Global-mean Vertical Tracer Mixing in Planetary Atmospheres I: Theory and Fast-rotating Planets

Most chemistry and cloud formation models for planetary atmospheres adopt a one-dimensional (1D) diffusion approach to approximate the global-mean vertical tracer transport. The physical underpinning of the key parameter in this framework, eddy diffusivity $K_{zz}$, is usually obscure. Here we analytically and numerically investigate vertical tracer transport in a 3D stratified atmosphere and predict $K_{zz}$ as a function of the large-scale circulation strength, horizontal mixing due to eddies and waves and local tracer sources and sinks. We find that $K_{zz}$ increases with tracer chemical lifetime and circulation strength but decreases with horizontal eddy mixing efficiency. We demarcated three $K_{zz}$ regimes in planetary atmospheres. In the first regime where the tracer lifetime is short compared with the transport timescale and horizontal tracer distribution under chemical equilibrium ($\chi_0$) is uniformly distributed across the globe, global-mean vertical tracer mixing behaves diffusively. But the traditional assumption in current 1D models that all chemical species are transported via the same eddy diffusivity generally breaks down. We show that different chemical species in a single atmosphere should in principle have different eddy diffusion profiles. In the second regime where tracer is short-lived but $\chi_0$ is non-uniformly distributed, a significant non-diffusive component might lead to a negative $K_{zz}$ under the diffusive assumption. In the third regime where the tracer is long-lived, global-mean vertical tracer transport is also largely influenced by non-diffusive effects. Numerical simulations of 2D tracer transport on fast-rotating zonally symmetric planets validate our analytical $K_{zz}$ theory over a wide parameter space.