Equations

This page translates the governing-equation part of the TARSA manuscript into the documentation format used by this site. The focus is the continuous model: what variable is transported, which processes are included, and why the resulting system is useful as a transparent forward operator.

Scope and intended applications

TARSA is a regional three-dimensional Eulerian transport model for generic tracers expressed as mass mixing ratios. In the current forward-model configuration, meteorology is prescribed externally, and the prognostic core contains only processes that are linear in tracer concentration for fixed meteorological forcing.

The implemented process set consists of:

• advection
• turbulence-driven vertical diffusion
• gravitational settling
• parameterized dry deposition
• parameterized wet deposition
• optional linear source terms for prescribed emissions or simple first-order chemistry

This scope is deliberate. TARSA is intended primarily as a reusable forward-operator framework for inverse problems, observationally constrained transport studies, and remote-sensing workflows. It is not designed as a full chemistry-aerosol system or as a maximal-process forecasting model. Nonlinear aerosol microphysics, gas-phase chemistry, moist convection, and radiative feedbacks are outside the prognostic core in the current version.

Governing transport equations

The transported variable is a generic tracer $C$, expressed as mass mixing ratio (kg of species per kg of moist air). Although the prognostic variable is a mixing ratio, the model is built around the flux (conservative) form of the air-mass-weighted tracer field $\rho C$. This is what makes TARSA mass consistent: a spatially uniform mixing ratio is preserved exactly under transport, including the discretely divergent winds that arise from prescribed ERA5 meteorology.

Two coupled balances define the continuous model. The air obeys the continuity equation

\[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{u}) = 0,\]

and the tracer mass density $\rho C$ obeys the conservative (flux-form) transport equation

\[\frac{\partial (\rho C)}{\partial t} + \frac{\partial (\rho u_x C)}{\partial x} + \frac{\partial (\rho u_y C)}{\partial y} + \frac{\partial \left[\rho\left(u_z - w_{\mathrm{sed}}\right)C\right]}{\partial z} = \frac{\partial}{\partial z}\!\left(\rho K_z \frac{\partial C}{\partial z}\right) + \rho E + \rho F_{\mathrm{dry}} + \rho F_{\mathrm{wet}},\]

where $(u_x, u_y, u_z)$ are the resolved wind components. TARSA discretizes this conservative form directly, because preserving the link between the discrete tracer-mass budget and the discrete air-mass budget is exactly the property a transport scheme needs to avoid spurious tracer sources or sinks in divergent flow. Dividing through by $\rho$ and subtracting $C$ times the air-continuity equation recovers the equivalent advective (non-conservative) form, but that equivalence holds only when the air-continuity equation is satisfied — which is why the flux form is the one advanced numerically.

The terms have the following meanings:

Nonlinear microphysical processes such as nucleation, coagulation, or condensational growth are not included in these equations. At the continuous-equation level, prescribed meteorology and prescribed linear source terms therefore give a linear tracer equation. This linearity does not automatically carry over to the discrete update: the default regularized Koren MUSCL advection option is nonlinear at the discrete level because its limiter depends on local solution gradients, whereas the first-order upwind configuration retains an affine one-step update. This distinction is made precise in Forward-operator structure below and matters for sensitivity analysis, collocation workflows, and inversion-oriented applications.

Mass consistency

A transport scheme is mass consistent (or constancy preserving) if a tracer that starts uniform in mixing ratio, $C \equiv c$, stays uniform under transport alone — independent of the wind divergence. Substituting $C \equiv c$ into the conservative tracer equation gives

\[\frac{\partial (\rho c)}{\partial t} + \nabla\cdot(\rho\mathbf{u}\,c) = c\left[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{u})\right] = 0,\]

so the tracer balance collapses onto the air-continuity equation and $C$ cannot drift away from $c$. TARSA reproduces this cancellation at the discrete level by advancing the pair $(\rho, \rho C)$ with one and the same set of face mass fluxes and recovering $C = \rho C/\rho$ after each step (see Numerical Methods). Prescribed ERA5 winds are not discretely non-divergent, so it is this discrete consistency — rather than the continuous identity above — that keeps the uniform-tracer benchmark (validation/VMR, after Sofiev et al., 2015, Fig. 15) at $\max|C-1| = 0$ to floating-point roundoff. The legacy volume-flux scheme instead produced a spurious $C^{n+1} = c - \Delta t\,c\,(\nabla\cdot\mathbf{u})$ drift, which the mass-consistent formulation removes.

Meteorological forcing and vertical coordinate

Real-case workflows use meteorological forcing from the ERA5 reanalysis. TARSA uses ERA5 pressure-level fields rather than native model-level fields: the 25 pressure levels used here reduce the three-dimensional request size by roughly $137/25 \approx 5.5$ relative to the full 137 model levels, and the pressure-level product is served from a fast online subset. This is a practical choice for the present regional applications, not a claim that pressure levels are optimal — the reduced near-surface resolution can affect boundary-layer gradients, deposition diagnostics, and shallow-plume transport, and model-level input is a natural future extension.

Horizontal wind components are read on pressure levels and interpolated to the TARSA grid as needed. ERA5 provides vertical motion as pressure velocity $\omega$ in Pa s$^{-1}$. TARSA converts this to physical vertical velocity using

\[u_z = -\frac{\omega}{g\rho},\]

with $g = 9.81$ m s$^{-2}$.

Air density is diagnosed from pressure, temperature, and humidity as

\[\rho = \frac{p_l}{R_d T_v},\]

where $R_d = 287.05$ J (kg K)$^{-1}$ is the gas constant for dry air and $T_v$ is the virtual temperature,

\[T_v = T(1 + 0.61q),\]

with $T$ the air temperature and $q$ the specific humidity.

Because ERA5 is defined on pressure levels, transport is carried out on a geometric-height vertical coordinate. The height $z$ at pressure level $p_l$ is computed relative to surface elevation $z_s$ as

\[z = z_s + \int_{p_l}^{p_s} \frac{R_d T_v}{g p'}\,dp',\]

where $p_s$ is the surface pressure. In practice, the integral is approximated numerically from the discrete ERA5 profiles.

Linear physical parameterizations

Vertical turbulent diffusion

Vertical turbulent mixing is represented in flux form:

\[\frac{1}{\rho}\frac{\partial}{\partial z}\left(\rho K_z \frac{\partial C}{\partial z}\right).\]

The eddy diffusivity $K_z$ is diagnosed from meteorological stability and wind shear using a simplified boundary-layer parameterization derived from the ECMWF framework. The model distinguishes stable, neutral, and convective regimes, but it does not implement the full EDMF treatment.

Sedimentation and deposition

The aerosol processes included in the current forward core are the ones that can be represented as linear operators acting on the transported tracer:

• gravitational settling
• dry deposition
• wet deposition

Nonlinear aerosol microphysics are not solved prognostically. Hygroscopic growth may be used diagnostically to adjust effective particle properties for sedimentation and deposition parameterizations, but it does not introduce nonlinear coupling in the transport update.

Forward-operator structure

One of the main design goals of TARSA is compatibility with repeated forward evaluation. For a fixed grid, time step, meteorological forcing, boundary-condition treatment, and solver configuration, the model advances the discrete tracer state through a one-step map

\[\mathbf{X}^{n+1} = \mathbf{F}_n(\mathbf{X}^n, \theta),\]

where $\mathbf{X}^n$ is the mixing-ratio state vector at step $n$ and $\theta$ may contain emissions, initial conditions, boundary values, or selected physical parameters.

Whether this map is affine depends on the advection option, and this must be kept separate from the statement that the continuous process set is linear:

• The first-order upwind replay configuration gives an affine sparse update $\mathbf{X}^{n+1} = \mathbf{A}_{\mathrm{step}}\,\mathbf{X}^n + \mathbf{q}(\theta)$ for fixed meteorology and boundary treatment, with a one-step operator $\mathbf{A}_{\mathrm{step}}$ that does not depend on the current tracer trajectory.
• The default regularized Koren MUSCL scheme is not globally affine in $\mathbf{X}^n$, because its limiter weights depend on the evolving tracer field through local gradient ratios. Its Jacobian is therefore trajectory-dependent, and the cubic-Hermite regularization only smooths selected positive-ratio limiter kinks that are relevant for local sensitivity calculations on nondegenerate stencils — it does not imply global differentiability of the full transport model.

When local derivatives are well defined along a reference solution, the corresponding tangent sensitivity has the compact form

\[\mathbf{S}^{n+1} = \mathbf{J}_n\,\mathbf{S}^n + \partial_\theta \mathbf{F}_n, \qquad \mathbf{J}_n = \frac{\partial \mathbf{F}_n}{\partial \mathbf{X}^n},\]

with $\mathbf{S}^n = \partial \mathbf{X}^n/\partial\theta$ and $\mathbf{J}_n$ evaluated along that trajectory. The tangent-linear, adjoint, and automatic-differentiation paths implement exactly this linearization; the scalar structural gradient check verifies that they agree to single precision for a deliberately simple emission-amplitude control.

Mass consistency is a separate property from global affineness. The density field $\rho$ is a prescribed parameter held frozen across a step; the scheme advances the pair $(\rho, \rho C)$ with one and the same face mass fluxes and recovers $C = \rho C/\rho$, which is what preserves a uniform mixing ratio exactly (the constancy invariant). This holds for the Koren MUSCL family regardless of whether its one-step Jacobian is trajectory-dependent, because for a uniform field the limiter selects the same interface state on both sides of every face. See Numerical Methods and the constant-VMR test.

Keeping the forward core explicit about what it does and does not do makes TARSA easier to configure, benchmark, interpret, and embed in retrieval, collocation, optimization, and inversion pipelines than a full chemistry-transport model.