Organization of mesoscale convective systems: 2. Linear theory

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 11,, doi:1.129/24jd545, 25 Organization of mesoscale convective systems: 2. Linear theory Anning Cheng Atmospheric Sciences, NASA Langley Research Center, Hampton, Virginia, USA Center for Atmospheric Sciences, Hampton University, Hampton, Virginia, USA Received 12 September 24; revised 1 February 25; accepted 18 February 25; published 4 June 25. [1] 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 2 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. Citation: Cheng, A. (25), Organization of mesoscale convective systems: 2. Linear theory, J. Geophys. Res., 11,, doi:1.129/24jd Introduction [2] Deep cumulus convection is frequently organized into mesoscale convective systems (MCSs), including squall lines (about 9% of all MCSs, according to Tsakraklides and Evans [22]) and roughly circular cloud masses (about 1% 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]. Copyright 25 by the American Geophysical Union /5/24JD545 [3] In a companion paper [Cheng, 25] (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, 21; Johnson et al., 25]. The reference layer is located below 4 mbar. It must be thicker than 2 mbar and the mean shear in the layer must be larger than 2 m s 1 per 1 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. 1of12

2 Figure 1. Schematic depiction of the reference layer, the updraft, and their interaction with downdraft and mesoscale inflow. [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 2 z, where g is the constant of 1@q gravity, S z = q, U z q 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 [197a] used a numerical method to investigate the same plane Couette flow. He found that the shearparallel linear MCSs developed regardless of the Richardson number. Asai [197b, 1972] further found that the shear-perpendicular disturbance can develop with small Richardson number. Asai [197a] 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 [197a, 197b, 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 197s, 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 lowlevel 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 shearperpendicular 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 Theory 2.1. Orientation, Width, and Propagation Speed [9] The nonlinear governing equations with the anelastic approximation are Table 1. Shear, R i, and N 1 2 for All Cases Discussed in Section 3 Case Name Shear, s 1 R i N 1 2,s 2 GATE slow movers GATE fast movers SCSMEX special case ARM case TOGA COARE case of12

3 nondimensionalization of all variables and parameters, as follows: ðx; yþ ¼ Lx*; ð y* Þ; z ¼ Hz*; ðu; v; wþ ¼ Uu*; ð v*; w* Þ; ð7þ q ¼ Qq*; r ¼ r*; ð8þ p r ¼ U 2 p* r* ; t ¼ L U t*; g ¼ U 2 L g*; N 1 2 ¼ gdq HQ N*2 1 ; ð9þ Figure 2. Schematic depiction of equivalent potential temperature sounding for type 1 and type þ þ þ w þ ¼ þ þ ¼ þ g ¼ ; r þ þ r ¼ ; ðln qþ þ ðln qþ þ ðln qþ þ wn @y g ¼ : ð5þ Here u, v, w, q, and r 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), N 1 2 is defined as N 2 1 ¼ g q e ; ð6þ where q e 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. [1] The WKBJ method [e.g., Morse and Feshbach, 1953] is used to solve (1) (5). The first step is to perform Figure 3. (a) Horizontal wind profile (solid line for u and dashed line for v) and (b) equivalent potential temperature soundings for Johnson and Keenan s [21] and Johnson et al. s [25] special case (line parallel to low-level shear) on 12 UTC, 23 May The shaded layer is the reference layer. 3of12

4 Figure 4. Radar image and observed shear vector of the reference layer for the ARM case. 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. Q and 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. [11] 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. [12] The small parameter used for the WKBJ expansion is b = h L, where h = z z, z is the height of the reference level, and z z is chosen such that b is on the order of 1 1. A general expansion can be written as: q* ¼ ðq* Þ ðþ þ bðq* Þ ðþ 1 þ b 2 ðq* Þ ðþ 2 þ...; ð1þ where q can be u, v, w, q, p, and r. superscript () represents the zeroth-order approximation, (1) represents the first-order approximation, (2) represents the secondorder approximation, and so on. Substituting (1) into the nondimensional equations, the zeroth-order system is the same set of equations as (1) (5) except that every term has a () superscript, which means these terms are zerothorder approximations near the reference level. [13] Although the zeroth-order system is similar to (1) (5), the physical meaning is totally different. The zerothorder 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. [14] The zeroth-order equations near the reference level are still nonlinear. They can be linearized by assuming that u ¼ u þ ðz z Þ and v ¼ v ðz z Þ; ð11þ where u and v are the components of horizontal velocity 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 1 density, H r ¼ r, is independent of the height. [15] The following equations can then be derived by using þ þ þ p r p r ¼ ; p q r q g ¼ þ þ w H r ¼ ; ð12þ ð13þ ð14þ ð15þ 4of12

5 2 6 4 [16] We seek solutions of the form u v w p q ^u ^v ¼ ^w exp½ðikxþ ð ly þ mz wtþš; kx; ly p; ^p 5 ^q mz p 2 : ð18þ With this form, a positive imaginary part of w 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 [197a, 197b, 1972], w must vanish at z = and z = H u, where H u is the height of the tropopause. One must have ^w(cos + i sin ) = and ^w(cos mh u + i sin mh u ) =, where mh u = p, but ^w cannot be zero, cos = 1, and cos p = 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. [17] Substituting (18) into (12) (16), the following equation can be obtained m ðku þ lv wþ 2 mk ðku þ lv wþ H r h þ i k 2 þ l 2 þ m 2 ð ku þ lv wþ 2 N1 2 k2 þ l 2 i ¼ : ð19þ Because m m 2 and k 2 + l 2 + m 2 H r m 2, the first term on the left hand side of (19) can be neglected. Thus (19) can then be simplified to k 2 þ l 2 þ m 2 iku ð þ lv wþ 2 mk ðku þ lv wþ in1 2 k2 þ l 2 ffi : ð2þ q q The same as Figure 3, except for the ARM case. q q q q þ N 2 1 g w ¼ ; ð16þ where v, q and r 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 ¼ ; ð17þ which is assumed from this point on. [18] Setting w w r + w i i, and separating the real and imaginary parts of (2), one obtains the following expressions for w r and w i, w i ¼ mk þ w r ¼ ku þ lv ; ð21þ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mk 2 4N1 2 ð k2 þ l 2 Þðk 2 þ l 2 þ m 2 Þ 2ðk 2 þ l 2 þ m 2 : Þ ð22þ 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 N 1 2, and the other has a negative N 1 2. According to (22), the one with negative N 1 2 5of12

6 Table 2. Speed and Orientation for ARM and TOGA COARE Cases Speed, a ms 1 Orientation, deg Observed Theory Observed Theory ARM case ( 7., 3.) ( 7.1, 3.1) TOGA COARE case 1 ( 6.5, 4.) ( 6., 4.6) TOGA COARE case 2 (.7,.5) N/A N/A N/A a Speed is given as line normal direction, line parallel direction. 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 N 2 1 : ð23þ In view of (23), the quantity under the square h root in (22) is i 2. always positive and at least as large as mk [19] Next, the components of the phase and group velocities are obtained from (21), as c px w r k ¼ u l þ v k ; c py ¼ w r k ¼ u l l þ v ; c ¼ u ; c gy r ¼ v ð24þ ð25þ 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 N 2 1 are constant in the reference layer, the location of the reference level determines the group and phase speeds of the MCSs Nonlinear MCSs [2] If there is no shear or very weak shear, (22) becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N1 2 ð w i ¼ k2 þ l 2 Þðk 2 þ l 2 þ m 2 Þ k 2 þ l 2 þ m 2 : ð26þ 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. [21] 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). [22] So far two necessary conditions have been identified for the reference level: N 1 2 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 Type 1 Linear MCSs: W i [23] An MCS that is neither amplifying nor decaying is characterized by w i. From (22) such a case can be expressed as mk þ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mk 2 4N1 2 ð k2 þ l 2 Þðk 2 þ l 2 þ m 2 Þ : ð27þ Equation (27) can be satisfied only when N 2 1 : In that case, (27) is reduced to k : ð28þ ð29þ 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 w i, 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 Type 2 Linear MCSs: W i > [24] When w i >, 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 N 2 1 < : ð3þ According to (22), the growth rate will depend on k, l, and m for a given set of shear and stratification values. Table 3. Wave Number k, Wavelength, and Growth Rate w i for All Type 2 Cases Case Name k, m 1 Wavelength, km w i,s 1 ARM case GATE faster movers TOGA COARE case of12

7 Figure 6. Radar image and observed shear vector of the reference layer for TOGA COARE case 1. [25] It is assumed that an observed MCS corresponds to the most rapidly growing mode. We can consider m,, and N 2 1 to be externally set. For given values of these external parameters, the growth rate is maximized ¼ ; ð31þ 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 k 2 þ m 2 ¼ 4N 2 1 mb2 : ( m þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 m 2 = 4N1 2 b2 ; ¼ : Substituting (22) into (31) and (32), and defining ð32þ After some algebra the following relation can be obtained, where k 2 m 2 ¼ 1 4R i; ð38þ b 2 k 2 þ l 2 þ m 2 ; the following is obtained after some algebra. ð33þ R i N : ð39þ ( k 2 þ l 2 þ m 2 mk sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 þ mk 2 = 4N1 2 b2 ðk 2 þ l 2 Þ ; ¼ 4kN 1 2 mb2 ; ð34þ ( l 4N1 2 mb2 þ 2k 2 mk sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi39 mk 2 = 4 4N1 2 b2 ðk 2 þ l 2 Þ5 ; ¼ : þ To satisfy (35), one must take l ¼ : ð35þ ð36þ Since R i, equation (38) always gives positive values of k 2, 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. [26] Substituting (36), (38) and (39) back into (22), the growth rate is given by a very simple expression: w i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4R i : ð4þ 2 As R i becomes more negative, the growth rate of a type 2 MCS increases. The growth rate is positive even R i =. 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 R i =. The modes are physically distinct. [27] Emanuel [1983] 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 7of12

8 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, N 1 2 = 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 [21]. Table 1 summarizes the observed shear, R i, and N 1 2 for all cases discussed in this section. Figure 7. The same as Figure 3, except for TOGA COARE case 1. Coriolis parameter or absolute vorticity (f) when the air parcel is displaced along a surface of constant angular momentum. Because f = in this study, obviously CSI is absent in the type 2 solution. [28] 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 [29] In this section, the linear theory is tested against observations from GATE (the Global Atmospheric Research 3.1. GATE [3] The GATE data are from Barnes and Sieckman [1984]. For their fast movers, the largest shear layer is between 85 mbar and 65 mbar, which is chosen as the reference layer (Figure 1 in part 1). N 2 1 = s 2 in this layer. The 65 mbar is chosen as the reference level. According to the criterion stated above for N 2 1, 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 85 mbar and 65 mbar is s 1, N 2 1 = s 2, and the reference level is at 65 mbar. So according to the critical N 2 1 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 An Unusual SCSMEX Case [31] In the tropics, the low-level jn 2 1 j tends to be large, and the middle-level jn 2 1 j tends to be small (see, for example, the equivalent potential temperature soundings given by Alexander and Young [1992] and LeMone et al. [1998]). 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 8 mbar (very unstable) and remains near neutral between 8 mbar and 5 mbar. This is consistent with the tropical observations of Alexander and Young [1992], LeMone et al. [1998], Johnson and Keenan [21], and Johnson et al. [25]. [32] An unusual case was noticed by Johnson and Keenan [21] and Johnson et al. [25], however, in an analysis of SCSMEX data for 23 May 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 [1992] and LeMone et al. [1998]. Figure 3a shows the wind profile for the unusual case. The largest wind shear layer is between 7 mbar and 9 mbar. This layer is chosen as the reference layer. The 7 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., 25]. For this special 8of12

9 Figure 8. Radar image and observed shear vector of the reference layer for TOGA COARE case 2. case, N 2 1 = s 2 which is less unstable than the critical N 2 1 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 ARM and TOGA COARE [33] 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 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. [34] The ARM case was observed on 26 June Figure 4 shows the orientation of the squall line as seen by radar. The size of the squall line is estimated to be 2 km 28 km. The largest shear layer is between 65 mbar and 85 mbar. This layer is thus chosen as the reference layer based on the data shown in Figure 5. The level at 65 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 w i, for this and the other cases discussed in this subsection. [35] The true group velocity of the mesoscale system was estimated from radar data. The radar images were examined every 1 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. [36] 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 It was a typical linear MCS case, which lasted from 9:3 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). [37] TOGA COARE case 2 occurred on 5 December 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 15 km 15 km. There is no layer with shear larger than 2 m s 1 per 1 mbar and thicker than 2 mbar below 4 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 highlevel 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 [38] 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, 21; Johnson et al., 25]. 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. [39] As mentioned earlier, Kuo [1963] and Asai [197a, 197b] 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 9of12

10 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. [4] 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 [197a, 197b]. To better understand the differences resulting from the different specifications of boundary conditions, the numerical method of Asai [197a, 197b] is used to solve for the growth rate of the solution for the reference level using a physically based specification of boundary conditions. [41] 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 w ¼ c u w ; ð41þ Figure 9. The same as Figure 3, except for TOGA COARE case 2. 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 [1972] 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 [1972]. Kuo [1963] and Asai [197a, 197b] did not obtain the shearperpendicular 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 [197a, 197b] theory, however, predicts the largest growth rate for the shearparallel MCSs. The explanation for such a contrast between the present and Asai s analyses may be twofold. First, the where c u 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 w ¼ c d w ; ð42þ where c d 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 c u and c d. [42] 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 [197a, 197b] numerical method. Three numbers are specified: Richardson number Ri = 1, Prandtl number Pr = 1, and Reynolds number Re = 1. The results are shown in Figure 1. The growth rate obtained from the new boundary condition is very large, 35 (nondimensional; Figure 1a), compared to that obtained from the rigid-wall boundary condition, 1 (nondimensional; Figure 1b). 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 1 of 12

11 Figure 1. Dimensionless growth rate as a function of wave number in x and y directions for (a) mixed-type upper and lower boundary condition and (b) rigid-wall upper and lower boundary condition. 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 [43] 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 [1992], LeMone et al. [1998], Johnson and Keenan [21], and Johnson et al. [25]. 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 1 mbar and unstable moist stratification over a minimum thickness of 2 mbar. The reference layer is supposed to drive the MCSs. [44] 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 N 1 2. 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 (w real, N 1 2 ) shear-parallel lines; the appropriate reference layer for these cases is usually in the middle troposphere. Type 2 solutions are amplifying (w complex, N 1 2 < ), 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. [45] 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, 21]. [46] Compared with Kuo s [1963] and Asai s [197a, 197b, 1972] work, the new linear theory can be applied to more general cases since constant vertical wind shear and N 1 2 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 [197a] 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. [47] The framework of the new linear model is different from that of Moncrieff and Green [1972], Moncrieff and Miller [1976], and Moncrieff [1992]. 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 11 of 12

12 new linear model does not explicitly address these currents. However, as pointed out in part 1, Moncrieff and Green s [1972] steering-level model and Moncrieff and Miller s [1976] propagating model exists when a nonlinear numerical model is used to simulate the line-parallel and lineperpendicular 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. [48] Acknowledgments. This study has been supported by National Science Foundation under grant ATM and by the U.S. Department of Energy s ARM Program under grant DE-FG3-95ER 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. References Alexander, G. D., and G. S. Young (1992), The relationship between EMEX mesoscale precipitation feature properties and their environmental characteristics, Mon. Weather Rev., 12, Asai, T. (197a), Three-dimensional features of thermal convection in a plane Couette flow, J. Meteorol. Soc. Jpn., 48, Asai, T. (197b), Stability of a plane parallel flow with variable vertical shear and unstable stratification, J. Meteorol. Soc. Jpn., 48, Asai, T. (1972), Thermal instability of a shear flow turning the direction with height, J. Meteorol. Soc. Jpn., 5, Barnes, G. M., and K. Sieckman (1984), The environment of fast- and slow-moving tropical mesoscale convective cloud lines, Mon. Weather Rev., 112, Cheng, A. (21), A theory of the mesoscale organization of moist convection and the associated vertical momentum transport, Ph.D. dissertation, 13 pp., Dep. of Atmos. Sci., Colo. State Univ., Fort Collins. Cheng, A. (25), Organization of mesoscale convective systems: 1. Numerical experiments, J. Geophys. Res., D15S11 doi:1.129/ 24JD5444. Emanuel, K. A. (1983), On assessing local conditional symmetric instability from atmospheric soundings, Mon. Weather Rev., 111, Garner, S. T., and A. J. Thorpe (1992), The development of organized convection in a simplified squall-line model, Q. J. R. Meteorol. Soc., 118, Johnson, R. H., and T. D. Keenan (21), Organization of oceanic convection during the onset of the 1998 east Asian summer monsoon, paper presented at 3th International Conference on Radar Meteorology, Am. Meteorol. Soc., Munich, Germany. Johnson, R. H., S. L. Aves, P. E. Ciesielski, and T. D. Keenan (25), Organization of oceanic convection during the onset of the 1998 east Asian summer monsoon, Mon. Weather Rev., 133, Kuo, H.-L. (1963), Perturbations of plane Couette flow in stratified fluid and origin of cloud streets, Phys. Fluids, 6, Lafore, J.-P., and M. W. Moncrieff (1989), A numerical investigation of the organization and interaction of the convective and stratiform regions of tropical squall lines, J. Atmos. Sci., 46, LeMone, M. A. (1983), Momentum flux by a line of cumulonimbus, J. Atmos. Sci., 4, LeMone, M. A., G. M. Barnes, and E. J. Zipser (1984), Momentum flux by lines of cumulonimbus over the tropical oceans, J. Atmos. Sci., 41, LeMone, M. A., E. J. Zipser, and S. B. Trier (1998), The role of environmental shear and thermodynamic conditions in determining the structure and evolution of mesoscale convective systems during TOGA COARE, J. Atmos. Sci., 55, Morse, P. M., and H. Feshbach (1953), Methods of Theoretical Physics, part II, pp , McGraw-Hill, New York. Moncrieff, M. W. (1992), Organized convective systems: Archetypal dynamical models, mass and momentum flux theory, and parameterization, Q. J. R. Meteorol. Soc., 17, Moncrieff, M. W., and J. S. A. Green (1972), The propagation and transfer properties of steady convective overturning in shear, Q. J. R. Meteorol. Soc., 98, Moncrieff, M. W., and M. J. Miller (1976), The dynamics and simulations of tropical cumulonimbus and squall lines, Q. J. R. Meteorol. Soc., 14, Petersen, W. A., R. C. Cifelli, S. A. Rutledge, B. S. Ferrier, and B. F. Smull (1999), Shipborne dual-doppler operations during TOGA COARE. Integrated observations of storm kinematics and electrification, Bull. Am. Meteorol. Soc., 8, Rotunno, R., J. B. Klemp, and M. L. Weisman (1988), A theory for strong, long-lived squall lines, J. Atmos. Sci., 45, Tsakraklides, G., and J. L. Evans (22), Global and regional diurnal variations of organized convection, J. Clim., 16, Wu, X., and M. Yanai (1994), Effects of vertical wind shear on the cumulus transport of momentum: Observations and parameterization, J. Atmos. Sci., 51, A. Cheng, Atmospheric Sciences, NASA Langley Research Center, Mail Stop 42, Hampton, VA 23681, USA. (a.cheng@larc.nasa.gov) 12 of 12