FluxCalculator

PinCFlow.FluxCalculator — Module

FluxCalculator

Module for flux calculation.

Provides functions for MUSCL reconstruction and flux computation.

apply_1d_muscl!(
    phi::AbstractVector{<:AbstractFloat},
    phitilde::AbstractMatrix{<:AbstractFloat},
    phisize::Integer,
    limiter_type::MCVariant,
)

Apply the Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) for reconstruction in one dimension.

The reconstruction to the left is given by

\[{\widetilde{\phi}}^\mathrm{L} = \begin{cases} \phi & \mathrm{if} \quad \phi = \phi_{i - 1} \quad \mathrm{or} \quad \phi = \phi_{i + 1},\\ \phi - \frac{1}{2} \eta \left(\frac{\phi_{i + 1} - \phi}{\phi - \phi_{i - 1}}\right) \left(\phi - \phi_{i - 1}\right) & \mathrm{else} \end{cases}\]

and that to the right is given by

\[{\widetilde{\phi}}^\mathrm{R} = \begin{cases} \phi & \mathrm{if} \quad \phi = \phi_{i - 1} \quad \mathrm{or} \quad \phi = \phi_{i + 1},\\ \phi + \frac{1}{2} \eta \left(\frac{\phi - \phi_{i - 1}}{\phi_{i + 1} - \phi}\right) \left(\phi_{i + 1} - \phi\right) & \mathrm{else}, \end{cases}\]

where

\[\eta \left(\xi\right) = \max \left[0, \min \left(2 \xi, \frac{2 + \xi}{3}, 2\right)\right]\]

is the monotonized-centered variant limiter.

Arguments

phi: Input vector.
phitilde: Output matrix with reconstructed values. The two columns of phitilde contain the reconstructions to the left and right. No reconstruction is computed for the first and last row of phitilde.
phisize: Length of the input vector phi.
limiter_type: Type of flux limiter to use.

PinCFlow.FluxCalculator.apply_3d_muscl! — Function

apply_3d_muscl!(
    phi::AbstractArray{<:AbstractFloat, 3},
    phitilde::AbstractArray{<:AbstractFloat, 5},
    nxx::Integer,
    nyy::Integer,
    nzz::Integer,
    limiter_type::AbstractLimiter,
)

Apply the Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) for reconstruction in three dimensions.

Arguments

phi: Input array.
phitilde: Output array with reconstructed values. The fourth dimension represents the directions in which the input was reconstructed and the fifth dimension the reconstructions to the left and right.
nxx: Size of phi in $\widehat{x}$-direction.
nyy: Size of phi in $\widehat{y}$-direction.
nzz: Size of phi in $\widehat{z}$-direction.
limiter_type: Type of flux limiter to use.

See also

PinCFlow.FluxCalculator.apply_1d_muscl!

PinCFlow.FluxCalculator.compute_flux — Function

compute_flux(
    usurf::AbstractFloat,
    phiup::AbstractFloat,
    phidown::AbstractFloat,
)::AbstractFloat

Compute and return the upstream flux from reconstructed values, based on the sign of the transporting velocity.

Arguments

usurf: Transporting velocity.
phiup: Upstream reconstruction for usurf > 0.
phidown: Downstream reconstruction for usurf > 0.

PinCFlow.FluxCalculator.compute_fluxes! — Function

compute_fluxes!(state::State, predictands::Predictands)

Compute fluxes by dispatching to specialized methods for each prognostic variable.

compute_fluxes!(state::State, predictands::Predictands, variable::Rho)

Compute the density fluxes in all three directions.

The fluxes are given by

\[\begin{align*} \mathcal{F}^{\rho, \widehat{x}}_{i + 1 / 2} & = \frac{\tau_{\widehat{x}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{x}}\right)\right] {\widetilde{\phi}}^\mathrm{R} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{x}}\right)\right] {\widetilde{\phi}}_{i + 1}^\mathrm{L}\right\},\\ \mathcal{F}^{\rho, \widehat{y}}_{j + 1 / 2} & = \frac{\tau_{\widehat{y}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{y}}\right)\right] {\widetilde{\phi}}^\mathrm{F} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{y}}\right)\right] {\widetilde{\phi}}_{j + 1}^\mathrm{B}\right\},\\ \mathcal{F}^{\rho, \widehat{z}}_{k + 1 / 2} & = \frac{\tau_{\widehat{z}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{z}}\right)\right] {\widetilde{\phi}}^\mathrm{U} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{z}}\right)\right] {\widetilde{\phi}}_{k + 1}^\mathrm{D}\right\}, \end{align*}\]

where

\[\begin{align*} \tau_{\widehat{x}} & = \left(J P_\mathrm{old}\right)_{i + 1 / 2} u_{\mathrm{old}, i + 1 / 2},\\ \tau_{\widehat{y}} & = \left(J P_\mathrm{old}\right)_{j + 1 / 2} v_{\mathrm{old}, j + 1 / 2},\\ \tau_{\widehat{z}} & = \left(J P_\mathrm{old}\right)_{k + 1 / 2} \widehat{w}_{\mathrm{old}, k + 1 / 2} \end{align*}\]

are the transporting velocities (weighted by the Jacobian) and $\widetilde{\phi}$ is the reconstruction of $\rho / P_\mathrm{old}$. More specifically, the superscripts $\mathrm{R}$, $\mathrm{L}$, $\mathrm{F}$, $\mathrm{B}$, $\mathrm{U}$ and $\mathrm{D}$ indicate reconstructions at the right, left, forward, backward, upward and downward cell interfaces of the respective grid points, respectively. Quantities with the subscript $\mathrm{old}$ are obtained from a previous state, which is partially passed to the method via predictands.

compute_fluxes!(state::State, predictands::Predictands, variable::RhoP)

Compute the density-fluctuations fluxes in all three directions.

The computation is analogous to that of the density fluxes.

compute_fluxes!(
    state::State,
    predictands::Predictands,
    model::Union{Boussinesq, PseudoIncompressible},
    variable::P,
)

Return in non-compressible modes.

compute_fluxes!(
    state::State,
    predictands::Predictands,
    model::Compressible,
    variable::P,
)

Compute the mass-weighted potential-temperature fluxes in all three directions.

The fluxes are given by

\[\begin{align*} \mathcal{F}^{P, \widehat{x}}_{i + 1 / 2} & = \left(J P_\mathrm{old}\right)_{i + 1 / 2} u_{\mathrm{old}, i + 1 / 2},\\ \mathcal{F}^{P, \widehat{y}}_{j + 1 / 2} & = \left(J P_\mathrm{old}\right)_{j + 1 / 2} v_{\mathrm{old}, j + 1 / 2},\\ \mathcal{F}^{P, \widehat{z}}_{k + 1 / 2} & = \left(J P_\mathrm{old}\right)_{k + 1 / 2} \widehat{w}_{\mathrm{old}, k + 1 / 2}. \end{align*}\]

compute_fluxes!(state::State, old_predictands::Predictands, variable::U)

Compute the zonal-momentum fluxes in all three directions.

The fluxes are first set to the advective parts

\[\begin{align*} \mathcal{F}^{\rho u, \widehat{x}}_{i + 1} & = \frac{\tau_{\widehat{x}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{x}}\right)\right] {\widetilde{\phi}}_{i + 1 / 2}^\mathrm{R} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{x}}\right)\right] {\widetilde{\phi}}_{i + 3 / 2}^\mathrm{L}\right\},\\ \mathcal{F}^{\rho u, \widehat{y}}_{i + 1 / 2, j + 1 / 2} & = \frac{\tau_{\widehat{y}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{y}}\right)\right] {\widetilde{\phi}}_{i + 1 / 2}^\mathrm{F} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{y}}\right)\right] {\widetilde{\phi}}_{i + 1 / 2, j + 1}^\mathrm{B}\right\},\\ \mathcal{F}^{\rho u, \widehat{z}}_{i + 1 / 2, k + 1 / 2} & = \frac{\tau_{\widehat{z}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{z}}\right)\right] {\widetilde{\phi}}_{i + 1 / 2}^\mathrm{U} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{z}}\right)\right] {\widetilde{\phi}}_{i + 1 / 2, k + 1}^\mathrm{D}\right\}, \end{align*}\]

with

\[\begin{align*} \tau_{\widehat{x}} & = \left[\left(J P_\mathrm{old}\right)_{i + 1 / 2} u_{\mathrm{old}, i + 1 / 2}\right]_{i + 1},\\ \tau_{\widehat{y}} & = \left[\left(J P_\mathrm{old}\right)_{j + 1 / 2} v_{\mathrm{old}, j + 1 / 2}\right]_{i + 1 / 2, j + 1 / 2},\\ \tau_{\widehat{z}} & = \left[\left(J P_\mathrm{old}\right)_{k + 1 / 2} \widehat{w}_{\mathrm{old}, k + 1 / 2}\right]_{i + 1 / 2, k + 1 / 2} \end{align*}\]

and $\widetilde{\phi}$ being the reconstruction of $\rho_{i + 1 / 2} u_{i + 1 / 2} / P_{\mathrm{old}, i + 1 / 2}$. If the viscosity is nonzero, the viscous parts (weighted by the Jacobian) are then added, i.e.

\[\begin{align*} \mathcal{F}^{\rho u, \widehat{x}}_{i + 1} & \rightarrow \mathcal{F}^{\rho u, \widehat{x}}_{i + 1} - \eta_{i + 1} \left(J \widehat{\Pi}^{11}\right)_{i + 1},\\ \mathcal{F}^{\rho u, \widehat{y}}_{i + 1 / 2, j + 1 / 2} & \rightarrow \mathcal{F}^{\rho u, \widehat{y}}_{i + 1 / 2, j + 1 / 2} - \eta_{i + 1 / 2, j + 1 / 2} \left(J \widehat{\Pi}^{12}\right)_{i + 1 / 2, j + 1 / 2},\\ \mathcal{F}^{\rho u, \widehat{z}}_{i + 1 / 2, k + 1 / 2} & \rightarrow \mathcal{F}^{\rho u, \widehat{z}}_{i + 1 / 2, k + 1 / 2} - \eta_{i + 1 / 2, k + 1 / 2} \left(J \widehat{\Pi}^{13}\right)_{i + 1 / 2, k + 1 / 2}. \end{align*}\]

Finally, if the diffusivity $\mu$ is nonzero, the diffusive parts (weighted by the Jacobian) are added, i.e.

\[\begin{align*} \mathcal{F}^{\rho u, \widehat{x}}_{i + 1} & \rightarrow \mathcal{F}^{\rho u, \widehat{x}}_{i + 1} - \mu_{i + 1} \left[J \widehat{\left(\boldsymbol{\nabla} u\right)}^{\widehat{x}}\right]_{i + 1},\\ \mathcal{F}^{\rho u, \widehat{y}}_{i + 1 / 2, j + 1 / 2} & \rightarrow \mathcal{F}^{\rho u, \widehat{y}}_{i + 1 / 2, j + 1 / 2} - \mu_{i + 1 / 2, j + 1 / 2} \left[J \widehat{\left(\boldsymbol{\nabla} u\right)}^{\widehat{y}}\right]_{i + 1 / 2, j + 1 / 2},\\ \mathcal{F}^{\rho u, \widehat{z}}_{i + 1 / 2, k + 1 / 2} & \rightarrow \mathcal{F}^{\rho u, \widehat{z}}_{i + 1 / 2, k + 1 / 2} - \mu_{i + 1 / 2, k + 1 / 2} \left[J \widehat{\left(\boldsymbol{\nabla} u\right)}^{\widehat{z}}\right]_{i + 1 / 2, k + 1 / 2}. \end{align*}\]

compute_fluxes!(state::State, old_predictands::Predictands, variable::V)

Compute the meridional-momentum fluxes in all three directions.

The fluxes are first set to the advective parts

\[\begin{align*} \mathcal{F}^{\rho v, \widehat{x}}_{i + 1 / 2, j + 1 / 2} & = \frac{\tau_{\widehat{x}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{x}}\right)\right] {\widetilde{\phi}}_{j + 1 / 2}^\mathrm{R} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{x}}\right)\right] {\widetilde{\phi}}_{i + 1, j + 1 / 2}^\mathrm{L}\right\},\\ \mathcal{F}^{\rho v, \widehat{y}}_{j + 1} & = \frac{\tau_{\widehat{y}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{y}}\right)\right] {\widetilde{\phi}}_{j + 1 / 2}^\mathrm{F} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{y}}\right)\right] {\widetilde{\phi}}_{j + 3 / 2}^\mathrm{B}\right\},\\ \mathcal{F}^{\rho v, \widehat{z}}_{j + 1 / 2, k + 1 / 2} & = \frac{\tau_{\widehat{z}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{z}}\right)\right] {\widetilde{\phi}}_{j + 1 / 2}^\mathrm{U} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{z}}\right)\right] {\widetilde{\phi}}_{j + 1 / 2, k + 1}^\mathrm{D}\right\}, \end{align*}\]

with

\[\begin{align*} \tau_{\widehat{x}} & = \left[\left(J P_\mathrm{old}\right)_{i + 1 / 2} u_{\mathrm{old}, i + 1 / 2}\right]_{i + 1 / 2, j + 1 / 2},\\ \tau_{\widehat{y}} & = \left[\left(J P_\mathrm{old}\right)_{j + 1 / 2} v_{\mathrm{old}, j + 1 / 2}\right]_{j + 1},\\ \tau_{\widehat{z}} & = \left[\left(J P_\mathrm{old}\right)_{k + 1 / 2} \widehat{w}_{\mathrm{old}, k + 1 / 2}\right]_{j + 1 / 2, k + 1 / 2} \end{align*}\]

and $\widetilde{\phi}$ being the reconstruction of $\rho_{j + 1 / 2} v_{j + 1 / 2} / P_{\mathrm{old}, j + 1 / 2}$. If the viscosity is nonzero, the viscous parts (weighted by the Jacobian) are then added, i.e.

\[\begin{align*} \mathcal{F}^{\rho v, \widehat{x}}_{i + 1 / 2, j + 1 / 2} & \rightarrow \mathcal{F}^{\rho v, \widehat{x}}_{i + 1 / 2, j + 1 / 2} - \eta_{i + 1 / 2, j + 1 / 2} \left(J \widehat{\Pi}^{12}\right)_{i + 1 / 2, j + 1 / 2},\\ \mathcal{F}^{\rho v, \widehat{y}}_{j + 1} & \rightarrow \mathcal{F}^{\rho v, \widehat{y}}_{j + 1} - \eta_{j + 1} \left(J \widehat{\Pi}^{22}\right)_{j + 1},\\ \mathcal{F}^{\rho v, \widehat{z}}_{j + 1 / 2, k + 1 / 2} & \rightarrow \mathcal{F}^{\rho v, \widehat{z}}_{j + 1 / 2, k + 1 / 2} - \eta_{j + 1 / 2, k + 1 / 2} \left(J \widehat{\Pi}^{23}\right)_{j + 1 / 2, k + 1 / 2}. \end{align*}\]

Finally, if the diffusivity $\mu$ is nonzero, the diffusive parts (weighted by the Jacobian) are added, i.e.

\[\begin{align*} \mathcal{F}^{\rho v, \widehat{x}}_{i + 1 / 2, j + 1 / 2} & \rightarrow \mathcal{F}^{\rho v, \widehat{x}}_{i + 1 / 2, j + 1 / 2} - \mu_{i + 1 / 2, j + 1 / 2} \left[J \widehat{\left(\boldsymbol{\nabla} v\right)}^{\widehat{x}}\right]_{i + 1 / 2, j + 1 / 2},\\ \mathcal{F}^{\rho v, \widehat{y}}_{j + 1} & \rightarrow \mathcal{F}^{\rho v, \widehat{y}}_{j + 1} - \mu_{j + 1} \left[J \widehat{\left(\boldsymbol{\nabla} v\right)}^{\widehat{y}}\right]_{j + 1},\\ \mathcal{F}^{\rho v, \widehat{z}}_{j + 1 / 2, k + 1 / 2} & \rightarrow \mathcal{F}^{\rho v, \widehat{z}}_{j + 1 / 2, k + 1 / 2} - \mu_{j + 1 / 2, k + 1 / 2} \left[J \widehat{\left(\boldsymbol{\nabla} v\right)}^{\widehat{z}}\right]_{j + 1 / 2, k + 1 / 2}. \end{align*}\]

compute_fluxes!(state::State, old_predictands::Predictands, variable::W)

Compute the vertical-momentum fluxes in all three directions.

The fluxes are first set to the advective parts

\[\begin{align*} \mathcal{F}^{\rho w, \widehat{x}}_{i + 1 / 2, k + 1 / 2} & = \frac{\tau_{\widehat{x}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{x}}\right)\right] {\widetilde{\phi}}_{k + 1 / 2}^\mathrm{R} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{x}}\right)\right] {\widetilde{\phi}}_{i + 1, k + 1 / 2}^\mathrm{L}\right\},\\ \mathcal{F}^{\rho w, \widehat{y}}_{j + 1 / 2, k + 1 / 2} & = \frac{\tau_{\widehat{y}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{y}}\right)\right] {\widetilde{\phi}}_{k + 1 / 2}^\mathrm{F} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{y}}\right)\right] {\widetilde{\phi}}_{j + 1, k + 1 / 2}^\mathrm{B}\right\},\\ \mathcal{F}^{\rho w, \widehat{z}}_{k + 1} & = \frac{\tau_{\widehat{z}}}{2} \left\{\left[1 + \mathrm{sgn} \left(\tau_{\widehat{z}}\right)\right] {\widetilde{\phi}}_{k + 1 / 2}^\mathrm{U} + \left[1 - \mathrm{sgn} \left(\tau_{\widehat{z}}\right)\right] {\widetilde{\phi}}_{k + 3 / 2}^\mathrm{D}\right\}, \end{align*}\]

with

\[\begin{align*} \tau_{\widehat{x}} & = \left[\left(J P_\mathrm{old}\right)_{i + 1 / 2} u_{\mathrm{old}, i + 1 / 2}\right]_{i + 1 / 2, k + 1 / 2},\\ \tau_{\widehat{y}} & = \left[\left(J P_\mathrm{old}\right)_{j + 1 / 2} v_{\mathrm{old}, j + 1 / 2}\right]_{j + 1 / 2, k + 1 / 2},\\ \tau_{\widehat{z}} & = \left[\left(J P_\mathrm{old}\right)_{k + 1 / 2} \widehat{w}_{\mathrm{old}, k + 1 / 2}\right]_{k + 1} \end{align*}\]

and $\widetilde{\phi}$ being the reconstruction of $\rho_{k + 1 / 2} w_{k + 1 / 2} / P_{\mathrm{old}, k + 1 / 2}$. If the viscosity is nonzero, the viscous parts (weighted by the Jacobian) are then added, i.e.

\[\begin{align*} \mathcal{F}^{\rho w, \widehat{x}}_{i + 1 / 2, k + 1 / 2} & \rightarrow \mathcal{F}^{\rho w, \widehat{x}}_{i + 1 / 2, k + 1 / 2} - \eta_{i + 1 / 2, k + 1 / 2} \left(J \Pi^{13}\right)_{i + 1 / 2, k + 1 / 2},\\ \mathcal{F}^{\rho w, \widehat{y}}_{j + 1 / 2, k + 1 / 2} & \rightarrow \mathcal{F}^{\rho w, \widehat{y}}_{j + 1 / 2, k + 1 / 2} - \eta_{j + 1 / 2, k + 1 / 2} \left(J \Pi^{23}\right)_{j + 1 / 2, k + 1 / 2},\\ \mathcal{F}^{\rho w, \widehat{z}}_{k + 1} & \rightarrow \mathcal{F}^{\rho w, \widehat{z}}_{k + 1} - \eta_{k + 1} \left[\left(J G^{13} \Pi^{13}\right)_{k + 1} - \left(J G^{23} \Pi^{23}\right)_{k + 1} - \Pi^{33}_{k + 1}\right]. \end{align*}\]

Finally, if the diffusivity $\mu$ is nonzero, the diffusive parts (weighted by the Jacobian) are added, i.e.

\[\begin{align*} \mathcal{F}^{\rho w, \widehat{x}}_{i + 1 / 2, k + 1 / 2} & \rightarrow \mathcal{F}^{\rho w, \widehat{x}}_{i + 1 / 2, k + 1 / 2} - \mu_{i + 1 / 2, k + 1 / 2} \left[J \widehat{\left(\boldsymbol{\nabla} w\right)}^{\widehat{x}}\right]_{i + 1 / 2, k + 1 / 2},\\ \mathcal{F}^{\rho w, \widehat{y}}_{j + 1 / 2, k + 1 / 2} & \rightarrow \mathcal{F}^{\rho w, \widehat{y}}_{j + 1 / 2, k + 1 / 2} - \mu_{j + 1 / 2, k + 1 / 2} \left[J \widehat{\left(\boldsymbol{\nabla} w\right)}^{\widehat{y}}\right]_{j + 1 / 2, k + 1 / 2},\\ \mathcal{F}^{\rho w, \widehat{z}}_{k + 1} & \rightarrow \mathcal{F}^{\rho w, \widehat{z}}_{k + 1} - \mu_{k + 1} \left[J \widehat{\left(\boldsymbol{\nabla} w\right)}^{\widehat{z}}\right]_{k + 1}. \end{align*}\]

compute_fluxes!(state::State, predictands::Predictands, tracer_setup::NoTracer)

Return for configurations without tracer transport.

compute_fluxes!(state::State, predictands::Predictands, tracer_setup::TracerOn)

Compute the tracer fluxes in all three directions.

The computation is analogous to that of the density fluxes.

compute_fluxes!(state::State, predictands::Predictands, variable::Theta)

Compute the potential temperature fluxes due to heat conduction (weighted by the Jacobian).

The fluxes are given by

\[\begin{align*} \mathcal{F}^{\theta, \widehat{x}}_{i + 1 / 2} & = - \lambda_{i + 1 / 2} \left\{\frac{J_{i + 1 / 2}}{\Delta \widehat{x}} \left[\left(\frac{P}{\rho}\right)_{i + 1} - \frac{P}{\rho}\right]\right.\\ & \qquad \qquad \qquad + \left.\frac{\left(J G^{13}\right)_{i + 1 / 2}}{2 \Delta \widehat{z}} \left[\left(\frac{P}{\rho}\right)_{i + 1 / 2, k + 1} - \left(\frac{P}{\rho}\right)_{i + 1 / 2, k - 1}\right]\right\},\\ \mathcal{F}^{\theta, \widehat{y}}_{j + 1 / 2} & = - \lambda_{j + 1 / 2} \left\{\frac{J_{j + 1 / 2}}{\Delta \widehat{y}} \left[\left(\frac{P}{\rho}\right)_{j + 1} - \frac{P}{\rho}\right]\right.\\ & \qquad \qquad \qquad + \left.\frac{\left(J G^{23}\right)_{j + 1 / 2}}{2 \Delta \widehat{z}} \left[\left(\frac{P}{\rho}\right)_{j + 1 / 2, k + 1} - \left(\frac{P}{\rho}\right)_{j + 1 / 2, k - 1}\right]\right\},\\ \mathcal{F}^{\theta, \widehat{z}}_{k + 1 / 2} & = - \lambda_{k + 1 / 2} \left\{\frac{\left(J G^{13}\right)_{k + 1 / 2}}{2 \Delta \widehat{x}} \left[\left(\frac{P}{\rho}\right)_{i + 1, k + 1 / 2} - \left(\frac{P}{\rho}\right)_{i - 1, k + 1 / 2}\right]\right.\\ & \qquad \qquad \qquad + \frac{\left(J G^{23}\right)_{k + 1 / 2}}{2 \Delta \widehat{y}} \left[\left(\frac{P}{\rho}\right)_{j + 1, k + 1 / 2} - \left(\frac{P}{\rho}\right)_{j - 1, k + 1 / 2}\right]\\ & \qquad \qquad \qquad + \left.\frac{\left(J G^{33}\right)_{k + 1 / 2}}{\Delta \widehat{z}} \left[\left(\frac{P}{\rho}\right)_{k + 1} - \frac{P}{\rho}\right]\right\}, \end{align*}\]

where $\lambda$ is the thermal conductivity (computed from state.namelists.atmosphere.thermal_conductivity).

Arguments

state: Model state.
predictands/old_predictands: The predictands that are used to compute the transporting velocities.
model: Dynamic equations.
variable: Flux variable.
tracer_setup: General tracer-transport configuration.

reconstruct!(state::State)

Reconstruct the prognostic variables at the cell interfaces of their respective grids, using the Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL).

This method calls specialized methods for each prognostic variable.

reconstruct!(state::State, variable::Rho)

Reconstruct the density.

Since the transporting velocity is $P \widehat{\boldsymbol{u}}$, the density is divided by $P$ before reconstruction.

reconstruct!(state::State, variable::RhoP)

Reconstruct the density fluctuations.

Similar to the density, the density fluctuations are divided by $P$ before reconstruction.

reconstruct!(state::State, variable::U)

Reconstruct the zonal momentum.

Since the transporting velocity is $P \widehat{\boldsymbol{u}}$, the zonal momentum is divided by $P$ interpolated to the respective cell interfaces before reconstruction.

reconstruct!(state::State, variable::V)

Reconstruct the meridional momentum.

Similar to the zonal momentum, the meridional momentum is divided by $P$ interpolated to the respective cell interfaces before reconstruction.

reconstruct!(state::State, variable::W)

Reconstruct the vertical momentum.

The vertical momentum is computed with compute_vertical_wind, set_zonal_boundaries_of_field! and set_meridional_boundaries_of_field!. Similar to the zonal and meridional momenta, the vertical momentum is divided by $P$ interpolated to the respective cell interfaces before reconstruction.

reconstruct!(state::State, tracer_setup::NoTracer)

Return for configurations without tracer transport.

reconstruct!(state::State, tracer_setup::TracerOn)

Reconstruct the tracers.

Similar to the density, the tracers are divided by $P$ before reconstruction.

Arguments

state: Model state.
variable: The reconstructed variable.
tracer_setup: General tracer-transport configuration.

See also