Tracer Transport Equations


\begin{DoxyImage}
\includegraphics[width=\textwidth,height=\textheight/2,keepaspectratio=true]{PPM_1d.png}
\doxyfigcaption{The 1-\/D finite volume advection of tracers. The reddish fluid will be in the cell at the end of the timestep.}
\end{DoxyImage}
 Given a piecewise polynomial description of the tracer concentration, the new tracer cell concentration is the average of the fluid that will be in the cell after a timestep.

\begin{eqnarray} \int_{x_{i-1/2}}^{x_{i+1/2}} A_i^{n+1} (x) dx = \int_{x_{i-1/2 - u \Delta t}}^{x_{i+1/2-u\Delta t}} A_i^{n} (x) dx &= \mbox{} \\ \int_{x_{i-1/2}}^{x_{i+1/2}} A_i^{n} (x) dx - \int_{x_{i+1/2 - u \Delta t}}^{x_{i+1/2}} A_i^{n} (x) dx &+ \int_{x_{i-1/2 - u \Delta t}}^{x_{i-1/2}} A_i^{n} (x) dx \end{eqnarray}

Fluxes are found by analytically integrating the profile over the distance that is swept past the face within a timestep.

\[ a_i^n = \frac{1}{\Delta x} \int_{x_{i-1/2}}^{x_{i+1/2}} A_i^n(x) dx \] \[ a_i^{n+1} = a_i^n - \frac{\Delta t}{\Delta x} (F_{i+1/2} - F_{i-1/2}) \] \[ F_{i+1/2} = \frac{1}{\Delta t} \int_{x_{i+1/2 - u \Delta t}}^{x_{i+1/2}} A_i^n(x) dx \] \[ F_{i-1/2} = \frac{1}{\Delta t} \int_{x_{i-1/2 - u \Delta t}}^{x_{i-1/2}} A_i^n(x) dx \]

With piecewise constant profiles, this approach give first order upwind advection. Higher order polynomials (e.\+g., parabolas) can give higher order accuracy.\hypertarget{Tracer_Transport_Equations_Multidimensional_Tracer_Advection}{}\section{Multidimensional Tracer Advection}\label{Tracer_Transport_Equations_Multidimensional_Tracer_Advection}
Using \char`\"{}\+Easter\textquotesingle{}s Pseudo-\/compressibility\char`\"{} (easter1993), we start with these basic equations for a tracer $\psi$\+:

\label{Tracer_Transport_Equations_ht-equation}%
\Hypertarget{Tracer_Transport_Equations_ht-equation}%
 \[ \frac{\partial h}{\partial t} + \vec{\nabla} \cdot (\vec{u}h) = 0 \equiv \frac{\partial h}{\partial t} + \vec{\nabla} \cdot (\vec{U}) \]

\[ \frac{\partial}{\partial t} (h \psi) + \vec{\nabla} \cdot (\vec{U}\psi) = 0 \]

\[ \frac{\partial \psi}{\partial t} + \vec{u} \cdot \vec{\nabla} \psi = 0 \]

We discretize the first of these equations in space\+:

\[ \frac{\partial h}{\partial t} = \frac{1}{\Delta x} \left(U_{i-\frac{1}{2},j} - U_{i+\frac{1}{2},j} \right) + \frac{1}{\Delta y} \left(V_{i, j-\frac{1}{2}} - V_{i,j+\frac{1}{2}} \right) \]

Using our monotonic one-\/dimensional flux\+:

\[ F_{i+\frac{1}{2},j} (\psi) = U_{i+\frac{1}{2},j} \psi_{i+\frac{1}{2},j} \]

we come up with an estimate based only on an update in the $x$ direction\+:

\[ \tilde{h}_{i,j} \tilde{\psi}_{i,j} = h^n_{i,j} \psi_{i,j} + \frac{\Delta t}{\Delta x} \left( F_{i-\frac{1}{2},j} (\psi^n) - F_{i+\frac{1}{2},j} (\psi^n) \right) \]

\[ \tilde{h}_{i,j} = h^n_{i,j} + \frac{\Delta t}{\Delta x} \left( U_{i-\frac{1}{2},j} - U_{i+\frac{1}{2},j} \right) \]

\[ \tilde{\psi}_{i,j} = \frac{\tilde{h}_{i,j} \tilde{\psi}_{i,j}}{\tilde{h}_{i,j}} \]

Next, we update in the $y$ direction\+:

\[ h^{n+1}_{i,j} \psi^{n+1}_{i,j} = \tilde{h}_{i,j} \tilde{\psi}_{i,j} + \frac{\Delta t}{\Delta y} \left( G_{i,j-\frac{1}{2}} (\tilde{\psi}) - G_{i,j+\frac{1}{2}} (\tilde{\psi}) \right) \]

\[ h^{n+1}_{i,j} = \tilde{h}_{i,j} + \frac{\Delta t}{\Delta y} \left( V_{i,j-\frac{1}{2}} - V_{i,j+\frac{1}{2}} \right) \]

\[ \psi^{n+1}_{i,j} = \frac{h^{n+1}_{i,j} \psi^{n+1}_{i,j}}{h^{n+1}_{i,j}} \]

\begin{DoxyItemize}
\item This method ensures monotonicity. Strang splitting can reduce directional splitting error. See \cite{easter1993}, \cite{durran2010} (section 5.\+9.\+4), and \cite{russell1981} .\end{DoxyItemize}
\begin{DoxyItemize}
\item Flux-\/form pseudo-\/compressibility advection is based on accumulated mass (or volume) fluxes, not velocities.\end{DoxyItemize}
\begin{DoxyItemize}
\item Additional pseudo-\/compressibility passes can be added to accommodate transports exceeding cell masses. Extra passes of tracer advection are used in M\+O\+M6 in the small fraction of cells where this is needed.\end{DoxyItemize}
\begin{DoxyItemize}
\item Explicit layered dynamics time-\/steps are limited by Doppler-\/shifted internal gravity wave speeds or inertial oscillations. Flow speeds in most of the ocean volume are much smaller than the peak internal wave speeds so that the advective time-\/steps can be longer.\end{DoxyItemize}
\begin{DoxyItemize}
\item Advective mass fluxes in M\+O\+M6 are often accumulated over multiple dynamic steps. The goal is that as we go to higher resolution, this tracer advection will remain stable at relatively long time-\/steps, allowing for the inclusion of many biogeochemical tracers without adding an undue burden in computational cost. \end{DoxyItemize}
