Fractal behavior drives resolution dependent vertical velocity fields

In the atmospheric science community, variable resolution models are a promising way to perform high resolution experiments without the cost of a global high resolution mesh. However, recent studies have shown that some models exhibit issues of resolution dependence that may make it impractical to use these models in a variable mesh configuration. In particular, a number of recent studies have demonstrated resolution dependent artifacts in clouds, precipitation, and moisture transport in some mainstream climate models. A theoretical understanding of this resolution dependence could give insight into model designs that behave consistently across resolutions. We develop a theoretical relationship that links the scaling properties of horizontal velocity to the resolution dependence of vertical velocity in a model with incompressible fluid dynamics. We show that the Community Atmosphere Model exhibits fractal scaling of its horizontal velocity fields and that the distribution of vertical velocity widens with decreasing grid spacing in accord with this theory. Our theory implies that if vertical grid spacing is held constant, while horizontal grid spacing decreases, the distribution of vertical velocity must widen in order to simultaneously accommodate the incompressibility constraint and the fractal nature of the horizontal velocity field. We hypothesize that this theoretical relationship is general enough that it should apply to any geophysical fluid model, with fractal velocity fields, that uses incompressibility as one of its core assumptions. Therefore, a large class of geophysical fluid models should exhibit a power law relationship between horizontal grid spacing and vertical (radial) velocity variance.