Scale interactions between the meso- and synoptic scales and the impact of diabatic heating

For both the meso- and synoptic scales, reduced mathematical models give insight into their dynamical behaviour. For the mesoscale, the weak temperature gradient approximation is one of several approaches, while for the synoptic scale the quasigeostrophic theory is well established. However, the way 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 address the scale interactions explicitly by developing a two-scale asymptotic model for the meso- and synoptic scales with two coupled sets of equations for the meso- and synoptic scales respectively. The mesoscale equations follow a weak temperature gradient balance and the synoptic-scale equations align with quasigeostrophic 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, different diabatic heating rates-representing the effect of precipitation-define different flow characteristics. With weak mesoscale heating relatable to precipitation rates of (sic)(6mm center dot h-1), the mesoscale dynamics resembles two-dimensional incompressible vorticity dynamics and the upscale impact of the mesoscale on the synoptic scale is only of a dynamical nature. With a strong mesosocale heating relatable to precipitation rates of O (60mm center dot h(-1)), divergent motions and three-dimensional effects become relevant for the mesoscale dynamics and the upscale impact also includes thermodynamical effects.