The anelastic system of equations (3-D momentum, continuity and thermodynamic energy) is used to investigate the organization of mesoscale convective systems (MCSs) using the reference layer concept presented in part 1. Latent heating is assumed to be proportional to the vertical velocity. The WKBJ method is used to solve this system of equations by perturbing the solution linearly from that at the reference level, which is the top of the reference layer. The reference layer is defined as a layer with maximum wind shear and unstable moist stratification over a minimum thickness of 200 mbar. The characteristics of the reference layer, such as the magnitude of the shear and moist stratification, determine the type of MCS' organization and associated properties in the analysis. This result agrees well with numerical simulations of linear MCSs presented in part 1 of this series of study. Three types of MCSs are identified from the linear theory: one is nonlinear and two are linear. The group speed of all three types turns out to be proportional to the mean wind at the reference level. Nonlinear MCSs occur when the vertical wind shear is weak and the stratification is unstable. Type 1 linear solutions are neutral, shear-parallel lines; their reference layers are usually in the middle troposphere. Type 2 linear solutions are amplifying, shear-perpendicular lines; their reference layers are usually in the lower troposphere. The theory predicts the widths and growth rates of the type 2 linear MCSs. The width of an amplifying MCS is determined as a simple function of the Richardson number, while the growth rate is a function of vertical wind shear and Richardson number. Tests of the linear theory using data from several field experiments show that it gives fairly realistic results for a variety of the observed MCSs.
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 Deep cumulus convection is frequently organized into mesoscale convective systems (MCSs), including squall lines (about 90% of all MCSs, according to Tsakraklides and Evans ) 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].
 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.
Kuo  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 Ri = gSzUz−2, where g is the constant of gravity, Sz = θ0−1, Uz = , θ0 is the undisturbed potential temperature, and U is the mean wind. Kuo  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).
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.
 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.  (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.
 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.
 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.
2.1. Orientation, Width, and Propagation Speed
 The nonlinear governing equations with the anelastic approximation are
Here u, v, w, θ, and ρ are the three components of velocity, the potential temperature, and the density, respectively. Rotation and dissipation are neglected, since these effects are not critical for determining the organization of mesoscale convective systems. In (5), N12 is defined as
where θe is the equivalent potential temperature of the environment. Equations (1)–(5) can describe anywhere of the convective system, but (1)–(5) are only used to describe the updraft region of the MCS and the effects of downdraft and subsidence on the updraft are treated as external forcing since the “target” of this linear theory is the updrafts. Equation (5) can be derived by assuming that the latent heat released by an MCS is proportional to the vertical velocity and that the updraft air is uniformly saturated.
 The WKBJ method [e.g., Morse and Feshbach, 1953] is used to solve (1)–(5). The first step is to perform nondimensionalization of all variables and parameters, as follows:
Here L and H are used to scale the horizontal and vertical dimensions of the organized mesoscale system, respectively, which are assumed to be of the same order. U is the scale of the environmental and cloud velocities. Θ and Π are the scales for potential temperature and density, respectively. Superscript “*” denotes a nondimensional variable. After substituting equations (7)–(9) into (1)–(5), the same set of equations as (1)–(5) are obtained except that every term is nondimensional.
 The second step is to perform a WKBJ expansion near the reference level, which can be any level in the reference layer. As discussed in part 1, the reference layer is usually located in the lower and middle troposphere. For simplicity, the top of the reference layer is chosen as the reference level. A schematic depiction of the convective system that is considered in this analysis is shown in Figure 1. Strong updrafts are produced by the low-level convergence and high-level divergence. The convergence and divergence are large near the surface and in the upper troposphere, but they become small near the middle troposphere. The updrafts and associated cloudy region can be described by the anelastic system of equations, while mesoscale inflows and downdrafts are treated as forcing.
 The small parameter used for the WKBJ expansion is β = , where h = z − z0, z0 is the height of the reference level, and z − z0 is chosen such that β is on the order of 10−1. A general expansion can be written as:
where q can be u, v, w, θ, p, and ρ. superscript “(0)” represents the zeroth-order approximation, “(1)” represents the first-order approximation, “(2)” represents the second-order approximation, and so on. Substituting (10) into the nondimensional equations, the zeroth-order system is the same set of equations as (1)–(5) except that every term has a “(0)” superscript, which means these terms are zeroth-order approximations near the reference level.
 Although the zeroth-order system is similar to (1)–(5), the physical meaning is totally different. The zeroth-order equations are valid near the reference level. If the solution obtained by the WKBJ expansion converges to the solution of the system, the solution from the reference level approximates the solution for the whole system. This is the basic assumption of the WKBJ method and is also a major assumption for this analytical study.
 The zeroth-order equations near the reference level are still nonlinear. They can be linearized by assuming that
where u0 and v0 are the components of horizontal velocity at the reference level, and are the components of the vertical shear of the mean wind, evaluated at the reference level. It is also assumed that the scale height of density, , is independent of the height.
 The following equations can then be derived by using (11):
where , and are the mean velocity, potential temperature and the mean density, respectively. The orientations of the x and y axes are in a sense arbitrary in these equations; horizontal “directions” have a meaning in this problem only with respect to the two externally specified vectors, namely the mean wind and the shear, both at the reference level. For convenience, the x direction is defined to be the direction of the reference level shear vector. This means that
which is assumed from this point on.
 We seek solutions of the form
With this form, a positive imaginary part of ω corresponds to a growing solution. Note that the convective system is assumed to be within an envelop of updraft. This form of solution also has some implications about the upper and lower boundary conditions. If a rigid wall is assumed at the upper and lower boundaries as by Asai [1970a, 1970b, 1972], w′ must vanish at z = 0 and z = Hu, where Hu is the height of the tropopause. One must have (cos 0 + i sin 0) = 0 and (cos mHu + i sin mHu) = 0, where mHu = π, but cannot be zero, cos 0 = 1, and cos π = −1, the rigid wall boundary condition thus cannot apply. For other types of boundary conditions, (18) can be used. Further discussions of the boundary condition will be presented in section 4.
 Substituting (18) into (12)–(16), the following equation can be obtained
Because ≈ 6.28 × 10−8 m−2 and k2 + l2 + m2 ≈ 1.18 × 10−6 m−2, the first term on the left hand side of (19) can be neglected. Thus (19) can then be simplified to
 Setting ω ≡ ωr + ωii, and separating the real and imaginary parts of (20), one obtains the following expressions for ωr and ωi,
According to (21), temporal oscillations at a fixed position are entirely due to the advection of the disturbance by the mean flow at the reference level. The plus sign is chosen before the discriminant in (22) because the more unstable mode is more interesting. If there are two levels in the troposphere, one has a positive N12, and the other has a negative N12. According to (22), the one with negative N12 grows faster. The organization of the MCS is expected to be determined by the level that has the largest growth rate. So a necessary condition for the reference level is
In view of (23), the quantity under the square root in (22) is always positive and at least as large as .
 Next, the components of the phase and group velocities are obtained from (21), as
The group speeds are simply equal to the mean wind at the reference level, while the phase speed also has a linear relationship with the mean wind at the reference level. Equations (24) and (25) are applied to the three types of solution to be discussed below. With the assumption that the shear and N12 are constant in the reference layer, the location of the reference level determines the group and phase speeds of the MCSs.
2.2. Nonlinear MCSs
 If there is no shear or very weak shear, (22) becomes
According to (26), the growth rate does not favor any horizontal wave number, i.e., the MCS has the same growth rate regardless of its horizontal orientation. So it is a nonlinear MCS. The clouds in this type of MCS are either organized in a circular shape or are randomly distributed.
 If the shear is very large, one cannot neglect it in (22). The growth rate is favored for alignment in one horizontal direction. This is a linear MCS. So the second necessary condition for the reference level is that it must have large vertical wind shear, because the growth rate with the shear is larger than that without the shear according to (22).
 So far two necessary conditions have been identified for the reference level: N12 ≤ 0 and large shear. Because the reference level is a part of reference layer, these two conditions apply to the reference layer. So it is not surprising that the two characteristics of the reference layer that are revealed by nonlinear cloud resolving model simulations presented in part 1 are consistent with the linear theory.
2.3. Type 1 Linear MCSs: ωi ≈ 0
 An MCS that is neither amplifying nor decaying is characterized by ωi ≈ 0. From (22) such a case can be expressed as
According to (29), the MCS is independent of x, i.e., the MCS is parallel to the shear. It is hypothesized that the shear-parallel linear MCSs observed in nature are those for which ωi ≈ 0, and (28) and (29) are satisfied. These shear-parallel systems are referred to “Type 1” MCSs. For Type 1, the dispersion relationship of the convective wave is (21). So the convective wave propagates with the basic divergence and convergence mechanism as shown in part 1 of this series of study since the growth rate of Type I is very small. During the first phase of the control run presented in part 1, the growth rate of the linear MCS is small and there is no cold pool. The new convective clouds are produced by the convergence induced by the interaction between the updrafts and the shear.
2.4. Type 2 Linear MCSs: ωi > 0
 When ωi > 0, the MCS is amplifying; these are “Type 2” MCSs. The disturbance energy can arise from either the potential energy or the kinetic energy of the mean flow, or both, but one expects
According to (22), the growth rate will depend on k, l, and m for a given set of shear and stratification values.
 It is assumed that an observed MCS corresponds to the most rapidly growing mode. We can consider m, , and N12 to be externally set. For given values of these external parameters, the growth rate is maximized for
This means that the most rapidly growing MCS is organized as a squall line perpendicular to the shear vector. With the use of (36), (34) is reduced to
After some algebra the following relation can be obtained,
Since Ri ≤ 0, equation (38) always gives positive values of k2, and it implies that the front-to-back horizontal scale of a type 2 MCS is never greater than its depth. This means that the updraft region of a linear MCS is very narrow.
 Substituting (36), (38) and (39) back into (22), the growth rate is given by a very simple expression:
As Ri becomes more negative, the growth rate of a type 2 MCS increases. The growth rate is positive even Ri = 0. This means that the disturbance can gain energy from the shear, without any contribution from buoyancy. Note that the type 2 solution does not reduce to the Type 1 solution when Ri = 0. The modes are physically distinct.
Emanuel  investigated the conditional symmetric instability (CSI) of convective systems. A convective system can also gain energy from the mean flow through CSI. However, the energy conversion of CSI depends on the Coriolis parameter or absolute vorticity (f) when the air parcel is displaced along a surface of constant angular momentum. Because f = 0 in this study, obviously CSI is absent in the type 2 solution.
 As shown in part 1, the shear-perpendicular linear MCSs can exist without the presence of a cold pool. The type 2 MCSs from the linear theory have similar physical characteristics (e.g., large instability and growth rate) as the shear perpendicular MCSs simulated with the nonlinear cloud resolving model presented in part 1. From now on, the linear theory is further tested against observations.
3. Observational Tests
 In this section, the linear theory is tested against observations from GATE (the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment), EMEX (Equatorial Mesoscale Experiment), ARM (the Atmospheric Radiation Measurements Program), TOGA COARE (Tropical Ocean–Global Atmospheres/Coupled Ocean Atmosphere Response Experiment), and SCSMEX (the South China Sea Monsoon Experiment). The shape, phase speed, and aspect ratio will be tested for GATE data. For all of the cases discussed here, N12 = −5 × 10−5 s−2 is used as a “critical” value to distinguish between Type 1 and Type 2 MCSs. Further tests are needed to determine whether this value can be applied more generally. More test cases are given by Cheng . Table 1 summarizes the observed shear, Ri, and N12 for all cases discussed in this section.
Table 1. Shear, Ri, and N12 for All Cases Discussed in Section 3
GATE slow movers
1.9 × 10−3
−1.2 × 10−5
GATE fast movers
4.0 × 10−3
−9.3 × 10−5
SCSMEX special case
2.0 × 10−3
−4.2 × 10−5
3.1 × 10−3
−1.2 × 10−4
TOGA COARE case 1
3.7 × 10−3
−1.47 × 10−4
 The GATE data are from Barnes and Sieckman . For their fast movers, the largest shear layer is between 850 mbar and 650 mbar, which is chosen as the reference layer (Figure 1 in part 1). N12 = −9.3 × 10−5 s−2 in this layer. The 650 mbar is chosen as the reference level. According to the criterion stated above for N12, the fast movers should be of type 2 MCSs. The shear was observed to be perpendicular to the line for the fast movers, so this is consistent with the linear theory. The theory predicts that the group speed of the fast movers is about 12 m s−1. This compares well with the observed propagation speed of 11.1 m s−1. For the slow movers the shear between 850 mbar and 650 mbar is −1.9 × 10−3 s−1, N12 = −1.2 × 10−5 s−2, and the reference level is at 650 mbar. So according to the critical N12 value, the slow movers are of Type 1 MCS. The group speed of the slow movers is estimated to be 2 m s−1. The observed propagation speed was about 2.2 m s−1.
3.2. An Unusual SCSMEX Case
 In the tropics, the low-level ∣N12∣ tends to be large, and the middle-level ∣N12∣ tends to be small (see, for example, the equivalent potential temperature soundings given by Alexander and Young  and LeMone et al. ). The linear theory therefore predicts that, in the tropics, Type 2 MCSs will develop when the low-level wind shear is strong, and that Type 1 MCSs, i.e., shear-parallel MCSs, will develop when the shear is concentrated at the middle levels (see Figure 2). The equivalent potential temperature decreases quickly below 800 mbar (very unstable) and remains near neutral between 800 mbar and 500 mbar. This is consistent with the tropical observations of Alexander and Young , LeMone et al. , Johnson and Keenan , and Johnson et al. .
 An unusual case was noticed by Johnson and Keenan  and Johnson et al. , however, in an analysis of SCSMEX data for 23 May 1998. In this case the low-level shear was parallel to the orientation of the linear MCS, which is considered to be special according to Alexander and Young  and LeMone et al. . Figure 3a shows the wind profile for the unusual case. The largest wind shear layer is between 700 mbar and 900 mbar. This layer is chosen as the reference layer. The 700 mbar is chosen as the reference level. Figure 3b shows the profile of the equivalent potential temperature for the case. The key point is that, in contrast to most other tropical soundings, there is a weakly unstable layer in the low troposphere and the air is dry aloft [Johnson et al., 2005]. For this “special” case, N12 = −4.2 × 10−5 s−2 which is less unstable than the critical N12 value. The special case therefore belongs to type 1. From the radar images (not shown), the growth rate for the special shear-parallel case was very small. This is to be expected for a “neutral” type 1 MCS.
3.3. ARM and TOGA COARE
 Discussed next are one case observed at the ARM Program's Southern Great Plains site, during June 1997, and two more that were observed in TOGA COARE from November 1992 to February 1993. The ARM case and the TOGA COARE case 1 are typical Type 2 linear MCSs, and the TOGA COARE case 2 is a nonlinear MCS.
 The ARM case was observed on 26 June 1997. Figure 4 shows the orientation of the squall line as seen by radar. The size of the squall line is estimated to be 20 km × 280 km. The largest shear layer is between 650 mbar and 850 mbar. This layer is thus chosen as the reference layer based on the data shown in Figure 5. The level at 650 mbar is chosen as the reference level. This is a Type 2 MCS (Table 1). Again, Table 2 shows that the linear theory produces a good agreement between the observed and predicted orientation and speed. Table 3 shows the predicted values of the wave number k, the corresponding wavelength, and the growth rate ωi, for this and the other cases discussed in this subsection.
Table 2. Speed and Orientation for ARM and TOGA COARE Cases
Speed is given as line normal direction, line parallel direction.
TOGA COARE case 1
TOGA COARE case 2
Table 3. Wave Number k, Wavelength, and Growth Rate ωi for All Type 2 Cases
4.48 × 10−3
1.06 × 10−2
GATE faster movers
3.09 × 10−3
9.07 × 10−3
TOGA COARE case 1
2.09 × 10−3
1.22 × 10−2
 The true group velocity of the mesoscale system was estimated from radar data. The radar images were examined every 10 min for an hour, and tracked the motion of the strong echo area. Table 2 shows the velocity and orientation for ARM case and TOGA COARE cases discussed in this section. In this case (and all the other cases), the predicted orientation angle and speed are in good agreement with the observations. The uncertainty of the orientation angle prediction is about 15°.
 Two cases from TOGA COARE IOPs (Intensive Observation Periods) are also used to test the linear theory. The TOGA COARE case 1 occurred on 9 February 1993. It was a typical linear MCS case, which lasted from 9:30 AM to 12:15 PM. The total rainfall was dominated by convective precipitation [Petersen et al., 1999]. Figure 6 shows the organized convective system as observed by the Colorado State University radar, which was located at 2°S, 156°E. There are two squall lines in the observed radar image. They have the same orientation. The reference level was chosen using the data plotted in Figure 7. The layer with maximum shear is between 775 mbar and 975 mbar, and is chosen as the reference layer. The 775 mbar level is the reference level. The instability is large (Table 1) in the reference layer, so this linear MCS is of Type 2, a shear-perpendicular MCS. Compared to the estimation from the radar data, the predicted line's speed and orientation are in good agreement (Table 2).
 TOGA COARE case 2 occurred on 5 December 1992. Figure 8 shows the organized clouds as observed by radar. The mesoscale system consists of individual convective cells, which are randomly arranged and do not appear to have any linear organization. The size of the nonlinear MCS is estimated to be 150 km × 150 km. There is no layer with shear larger than 2 m s−1 per 100 mbar and thicker than 200 mbar below 400 mbar (Figure 9). Therefore no reference layers exist in this sounding. Although the shear between 325 mbar and 525 mbar has a magnitude of 4.1 m s−1, the stratification is stable. It seems that the high-level shear does not have much effect on the organization of the MCS for this case. So, nonlinear MCS can be predicted from the linear theory.
4. Comparison With Previous Work
 On the basis of the reference layer concept inferred from the simulations of shear-parallel and shear-perpendicular MCSs by a nonlinear 3-D cloud-resolving model, the new linear theory can be applied to almost any type of vertical profiles of winds and thermodynamic soundings from observations [e.g., Alexander and Young, 1992; LeMone et al., 1998; Johnson and Keenan, 2001; Johnson et al., 2005]. The theoretical predictions of type of MCS organization are consistent with the nonlinear simulations presented in part 1 and agree remarkably well with tropical and midlatitude observations. Despite of the apparent success of the new theory, it should be compared with the earlier studies so that the new theory can be accepted in a proper context. Limitations of the earlier studies and the present study can be better understood from such a comparison.
 As mentioned earlier, Kuo  and Asai [1970a, 1970b] studied a linear system very similar to that described by (12)–(16). They assumed a constant shear of horizontal wind and a constant gradient of potential temperature throughout the troposphere. Rigorously speaking, their results can only be applied to atmosphere satisfying such conditions, which seldom occurs in nature, while the present study can be generally applied to any situation due to its adoption of the reference layer concept. Asai  studied a thermal instability of a shear flow that changes direction with height. The type 2 MCS of the present study can be compared well to Figures 5 and 6 of Asai . Kuo  and Asai [1970a, 1970b] did not obtain the shear-perpendicular linear MCSs in their pioneering analyses even though this type of linear MCSs is commonly observed and simulated. From the modeling results presented in part 1, a shear-parallel linear MCS exists when the reference layer is weakly unstable and the growth rate is small during the first phase of its life cycle. Asai's [1970a, 1970b] theory, however, predicts the largest growth rate for the shear-parallel MCSs. The explanation for such a contrast between the present and Asai's analyses may be twofold. First, the fundamental concepts and approaches are different, in particular, the WKBJ method is used to analyze the perturbation from the reference level in the present analysis. Second, the different specifications of the boundary conditions contribute to the different solutions of the linear systems, which is further explained below in details.
 As discussed in section 2, the WKBJ method does not explicitly specify the boundary condition of the reference layer. The form of the solution (18) assumed for the reference level does not satisfy the rigid-wall boundary condition used by Asai [1970a, 1970b]. To better understand the differences resulting from the different specifications of boundary conditions, the numerical method of Asai [1970a, 1970b] is used to solve for the growth rate of the solution for the reference level using a physically based specification of boundary conditions.
 Upper and lower boundary conditions that may be used to express the interaction among the updraft, downdraft and mesoscale inflow of the reference layer can be derived according to Figure 1. If the effects of either downdraft or mesoscale inflow are considered, mass is infused into the reference layer from the nonconvergence level. This effect can be expressed by
where cu is a constant with the unit of m−1. Equation (41) represents a mixed type of boundary conditions. This upper boundary condition is obtained by assuming that the effects of either downdraft or mesoscale inflow are proportional to the strength of the updraft. With (41), the specific effects of the downdraft and mesoscale inflow, however, cannot be totally determined. A lower boundary condition of the reference layer is needed to totally determine these effects, which can be written as
where cd is a constant with the unit of m−1. Equation (42) can be derived by assuming that the low-level convergence is proportional to the strength of the updraft. As discussed in part 1 and section 2 of this study, the reference layer determines the organization of MCSs. The MCSs, in turn, influence the reference layer through external factors such as downdraft and mesoscale inflow, which are now implicitly expressed by (41) and (42) for the linear theory. The magnitude of these influences are controlled by parameters cu and cd.
Equations (41) and (42) are now used as the upper and lower boundary conditions to numerically solve for the growth rate of the linear system presented in section 2, using Asai's [1970a, 1970b] numerical method. Three numbers are specified: Richardson number Ri = −10, Prandtl number Pr = 1, and Reynolds number Re = 100. The results are shown in Figure 10. The growth rate obtained from the new boundary condition is very large, 350 (nondimensional; Figure 10a), compared to that obtained from the rigid-wall boundary condition, 10 (nondimensional; Figure 10b). The largest growth rate occurs at nearly zero wave number in y direction with the new boundary condition, whereas it occurs at nearly zero wave number in x direction with the rigid-wall boundary condition. This means that the shear-perpendicular linear MCSs are more likely obtained with the new boundary condition. The shear-parallel linear MCSs are more likely obtained with the rigid-wall boundary condition, in agreement with Asai's results. These numerical results also suggest that the WKBJ linear solution for type 2 MCS is consistent with the numerical results with explicitly imposed external factors.
5. Summary and Discussions
 The anelastic system of equations (3-D momentum, continuity, and thermodynamic energy) for the problem of MCSs have been used to investigate the organization of MCSs, based upon the reference layer concept presented in part 1. Latent heating is assumed to be proportional to the vertical velocity. The WKBJ method is used to solve this system of equations. An MCS is assumed to have a consistent organization through out the whole troposphere and the solution of the system is determined by a reference layer. The reference layer is derived from the definitions of low-level shear and the middle-level shear from Alexander and Young , LeMone et al. , Johnson and Keenan , and Johnson et al. . The reference level is chosen to be the top level of the reference layer. The reference layer is defined as a layer with maximum wind shear at least 2 m s−1 per 100 mbar and unstable moist stratification over a minimum thickness of 200 mbar. The reference layer is supposed to drive the MCSs.
 Three types of MCSs have been identified: a nonlinear MCS and two linear MCSs. The phase and group speed of all disturbances turn out to be proportional to the mean wind at the reference level. A nonlinear MCS occurs when the vertical wind shear is weak and N12 ≤ 0. Once the area of an MCS is known, the wavelength of the MCS in both the x and y directions can easily be calculated. Type 1 solutions are neutral (ω real, N12 ≈ 0) shear-parallel lines; the appropriate reference layer for these cases is usually in the middle troposphere. Type 2 solutions are amplifying (ω complex, N12 < 0), and shear-perpendicular lines; the reference layer is usually in the lower troposphere. The theory predicts the widths and growth rates of the Type 2 disturbances. The width of an amplifying squall line is determined as a simple function of the Richardson number, while the growth rate is a function of the vertical wind shear and Richardson number. If an elliptical shape is assumed for type 2 MCS, the length of such an MCS is known once the area of the MCS is obtained.
 The theory has been tested using observations from GATE, ARM, EMEX, SCSMEX, and TOGA COARE. The theory gives encouraging predictions of the orientation and speed of organized convection that are similar to those observed. This information is potentially useful for the parameterization of convective momentum transport, including the effects of the perturbation pressure gradient force [e.g., Wu and Yanai, 1994; Cheng, 2001].
 Compared with Kuo's  and Asai's [1970a, 1970b, 1972] work, the new linear theory can be applied to more general cases since constant vertical wind shear and N12 are only required in a finite depth, which is almost always true in nature. With the mixed type upper and lower boundary condition, the system is solved using Asai's [1970a] method and a shear-perpendicular mode is identified, which has a growth rate 35 times larger than that of the shear-parallel mode that is resulted from using the rigid-wall upper and lower boundary conditions. This mode exists with various Richardson numbers. Furthermore, the new theory provides the size information of MCSs, which is useful for the parameterization of the effects of the MCSs.
 The framework of the new linear model is different from that of Moncrieff and Green , Moncrieff and Miller , and Moncrieff . Moncrieff and colleagues emphasized that an MCS consists of a rising current, a second current descending from the rear of the system, and an anvil outflow at the front of the system. The new linear model does not explicitly address these currents. However, as pointed out in part 1, Moncrieff and Green's  steering-level model and Moncrieff and Miller's  propagating model exists when a nonlinear numerical model is used to simulate the line-parallel and line-perpendicular MCSs. So the linear model does bridge the simple solution with those complicated cloud-resolving model simulations and provides a potential abilities for a more realistic parameterization of convective systems.
 This study has been supported by National Science Foundation under grant ATM-9812384 and by the U.S. Department of Energy's ARM Program under grant DE-FG03-95ER61968. Data were kindly provided by Mike Splitt of the University of Utah, Minghua Zhang of State University of New York at Stony Brook, and Richard Cederwall of Lawrence Livermore National Laboratory. I thank Paul Hein, Walt Petersen, and Steven Rutledge for providing the CSU-MIT radar images for TOGA COARE. The author also benefited from discussions with D. Randall, R. Johnson and W. Schubert of Colorado State University, and Kuan-Man Xu of NASA Langley Research Center.