\partial t}+\nabla \cdot \vec{F}(\vec{U}) = \vec{S}(\vec{U}),\]
\[\begin{aligned} \vec{U} =\left[ \begin{array}{c} \rho \\ \rho \vec{v} \\ E \\ \rho X_n \\ E_g \\ \vec{F}_g \\ \rho_{\mathrm{d},k} \\ \rho_{\mathrm{d},k} \vec{v}_{\mathrm{d},k} \end{array}\right], \; \vec{F}(U) = \left[ \begin{array}{c} \rho \vec{v} \\ \rho \vec{v} \otimes \vec{v} + \left(p + \frac{1}{2} B^2\right)\mathsf{I} - \vec{B} \otimes \vec{B} \\ \left(E + p + \frac{1}{2} B^2\right)\vec{v} - (\vec{v} \cdot \vec{B}) \vec{B} \\ \rho X_n \vec{v} \\ \vec{F}_g \\ c^2 \mathsf{P}_g \\ \rho_{\mathrm{d},k} \vec{v}_{\mathrm{d},k} \\ \rho_{\mathrm{d},k} \vec{v}_{\mathrm{d},k} \otimes \vec{v}_{\mathrm{d},k} \end{array}\right], \; \vec{S}(U)=\left[ \begin{array}{c} 0 \\ \sum_g \vec{G}_g + \rho \vec{g} + \sum_{k=1}^{N_{\mathrm{dust}}} \rho_{\mathrm{d},k} \frac{\vec{v}_{\mathrm{d},k} - \vec{v}}{T_{\mathrm{s},k}} \\ c \sum_g G^0_{g} + \rho \vec{v} \cdot \vec{g} + \mathcal{H} - \mathcal{C} + \sum_{k=1}^{N_{\mathrm{dust}}} \rho_{\mathrm{d},k} \frac{\vec{v}_{\mathrm{d},k} - \vec{v}}{T_{\mathrm{s},k}} \cdot \vec{v} + \omega \sum_{k=1}^{N_{\mathrm{dust}}} \rho_{\mathrm{d},k} \frac{(\vec{v}_{\mathrm{d},k} - \vec{v})^{2}}{T_{\mathrm{s},k}} \\ \rho \dot{X}_n \\ - c G^0_{g} \\ - c^2 \vec{G}_g \\ 0 \\ \rho_{\mathrm{d},k} \frac{\vec{v} - \vec{v}_{\mathrm{d},k}}{T_{\mathrm{s},k}} \end{array}\right], \end{aligned}\]

where the total fluid energy is

\[ E = \rho e + \frac{1}{2} \rho v^2 + \frac{1}{2} B^2 \, , \]

and the face-centered magnetic field is evolved according to the ideal MHD induction equation:

\[ \frac{\partial \vec{B}}{\partial t} - \nabla \times \left(\vec{v} \times \vec{B}\right) = 0. \]

Quokka also solves the non-conservative auxiliary internal energy equation:

\[\begin{aligned} \frac{\partial (\rho e_{\text{aux}})}{\partial t} = - \nabla \cdot (\rho e_{\text{aux}} \vec{v}) - p \nabla \cdot \vec{v} + S_{\text{rad}} + \mathcal{H} - \mathcal{C} + \omega \sum_{k=1}^{N_{\mathrm{dust}}} \rho_{\mathrm{d},k} \frac{(\vec{v}_{\mathrm{d},k} - \vec{v})^{2}}{T_{\mathrm{s},k}}, \\ \Delta S_{\text{rad}} = \int \sum_g c G^0_g \ dt - \frac{1}{2} \Delta \left(\rho v^2 \right), \end{aligned}\]

and the gravitational Poisson equation:

\[\begin{aligned} \nabla^2 \phi = -4 \pi G \left( \rho + \sum_i \rho_i \right), \\ \vec{g} \equiv -\nabla \phi, \end{aligned}\]

where

  • \(\rho\) is the gas density,
  • \(\vec{v}\) is the gas velocity,
  • \(E\) is the total fluid energy density, including magnetic energy when MHD is enabled,
  • \(\rho e_{\text{aux}}\) is the auxiliary gas internal energy density,
  • \(X_n\) is the fractional concentration of species \(n\),
  • \(\dot{X}_n\) is the chemical reaction term for species \(n\),
  • \(\mathcal{H}\) is the optically-thin volumetric heating term (radiative and chemical),
  • \(\mathcal{C}\) is the optically-thin volumetric cooling term (radiative and chemical),
  • \(p(\rho, e)\) is the gas pressure derived from a general convex equation of state,
  • \(\vec{B}\) is the magnetic field,
  • \(\mathsf{I}\) is the identity tensor,
  • \(E_g\) is the radiation energy density for group \(g\),
  • \(F_g\) is the radiation flux for group \(g\),
  • \(\mathsf{P}_g\) is the radiation pressure tensor for group \(g\),
  • \(G_g\) is the radiation four-force \([G^0_g, \vec{G}_g]\) due to group \(g\),
  • \(\Delta S_{\text{rad}}\) is the change in gas internal energy due to radiation over a timestep,
  • \(\phi\) is the Newtonian gravitational potential,
  • \(\vec{g}\) is the gravitational acceleration,
  • \(\rho_i\) is the mass density due to particle \(i\),
  • \(\rho_{\mathrm{d},k}\) is the dust mass density for dust species \(k\) (\(k \in [1, N_{\mathrm{dust}}]\)),
  • \(\vec{v}_{\mathrm{d},k}\) is the dust velocity for dust species \(k\),
  • \(T_{\mathrm{s},k}\) is the aerodynamic stopping time for dust species \(k\),
  • \(\omega\) is the fraction of frictional heating deposited into the gas.

Note that since work done by radiation on the gas is included in the \(c \sum_g G^0_g\) term, \(S_{\text{rad}}\) is not the same as \(c \sum_g G^0_g\).

Collisionless particles

Quokka solves the following equation of motion for collisionless particles:

\[\frac{d^2 \vec{x}_i}{d t^2} = \vec{g} ,\]

where \(\vec{x}_i\) is the position vector of particle \(i\).