22.8.8 Solving for the tracer

The tracer equation is solved using a variety of time differencing schemes (leapfrog, forward, Euler backward) as outlined in Section 21.4. A complication arises due to the vertical diffusion term which may be solved explicitly or implicitly. The implicit treatment is appropriate when large vertical diffusion coefficients (or small vertical grid spacing) limit the time step. The following discussion pertains to the leapfrog scheme. The other schemes amount to changing $2\Delta t$ to $\Delta t$ in what follows.