Modeling the Interaction between the Quasi - Geostrophic Vertical 1 Motion and Convection in a Single Column

4 A single-column modeling approach is proposed to study interaction between convection and 5 large-scale dynamics using the quasi-geostrophic (QG) framework. Vertical motion is repre6 sented by the QG omega equation with the diabatic heating term included. This approach 7 extends the notion of “parameterization of large scale dynamics”, previously applied in the 8 tropics using the weak temperature gradient approximation and other comparable meth9 ods, to the extratropics, where balanced adiabatic dynamics plays a larger role in inducing 10 large-scale vertical motion. The diabatic heating term in the QG-omega equation repre11 sents the feedback from convection, coupling the convection and large-scale vertical motion. 12 The strength of the coupling depends on the characteristic wavelength of the large-scale 13 disturbances, a free parameter in the system. 14 This approach is demonstrated using two representations of convection: a single-column 15 model with a convective parameterization, and linear response functions derived by Z. Kuang 16 from a large set of cloud-resolving simulations. The results are qualitatively similar in both 17 cases, though the linear response functions allow for a more thorough analysis of the system 18 dynamics. The behavior of convection that is strongly coupled to large-scale vertical motion 19 is significantly different from that in the uncoupled case in which large-scale dynamics is 20 not present. The positive feedback of the diabatic heating on the large-scale vertical motion 21 reduces the stability of the system, extends the decay time scale after initial perturbations, 22 and increases the amplitude of the convective response to transient large-scale perturbations 23 or imposed forcings. The diabatic feedback of convection on vertical motion is strongest for 24 horizontal wavelengths roughly between 2000 km and 1000 km. 25