## 1. Introduction

[2] Deep cumulus convection is frequently organized into mesoscale convective systems (MCSs), including squall lines (about 90% of all MCSs, according to *Tsakraklides and Evans* [2002]) and roughly circular cloud masses (about 10% of MCSs). In this study, squall lines are called “linear MCSs,” and circular cloud masses “nonlinear MCSs.” No existing cumulus parameterization used in large-scale models can predict the existence of an MCS, let alone its “organization” in terms of its shape, orientation, size, and propagation speed. Such predictions would be of both scientific and practical interests. For example, for a given large-scale average precipitation rate, a slowly moving convective system that covers a small area tends to produce relatively heavy rainfall over only a portion of a region. Mesoscale organization is also very important for determining the vertical momentum transport by a convective system [e.g., *LeMone et al.*, 1984].

[3] In a companion paper [*Cheng*, 2005] (hereinafter referred to as part 1), a nonhydrostatic, fully compressible three dimensional cloud resolving model have been used to show that the orientation of an MCS is determined by the shear and moist stratification in a reference layer. The definition of the reference layer is based on observations [e.g., *Alexander and Young*, 1992; *LeMone et al.*, 1998; *Johnson and Keenan*, 2001; *Johnson et al.*, 2005]. The reference layer is located below 400 mbar. It must be thicker than 200 mbar and the mean shear in the layer must be larger than 2 m s^{−1} per 100 mbar. If there are two layers satisfying such conditions, the lower layer is chosen as the reference layer. Part 1 also shows that the cold pool and condensation do not affect the overall orientation of the linear MCSs, although they play an important role in determining the detailed structures of the simulated MCSs. These results drastically simplify the problem of the organization of MCSs: the analytical solution can be found based upon a linearized dry system. In this paper, a linear model is proposed and solved and the results are compared with those from part 1 and with observations.

[4] *Kuo* [1963] studied convection in the presence of shear using linearized equations. His approach was purely analytical. He found that a shear-parallel linear MCS tends to develop in a plane Couette flow when the Richardson number is less than zero and greater than −2. The Richardson number is defined as *R*_{i} = *gS*_{z}*U*_{z}^{−2}, where *g* is the constant of gravity, *S*_{z} = θ_{0}^{−1}, *U*_{z} = , θ_{0} is the undisturbed potential temperature, and *U* is the mean wind. *Kuo* [1963] did not rule out the possibility of shear-perpendicular systems when the Richardson number is very large and negative (very unstable or very weak shear or both).

[5] *Asai* [1970a] used a numerical method to investigate the same plane Couette flow. He found that the shear-parallel linear MCSs developed regardless of the Richardson number. *Asai* [1970b, 1972] further found that the shear-perpendicular disturbance can develop with small Richardson number. *Asai* [1970a] also found that the linear MCS moves with the speed of the midlevel mean flow. He concluded that the conversion of kinetic energy from the mean flow to perturbation flow is favored when the linear MCS is parallel to the shear. *Asai* [1970a, 1970b, 1972] used the rigid upper and lower boundary conditions since it was assumed that the Couette flow covers the entire vertical domain.

[6] After nonlinear cloud-resolving models were developed in the late 1970s, some new factors that can influence the orientation of MCSs were identified. *Rotunno et al.* [1988] (hereinafter referred to as RKW) pointed out that the cold surface outflow from an old cell can lift environmental boundary layer air to its level of free convection, and that this effect can be enhanced by the low-level shear. They emphasized that the interactions of the cold pool and low-level shear play a central role in the organization of an MCS. Others [e.g., *LeMone*, 1983; *LeMone et al.*, 1984; *Lafore and Moncrieff*, 1989; *Garner and Thorpe*, 1992] reported that the mesoscale circulations such as mesoscale inflow, perturbation pressure gradient force and downdraft are also important. However, the connections between the numerical simulations and linear theories did not receive much attention.

[7] This series of study attempts to relate the linear theories to the results from numerical simulations. Part 1 presents numerical simulations of shear-parallel and shear-perpendicular MCSs, which reveal the important role of wind shear in the reference layer in determining the orientation of MCSs. The major objective of the present study is twofold: present a new linear theory based upon the WKBJ analysis of the perturbation from the reference level and validate the theoretical prediction of the characteristics of MCSs against observations from several field experiments.

[8] The rest of the paper is organized as follows. Section 2 presents a simple linear theory for the MCSs. Section 3 presents results of tests of the linear theory against both midlatitude and tropical observations. Discussion and comparison with previous linear theories are presented in section 4. Summary and conclusions are given in section 5.