I have a 2D velocity field, $(u, w)$, in the $x-z$ plane defined by a streamfunction, $z^\star$, such that
\begin{align} u = \frac{\partial z^\star}{\partial z}, w &= - \frac{\partial z^\star}{\partial x} \end{align} The streamfunction is also used to define the coordinate transform, $z^\star = H \frac{z - h(x)}{H - h(x)}$, where $H$ is a constant and $h(x)$ defines the height of the bottom surface.
The advection equation of a scalar field, $\phi$, is \begin{align} \frac{\partial \phi}{\partial t} + \frac{\partial u \phi}{\partial x} + \frac{\partial w \phi}{\partial z} = 0 \end{align} which can be rewritten in transformed coordinates as \begin{align} \frac{\partial \phi}{\partial t} + \frac{\partial u \phi}{\partial x} + \frac{\partial u \phi}{\partial z^\star} \frac{\partial z^\star}{\partial x} + \frac{\partial w \phi}{\partial x}\frac{\partial x}{\partial z} + \frac{\partial w \phi}{\partial z^\star} \frac{\partial z^\star}{\partial z} &= 0 \end{align} Using the fact that $\partial x/\partial z = 0$, and using the product rule and the definitions of $u$ and $w$ this simplifies to \begin{align} \frac{\partial \phi}{\partial t} + \frac{\partial u \phi}{\partial x} &= 0 \end{align} Once again, using the product rule and the definition of $u$ and $z^\star$ gives \begin{align} \frac{\partial \phi}{\partial t} + \frac{H}{H-h(x)} \frac{\partial \phi}{\partial x} + \frac{H}{\left(H - h(x)\right)^2}\phi &= 0 \end{align}
I'm trying to solve for $\phi$ but I'm unsure how to proceed.
I'm not going to get into transformed coordinates in this answer, but hopefully this answer will help. The domain of dependence of any point, $(x_a,z_a)$, at time $t$ when solving the linear advection equation is a single point. Let's call it $(x_d, z_d)$. So in order to find $\phi(x_a, z_a, t)$, all you need to do is find the domain of dependence, $(x_d, z_d)$ and then $\phi(x_a, z_a, t)=\phi_0(x_d, z_d)$ where $\phi_0$ are the initial conditions. So let's start by trying to find $(x_a,z_a)$, at time $t$ from $(x_d, z_d)$: $$ x_a = x_d + \int_0^t u(x,z)\ dt $$ and $\Psi$ is the streamfunction so that $u=\Psi_z$. This equation could be integrated numerically. But that won't give us an analytic solution. Instead, we need to transform the coordinates so that $u(x^*,z^*)$ and $z^*=$const along the streamline. Then the integration will start to look more manageable. We will then need to work out how to replace $u(x^*)$ with $u(t)$. Then I think the integration might be possible analytically.
Hope this helps
Hilary