# Equations

This follows what is *currently* implemented in `examples`

: it should be kept up-to-date as code is modified. If you think something *should* be changed (but hasn't been), please add a note.

This describes the ClimaAtmos model equations and its discretizations. Where possible, we use a coordinate invariant form: the ClimaCore operators generally handle the conversions between bases internally.

## Prognostic variables

- $\rho$:
*density*in kg/m³. This is discretized at cell centers. - $\boldsymbol{u}$
*velocity*, a vector in m/s. This is discretized via $\boldsymbol{u} = \boldsymbol{u}_h + \boldsymbol{u}_v$ where- $\boldsymbol{u}_h = u_1 \boldsymbol{e}^1 + u_2 \boldsymbol{e}^2$ is the projection onto horizontal covariant components (covariance here means with respect to the reference element), stored at cell centers.
- $\boldsymbol{u}_v = u_3 \boldsymbol{e}^3$ is the projection onto the vertical covariant components, stored at cell faces.

*energy*, stored at cell centers; can be either:- $\rho e$:
*total energy*in J/m³ - $\rho e_\text{int}$:
*internal energy*in J/m³

- $\rho e$:
- $\rho \chi$:
*other conserved scalars*(moisture, tracers, etc), again stored at cell centers.

## Operators

We make use of the following operators

### Reconstruction

- $I^c$ is the face-to-center reconstruction operator (arithmetic mean)
- $I^f$ is the center-to-face reconstruction operator (arithmetic mean)
- $WI^f$ is the center-to-face weighted reconstruction operator
- $WI^f(J, x) = I^f(J*x) / I^f(J)$, where $J$ is the value of the Jacobian for use in the weighted interpolation operator

- $U^f$ is the 1st or 3rd-order center-to-face upwind product operator # fix link

### Differential operators

- $\hat{\mathcal{D}}_h$ is the discrete horizontal spectral weak divergence.
- $\mathcal{D}^c_v$ is the face-to-center vertical divergence.

Add vertical diffusive tendencies (including surface fluxes)

- $\mathcal{G}_h$ is the discrete horizontal spectral gradient.
- $\mathcal{G}^f_v$ is the center-to-face vertical gradient.
- the gradient is set to 0 at the top and bottom boundaries.

- $\mathcal{C}_h$ is the curl components involving horizontal derivatives
- $\mathcal{C}_h[\boldsymbol{u}_h]$ returns a vector with only vertical
*contravariant*components. - $\mathcal{C}_h[\boldsymbol{u}_v]$ returns a vector with only horizontal
*contravariant*components.

- $\mathcal{C}_h[\boldsymbol{u}_h]$ returns a vector with only vertical
- $\hat{\mathcal{C}}_h$ is the weak curl components involving horizontal derivatives
- $\mathcal{C}^f_v$ is the center-to-face curl involving vertical derivatives.
- $\mathcal{C}^f_v[\boldsymbol{u}_h]$ returns a vector with only a horizontal
*contravariant*component. - the curl is set to 0 at the top and bottom boundaries.
- We need to clarify how best to handle this.

- $\mathcal{C}^f_v[\boldsymbol{u}_h]$ returns a vector with only a horizontal

### Projection

- $\mathcal{P}$ is the direct stiffness summation (DSS) operation, which computes the projection onto the continuous spectral element basis.

## Auxiliary and derived quantities

$\boldsymbol{\Omega}$ is the planetary angular velocity. We use either:

- a
*shallow atmosphere*approximation, with`math \boldsymbol{\Omega} = \Omega \sin(\phi) \boldsymbol{e}^v`

where $\phi$ is latitude, and $\Omega$ is the planetary rotation rate in rads/sec (for Earth, $7.29212 \times 10^{-5} s^{-1}$) and $\boldsymbol{e}^v$ is the unit radial basis vector. This implies that the horizontal contravariant component $\boldsymbol{\Omega}^h$ is zero. - a
*deep atmosphere*, with`math \boldsymbol{\Omega} = (0, 0, \Omega)`

i.e. aligned with Earth's rotational axis.

- a
$\tilde{\boldsymbol{u}}$ is the mass-weighted reconstruction of velocity at the interfaces: by interpolation of contravariant components

\[\tilde{\boldsymbol{u}} = WI^f(\rho J, \boldsymbol{u}_h) + \boldsymbol{u}_v\]

$\bar{\boldsymbol{u}}$ is the reconstruction of velocity at cell-centers, carried out by linear interpolation of the covariant vertical component:

\[\bar{\boldsymbol{u}} = \boldsymbol{u}_h + I_{c}(\boldsymbol{u}_v)\]

$\Phi = g z$ is the geopotential, where $g$ is the gravitational acceleration rate and $z$ is altitude above the mean sea level.

$\boldsymbol{b}$ is the reduced gravitational acceleration

\[\boldsymbol{b} = - \frac{\rho - \rho_{\text{ref}}}{\rho} \nabla \Phi\]

$\rho_{\text{ref}}$ is the reference state density

$K = \tfrac{1}{2} \|\boldsymbol{u}\|^2$ is the specific kinetic energy (J/kg), reconstructed at cell centers by

\[K = \tfrac{1}{2} (\boldsymbol{u}_{h} \cdot \boldsymbol{u}_{h} + 2 \boldsymbol{u}_{h} \cdot I_{c} (\boldsymbol{u}_{v}) + I_{c}(\boldsymbol{u}_{v} \cdot \boldsymbol{u}_{v})),\]

where $\boldsymbol{u}_{h}$ is defined on cell-centers, $\boldsymbol{u}_{v}$ is defined on cell-faces, and $I_{c} (\boldsymbol{u}_{v})$ is interpolated using covariant components.

$p$ is air pressure, derived from the thermodynamic state, reconstructed at cell centers.

$p_{\text{ref}}$ is the reference state pressure. It is related to the reference state density by analytical hydrostatic balance: $\nabla p_{\text{ref}} = - \rho_{\text{ref}} \nabla \Phi$.

$\boldsymbol{F}_R$ are the radiative fluxes: these are assumed to align vertically (i.e. the horizontal contravariant components are zero), and are constructed at cell faces from RRTMGP.jl.

$\nu_u$, $\nu_h$, and $\nu_\chi$ are hyperdiffusion coefficients, and $c$ is the divergence damping factor.

No-flux boundary conditions are enforced by requiring the third contravariant component of the face-valued velocity at the boundary, $\boldsymbol{\tilde{u}}^{v}$, to be zero. The vertical covariant velocity component is computed as

\[\tilde{u}_{v} = \tfrac{-(u_{1}g^{31} + u_{2}g^{32})}{g^{33}}.\]

## Equations and discretizations

### Mass

Follows the continuity equation

\[\frac{\partial}{\partial t} \rho = - \nabla \cdot(\rho \boldsymbol{u}) + \rho \mathcal{S}_{qt}.\]

This is discretized using the following

\[\frac{\partial}{\partial t} \rho = - \hat{\mathcal{D}}_h[ \rho \bar{\boldsymbol{u}}] - \mathcal{D}^c_v \left[WI^f( J, \rho) \tilde{\boldsymbol{u}} \right] + \rho \mathcal{S}_{qt}\]

with the

\[-\mathcal{D}^c_v[WI^f(J, \rho) \boldsymbol{u}_v]\]

term treated implicitly (check this)

### Momentum

Uses the advective form equation

\[\frac{\partial}{\partial t} \boldsymbol{u} = - (2 \boldsymbol{\Omega} + \nabla \times \boldsymbol{u}) \times \boldsymbol{u} - \frac{1}{\rho} \nabla (p - p_{\text{ref}}) + \boldsymbol{b} - \nabla K\]

#### Horizontal momentum

By breaking the curl and cross product terms into horizontal and vertical contributions, and removing zero terms (e.g. $\nabla_v \times \boldsymbol{u}_v = 0$), we obtain

\[\frac{\partial}{\partial t} \boldsymbol{u}_h = - (2 \boldsymbol{\Omega}^h + \nabla_v \times \boldsymbol{u}_h + \nabla_h \times \boldsymbol{u}_v) \times \boldsymbol{u}^v - (2 \boldsymbol{\Omega}^v + \nabla_h \times \boldsymbol{u}_h) \times \boldsymbol{u}^h - \frac{1}{\rho} \nabla_h (p - p_{\text{ref}}) - \nabla_h (\Phi + K),\]

where $\boldsymbol{u}^h$ and $\boldsymbol{u}^v$ are the horizontal and vertical *contravariant* vectors. The effect of topography is accounted for through the computation of the contravariant velocity components (projections from the covariant velocity representation) prior to computing the cross-product contributions.

This is stabilized with the addition of 4th-order vector hyperviscosity

\[-\nu_u \, \nabla_h^2 (\nabla_h^2(\boldsymbol{\overline{u}})),\]

projected onto the first two contravariant directions, where $\nabla_{h}^2(\boldsymbol{v})$ is the horizontal vector Laplacian. For grid scale hyperdiffusion, $\boldsymbol{v}$ is identical to $\boldsymbol{\overline{u}}$, the cell-center valued velocity vector.

\[\nabla_h^2(\boldsymbol{v}) = \nabla_h(\nabla_{h} \cdot \boldsymbol{v}) - \nabla_{h} \times (\nabla_{h} \times \boldsymbol{v}).\]

The $(2 \boldsymbol{\Omega}^h + \nabla_v \times \boldsymbol{u}_h + \nabla_h \times \boldsymbol{u}_v) \times \boldsymbol{u}^v$ term is discretized as:

\[\frac{I^c\{(2 \boldsymbol{\Omega}^h + \mathcal{C}^f_v[\boldsymbol{u}_h] + \mathcal{C}_h[\boldsymbol{u}_v]) \times (I^f(\rho J)\tilde{\boldsymbol{u}}^v)\}}{\rho J}\]

where

\[\omega^{h} = (\nabla_v \times \boldsymbol{u}_h + \nabla_h \times \boldsymbol{u}_v)\]

The $(2 \boldsymbol{\Omega}^v + \nabla_h \times \boldsymbol{u}_h) \times \boldsymbol{u}^h$ term is discretized as

\[(2 \boldsymbol{\Omega}^v + \mathcal{C}_h[\boldsymbol{u}_h]) \times \boldsymbol{u}^h\]

and the $\frac{1}{\rho} \nabla_h (p - p_h) + \nabla_h (\Phi + K)$ as

\[\frac{1}{\rho} \mathcal{G}_h[p - p_{\text{ref}}] + \mathcal{G}_h[\Phi + K] ,\]

where all these terms are treated explicitly.

The hyperviscosity term is

\[- \nu_u \left\{ c \, \hat{\mathcal{G}}_h ( \mathcal{D}(\boldsymbol{\psi}_h) ) - \hat{\mathcal{C}}_h( \mathcal{C}_h( \boldsymbol{\psi}_h )) \right\}\]

where

\[\boldsymbol{\psi}_h = \mathcal{P} \left[ \hat{\mathcal{G}}_h ( \mathcal{D}(\boldsymbol{u}_h) ) - \hat{\mathcal{C}}_h( \mathcal{C}_h( \boldsymbol{u}_h )) \right]\]

#### Vertical momentum

Similarly for vertical velocity

\[\frac{\partial}{\partial t} \boldsymbol{u}_v = - (2 \boldsymbol{\Omega}^h + \nabla_v \times \boldsymbol{u}_h + \nabla_h \times \boldsymbol{u}_v) \times \boldsymbol{u}^h - \frac{1}{\rho} \nabla_v (p - p_{\text{ref}}) - \frac{\rho - \rho_{\text{ref}}}{\rho} \nabla_v \Phi - \nabla_v K .\]

The $(2 \boldsymbol{\Omega}^h + \nabla_v \times \boldsymbol{u}_h + \nabla_h \times \boldsymbol{u}_v) \times \boldsymbol{u}^h$ term is discretized as

\[(2 \boldsymbol{\Omega}^h + \mathcal{C}^f_v[\boldsymbol{u}_h] + \mathcal{C}_h[\boldsymbol{u}_v]) \times I^f(\boldsymbol{u}^h) ,\]

and the $\frac{1}{\rho} \nabla_v (p - p_{\text{ref}}) - \frac{\rho - \rho_{\text{ref}}}{\rho} \nabla_v \Phi - \nabla_v K$ term as

\[\frac{1}{I^f(\rho)} \mathcal{G}^f_v[p - p_{\text{ref}}] - \frac{I^f(\rho - \rho_{\text{ref}})}{I^f(\rho)} \mathcal{G}^f_v[\Phi] - \mathcal{G}^f_v[K] ,\]

with the latter treated implicitly.

This is stabilized with the addition of 4th-order vector hyperviscosity

\[-\nu_u \, \nabla_h^2 (\nabla_h^2(\boldsymbol{\overline{u}})),\]

projected onto the third contravariant direction.

### Total energy

\[\frac{\partial}{\partial t} \rho e = - \nabla \cdot((\rho e + p) \boldsymbol{u} + \boldsymbol{F}_R) + \rho \mathcal{S}_{e},\]

which is stabilized with the addition of a 4th-order hyperdiffusion term on total enthalpy:

\[- \nu_h \nabla \cdot \left( \rho \nabla^3 \left(\frac{ρe + p}{ρ} \right)\right)\]

This is discretized using

\[\frac{\partial}{\partial t} \rho e \approx - \hat{\mathcal{D}}_h[ (\rho e + p) \bar{\boldsymbol{u}} ] - \mathcal{D}^c_v \left[ WI^f(J,\rho) \, \tilde{\boldsymbol{u}} \, I^f \left(\frac{\rho e + p}{\rho} \right) + \boldsymbol{F}_R \right] - \nu_h \hat{\mathcal{D}}_h( \rho \mathcal{G}_h(\psi) ).\]

where

\[\psi = \mathcal{P} \left[ \hat{\mathcal{D}}_h \left( \mathcal{G}_h \left(\frac{ρe + p}{ρ} \right)\right) \right]\]

Currently the central reconstruction

\[- \mathcal{D}^c_v \left[ WI^f(J,\rho) \, \tilde{\boldsymbol{u}} \, I^f \left(\frac{\rho e + p}{\rho} \right) \right]\]

is treated implicitly.

The Jacobian computation should be updated so that the upwinded term

\[- \mathcal{D}^c_v\left[WI^f(J, \rho) U^f\left(\boldsymbol{u}_v, \frac{\rho e + p}{\rho} \right)\right]\]

is treated implicitly.

### Scalars

For an arbitrary scalar $\chi$, the density-weighted scalar $\rho\chi$ follows the continuity equation

\[\frac{\partial}{\partial t} \rho \chi = - \nabla \cdot(\rho \chi \boldsymbol{u}) + \rho \mathcal{S}_{\chi}.\]

This is stabilized with the addition of a 4th-order hyperdiffusion term

\[- \nu_\chi \nabla \cdot(\rho \nabla^3(\chi))\]

This is discretized using

\[\frac{\partial}{\partial t} \rho \chi \approx - \hat{\mathcal{D}}_h[ \rho \chi \bar{\boldsymbol{u}}] - \mathcal{D}^c_v \left[ WI^f(J,\rho) \, U^f\left( \tilde{\boldsymbol{u}}, \frac{\rho \chi}{\rho} \right) \right] - \nu_\chi \hat{\mathcal{D}}_h ( \rho \, \mathcal{G}_h (\psi) )\]

where

\[\psi = \mathcal{P} \left[ \hat{\mathcal{D}}_h \left( \mathcal{G}_h \left( \frac{\rho \chi}{\rho} \right)\right) \right]\]

Currently the central reconstruction

\[- \mathcal{D}^c_v \left[ WI^f(J,\rho) \, \tilde{\boldsymbol{u}} \, I^f\left( \frac{\rho \chi}{\rho} \right) \right]\]

is treated implicitly.

The Jacobian computation should be updated so that the upwinded term

\[- \mathcal{D}^c_v\left[WI^f(J, \rho) U^f\left(I^f(\boldsymbol{u}_h) + \boldsymbol{u}_v, \frac{\rho \chi}{\rho} \right) \right]\]

is treated implicitly.

## Microphysics source terms

Sources from cloud microphysics $\mathcal{S}$ represent the transfer of mass between the working fluid (dry air, water vapor cloud liquid and cloud ice) and precipitation (rain and snow), as well as the latent heat release due to phase changes.

The scalars $\rho q_{rai}$ and $\rho q_{sno}$ are part of the state vector when running simulations with 1-moment microphysics scheme, and represent the specific humidity of liquid and solid precipitation (i.e. rain and snow).

\[q_{rai} := \frac{m_{rai}}{m_{dry} + m_{vap} + m_{liq} + m_{ice}}\; , \;\;\;\; q_{sno} := \frac{m_{sno}}{m_{dry} + m_{vap} + m_{liq} + m_{ice}}\]

The different source terms are provided by CloudMicrophysics.jl library and are defined as the change of mass of one of the cloud condensate or precipitation species normalised by the mass of the working fluid. See the CloudMicrophysics.jl docs for more details.

Throughout the rest of the derivations we are assuming that the volume of the working fluid is constant (not the pressure). This is strange for phase changes and needs more thinking.

### Case 1: Mass of the working fluid is changed

When the phase change is happening within the working fluid (for example condensation from water vapor to liquid water), there is no change to any of the state variables. Considering the transition from $x \rightarrow y$ where $x$ is either water vapor, cloud liquid water or cloud ice and $y$ is either rain or snow

\[\mathcal{S}_{x \rightarrow y} := \frac{\frac{dm_x}{dt}}{m_{dry} + m_{vap} + m_{liq} + m_{ice}}\]

\[\frac{d}{dt} \rho = \frac{d}{dt} \rho q_{tot} = \rho \mathcal{S}_{x \rightarrow y} = - \frac{d}{dt} \rho q_y\]

\[\frac{d}{dt} \rho e = \rho \mathcal{S}_{x \rightarrow y} (I_{y} + \Phi)\]

where $I_{y}$ is the internal energy of the $y$ phase. This formula applies to the majority of microphysics processes. Namely, it is valid for processes where $T=const$ such as autoconversion and accretion between species of the same phase. It is also valid for rain evaporation, deposition/sublimation, and accretion of cloud water and snow in temperatures below freezing (which result in snow).

### Case 2: Phase change outside of the working fluid

For cases where both $x$ and $y$ are not part of the working fluid (melting of snow, freezing of rain)

\[\mathcal{S}_{x \rightarrow y} := \frac{\frac{dm_{x}}{dt}}{m_{dry} + m_{vap} + m_{liq} + m_{ice}}\]

\[\frac{d}{dt} \rho q_{x} = - \frac{d}{dt} \rho q_{y} = \rho \mathcal{S}_{x \rightarrow y}\]

\[\frac{d}{dt} \rho = \frac{d}{dt} \rho q_{tot} = 0\]

\[\frac{d}{dt} \rho e = - \rho \mathcal{S}_{x \rightarrow y} L_{f}\]

where $L_f$ is the latent heat of fusion. The sign in the last equation assumes $x$ stands for rain and $y$ for snow.

### Additional cases

Accretion of cloud ice by rain results in snow. This process combines the effects from the loss of working fluid $q_{ice}$ (described by case 1) and the phase change from rain to snow (described by case 2).

Accretion of cloud liquid by snow in temperatures above freezing results in rain. It is assummed that some fraction $\alpha$ of snow is melted during the process and both cloud liquid and melted snow are turned into rain.

\[\mathcal{S}_{acc} := \frac{\frac{dm_{liq}}{dt}}{m_{dry} + m_{vap} + m_{liq} + m_{ice}}\]

\[\frac{d}{dt} \rho = \frac{d}{dt} \rho q_{tot} = \rho S_{acc}\]

\[\frac{d}{dt} \rho q_{sno} = \rho \alpha S_{acc}\]

\[\frac{d}{dt} \rho q_{rai} = - \rho (1 + \alpha) S_{acc}\]

\[\frac{d}{dt} \rho e = \rho \mathcal{S}_{acc} ((1+\alpha) I_{liq} - \alpha I_{ice} + \Phi)\]

### Precipitation temperature

Precipitation is assumed to have the same temperature as the surrounding air $T_a$. The corresponding energy sink associated with heat transfer between air and precipitating species can be written as

\[\frac{d}{dt} \rho e = - \rho q_p (\boldsymbol{u} - v_p) c_p \nabla T_a\]

where $q_p$, $\boldsymbol{u}$, $v_p$, $c_p$ are the precipitation specific humidity, air velocity, precipitation terminal velocity assumed to be along the gravity axis, specific heat of precipitating species.

We should consider replacing $T_a$ with some approximation of wet bulb temperature.

### Stability and positivity

All source terms are individually limited such that they don't exceed the available tracer specific humidity divided by a coefficient $a$.

\[\mathcal{S}_{x \rightarrow y} = min(\mathcal{S}_{x \rightarrow y}, \frac{q_{x}}{a \; dt})\]

This will not ensure positivity because the sum of all source terms, combined with the advection tendency, could still drive the solution to negative numbers. It should however help mitigate some of the problems. The source terms functions treat negative specific humidities as zeros, so the simulations should be stable even with small negative numbers.

We do not apply hyperdiffusion for precipitation tracers.