A two-scale model for the meso- and synoptic scales

For both the meso- and synoptic scale, reduced models exist that give insight into atmospheric dynamics. For the mesoscale, the weak temperature gradient approximation is one of several approaches, while for the synoptic scale,  the quasi-geostrophic theory is well established. However, how these two scales interact with each other, is usually not included in such reduced models, thereby limiting our current perception of flow dependent predictability and upscale error growth.

Here, we explicitly address the scale interactions by developing a two-scale asymptotic model for the meso- and synoptic scales with two, coupled sets of equations for the meso- and synoptic scale, respectively. The mesoscale equations follow a weak temperature gradient balance and the synoptic scale equations align with quasi-geostrophic theory. Importantly, the equation sets are coupled via scale interaction terms: eddy correlations of mesoscale variables impact the synoptic potential vorticity tendency and synoptic variables force the mesoscale vorticity (for instance due to tilting of synoptic scale wind shear). Furthermore, we impose different diabatic heating rates as proxies for the effect of latent heating on the different scales and distinguish between a weak, heating regime with O (4K/3.5h)  mesoscale heating rates and a strong heating regime with O (40K/3.5h) mesoscale heating rates. With weak heating, the upscale impact of the mesoscale on the synoptic scale is only of dynamical nature. With strong heating, the upscale impact also includes thermodynamical effects.
The scale interactions and the imposed diabatic heating can provide new insight into atmospheric dynamics, flow dependent error growth characteristics and predictability.