15 FEBRUARY 1997 OLSSON AND COTTON 479

Balanced and Unbalanced Circulations in a Primitive Equation Simulation of a Midlatitude MCC. Part II: Analysis of Balance

PETER Q. OLSSON AND WILLIAM R. COTTON Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado (Manuscript received 7 August 1995, in ®nal form 27 June 1996)

ABSTRACT A nonlinear balance condition, which permits the diagnosis of both balanced divergent and nondivergent ¯ows, is presented. This analysis approach is applied to the results of a numerical simulation of a midlatitude mesoscale convective complex (MCC) to assess the degree of balance of these and similar convective weather systems. It is found that, to a large extent, the simulated MCC represents a highly balanced ¯uid system. The non- divergent component of the MCC wind ®eld was found to be largely balanced from the time of initial convection to dissipation. Perhaps more surprisingly, the storm-induced divergent model winds are also balanced to a fair degree, though certainly less so than the nondivergent ¯ow. Further, the balanced divergent ¯ow makes up a signi®cant portion of the total balanced ¯ow in some regions of the MCC. System-scale divergence pro®les of the model and balanced winds are compared and found to agree reasonably well, especially in the growth and mature stages of the MCC. Within a stationary averaging volume enclosing the MCC, the greatest disparity between the model and balanced circulations is found in the downward vertical motion. The model downward mass ¯ux signi®cantly exceeds the balanced downward ¯ux at most times during the simulation, suggesting that the process of mass adjustment due to convective heating is largely dominated by unbalanced fast-manifold processes, such as inertia± gravity waves. The unbalanced ¯ow is found to be composed largely of divergent circulations of periodic nature (i.e., gravity waves). The appearance and characteristics of these features are found to be in good agreement with current theoretical predictions regarding the atmospheric response to convective heating and associated compensating subsidence.

The (modi®ed) Rossby radii (␭R) for two lowest-order gravity wave modes are calculated. The mesoscale n convective vortex (MCV) within the storm is larger than ␭R for all but the gravest mode. The ␭R for the mature nϭ1 storm as an ensemble also indicates a good degree of balance with ␭R scaling similar to the MCC and larger

values of n scaling smaller than the system as a whole. These ␭R values strongly suggest that this simulated MCC represents an inertially stable balanced mesoscale convective system.

1. Introduction tical in nature, especially if they occur in an environment with background rotation. Much of what is considered ``meteorologically sig- In the middle latitudes, mesoscale convective com- ni®cant'' atmospheric motion can be described by a sub- plexes (MCCs) represent an extreme case of a large- set of the primitive equations, generically termed ``bal- amplitude energetic perturbation to the background en- anced models.'' These balanced models, while several vironment. While substantial transients are expected in number and of varying complexity and completeness, (and observed), a separate class of nonradiating, bal- have a common de®ning attribute. They neglect ``fast anced motions also evolve. These motions, and the al- manifold'' behavior: those atmospheric motions, typi- tered thermal structure that balances them, are conve- ®ed by gravity and acoustic waves, that have short niently contained in a single quantity, the potential vor- timescales and often exhibit oscillatory behavior. While ticity (PV). this fast manifold behavior is ubiquitous, it generally In recent years, balanced modeling has been applied represents transient responses to changes in the ener- to MCCs and smaller mesoscale convective systems getic balance of the atmosphere. The more long-lived, (MCSs). Schubert et al. (1989) used a two-dimensional ``slow-manifold'' atmospheric motions tend to be vor- semigeostrophic (SG) model to study the balanced ef- fects of an imposed heating function typical of a long squall line. Using the same model, Hertenstein and Schubert (1991) extended that study to include the heat- Corresponding author address: Dr. Peter Q. Olsson, Geophysical Institute, University of Alaska Fairbanks, P.O. Box 757320, Fair- ing contribution from a trailing stratiform region typical banks, AK 99775-7320. of that observed in an MCC. The results showed quite E-mail: [email protected] reasonable agreement with observations, especially con-

᭧1997 American Meteorological Society 480 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54 sidering the constraints of the two-dimensional SG mod- sating subsidence. The ®nal section contains a summary el and the lack of ambient wind shear. of the results and conclusions. Raymond and Jiang (1990) used the more exact non- linear balance (NLB) model in three dimensions to con- 2. Balance approximations sider the effects of a PV anomaly similar to that which may be expected in a mature midlatitude MCC. They In the context of balanced models, the term ``balance'' demonstrated the ability of the balanced circulations as- generally implies a diagnostic relationship, typically be- sociated with such a PV distribution and background tween the wind and mass ®elds. As justi®ed by scaling environmental shear to cause mesoscale upward motion arguments, such a constraint is assumed to be approx- and maintain or regenerate convection. Raymond (1992) imately valid over some range of ¯ow parameters. In- extended his NLB model to include advection by the corporation of a balance approximation in the system irrotational wind and modeled several idealized cases of governing equations results in a smaller solution space. Furthermore, the balance relationship provides of forced and unforced ¯ows with Rossby Number (Ro) of order one and also cases with no background rotation an invertibility condition for the PV or some similar → quantity (e.g., quasigeostrophic pseudopotential vortic- (fϭ0, Ro ϱ). A balanced model simulation of an idealized midla- ity, Charney and Stern 1962; the potential pseudodens- titude MCS in three dimensions was performed by Jiang ity, Schubert et al. 1989). Therefore, from a given PV and Raymond (1995), who used the semibalanced model distribution and appropriate boundary conditions, the of Raymond (1992) to simulate the later (stratiform pre- wind and mass ®elds may be determined, at least in cipitation) stages of a mature MCS. An alternative ap- principle. The agreement between the diagnosed ``bal- proach to understanding the role balanced dynamics anced'' mass and wind ®elds and their observed coun- play in mesoscale convection was demonstrated by Da- terparts provides a measure of the accuracy or validity vis and Weisman (1994, hereafter DW), who used PV of the balance approximation, that is, the degree of bal- inversion to analyze a simulated mesoscale convective ance of a given ¯uid system at a given time. A similar vortex (MCV) produced within an idealized squall line. comparison can be made between balanced ®elds and This use of PV inversion and the NLB equation in a their predicted counterparts obtained from the integra- diagnostic or ®ltering sense (as compared to a predictive tion of a PE model by inverting the PV computed from model) permitted diagnosis of the balanced nondiver- the model output. The latter approach is used in this gent component of the total ¯ow computed by the cloud- study. resolving primitive equation (PE) model. DW deter- Quasigeostrophy (Charney 1948), the oldest and most mined that while the perturbation ¯ow comprising the well-known balance approximation, is based on geo- MCV was largely balanced, its formation could not be strophic balance, a simple proportional relationship be- described within the constraints of balanced dynamics tween the horizontal winds and pressure gradient. Sev- as de®ned in that paper. DW and Jiang and Raymond eral other, more general, balance approximations have (1995) substantially furthered the understanding of been proposed and incorporated into balanced models. MCSs as balanced systems, freeing the conceptual bal- NLB is one of the most general of those systems that anced model of the MCS from the quasi-axisymmetric may be appropriately applied to ¯uids with signi®cant vortex framework (e.g., Raymond and Jiang 1990) by rotation. Scaling arguments have been advanced to dem- considering the three-dimensional PV structure of the onstrate the validity of NLB in a variety of ¯ows (e.g., evolving system in a sheared ¯ow. McWilliams and Gent 1980; McWilliams 1985; Ray- In a companion paper Olsson and Cotton (1997, here- mond 1992). after referred to as Part I) presented the results of PE To obtain the NLB condition the horizontal wind, Vh, simulation of a MCC, which occurred over the central is decomposed into nondivergent and irrotational com- United States on 23±24 June 1985. This paper, with its ponents. This is accomplished by de®ning a stream- emphasis on the balanced dynamics of the simulation, function, ⌿, and a velocity potential, ␹, with the fol- presents a complementary analysis of that work and lowing properties: ϫ⌿k ١ ١␹ Ϫ addresses the question: Does the MCC represent a bal- Vh ϭ (u, ␷, 0) ϭ anced ¯uid-dynamical system? The approach used here ␹ץ ␹ץ to address this issue is similar to that of DW except that ,(١␹ϭ , ϭ(u,␷ y ␹␹ץxץthe de®nition of the balanced ¯ow is extended to include ΂΃ the balanced or slaved secondary divergent circulations ⌿ץ ⌿ץ diagnosed from the NLB ␻ equation. The following (ϫ (Ϫ⌿k) ϭϪ , ϭ(u,␷ ١ x ⌿ ⌿ץyץsection contains a discussion of the NLB equation, the ΂΃ associated PV approximation, and the ␻ equation con- 2 ,␦Vh ϭ´sistent with this level of balance approximation. In sec- ٌ␹ϭ١ ,١ϫV ϭ␨´tion 3 results of the balanced analysis are presented. In ٌ⌿ϭ2 k section 4, these results are discussed and compared to h (ϫ⌿k)ϭ١ϫ١␹ ϭ0, (1 ١)´١ -current theories of adjustment to balance and compen 15 FEBRUARY 1997 OLSSON AND COTTON 481

where ␨ is the relative vorticity, ␦ the horizontal diver- Equations (2) and (3) represent the invertibility con- the horizontal gradient operator. Note that dition for the NLB balance system. In simpler balance ١ gence, and this decomposition has the ambiguity that ¯ow that is systems, such as quasigeostrophic (QG) balance, it is both nondivergent and irrotational may be represented possible to reduce the balance equation (in that case, by either ⌿ or ␹. To remove this ambiguity the following simply the de®nition of the geostrophic wind) to the convention is used: all ¯ow that is both irrotational and functional form ⌿ϭG(⌽), reducing the invertibility nondivergent is represented by the streamfunction, ⌿. condition to a single equation. The convergence con- Thus, all ¯ow represented by ␹ is strictly divergent and dition for this system in this application requires that

(u␹, ␷␹) shall hereafter be referred to as the divergent PV Ͼ 0 everywhere. To meet that condition, negative ¯ow. PV values were given a very small positive value before The NLB approximation for frictionless ¯ow inversion as done in Davis and Emanuel (1991).

⌿ 22ץ ⌿22ץ⌿ץ ١⌿) ϩϪ2 ϭ0, (2) a. An ␻ equation for the NLB system f )´١ ⌽Ϫ2ٌ yץxץy΂΃ץx22ץ[] where ⌽ϭgz is the geopotential, is made at the level of The PV inversion discussed above yields no repre- the differentiated momentum equations and neglects ␦ sentation of vertical motion. Latent heat release with its t in the divergence equation. Therefore, NLB related buoyancy production is the most signi®cant localץ/␦ץ and is exact for nondivergent ¯ows, such as gradient wind forcing mechanism for vertical motion. Compared to balance and deformation wind ®elds. other precipitating mesoscale systems, heating rates in To obtain an invertibility condition it is necessary to MCSs are quite large (Houze 1989), and the divergent de®ne the appropriate approximation to the exact Ertel circulation (e.g., ascending front-to-rear and descending PV: rear-to-front ¯ow branches) coupled with the diabati- cally forced vertical motions are important factors in 1 overall MCS organization. Clearly, any balanced theory → (␪ ١´q ϭ (Z ␳ 3 of mesoscale convection is incomplete without some representation of these secondary circulations. ⌽ The full primitive equations allow for a rich varietyץ ⌿ 22ץ ⌽ץ ⌿22ץ g␬␲ q ϭϪ ϩ y of responses to such heat release. Acoustic and gravityץ␲ץ yץ␲ץ xץ␲ץ xץ␲ץ] (nlb p(␲ waves, loosely termed ``fast manifold'' processes, are -⌽2 characterized by divergent ¯ow and relatively fast osץ 2 Ϫ(ٌ⌿ϩh f) , (3) cillations, and act to radiate energy away from the heat [␲ 2ץ source. In addition to these fast manifold modes, the where the vertical coordinate is the Exner function primitive equations also admit a large-scale nonradia- ␬ ␲(p)ϭcp(p/po) , ␳ is the density, ␪ the potential tem- tive or ``slow-manifold'' divergent response. It is perature, p the pressure, R the gas constant and cp the through this subset of divergent motions that balance is speci®c heat capacity at constant volume; ␬ϭR/cp, and achieved and maintained in a balanced model. Z is the vector absolute vorticity. This approximate form The ␻-equation for the NLB system is formed in a of the full Ertel PV equation neglects: manner similar to that of its QG analog. The hydrostatic vertical vorticity equation ● vertical shear of the horizontal divergent winds, V␹, ␨ץ ␨ץ ,and ␨ϩf)Ϫ␻)١´ ␨ ϩ f ) ϭϪv)١´ ϩ V ␲ץ t ⌿ ␹ץ -horizontal shear of the vertical velocity, a conse ● quence of the hydrostatic approximation: (a) (b) (c) ⌽ץ ␷ץ ␻ץ uץ ␻ץ (␪ ϭϪ . (4 ,␲ Ϫ ␦(␨ ϩ f ) ϩϪץ ␲ץxץ␲ץyץ΂΃ In principle, there are an in®nite number of combi- (d) (e) nations of ⌽(x,y,␲) and ⌿(x,y,␲), which satisfy a given (5) qnlb in (3). Initially, the diabatic heating perturbs the PV largely through changes in ␪(x,y,␲) (and hence to ⌽ indicates that local changes in the (relative) vorticity through the hydrostatic approximation). However, the may be brought about by horizontal advection of ab- (١␪ is solute vorticity by (a) the nondivergent wind, and (b partitioning of PV between the vorticity and uniquely de®ned by the NLB equation. The balance the divergent wind, (c) vertical advection of relative condition, in the general case, will not partition PV with vorticity, (d) horizontal convergence, and (e) tilting of the whole perturbation in the ␪ distribution. Evidently, horizontal vorticity into (or away from) the vertical. the balance condition puts a signi®cant constraint on Casting the continuity equation in terms of the ve- how the system can adjust. locity potential, ␹ 482 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54

-horizontal divergent ¯ow and the balanced vertical mo ץ (␲Ϫ␮ (␲␻␮ )ϭϪٌ2␹ϭϪ␦, (6 .␲ tionץ The terms within the last pair of square brackets in where ␮ ϭ 1/␬Ϫ1, indicates that for any nontrivial (9) are functions of time, and determine how the vertical ␻(␲), divergent ¯ows must be considered. Therefore, motion is related to terms of O(␹) must be retained to diagnose nontrivial secondary circulations. For the vorticity equation (5), ● the spatially varying coriolis torque interacting with the only approximation involves neglecting the tilting time-varying sheared zonal ¯ow, of the divergent horizontal winds in (e): ● time-varying vertically sheared nondivergent ageos- trophic ¯ow, and .␷ ● the spatial distribution of the diabatic warming rateץ ␻ץu⌿⌿ץ ␻ץ ␷ץ ␻ץ uץ ␻ץ ϪഠϪ ␲ץxץ␲ץyץ␲ץxץ␲ץyץ ΂΃΂΃ In our diagnostic analysis of the numerical simulation, this last term (diabatic warming rate) was frequently ␷ץ␻ץu␹␹ץ␻ץ ϩϪ; (7) quite large in regions where precipitation was occurring ␲ץxץ␲ץ␷ץ ΂΃and typically dominated the ␻ forcing there. the last term within parentheses (...) → 0, while the Including a divergent component to the de®nition of exact form of the thermodynamic equation the ``balanced'' wind is somewhat of a departure from conventional terminology. However, the nonradiating or ␪ץ ␪ץ ١␪ ϩ ␻ ϭ Q, (8) ``slaved'' component of the divergent winds diagnosable´ϩ Vh ␲ in a nonlinear balance model play an important role inץ tץ with Q being the diabatic warming rate, is retained. advection (Raymond 1992) and other dynamical pro- ␲ (balanced cesses. In light of the preceding discussion of the roleץ/ץ␲(2) Ϫ fץtץ/2ץBy combining ٌ2 (8) Ϫ vorticity equation), the NLB omega equation is ob- that the balanced divergent ¯ow and associated vertical tained: motion play in achieving and maintaining balance, the authors feel that the term ``slaved'' can be misleading .⌿ and prefer the term ``balanced'' divergent ¯owץץ (f␩␲Ϫ␮(␲␻␮) ϩٌ2(␴␻ ␲ The two sets of equations outlined here completelyץ ␲ץ ΂΂ ΃ ΃ de®ne the three-dimensional balanced ¯ow (both non- ⌿ 2divergent and divergent) through ⌿, ␹, and ␻, and theץ ␩␮ץ ␻ץ⌿ץ 22␻ץ⌿ץץ Ϫ f ϩϪf+␻balanced mass ®eld through ⌽. From these quantities, ␲2ץ ␲␲ץyץyץxץxץyץxץ␲ץ []΂΃΂΃all the other relevant balanced dynamical properties of -the ¯uid system can be deduced. Solution of these equa ץ 2 ١␩) tion systems is outlined in appendix A and a description´١␪)ϩf (V´ϭٌ(Vhh ␲ of the analysis grid, that is, the data volume upon whichץ[] .the PV inversion is performed, is given in appendix B ץ ⌿22ץ ϩϪ␤ Ϫ J(t)Ϫٌ2Q(t), At this point it is worthwhile to consider the potential tshortcomings of this balance system in the context ofץ␲ץtץyץxץ[] (9) the MCC. As discussed in Part I, several studies have where the term J(t) is the Jacobian term in square braces found signi®cant production of horizontal vorticity near y, the top of the low-level cold pool or gravity currentץ/fץin (2), ␩ is the vertical absolute vorticity, ␤ ϭ -␲is a measure of static sta- commonly found preceding the leading line of convecץ/␪ץ␲2 ϭϪץ/2⌽ץand ␴ ϭ bility. tion in a maturing squall line or MCC. This cold out¯ow The continuity equation (6) and the NLB ␻ equation is typically divergent in nature and therefore projects (9) represent a coupled system in ␻ and ␹, the vertical largely on V␹. Such horizontal vorticity may be tilted and horizontal components of the balanced divergent vertically by convective updrafts, this mechanism being wind. Similar to the coupled system in ⌽ and ⌿,a instrumental in the development of some simulated solution [␻(x,y,␲), ␹(x,y,␲)], can in principle be found MCVs (e.g., DW). As is discussed by DW and can be that simultaneously satis®es these equations and accom- seen in (3), however, it is precisely this vorticity that is panying boundary conditions. The ␻ equation describes not included in qnlb. The NLB vorticity equation also how, through a great variety of simultaneous interac- neglects this tilting as a source term, as seen in (7). tions, the divergent part of the ¯ow adjusts with the PV The above discussion would suggest that, for systems and balance equation. The group of terms in (9), which where tilting of storm-generated horizontal vorticity due are functions of Vh, the total horizontal wind (and hence to vertical shear of the divergent horizontal winds is functions of ␹), adjust the differential horizontal ad- instrumental in MCV production, a prognostic model vection of vorticity and ␪ with the spatial distribution based on NLB would indeed fail to evolve an MCV of ␻. The continuity equation simultaneously maintains correctly. This is much less of a problem in the diag- the conservation of mass as a function of the balanced nostic application of NLB here since the error caused 15 FEBRUARY 1997 OLSSON AND COTTON 483

TABLE 1. Summary of de®nitions of the various two-dimensional wind ®elds used in the discussion of the balanced analysis. See Table 1 for a de®nition of ``divergent'' and ``nondivergent'' in terms of the velocity potential ␹ and the streamfunction ⌿. COMPONENTS OF VECTOR WINDS

Vu ϭ Vm Ϫ Vb Vu␹ ϭ Vm␹ Ϫ Vb␹ Vu⌿ ϭ Vm⌿ Ϫ Vb⌿

Vm ϭ Vm⌿ ϩ Vm␹ Vb ϭ Vb⌿ ϩ Vb␹ Vu ϭ Vu⌿ ϩ Vu␹

Vb

Vm model wind balanced wind Vu unbalanced wind

Vm⌿ model nondiv. wind Vb⌿ balanced nondiv. wind Vu⌿ unbal. nondiv. wind

Vm␹ model diverg. wind Vb␹ balanced diverg. wind Vu␹ unbal. diverg. wind

by systematic neglect of V␹ in qnlb does not accumulate June 1985 (designated 23/2000) along a quasi-stationary over several hours. front in central Iowa. Over the next few hours, con- vection developed along and south of the front in Iowa and SE Nebraska (see Fig. 6 in Part 1 for an overview b. De®nitions of balanced winds and related of the observed and simulated storm movement and quantities location). By 24/0000 the storm had achieved MCC For the purpose of discussing the results of the bal- status, and an MCV had formed at the eastern end of anced analysis, it is useful to de®ne several wind ®elds. the E±W oriented convective line in SE Iowa. The MCC It is understood in the ensuing discussion that the vector slowly propagated to the south, and by 24/0600 the winds are the horizontal winds, unless otherwise stated. dissipating storm extended across central Missouri west- Vector ®eld decomposition can be achieved in a num- ward into east-central Kansas. During the mature and ber of different and useful (and probably confusing!) dissipation stages of the MCC, a second, smaller MCV ways. The main interest here is in the balanced (Vb) and developed in far eastern Kansas. At the end of the sim- unbalanced (Vu) winds, where Vb is the sum of the bal- ulation (24/1200), the stratiform cloud shield was dis- anced nondivergent and divergent components (Vb⌿ and sipating, but a strong remnant PV anomaly still existed Vb␹) as derived from the balanced analysis. Similar to in the middle troposphere. the de®nition of the ageostrophic winds in the context Figure 1a shows midtropospheric Vm just before the of geostrophic balance, the (nonlinearly) unbalanced Vu ®rst convection occurred. Figure 1b shows Vu, the un- is de®ned as the residual balanced component of the model winds both at 23/1900. (Note: the wind vector scale difference be- V ϭ V Ϫ V . (10) u m b tween panels, with the maximum model winds about 29 Ϫ1 The sum of Vu and Vb constitutes the complete, two- ms and the maximum unbalanced wind magnitude Ϫ1 dimensional vector wind ®eld (Vm), as produced in the about 1.5 m s .) Only in a few locations does the un- model. For the sake of intercomparison, it is helpful to balanced wind exceed a few percent of the model wind compute the nondivergent and divergent components of ®eld. A similar result is found at other analysis levels. the model winds. This is achieved by computing the Before the onset of convection, the balance approxi- vertical vorticity and divergence from the model data, mation is evidently quite accurate. and solving the Poisson equations de®ning the vorticity Figures 2a and 2b show vector plots of storm relative and divergence in terms of ⌿ and ␹, respectively. The Vm and Vb (the balanced winds), respectively, after sum of the decomposed winds, Vm⌿ ϩ Vm␹, is found to about 3 h of convection. The MCV circulation in south- agree well with the model winds Vm. The same approach central Iowa appears clearly in both plots. The MCV as in (10) is then used to de®ne the unbalanced non- signature in Vb is not surprising, since, to maintain tem- divergent winds, Vu⌿, and unbalanced divergent winds, poral coherence, this strong long-lived vortical circu- Vu␹. Table 1 contains a summary of these de®nitions. lation (Part I) requires a state of near-cyclostrophic bal- In a like manner to (10), an unbalanced vertical ve- ance (a limiting case of nonlinear balance) in order to locity (␻u) in the Exner function vertical coordinate sys- maintain temporal coherence. Overall, the winds agree tem may be de®ned quite well, though there are some differences. The major

difference between Vm and Vb is due to the unbalanced ␻u ϭ ␻m Ϫ ␻b, (11) divergent winds, Vu␹, which typically comprise over where ␻b satis®es both the continuity equation (6) for 80% of Vu at this level. the balanced velocity potential ␹b, and the NLB omega Another feature of interest evident in Fig. 2 is the equation (9). anticyclonic turning of Vm, NNE of the MCV, which is

not captured in Vb. Recall that the PV was ®ltered pre- vious to inversion to ensure that the convergence con- 3. Results of the balanced analysis dition for the ⌽, ⌿ solver, PV Ͼ 0 was met. In Part I As shown in Part I, the ®rst convection associated it was shown that narrow bands of negative PV were with the simulated MCC initiated around 2000 UTC 23 generated to the north of the convective elements. The 484 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54

FIG. 2. (a) The storm-relative model winds Vm and (b) the storm-

relative balanced winds Vb on the 771-mb surface at 23/2200. The maximum wind vector represents a speed of 20 m sϪ1. Storm motion Ϫ1 Ϫ1 for this and subsequent ®gures is Us ϭ 7ms ,Vs ϭϪ5ms , unless otherwise stated. Note also that in this and later ®gures with wind vectors, the vectors are plotted at every other model grid point in both dimensions. FIG. 1. (a) The model winds at 23/1900 on the 535-mb pressure level. The maximum wind vector represents a speed of 29 m sϪ1. (b) The unbalanced winds at the same time and level. The maximum wind vector represents a speed of 1.5 m sϪ1. For comparison, the scaling between a and b is 10:1 (a vector in panel a would be 10 At later times, the model ¯ow still remains reasonably times larger in panel b). The ring in Fig. 1a shows the horizontal location of the averaging domain discussed later. balanced. Figure 3 shows the wind vectors Vb and Vu on the 535-mb surface, when the storm is mature. The

slight anticyclonic tendency of the unbalanced winds Vu ®ltering process by necessity caused these regions to be (Fig. 3b) is still evident (again note the difference in given very weak positive values of PV. This results in wind vector scales between panels). The rear-in¯ow jet a local biasing of the PV inversion solution toward pos- shows up well in the northerly component of the bal- itive vorticity and represents a fundamental weakness anced winds (Fig. 3a) over Iowa. Here, Vu also shows of the balanced model paradigm. However, since the PV a weak (ϳ 4msϪ1) ¯ow along the Iowa/Illinois border. inversion in a sense represents a double integration, it This unbalanced feature is rather atypical of the unbal- acts as a smoothing operator and mitigates the effects anced ¯ow as a whole in that it was composed largely of PV ®ltering, especially if the regions of negative PV of Vu⌿, the nondivergent component of the unbalanced are relatively con®ned as is the case here. circulation. 15 FEBRUARY 1997 OLSSON AND COTTON 485

TABLE 2. The ␲ levels in the analysis domain and their mb equiv- alents. The fourth column lists, for reference, the associated geometric heights of each surface for a point near the midpoint of the Iowa/ Missouri border at 23/1800.

Analysis ␲ Pressure Height level (J KϪ1 kgϪ1) (mb) (m) 1 963.0 864.3 1387.8 2 932.1 771.3 2369.8 3 901.3 685.6 3360.2 4 870.4 607.0 4356.5 5 839.5 535.0 5361.3 6 808.7 469.3 6375.1 7 777.8 409.6 7396.8 8 746.9 355.5 8425.8 9 716.1 306.7 9462.6 10 685.2 262.9 10 514.0 11 654.3 223.8 11 586.4 12 623.5 189.0 12 686.7 13 592.6 158.3 13 828.3 14 561.7 131.3 15 017.8 15 530.9 107.7 16 257.1 16 500.0 87.4 17 555.1

unit mass (ᑬ) associated with the various divergent and wind ®elds:

(␳/2)V ´V d⍀ ͵ ii ⍀ ᑬi ϭ , (12) ͵ ␳ d⍀ ⍀ where ⍀ represents a volume of integration and ␳ the pseudodensity appropriate to the coordinate system. Us-

ing the appropriate wind ®elds in (12), ᑬ;m, ᑬ;u, ᑬu⌿, and ᑬu␹ were computed at half-hour intervals in an av- eraging volume whose horizontal projection is shown in Fig. 1a.2 The (pressure) depth of the volume was that of the analysis domain (refer to Table 2 in appendix B). This stationary volume, with a radius of 450 km, was FIG. 3. Both panels at 24/0430 on the 535-mb level. (a) The bal- of suf®cient horizontal extent to contain most of the anced winds Vb. The maximum wind vector represents a speed of 30 vertical motion directly related to the MCC during its Ϫ1 Ϫ1 ms . (b) The unbalanced winds Vu, maximum speed of 6.7 m s . lifetime. Figure 4 shows the time evolution of the various ᑬs

normalized by ᑬm, the model kinetic energy. At 23/1900 (Ϫ5 h on the ordinate in the ®gure), the top curve of Bulk characteristics the main plot in Fig. 4 shows that ᑬu was only about 1) VOLUME AVERAGES 1% the magnitude of ᑬm, again in good agreement with the quasi-balance assumption. With the onset of con- To gain an understanding of the atmospheric balance of the storm region as a whole, it is useful to de®ne some bulk measure of the relative signi®cance of the various wind ®elds by de®ning an average mass-weight- winds, Vm. Due to the quadratic nature of (12), the sum of the un- balanced and balanced ᑬs, (ᑬ ,ᑬ ), does not equal the total ᑬ com- ed mean square, or kinetic energy-type measure1 per u b m puted from the model winds. 2 Several volume sizes were tried. Unless stated, the results pre- sented in this section were relatively insensitive to the size of the averaging volume, though of course very large domains showed a 1 It should be borne in mind that the only true kinetic energy, in smaller amplitude since they included more undisturbed model cir- the physical sense, is that associated with the total model horizontal culations. 486 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54

FIG. 4. Here, ᑬ as de®ned in Eq. (12) normalized by the volume- FIG. 5. Area-integrated upward and downward vertical mass ¯ux average kinetic energy per unit mass, ᑬ . The top (solid line) is ᑬ m u (1010 kg sϪ1) through the 535-mb surface. The solid lines indicate the associated with the total unbalanced ¯ow, V . The dotted line with u ¯uxes by ␻ and dotted lines, mass ¯uxes by ␻ . The time axis is ®lled circles is ᑬ , associated with the unbalanced divergent ¯ow, m b u␹ the same as in Fig. 4. The averaging area is indicated in Fig. 1a. Vu␹. The bottom (dashed) line is ᑬu⌿, associated with the unbalanced nondivergent ¯ow Vu⌿. The inset plot in the upper-left hand corner displays ᑬ associated with the residual: VRϭ Vm Ϫ Vb⌿, that is, the difference between the model winds and the balanced nondivergent Vb␹, is, at least locally, an important component of the winds. The top curve in the main ®gure, ᑬu/ᑬm, is plotted in the inset (bottom curve) for comparison. Note the scale difference on the total balanced winds. (Recall that ᑬu is calculated using abcissae. Refer to Fig. 1a for the location of the averaging volume. the residual Vm Ϫ Vb⌿ Ϫ Vb␹. This is the top curve in the main ®gure and also appears for reference in the inset ®gure as the lower curve.) Some balanced models (e.g., Raymond and Jiang 1990) neglect advection by vection, the ratio ᑬu/ᑬm quickly grew to several percent. V , the balanced divergent wind. Any balanced model At the time of maximum extent of the storm, between b␹ that neglects advection by Vb␹ would be expected to 24/0200 and 24/0400, ᑬu/ᑬm reached almost 12%. As exhibit signi®cant errors in regions of strong convection. the storm dissipated, the ratio slowly fell, decreasingly more or less linearly after 24/0730, when the convection had ceased. By the end of the simulation (24/1200), ᑬu/ 2) AREA-INTEGRATED VERTICAL FLUXES IN THE ᑬm was still over 6%, several times larger than the pre- MIDTROPOSPHERE convective value at 23/1900. The dotted (middle) curve in Fig. 4, with the ®lled- The area-integrated vertical mass ¯ux through the circle symbols, shows the evolution of ᑬu␹ associated midtroposphere provides a good bulk measure of the with Vu␹, while the dashed (bottom) curve shows ᑬu⌿, consistency between the model-produced vertical mo- associated with the Vu␺. Throughout the time series, ᑬu␹/ tion, ␻m, and the balanced vertical motion, ␻b. Figure ᑬm closely tracks ᑬu/ᑬm, both in amplitude and trend. 5 shows the area-integrated upward and downward mass

In comparison, ᑬu⌿/ᑬm does increase, but only gradu- ¯ux through the 535-mb surface as a function of time. ally, exceeding 2% for just a short period (24/0600 → What is most apparent in this ®gure is that the upward

24/1000) near the end of the storm. ¯ux from both ␻m and ␻b exceeds the downward ¯ux The unbalanced ᑬ evolution shows several interest- during almost the entire lifetime of the storm.3 The pos- ing features about the bulk ¯ow behavior of the storm, itive vertical mass ¯uxes, ␾mϩ ϭ⌺␳␻mϩ, ␾bϩ ϭ⌺␳␻bϩ, the most obvious (and anticipated) result being that the are almost identical during the ®rst 3 h of convection. total unbalanced ¯ow is largely divergent. A signi®cant After 23/2300, ␾mϩ exceeds ␾bϩ for the duration of the pulsing is apparent as a ripple superimposed on the gen- simulation with the largest values of both ␾mϩ and ␾bϩ eral trend of ᑬu/ᑬm. Even in the earliest stages of the occurring between 24/0000 and 24/0130, and then de- storm development, this pulsing is apparent. This is not present in ᑬu⌿/ᑬm and it is speculated that this higher frequency oscillatory behavior is likely associated with gravity wave motions (see related discussion in section 3 As the storm matured, a large anvil cloud developed to the north and east of the convective line as a result of the considerable westerly 4c). A similar pulsing was found in the two-dimensional shear above 6000 m. The evaporative cooling of over Ϫ10 C hϪ1 MCC modeling studies of Tripoli and Cotton (1989). below this anvil cloud forced mesoscale downdrafts. Due to the east- The inset in the upper-left hand corner of Fig. 4 shows ward dislocation of these downdrafts relative to the convective up- the ᑬ associated with the quantity V Ϫ V , the dif- drafts, an increasingly large amount of diabatically forced downward m b⌿ motion occurred outside the averaging domain after 24/0700. Before ference between the model winds and the balanced non- this time, almost all the vertical motion directly induced by diabatic divergent winds. This quantity has a maximum ap- processes occurred within the domain. This effect can be seen in the proaching 30%. Obviously the balanced divergent ¯ow decreasing magnitude of ␾bϪ in Fig. 6 after 24/0700. 15 FEBRUARY 1997 OLSSON AND COTTON 487

FIG. 6. Two-hour average horizontal mass ¯ux divergence pro®les (10Ϫ5 kg mϪ3 sϪ1) at (a) 24/0000, (b) 24/0400, and (c) 24/0800. In each of the ®gures the solid line is the mass ¯ux divergence from the model, while the dashed line is the balanced mass ¯ux divergence. The vertical axis is meters AGL. (d) The area-averaged heating function, Q(z) (ЊChϪ1), for the same three times as the divergence pro®les.

creasing only gradually until around 24/0600, as the in heating function, dQ(z)/dt, has been shown to produce convective line weakens. gravity waves (Emanuel 1983; Mapes 1993, hereafter

It is interesting to note that the residual unbalanced M93). The increase in both ␾uϩ and the unbalanced upward mass ¯ux, downward ¯ux, ␾uϪ, correlates well with the develop- ment of a much more complex Q(z) (Fig. 6d). By ␾ ϭ ␾ Ϫ ␾ , uϩ mϩ bϩ 24/0800, Q(z) is dominated by evaporative cooling and increases from 24/0100 onward until 24/0700, well into ␾uϩ slowly decreases as the waves propagate out of the the dissipating stage of the MCC. During this time, the averaging domain. stratiform region of the storm is enlarging. Concur- The downward mass ¯uxes in Fig. 5, ␾mϪ and ␾bϪ, rently, below the stratiform region, the rear in¯ow is show considerably less consistency than their positive developing as precipitation evaporates into the dry mid- counterparts. The downward model mass ¯ux, ␾mϪ, rap- level air. This broad region of cooling modi®es the mean idly increases from the time of the ®rst convection until vertical pro®le of the heating function, Q(z). A change 23/2200, corresponding to the brief period when 488 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54

␾bϩ Ͼ␾mϩ. After this time, ␾mϪ dramatically weakens, curred over a very thin level, just at the tropopauseÇ, remaining more or less constant through the rest of the creating the sharply peaked maximum of Dm. storm lifetime. By contrast, ␾bϪ gradually increases in magnitude from the time of ®rst convection until 4. Discussion 24/0130 and then slowly returns to near zero by the end of the simulation. The time evolution of ␾bϪ generally Historically, balance approximations were referred to resembles a smaller amplitude, inverse image of its pos- as ®ltering approximations, as they ``®ltered'' out be- itive counterpart ␾bϩ. havior (e.g., gravity waves, acoustic waves, Lamb waves) deemed undesirable in a numerical model (Hal- tiner and Williams 1980; Holton 1992). The neglect 3) AVERAGED DIVERGENCE PROFILES (®ltering) of gravity wave oscillations by balanced mod- els is equivalent to the assumption that the gravity wave Before leaving the discussion of the divergent ¯ow, → it is instructive to consider some vertical pro®les of the phase speed (c) ϱ, that is, that gravity waves radiate average horizontal mass ¯ux divergence, away immediately. This does not mean that the effects of gravity wave propagation in a rotating ¯uid are not ␳V dA contained within the theory, but rather that the transient´١ ∫ DÅ (z) ϭ i i ϭ m, b (13) ∫ dA adjustment process happens instantaneously, resulting in only the slow-manifold, balanced ¯ow. The scale over for each of the levels in the averaging volume. Figure which these adjustments occur is given by the Rossby

6 shows such 2-h time-average pro®les centered at radius (or ``radius of deformation''), ␭R. The difference 24/0000, 24/0400, and 24/0800. The solid line in each between the balanced and model circulations in the pres- panel is the model mass-¯ux divergence, Dm, and the ent application largely represents transient or fast man- dashed line, Db, the mass-¯ux divergence associated ifold processes, typically of an oscillatory nature, which with the balanced wind. have no representation in the balance paradigm. Some The ®rst two times (24/0000 and 24/0400) reveal sev- of those processes are considered here. eral of the same features. As expected, both Dm and Db show convergence at lower levels, with divergence in a. Compensating subsidence the upper half of the troposphere. The level of maximum convergence (ϳ2500 m) is the same for both Dm and The differing nature of the upward and downward Db. Here,Db consistently shows a level of maximum midtropospheric ¯uxes discussed in section 3 suggests divergence somewhat lower and weaker than Dm; Db that the model dynamics forcing upward motion within also shows weaker convergence at the lowest level the averaging domain are different from those that drive throughout the convective portion of the storm. In gen- the downward motion. The relatively close agreement eral, however, the balanced divergence and model di- between ␾mϩ and ␾bϩ indicates that the main driving vergence pro®les at 24/0000 and 24/0400 are in good process for updraft motion, latent heat release resulting agreement and show strong similarities to observed from cloud microphysical processes, is instantaneously MCS divergence pro®les (e.g., Fig. 2 of M93 and Fig. well-represented in the NLB ␻-equation solution. By

12 of Cotton et al. 1989). contrast, the inconsistency between ␾mϪ and ␾bϪ indi- At 24/0800 (Fig. 6c), Dm and Db show much poorer cates that the dynamics driving downward motion with- agreement. By this time, the latent heating has largely in the averaging domain are relatively poorly repre- ceased, with the exception of two con®ned regions near sented in that solution. The location of the balanced the decadent squall line. The average heating function, downdrafts causing the modest ␾bϪ seen in Fig. 5 cor- Q(z) ϭ d␪/dt (Fig. 6d) computed from the numer- relate well with the evaporative cooling behind, and to ical simulation has changed signi®cantly from 24/0400 the east, of the convective line. The quite different be- to 24/0800. The level of maximum convergence for the havior of ␾mϪ indicates that the PE model is responding model ¯ow has moved upward to 5000 m from 2500 to more than the direct evaporative cooling felt by the m earlier. Here, Dm is now positive (divergent) at the balanced ␻ equation. lowest level, while Db shows its strongest convergence The difference between ␾mϩ and ␾mϪ graphically il- there. The model surface wind ®elds (below the bottom lustrates a well-known and important aspect of so-called of the balanced-analysis domain) have a strong ¯ow of compensating subsidence (i.e., that downward motion relatively cool air away from the storm. necessary to maintain continuity in a large-scale inte-

Throughout the evolution of the simulated MCC, Vb grated sense)Ðthe mass ¯ux locally forced in buoyantly showed a systematic tendency to have a lower level of driven updrafts results in downward motion on a much maximum divergence relative to Vm. While the diver- greater scale. The difference between ␾mϪ and ␾bϪ high- gence pro®les show a very pronounced maximum in lights another point: The mechanisms by which com-

Dm, which remains at a height of about 11 500 m, Db pensating subsidence occur are largely fast manifold shows a less pronounced maximum at about 10 000 m. processes not contained in balanced model theories, In the PE model, the detrainment from the updrafts oc- which exclude inertia-gravity and acoustic wave motion. 15 FEBRUARY 1997 OLSSON AND COTTON 489

The nature of compensating subsidence has received balance approximation is valid, the balanced wind and recent attention in several studies (e.g., Bretherton and mass ®elds at any give time could be expected to rep- Smolarkiewicz 1989; Nicholls et al. 1991; Schmidt and resent the ®nal adjusted state (which may or may not Cotton 1990; M93). Results of Bretherton and Smolar- include such an area contraction) after all transients, kiewicz (1989) and Nicholls et al. (1991), for a non- such as buoyancy bores, have passed. However, since rotating atmosphere at rest, have shown that the dis- the transient motions are not represented in the NLB turbances forced by the initiation or termination of an model, any meteorological phenomena that inherently imposed heating function (i.e., a nonzero dQ/dt as dis- depend on, or are triggered by, the transient motions cussed above) have the propagation mode associated themselves would not be captured in a balanced model with gravity waves. Therefore, results from gravity simulation. Studies in the literature have found the os- wave theory are applicable. The predicted horizontal cillatory nature of gravity waves to mediate several phe- gravity wave propagation speed for a wave with vertical nomena such as squall line propagation (e.g., Cram et wavenumber m, al. 1992) and convective initiation (e.g., Crook 1987; Zhang and Fritsch 1988b; Crook et al. 1990; Schmidt c ϭ N/m ϭ NH/n␲, (14) x and Cotton 1990). A balanced model, by ®ltering out is indeed found by Nicholls et al. In (14), H is a scale fast-manifold oscillatory motion, will have several po- height, n a positive integer, and N the Brunt±VaÈisaÈlaÈ tential limitations when simulating phenomena that rely frequency. on rapid vertical parcel displacements. Unlike classical linearly superimposed gravity waves, such propagating ``buoyancy bores'' do not have a spa- b. Some further aspects of the unbalanced ¯ow tial or temporal periodicity and do leave a permanent net horizontal displacement in the ¯uid through which Potential vorticity is a conserved property in the in- they propagate (M93). These bores then must transport version process in the sense that the resulting balanced (rearrange) mass in a strati®ed ¯uid, and evidently rep- wind and thermal structure yield the same PV produced resent the agents of compensating subsidence, enabling by the model. This does not, however, mean that the the ¯uid to recover a state of mass equilibrium. Because same wind and thermal ®elds are recovered. Rather, the of their gravity-wave-like nature, such bores would not agreement depends on how well the balance assumption be explicitly represented in balanced models, yet the is met. (The ®ltering of negative PV also causes a minor volume integral of (6), the continuity equation, still must difference, because the model ®elds, Vm and ␪m no lon- be satis®ed, and the compensating subsidence must al- ger agree with the positive-de®nite PV ®eld used in the most immediately occur in a more global sense, being inversion.) Figure 7a (23/2200) shows the midtropo- spread over a much larger domain. The midtropospheric spheric wind vectors and relative vorticity (␨m), while ␻b does show evidence of weak subsidence across a Fig. 7b shows the nondivergent component of the un- very large region of analysis domain (much larger than balanced winds, Vu⌿, and the associated unbalanced vor- the averaging volume) over which the stronger directly ticity, diabatically induced vertical motions are superimposed. ␨ ϭ ␨ Ϫ ␨ , (15) In an in®nite, inviscid, nonrotating domain, the buoy- u m b

١؋ (10). The Vu⌿ ®eld shows a de®nite ancy bore would radiate to in®nity, leaving no trace of found by taking itself in the winds or temperature structure of the ¯uid. anticyclonic ¯ow, in agreement with ␨u in Fig. 7b, im- Gravity waves in a rotating ¯uid, however, become mod- plying that Vb is more cyclonic than Vm. For PV to be i®ed (deformed) as they propagate away from their maintained, this would then necessitate that the balanced source, with ␭R being the length scale over which this ␪ ®eld is less stable than the model ␪ ®eld. Figure 7c occurs. The irreversible, horizontal displacement of the shows the difference Brunt±VaÈisaÈlaÈ frequency, ¯uid caused by the passage of the bore implies a ver- N ϭ N Ϫ N . (16) tically varying horizontal motion that in turn, requires u m b a balancing horizontal temperature gradient (Gill 1982). .

The buoyancy required by the balancing horizontal ther- As expected, Nu Ͼ 0 over most of this region, which mal structure comes from the bore itself. Hence, the implies Nb Ͻ Nm, that is, the balanced ␪ ®eld is less bore becomes, in a sense, absorbed, as it propagates stable than the model ®elds. outward, leaving energy behind in the balancing or ad- This difference between the model and balanced ®elds justment process. The rotational dynamics that scale must result from a number of factors besides the ®ltering with ␭R are contained in the NLB equation set. This process since, at this early time in the storm evolution, suggests a length scale or ``e-folding'' length governing only a very small region of weak negative PV is found. z Kץ/V␹ץ the balanced compensating subsidence in the balanced The NLB approximation to PV assumes that ,z, which is typically a poor assumption withinץ/V␺ץ .␻ equation In the idealized approach of M93 the passage of the and ahead of, the convective line (e.g., Skamarock et buoyancy bore caused ``area contraction,'' that is, a al. 1994). The hydrostatic assumption is also included large-scale low-level convergence. To the extent that the in the NLB PV approximation, eliminating horizontal 490 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54

¯ow accelerating into the storm from the north has a strong divergent component that increases with height in the lower- and midtroposphere. Again, this contri- bution to PV is neglected in the NLB approximation. Still, even in light of the differences discussed above, the rapidly developing storm system is balanced to a remarkable degree.

c. Use of NLB diagnostics as a gravity wave ®lter As mentioned earlier the unbalanced ¯ow represents the fast manifold residual of the balanced ®ltering pro- cess. Before the divergent motions become complicated by interfering waves of differing wavelengths, the ®l- tering capacity in this application is quite evident, even in the unbalanced horizontal winds. A clear example of this is seen in Fig. 8 (23/2200)

which shows ␻u on the 607-mb surface. The contouring has been limited to the range Ϫ0.5 ϫ 10Ϫ3 to 5 ϫ 10Ϫ3 m2 sϪ3 KϪ1 to emphasize the periodic motion. At this

time, the mesoscale ␻u is the combination of two su- perimposed nearly radially symmetric wave patterns. The western source has not been in existence as long and as a result the ®rst outgoing wave has not propa- gated out as far as the eastern pattern. The eastern pat- tern is the eastward-advecting remnant of the ®rst short- lived convective pulse (Part I), which ®red over central Iowa shortly before 23/2000. The ®ltering ability of the balanced analysis is useful in comparing some aspects of the unbalanced ¯ow be- havior to that predicted by gravity wave theory. An example is the problem of convective mass adjustment or compensating subsidence touched upon in section 4a. M93 proposed that a heating pro®le typical of tropical MCSs,

Q(z) ϭ Q0[sin(␲z/H) ϩ sin(2␲z/H)/2], (17) would produce a divergence pro®le of the form FIG. 7. All plots at 23/2200 on the 535-mb surface. (a) The model Ϫ␲␳Q0 winds, Vm, and relative vorticity ␨m. The maximum wind vector rep- D(Q, z) ϭ [cos(␲z/H) Ϫ cos(2␲z/H)/2]. (18) resents a speed of 27 m sϪ1. The contour interval for the vorticity is HN 2 1 ϫ 10Ϫ4 sϪ1. (b) The unbalanced nondivergent winds, V , and the u⌿ In (17) and (18) ␲ ഠ 3.14159, N is the Brunt±VaÈisaÈlaÈ associated unbalanced vorticity ␨u. Here, the maximum wind speed is 2.2 m sϪ1, and the contour interval is 0.5 ϫ 10Ϫ5 sϪ1. (c) The frequency, and H is the depth of the domain. A sketch difference Brunt±VaÈisaÈlaÈ frequency Nu as de®ned by (16), contour of Q(z) and D(z) appear to the far left in Fig. 9. The interval of 8 ϫ 10Ϫ4 sϪ1. shape of the idealized divergence, D(z), agrees quite well with the lower two-thirds of the ``early-MCC'' di- vergence pro®le in Fig. 6a. y). With vertical Instantaneously imposing Q in a strati®ed atmosphereץ/␻ץ ,xץ/␻ץ) shear of the vertical winds velocities in the convective updrafts of a few meters per should result in a gravity wave motion corresponding second in close proximity to downdrafts of similar to two lowest-order vertical modes. The gravest mode, strength, this can also be a poor approximation near the of vertical wavenumber 1/2 (M93's lϭ1 bore), resulting squall line. There is a high spatial correlation between from the ®rst term in (17) would be a half sine wave the extrema in ␨u, Nu, and ␻ (not shown) at this level, of negative vertical motion, Ϫw0sin(␲z/H), while the that is, these extrema are located in the region near the second mode, with vertical wavenumber 1 would consist updraft and downdraft cores, where the motion is likely of an up-and-down couplet. nonhydrostatic. The difference between ␨m and ␨b locally Using the equations developed by Bretherton and exceeds 10%. This difference decreases rapidly with Smolarkiewicz (1989), M93 calculated the displacement distance to the south but less rapidly to the north. The of a surface, ⌬z(r,t), for the radially symmetric case. 15 FEBRUARY 1997 OLSSON AND COTTON 491

FIG. 8. The unbalanced vertical velocity (␻b ϭ d␲/dt) at 23/2200 on the 607-mb surface. The contour level is 5 ϫ 10Ϫ4 m2 sϪ3 KϪ1. The contouring has been limited to the range Ϫ5 ϫ 10Ϫ3→5 ϫ 10Ϫ3 to emphasize the wavelike vertical motion at distances well removed from the convective region. The ``ϩ'' and ``Ϫ'' signs denote regions of upward and downward motion, respectively. (In the ␲ vertical coordinate, positive values imply negative vertical motion). The bold double ended arrow indicates the approximate distance the leading wave- front has propagated in the 3-h period since convection began.

After a propagation time ␶, the bores have traveled L 23/2200 is an almost constant 1.42ϫ10Ϫ2 sϪ1. The prop- and 2L, where L ϭ c2␶, c2 being the propagation speed agation speed of a linear gravity wave is given by (14). of the higher-order bore. Using N ϭ 10Ϫ2 sϪ1 and c ϭ Using these values in (14), the inverse of the vertical 50 m sϪ1, M93 found 2L to be about 550 km (see his wavenumber is mϪ1 ϭ 3584 m, giving a value for H of Fig 4). 11.3 km. From Fig. 4a in Part I, the tropopause height The clear gravity wave pattern in Fig. 8 permits an in the southern end of the cross section can be estimated application of this theory to the PE simulation. From from the PV gradient as about 11 km, which agrees very the center of the pattern southward, the leading edge of well with the computation of H above. The vertical mo- the storm-induced vertical motion is about 565 km. The tion depicted in Fig. 9 closely matches that along the convection has been active for about 3 h, giving a prop- heavy line in Fig. 8, with subsidence from about 300 agation speed of 51 m sϪ1. The Brunt±VaÈisaÈlaÈ frequency km south to 565 km south, and a rapid shift to positive on the 607-mb surface south of the storm region at vertical motion at about the 300-km point. A plot similar to Fig. 8 on the 356-mb surface (not shown) shows a broad subsidence in qualitative agreement with Fig. 9, though at this level the radial pattern evident at 607 mb is quite distorted by the strongly sheared ¯ow. The more detailed wave structure closer to the storm is due to higher wavenumber oscillations with slower propagation speeds. Using (14) with nϭ2 gives a hor- izontal phase speed of 25.5 m sϪ1 for the second mode, with a corresponding wavelength of about 280 km, as Ϫ1 FIG. 9. Schematic depiction of the results of M93. The line plot compared to 25.0 m s and 250 km, respectively, for above rϭ0 shows the heating function Q(z) (solid). The dotted line nϭ2 in M93. These values are similar to those in Zhang in the line plot shows the resulting divergence pro®le D(Q,z) expected and Fritsch (1988b), where internal waves associated in the region 0 Ͻ r Ͻ L. After a propagation time ␶,thelϭ2 and with a simulated MCS had 25±30 m sϪ1 phase speeds lϭ1 bores have traveled L and 2L, respectively. The horizontal dashed line indicates the relative displacement of a surface in the lower half and wavelengths of 250±300 km. Considering that the of the domain. Adapted from Fig. 3 of M93. work of M93 was highly idealized, with horizontally 492 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54 homogeneous initial conditions and no shear, topogra- released energy is con®ned locally to within the storm, phy, or Coriolis force, the close agreement with this PE and is projected into the rotational ¯ow. In support of simulation of an MCC is notable. their dynamical de®nition, Cotton et al. (1989) cite ob- servational studies (e.g., McAnelly and Cotton 1986, 1989) showing that as an MCC matures, the system d. The Rossby radius revisited becomes less divergent and the vertical motion less pro- In an attempt to de®ne a metric that determines the nounced, in agreement with the reasoning of Ooyama dynamic size (and implicitly, the relative state of bal- (1982) that the balanced dynamically large system is ance) of an MCC, Cotton et al. (1989) proposed a dy- more characteristically two-dimensional. namic de®nition for MCCs: This argument is supported by the ®ndings of Schu- A mature MCC represents an inertially stable mesoscale bert et al. (1980). Using a linearized axisymmetric vor- convective system that is nearly geostrophically balanced and tex model, they found that convective-type heating in whose horizontal scale is comparable to or greater than ␭ . a basic state with no relative vorticity produced a re- R sponse dominated by inertia±gravity waves, with very From the discussion of balance approximations in little energy going into the rotational spectrum of modes. section 2 it should be apparent that the use of the term When the basic state was one in which the inner region ``geostrophic'' by Cotton et al. in this de®nition is too had positive relative vorticity, and hence a greater in- restrictive. Geostrophic balance implies that (at the level ertial stability, the heating became much more ef®cient of the f-plane approximation) the nonlinear term in in producing a balanced vortical ¯ow. In the former square brackets in (2) is zero, a manifestly poor ap- case, energy was radiated away, whereas in the latter proximation in the application to MCCs. With the re- case the energy was largely con®ned to the storm-scale striction to geostrophy relaxed, however, the above def- circulations. Hack and Schubert (1986), using an axi- inition still provides a useful dynamic metric. symmetric PE model, further quanti®ed this argument Cotton et al. (1989) used a form of the Rossby radius by demonstrating that the nonlinear effects of diabatic (Schubert and Hack 1982; Frank 1983) that accounts heating on acceleration of the tangential winds depend for locally enhanced vorticity: signi®cantly on the immediate state of the large-scale c circulation. ␭ ϭ n , (19) Zhang and Fritsch (1988a) suggest that the convective R 1/2 Ϫ11/2 (␨ϩf)(2VR ϩ f ) moistening of the atmosphere in a region of positive where ␨ is the relative vorticity, V the tangential velocity relative vorticity produces a further stabilizing in¯uence at some radius R, f the Coriolis parameter, and cn the through what they term the ``virtual temperature effect.'' phase speed of the nth mode pure gravity wave. The They argue that a 1ЊC increase in Tv through the virtual quantity in the denominator of (19) is the square root effect could account for a signi®cant portion of the geo- of the inertial stability parameter (Schubert and Hack potential perturbation observed in long-lived midlati-

1982). The form of ␭R in (19) reduces to the more fa- tude vortices and that this may be especially important miliar result (Gill 1982) in the genesis stage of MCVs since, unlike sensible heating, the local effects of moistening (virtual warm- ␭ ϭ c /f, (20) R n ing) cannot be dissipated by gravity±inertia waves. in the absence of signi®cant relative vorticity and cur- The work of DW offers a contrasting viewpoint of vature, that is, as ␨ → 0 and R → ϱ. This form has the the importance of large-scale inertial stability to vortex clear interpretation as the distance a wave would travel formation and longevity. The numerical simulations of during the rotational timescale 1/f. Skamarock et al. 1994 and DW demonstrated that bal- MCSs typically develop a strong positive vorticity anced squall line ``end vortices'' can form even in the maximum (and associated PV anomaly) in the lower absence of background absolute vorticity. DW suggest and middle troposphere (e.g., Schubert and Hack 1982; that the signi®cance of inertial stability depends on Raymond and Jiang 1990; Hertenstein and Schubert scale, and that for scales less than 100 km, existence of 1991; Fritsch et al. 1994; Hertenstein 1996; Part I). weak low-level shear is a more dominant factor. Cotton et al. (1989) argue that as an MCC develops this Clearly, the viewpoint of DW not withstanding, in- maximum, it achieves greater inertial stability, and, ertial stability provides a tremendously important con- hence, a smaller ␭R. The inertial stability (the horizontal straint on the motions excited in the atmosphere as a analog of static stability) provides a measure of the hor- response to thermal forcing. Unfortunately, the previous izontal restoring force resisting lateral displacements. theoretical studies have used idealized geometries (i.e., Enhancement of the inertial stability is often referred to axisymmetry) and other approximations and simpli®- as ``stiffening'' (Ooyama 1982; Schubert and Hack cations such as ideal vortex structure and linearization

1982). For systems smaller than ␭R, the energy (here in that may lead to analytic solutions, but whose appli- the form of CAPE) released in the deep convection ex- cability to the fully three-dimensional MCC is not well cites gravity waves, which effectively radiate energy de®ned. Many, if not most, MCCs have a linear structure from the system. For MCSs larger than ␭R, more of the in the leading edge of the convection. Regrettably, no 15 FEBRUARY 1997 OLSSON AND COTTON 493 fully three-dimensional theory of inertial stability exists mode. This result is not surprising, in light of the good for the general case. agreement between Vm and Vb in the vortical circulation To apply (19) to their composite MCC, Cotton et al. in Fig. 2. Typical values for the mature storm as a whole (1989) chose R to be ϳ320 km, corresponding to the might be R ഠ 250 km, ␨ ഠ 1.5 ϫ 10Ϫ4 sϪ1, f ϭ 1.0 ϫ radius of the cloud shield with temperature Յ Ϫ33ЊC. 10Ϫ4 sϪ1, V ഠ 15 m sϪ1. Assuming the phase speeds

A problem arises in calculating the appropriate gravity- above are appropriate, ␭R ϭ 214 km for the gravest inertia phase speed from (14) to use in these calculations mode and ␭R ϭ 107 km for the second vertical mode. of ␭R, since the Brunt±VaÈisaÈlaÈ frequency shows con- The circulations attributable to the MCC certainly siderable temporal and spatial variation. Further, there exceed 200 km in scale. This analysis also suggests that are different phase speeds (and hence a distinct ␭R) for all higher order vertical modes would be strongly mod- each vertical mode. Phase speeds can vary from ϳ3 → i®ed before they could propagate a great distance from 300msϪ1 (Schubert et al. 1980), giving a wide range the storm, while the ®rst mode would radiate a signif- of values for the different ␭R. Cotton et al. chose a value icant amount of energy away from the MCC. These ␭R of 30 m sϪ1 based on results of a two-dimensional sim- calculations, while imprecise, further support the con- ulation by Tripoli and Cotton (1989). Using these values clusion that the MCC in this study is, to a large extent, at a latitude of 40Њ, they computed ␭R ϭ 300 km for a balanced system. the composite MCC. This ␭R was slightly smaller than their cloud shield radius R, of about 320 km. Other 5. Summary and conclusions researchers have found similar values for the Rossby radius (e.g., Johnson and Bartels 1992). This and a companion paper (Part I) are an attempt

While the interpretation of ␭R is fairly clear for simple to resolve the question: Does the MCC represent a bal- systems, caution needs to be exercised in interpreting anced ¯uid-dynamical system? The answer to that ques- the calculated values for ␭R in Cotton et al. (1989) and tion is, to a large extent, yes, at least for this simulated in the present context since both the static stability and case. The nondivergent component of the PE model vorticity generally show considerable variation within winds was found to consist, to a great degree, of bal- the domain. The approach used here is to de®ne ``typ- anced ¯ow, Vb⌿. Further, this was not a state that was ical'' values of the necessary variables in the appropriate achieved after some period of adjustment. Rather, the context, and other choices can arguably be equally valid. nondivergent winds, throughout the MCC lifetime, from

It should also be borne in mind that a different ␭R exists initial convection to dissipation, remained in a nearly for each mode of vertical wave structure. The resultant balanced state. This almost ``instantaneous adjustment''

␭R(m) may be considered a scale or ``e-folding'' length was also observed by DW in their balanced diagnosis appropriate to that mode. Some adjustment still occurs of an idealized squall line and associated MCV. Perhaps for distances greater than ␭R. more surprisingly, the storm-induced divergent model The analysis of the unbalanced ¯ow presented earlier winds also remained balanced to a fair degree, though offers some speci®c guidance in the computation of ␭R certainly less so than the nondivergent ¯ow. Further, the in our study. While several factors can locally in¯uence balanced divergent ¯ow comprises a signi®cant portion the value of cn, the calculations of ␭R discussed below of the total balanced ¯ow in some regions of the MCC. assume that cn as calculated for the large-scale envi- Gravity waves, induced by the intense heating, consist ronment is appropriate. The suitability of this choice is almost entirely of divergent ¯ow, which is unbalanced. discussed in appendix C. Using the more conventional The good agreement between model and balanced up- notation for ␭R in (20) (a deformation radius appropriate ward vertical mass ¯ux, together with the periodic wave- to the large-scale storm environment) and the phase like appearance of ␻u and Vu␹ and the known ®ltering speeds found earlier, yields values of 509 km and 255 characteristics of the balance model, suggests that the km for the ®rst 2 modes, respectively. Where the vor- bulk of the three-dimensional unbalanced divergent mo- ticity is signi®cantly perturbed by the MCC, ␭R as com- tion may be attributed to gravity waves. puted using (19) will be smaller. The PE simulation Within the averaging volume, the greatest disparity discussed in Part I indicates that an MCC has several between the model and balanced circulations was found important structures, each having a relevant length scale. in the downward vertical motion. Both ␾bϪ and ␾mϪ, The vorticity can also vary by several hundred percent, the integrated downward mass ¯uxes, were considerably even in somewhat radially shaped features. Still, (19) smaller than their upward counterparts, indicating that may be applied, at least heuristically, with some guid- the subsidence compensating both ␾bϩ and ␾mϩ occurred ance from the balanced analysis. over an area much greater than the averaging volume. Several of the features within the MCC are clearly The model downward mass ¯ux exceeded the balanced balanced. For example, the MCV, which formed early downward ¯ux by a factor of 2 at almost all times during in the storm, has R ഠ 100 km, and ␨ ഠ 3.5 ϫ 10Ϫ4 sϪ1 the lifetime of the simulated MCC. This suggests that as an average over the storm-relative closed circulation. the process of mass adjustment due to convective heat- Ϫ4 Ϫ1 Ϫ1 Using f ϭ 1.0 ϫ 10 s and V ഠ 10 m s , ␭R ϭ 138 ing is largely dominated by unbalanced fast-manifold km for the n ϭ1 mode and ␭R ϭ 69 km for the second processes, such as inertia±gravity waves, a theory sup- 494 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54 ported by recent research on the convective adjustment general, and other MCSs, it would be useful to apply process. This adjustment process, which occurs on a the balanced analysis used in this paper to other MCC scale much larger than the more con®ned upward ¯ux, simulations. In particular, it would be illuminating to has a much larger timescale and the instantaneous ad- analyze storms developing in differing environments justment assumed in the balanced system is evidently and at different latitudes. less valid for this process.

The ␻u and divergence ®elds showed strong evidence Acknowledgments. The authors are grateful for the of the predicted domination by vertical waves 1 and 2. thoughtful advice and encouragement offered from sev- The (modi®ed) Rossby radii for these two lowest-order eral parties, in particular Rolf Hertenstein, Ute Olsson, modes were calculated and it was shown that the MCV Scott Rafkin, and Dr. Robert Walko. Special thanks go within the storm was larger than ␭R for all waves with to Drs. Wayne Schubert and Michael Montgomery, who n Ͼ 1. The ␭R for the mature storm as an ensemble also were instrumental in the development and understanding nϭ1 predicted a good degree of balance with ␭R scaling of many of the balanced dynamical concepts used here- similarly to the MCC. Based on the results presented in. Appreciation is also extended to Drs. Hongli Jiang here, we suggest the following re®nement of the dy- and J. M. Fritsch and to Ray McAnelly and two anon- namic de®nition for MCCs put forth by Cotton et al. ymous reviewers for their careful and constructive re- (1989): views of this manuscript. This work was supported by A mature MCC represents an inertially stable mesoscale the National Science Foundation through Grant ATM- convective system that is in a nearly balanced dynamical state 9118963. and whose horizontal scale is comparable to or greater than a locally de®ned ␭R. APPENDIX A A question related to the representativeness of the Details of the Numerical Solution MCC as a balanced system is the question: Can a prog- nostic balanced model faithfully represent the full evo- The solution of the balanced ⌿,⌽ equation set used lution of an MCC? The results presented here and in here is quite similar to that found in Davis and Emanuel Part I, when contrasted with those of DW, suggest that (1991, hereafter referred to as DE). With moderate forc- the answer might signi®cantly depend on the large-scale ing, either one of these equations is readily solved using environment and initial conditions of the simulation. In a standard method such as successive over-relaxation the case considered by DW, the storm generation of (SOR). Solving the system as a whole, however, proves horizontal vorticity by the divergent cold pool, dV␹/dz, to be much more dif®cult. The application of this bal- a term neglected in the NLB approximation, was es- ance model to the MCS regime is aggravated by the sential in the development of the vortical structure seen extremely large shears and strong temperature gradients in their simulation. As discussed in Part I, the MCV in (and hence large PV gradients) generated locally by the the PE simulation analyzed here did not result from the deep convection. While, at least in principle, a balanced same mechanism as in DW. Instead this MCV formed solution exists to this PV distribution, the nonlinear by convergence/stretching of ambient vertical vortici- terms are so dominant that success in achieving con- tyÐa mechanism well-represented within the approxi- vergence with the numerical method becomes very de- mations inherent in the NLB systemÐand as such pendent on the initial guess. The initial nonlinear error should be more realizable in a prognostic NLB model. often forces the solution out of its attractor basin within Any de®nitive resolution to this question, however, will the ®rst cycle. require the application of prognostic balanced models For times when strong convection was not present, to MCS simulations in a variety of realistic environ- an adequate initial guess was simply the primitive equa- ments. For any situation in which the subsequent evo- tion solution interpolated onto the analysis grid. When lution of the ¯uid state depends largely on the mechanics deep convection was present, the initial guess consisted of the transient adjustment process, however, (e.g., the of a combination of the balanced solution at an earlier lifting of a parcel beyond its level of free convection time and the PE solution at the current time while by gravity wave oscillations) a balanced model would boundary conditions were derived (by necessity) en- not be expected to perform well. tirely from the current PE solution. During periods The results found in this study present a picture of where domainwide diurnal temperature changes were the ``signature'' an MCC leaves in its environment after occurring, this method was applied only to the stream- the decay of transients. Hopefully, this information will function since the geopotential was systematically be found useful in parameterizing the effects of MCCs changing over large areas. At some times an additional and MCSs in general circulation and climate models nine point Gaussian smoothing of the initial guess ®elds where such features are typically on the order of a single was required to achieve convergence. grid cell. It is often tempting to overgeneralize the re- In addition to the carefully constructed initial guess sults of a particular case study. To determine the ap- discussed above, a few other adjustments were neces- plicability of the results presented above to MCCs in sary to ensure convergence. During the ®rst few cycles 15 FEBRUARY 1997 OLSSON AND COTTON 495

␲2 This divergence was also used as the ®rst guess in theץ/2⌽ץ) of the inversion process, unstable lapse rates Ͻ 0) would occasionally appear as the geopotential cyclic-iteration procedure. made relatively large adjustments to a balanced state, A much more detailed discussion of the solution tech- creating a numerical instability. When this occurred, the nique can be found in Olsson (1994). static stability was set to a very small positive value. As the solver converged to a balanced ⌽ solution this problem disappeared. APPENDIX B Another consequence inherent in the solution of this The Analysis Grid inversion problem is a ``ringing'' as the system ap- The analysis grid corresponds to a subvolume of the proaches convergence and oscillates between two so- model grid 2 (see Fig. 1, Part I, for locations of the lutions. This oscillation is damped by using an under- model grids). To obtain this subvolume, the lateral relaxation method after each cycle. Underrelaxation af- boundary points of grid 2 were discarded, and to avoid ter the ®rst few cycles was also useful in dampening possible nonphysical noise due to the upper boundary, out the positive feedback runaway behavior due to the the top ®ve model levels were discarded. As the diag- initial guess error. Iversen and Nordeng (1984) found a nostic system assumes frictionless ¯ow, it would also comparable behavior with their family of balanced prog- be desirable to eliminate points in the boundary layer nostic models. DE reported using a similar damping since friction effects are generally most signi®cant near method in their diagnostic system. the surface. The boundary layer in this simulation was, Appropriate boundary conditions are an essential part at times, several kilometers deep. As a compromise, the of the PV inversion (Hoskins et al. 1985). Balanced bottom three model levels of grid 2 were also discarded. model simulations are often formulated such that the Once this subset of model data was obtained for each boundary conditions are simple to apply. When invert- analysis time (23/1800 to 24/1200 at 30-min intervals, ing observed or PE model-generated data, the speci®- all times UTC), the bottom level at each time was cation of boundary conditions can be much more dif- searched to ®nd the minimum ␲ value, and the top level ®cult. was searched for the maximum ␲ level. This de®ned Boundary conditions as described in DE were used the range of ␲ (␲ and ␲ , respectively) common to all for the PV inversion with ⌿ and ⌽ from the PE sim- b t grid columns in the model volume (x,y,␴ ) subset, and ulation speci®ed on lateral boundaries. The same ap- z determined the bottom and top of the analysis (x,y,␲) proach was used for horizontal (top and bottom) bound- domain. With ␲ and ␲ determined, a clamped cubic ary conditions for ⌿. A Neumann horizontal boundary b t spline z → ␲ interpolation (Burden et al. 1981) of each ␲ ϭϪ␪, withץ/⌽ץ condition was used for ⌽, namely ®eld in the dataset was performed on each vertical col- ␪ on these boundaries speci®ed from the PE model re- umn to place the model data on 16 evenly spaced con- sults. For the ␹, ␻ solver, boundary conditions consistent stant-␲ surfaces. The model grid 2 horizontal grid spac- with the PE model are somewhat harder to derive since ing (Part I) was retained. The result was an analysis a relationship similar to (2.4) in DE does not exist for domain, 78 points E±W and 63 points N±S, with a hor- the divergent wind. In general, to maintain mass con- izontal spacing of 25 km, and 16 ␲ levels, ranging from tinuity in a domainwide sense, some divergent wind is ␲ ϭ 963JKϪ1 kgϪ1 to ␲ ϭ 500JKϪ1 kgϪ1, separated required at each level. A boundary condition of ␹ ϭ 0 b t by a constant ⌬␲ of 30.87 J KϪ1 kgϪ1. Table 2 lists the at the lateral boundaries was used for these calculations. ␲ levels, their millibar equivalents and a representative This boundary condition has the effect of eliminating geometric height for each level, which was taken from any tangential component of the divergent wind ®eld a point near the center of the storm region at 23/1800. along the boundaries. This side effect was found to be The highest terrain and hence the lowest ␲ on the ␴ negligible when interpreting the results in the interior z ϭ 4 level of grid 2 is found along the western boundary of the domain. and in the Appalachian, Ozark, and Smoky Mountains. The upper and lower horizontal boundaries present Southern Iowa and northern Missouri, the region over more of a problem. The upper boundary is above 100 which the storm occurred, has signi®cantly lower ter- mb, well into the lower stratosphere. Due to the very rain, making the bottom of the analysis domain well (ϳ stable nature of the stratosphere, it is reasonable to as- 1 km) above the surface in the region of greatest interest sume that most of the vertical motion seen there is due and mitigating surface frictional effects ignored in the to gravity waves, and a rigid lid (␻ϭ0) is justi®ed. The balanced analysis. bottom of the analysis domain (a constant pressure sur- face) is not the earth's surface and boundary condition of ␻ϭ0isnot justi®ed here. Indeed, a signi®cant amount APPENDIX C of the convergence feeding the convective updrafts is Local Variation of cn found very near the surface where the high ␪e values are found. To derive ␻(x,y) at the lower boundary, (6) Within the region of a warm-core vortex with its large was integrated downward using ␻top ϭ 0 and the two- PV, the static stability is typically greater than that of dimensional divergence obtained from the PE model. the surrounding environment (Raymond and Jiang 496 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 54

1990). In the present case, the Brunt±VaÈisaÈlaÈ frequency, Cram, J. M., R. A. Pielke, and W. R. Cotton, 1992: Numerical sim- N, de®ned by ulation of an analysis of a prefrontal squall line. Part II: Prop- agation of the squall line as an internal gravity wave. J. Atmos. Sci., 49, 209±225. ln␪ gdT ץ N2ϭg ϭϩ⌫, (C1) Crook, N. A., 1987: Moist convection at a surface cold front. J. Atmos. .zTdz d Sci., 44, 3469±3494ץ ΂΃ ÐÐ, R. E. Carbone, M. W. Moncreiff, and J. W. Conway, 1990: The has been increased in the MCV by about 30% over the generation and propagation of a nocturnal squall line. Part II: surrounding environment. Here, ⌫ in (C1) represents Numerical simulations, Mon. Wea. Rev., 118, 50±65. d Davis, C. A., and K. A. Emanuel, 1991: Potential vorticity diagnostics the dry adiabatic lapse rate and T the environmental of cyclogenesis. Mon. Wea. Rev., 119, 1929±1953. temperature. Since the phase speed cn ϱ N, from (19) ÐÐ, and M. L. Weisman, 1994: Balanced dynamics of mesoscale or (20) one might expect a dilatation of ␭R in the region vortices in simulated convective systems. J. Atmos. Sci., 51, 2005±2030. of enhanced static stability due to a locally larger cn there. Durran, D. R., and J. B. Klemp, 1982: On the effects of moisture on the Brunt±VaÈisaÈlaÈ frequency. J. Atmos. Sci., 39, 2152±2158. Moisture saturation is another factor effecting the Emanuel, K. A., 1983: Elementary aspects of the interaction between static stability. For a saturated parcel displaced verti- cumulus convection and the large-scale environment. Mesoscale cally, latent heating (cooling) partially compensates the Meteorology: Theories, Observation and Models, D. K. Lilly temperature change due to adiabatic expan- and T. Gal-Chen, Eds., D. Riedel, 551±575. sion(compression). Since a saturated parcel (ignoring Frank, W. M., 1983: The cumulus parameterization problem. Mon. Wea. Rev., 111, 1859±1871. precipitation effects) conserves equivalent potential Fritsch, J. M., J. D. Murphy, and J. S. Kain, 1994: Warm core vortex temperature, ␪e, rather than ␪, a moist Brunt±VaÈisaÈlaÈ ampli®cation over land. J. Atmos. Sci., 51, 1780±1807. frequency, Nm, given by Gill, A. E., 1982: Atmosphere±Ocean Dynamics. Academic Press, 662 pp. -ln␪ gdT Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of at ץ 2 g Nmmϭg ϭϩ⌫, (C2) mospheric vortices to heating by organized cumulus convection. .zTdz΂΃ J. Atmos. Sci., 43, 1559±1573ץ Haltiner, G. J., and R. T. Williams, 1980: Numerical Prediction and with ⌫m the saturated (with respect to water) adiabatic Dynamic Meteorology. 2d ed. Wiley and Sons, 477 pp. lapse rate, is obtained (Durran and Klemp 1982). Com- Hertenstein, R. F. A., 1996: Evolution of potential vorticity associated parison of (C1) and (C2) shows that the difference be- with mesoscale convective systems. Dept. of Atmos. Sci. Paper 599, 177 pp. [Available from Department of Atmospheric Sci- tween N and Nm is due the differing values of ⌫d and Ϫ1 ence, Colorado State University, Fort Collins, CO 80523-1371.] ⌫m(T,p), the latter ranging from about 3.75 K gpkm ÐÐ, and W. H. Schubert, 1991: Potential vorticity anomalies as- Ϫ1 to the dry adiabatic value, 9.76 K gpkm in the tro- sociated with squall lines. Mon. Wea. Rev., 119, 1663±1672. posphere (Iribarne and Godson 1981). This saturated Holton, J. R., 1992: An Introduction to Dynamic Meteorology. Ac- ademic Press, 507 pp. adiabatic effect, by decreasing cn compared to that in an unsaturated atmosphere, would act to decrease ␭ . Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the R use and signi®cance of isentropic potential vorticity maps. Quart. Chen and Frank (1993), in a numerical study of ide- J. Roy. Meteor. Soc., 111, 877±946. alized MCVs, found signi®cant contraction of ␭R when Houze, R. A., Jr., 1989: Observed structure of mesoscale convective using Nm rather than N in their calculations. systems and implications for large scale heating. Quart. J. Roy. In the present case, we found that these two effects Meteor. Soc., 115, 425±461. largely cancelled, and our calculations of c for the Iribarne, J. V., and W. L. Godson, 1981: Atmospheric Thermodynam- n ics. D. Reidel, 259 pp. storm-perturbed region did not differ signi®cantly from Iversen, T., and T. E. Nordeng, 1984: A hierarchy of nonlinear ®ltered the larger-scale value. For this reason, and due to the models-numerical solutions. Mon. Wea. Rev., 112, 2048±2059. approximate scaling nature of ␭R, the larger-scale values Jiang, H., and D. J. Raymond, 1995: Simulation of a mature mesoscale for c are used here. convective system using a nonlinear balance model. J. Atmos. n Sci., 52, 161±175. Johnson, R. H., and D. L. Bartels, 1992: Circulations associated with REFERENCES a mature-to-decaying midlatitude mesoscale convective system. Part II: Upper-level features. Mon. Wea. Rev., 120, 1301±1320. Bretherton, C. S., and P. K. Smolarkiewicz, 1989: Gravity waves, Mapes, B. E., 1993: Gregarious tropical convection. J. Atmos. Sci., compensating subsidence, and detrainment around cumulus 50, 2026±2037. clouds. J. Atmos. Sci., 46, 740±759. McAnelly, R. L., and W. R. Cotton, 1986: Meso-␤ scale character- Burden, R. L., D. J. Faires, and A. C. Reynolds, 1981: Numerical istics of an episode of meso-␣ scale convective complexes. Mon. Analysis. Prindle, Weber & Schmidt, 598 pp. Wea. Rev., 114, 1740±1770. Charney, J. G., 1948: On the scale of atmospheric motions. Geofys. ÐÐ, and ÐÐ, 1989: The precipitation lifecycle of mesoscale con- Publ., 17, 1±17. vective complexes over the central United States. Mon. Wea. ÐÐ, and M. F. Stern, 1962: On the stability of internal baroclinic Rev., 117, 784±808. jets in a rotating atmosphere. J. Atmos. Sci., 19, 159±172. McWilliams, J. C., 1985: A uniformly valid model spanning the re- Chen, S. S., and W. M. Frank, 1993: A numerical study of the genesis gimes of geostrophic and isotropic, strati®ed turbulence: Bal- of extratropical convective mesovortices. Part I: Evolution and anced turbulence. J. Atmos. Sci., 42, 1773±1774. dynamics. J. Atmos. Sci., 50, 2401±2426. ÐÐ, and P. R. Gent, 1980: Intermediate models of planetary cir- Cotton, W. R., M.-S. Lin, R. L. McAnelly, and C. J. Tremback, 1989: culations in the atmosphere and ocean. J. Atmos. Sci., 37, 1657± A composite model of mesoscale convective complexes. Mon. 1678. Wea. Rev., 117, 765±783. Nicholls, M.E., R. A. Pielke, and W. R. Cotton, 1991: Thermally 15 FEBRUARY 1997 OLSSON AND COTTON 497

forced gravity waves in an atmosphere at rest. J. Atmos. Sci., adjustment in an axisymmetric vortex. J. Atmos. Sci., 37, 1464± 48, 2561±2572. 1484. Olsson, P. Q., 1994: Evolution of balanced ¯ow in a simulated me- ÐÐ, S. R. Fulton, and R. F. A. Hertenstein, 1989: Balanced atmo- soscale convective complex. Dept. of Atmos. Sci. Paper 570, spheric response to squall lines. J. Atmos. Sci., 46, 2478±2483. 177 pp. [Available from Department of Atmospheric Science, Schmidt, J. M., and W. R. Cotton, 1990: Interactions between upper Colorado State University, Fort Collins, CO 80523-1371.] and lower tropospheric gravity waves on squall line structure ÐÐ, and W. R. Cotton, 1997: Balanced and unbalanced circulations and maintenance. J. Atmos. Sci., 47, 1205±1222. in a primitive equation simulation of a midlatitude MCC. Part Skamarock, W. C., M. L. Weisman, and J. B. Klemp, 1994: Three- I: The numerical simulation. J. Atmos. Sci., 54, 457±478. dimensional evolution of simulated long-lived squall lines. J. Ooyama, K., 1982: Conceptual evolution of the theory and modeling Atmos. Sci., 51, 2563±2584. of the tropical cyclone. J. Meteor. Soc. Japan, 60, 369±380. Tripoli, G., and W. R. Cotton, 1989: A numerical study of an observed Raymond, D. J., 1992: Nonlinear balance and potential vorticity orogenic mesoscale convective system. Part II: Analysis of gov- thinking at large Rossby number. Quart. J. Roy. Meteor. Soc., erning dynamics. Mon. Wea. Rev., 117, 301±324. 118, 987±1015. Zhang, D.-L., and J. M. Fritsch, 1988a: Numerical sensitivity of vary- ÐÐ, and H. Jiang, 1990: A theory for long-lived mesoscale con- ing model physics on the structure evolution and dynamics of vective systems. J. Atmos. Sci., 47, 3067±3077. two mesoscale convective systems. J. Atmos. Sci., 45, 261±293. Schubert, W. H, and J. J. Hack, 1982: Inertial stability and tropical ÐÐ, and ÐÐ, 1988b: Numerical simulation of the meso-␤ structure cyclone development. J. Atmos. Sci., 39, 1687±1697. and evolution of the 1977 Johnstown ¯ood. Part III: Internal ÐÐ, ÐÐ, P. L. Silva Diaz, and S. R. Fulton, 1980: Geostrophic gravity waves and the squall line. J. Atmos. Sci., 45, 1252±1268.