generation and propagation of african squall lines
TRANSCRIPT
Quart. J . R. Met. SOC. (1984), 110, pp. 695-721 551.515.81S1.553.8
Generation and propagation of African squall lines By DAVID BOLTON
Atmospheric Physics Group, Imperial College
(Received 21 December 1982; revised 5 August 1983)
SUMMARY A study of squall lines affecting Minna, Nigeria during 1974-1976 emphasizes two particular aspects of
their dynamics: that a strong jet around 650mb is essential for their development, and that the forward edge of the squall line moves slightly faster than the maximum jet speed. A linear model of the squall line as a superposition of normal modes of an unstably stratified sheared flow sheds light on these observations. If the flow profile has a mid-level maximum, constructive interference of normal modes gives rise to an expanding circle of convection with an arc front moving slightly faster than the maximum mean flow speed, followed by individual cells. If, however, shear is unidirectional, there is no arc front, and a line of cells is formed moving at the middle level mean flow speed. These two cases are suggested as models for tropical and mid-latitude squall lines respectively, while a further case provides a possible model for storm-splitting.
1. OBSERVATIONS OF SQUALL LINES AT MINNA
To a surface observer, a squall line (SL) in West Africa may be identified on approach by appearance of a dark heavy band of cumulonimbus, at the forward edge of which is a long roll of low cloud (Hamilton and Archbold 1945). In contrast, local thunderstorms do not normally exhibit these features, have a smaller horizontal extent, and are generally less severe. On satellite pictures, SLs may be distinguished from other cloud systems by their compact oval shape and very high brightness (Fortune 1980)- factors used, along with explosive growth, in identification of SLs in a study by Martin and Schreiner (1981).
Bolton (1981) made a study of SLs at the newly opened upper air station at Minna (9'37" 6"34'E) for the period June 1974 to August 1976. As the approaching bank of cumulonimbus with its roll cloud is not reported in routine surface observations, iden- tification of SLs had to be based on their severity and horizontal extent. Four features were selected: (i) maximum squall speed at least 12 4 s ; (ii) rainfall at least 5 mm; (iii) storm duration at least three hours; (iv) pressure jump at least 0.5 mb.
Squall lines were identified as those thunderstorms exhibiting (i) to (iv) at Minna, and (i) to (iii) in at least one of the surface stations within a 300 km radius of Minna (Jos, Kano, Kaduna, Lokoja, Bida, Yelwa and Ilorin), though the conditions at Minna were somewhat relaxed in cases where there were a number of clear reports at other stations. Eighty SLs were thus identified, while in another nineteen cases of possible SLs the data were judged to be inconclusive. The criteria adopted may seem somewhat weak, but as a typical SL is of length 300 km (Eldridge 1957) it would not be expected to pass all the stations; also little rain would fall at a station unless a cumulonimbus cell passed directly overhead. Porter ef al. (1955), in a study of SLs in the United States, used a similar, but rather more stringent set of criteria, this being possible because of the greater density there of the surface station network.
During GATE it was possible to supplement the surface data with pictures from Synchronous Meteorological Satellite SMS-1, and these indicated that many of the SLs were generated over the Jos plateau, a region of high ground (around 100Om) to the north-east of Minna; Reed and Jaffe (1981) pointed out the importance of high ground in SL generation. The satellite pictures also assisted in estimation of the velocity of motion of the SLs and the mean for this GATE period, 17.5 m / s from 083", will be used subsequently as the reference mean (it differed slightly from the 16.4 m / s , 079" obtained for the full 27-month period). Satellite pictures were, however, insufficient to pick out
695
696 D. BOLTON
all the SLs identified by the surface observations, as some were associated with cloud masses with a diffuse leading edge and showing very little growth, which Martin and Schreiner would reject as SLs. Possibly such SLs formed in a larger cloud cluster, or had their anvils blown ahead of the squall front by strong easterly upper-level winds.
Using the eighty SLs identified at Minna, it was possible, using only routine upper air soundings, to infer conditions before and after passage of a SL, and even to produce a composite time-height section over the land which reproduced qualitatively some of the results, including velocity and divergence patterns, that Gamache and Houze (1982) obtained from intensive radar observations over the ocean. Two sets of results are of particular relevance to the present mathematical model, and will now be presented.
(a ) Conditions for occurrence of squall lines For the GATE period 10 June to 22 September 1974, mean Minna soundings at
1200 and 0000 GMT for days on which SLs did not occur are shown in Figs. 1. Bars of length equal to the standard error, e, of the difference of the means are drawn, two means being regarded as significantly different at the 95% confidence level if they differ by at least 2e (see for instance Spiegel 1972).
Figures l(a) and (b) show mean wind components along and perpendicular to the GATE mean direction of SL motion. A strong 150 mb tropical easterly jet is evident at both 1200 and 0000 h, while the African easterly jet maximum at 650 mb is in both cases much weaker. A similar wind profile was obtained by Frank (1978) for cloud clusters over the ocean. Surprisingly there is a statistically significant increase in the ‘southerly’ component in the middle troposphere during the day, possibly the ageostrophic flow generated by temperature differences due to the greater energy absorption during the day, and emission at night, of the cloud tops as compared to the clear air to the north (McBride and Gray 1980).
One other variable is necessary for the squall line model: parcel excess log(potentia1 temperature), given by
&p = In( ~pparcel/~environrnent) (1)
I
I - -20 -15 -10 - 5
E
,100
ME3
,400
-700
5 m/s W
,
AFRICAN SQUALL LINES 697
where OPparcel is found by taking a parcel dry adiabatically from an inflow level of 950 mb to its lifting condensation level (given by Eq. (22) of Bolton 1980a), then moist adi- abatically above ( Oparcel being found here by inverting Eq. (43) of Bolton 1980a). The 0000 h profile (Fig. l(c)) shows, at all levels, less buoyancy than at 1200 h, and this is particularly significant below 800mb, where there is a strong negative buoyancy at 0000 h, but very little negative buoyancy at 1200 h. Thus in the afternoon on non-SL days there is very little hindrance to small-scale convection, and consequently little opportunity for formation of the warm moist surface layer necessary for SL development.
Most of the soundings available at Minna were for midday, so for the 60 occasions in the period June 1974 to August 1976 when soundings were available both before and after passage of a SL, the means before and after soundings were found (Figs. 2). Error bars show standard errors, e , of the means of the differences between the soundings.
The wind profiles are again taken along and perpendicular to the GATE mean direction of propagation. The ‘westerly’ means (Fig. 2(a)) differ from those on non-SL days in that the upper-level tropical easterly jet is much weaker, while the African easterly jet has a much stronger maximum at 650mb, a situation similar to that found over the oceans in Frank’s (1978) comparison of cloud clusters with SLs. The African easterly jet is actually stronger than the tropical easterly jet both in the means and in most of the individual soundings, so that a necessary condition for occurrence of a SL is a strong middle-level wind maximum, preferably stronger than any upper-level wind. The ‘southerly’ means, however, show very little shear. After passage of the SL, in contrast to the sharp increase in low-level shear observed in VIMHEX (Betts et al. 1976), very little change is evident here in the ‘westerly’ mean, suggesting that wind profiles return to pre-squall conditions in substantially less than 24 hours. Surprisingly, however, there is a significant increase in the ‘southerly’ mean between 600 and 300 mb, which could be caused by retention of southerly momentum of low-level air rising in updraughts, or by movement of easterly wave troughs past Minna since, according to Payne and McGarry (1977), there is a preferred generation of SLs ahead of the troughs.
m /5
Figure 2. Minna means before and after squall line for 24-h interval. (a) ‘Westerly’ wind component u ( m / s ) ; (b) ‘southerly’ wind component u ( 4 s ) ; (c) parcel excess log(potentia1 temperature) a&. Full line: before; dashed line: after; dash-dot line in (a): mean speed of squall line; dotted line: non-squall mean; bars
denote one standard error for mean of difference.
698 D. BOLTON
The pre-squall d@p profile (Fig. 2(c)) shows a much greater negative buoyancy below 700 mb than does the non-SL mean, but significantly greater positive buoyancy above, so that conditions are ideal for accumulation of potential energy in a surface layer as a result of solar heating. The post-squall sounding shows even greater negative buoyancy below 700 mb, so it is not surprising that convection is suppressed after passage of a SL. There is, however, an overall net positive area, which, taken with the restoration of pre-squall wind conditions, means that occurrence of a second SL within a relatively short time is not unlikely; for the eighty SLs studied, for instance, there were nine occasions on which SLs occurred within 24 hours of each other.
( b ) Estimates of storm velocity Various workers have suggested methods for estimation of storm speeds from other
(i) Gurka (1976) noted that the storm speed is nearly equal to the maximum speed of
(ii) Moncrieff and Miller (1976) gave a formula modified by Fernandez and Thorpe
parameters, for example:
the wind squall;
(1979) to
c = D + 0*32(APE)'D (2) where c is the storm speed, a the component of the mean tropospheric wind in the direction of progation, and
APE = [' gd@, dz, (3)
d@p being parcel excess log(potentia1 temperature), as in (l), z1 the level of inflow (taken in this case to be 0.5 km), and z2 the equilibrium level of parcel (level where d@p = 0), all variables being calculated from a pre-squall sounding;
(iii) Fernandez (1980) showed observationally that storm velocity is close to the maximum mid-level environmental wind velocity. When these methods are used to estimate the speed of Minna SLs, errors are
typically several m/s, similar to those found by Fernandez (1980, 1982) for SLs over Venezuela, the east Atlantic and West Africa. But what is really required is to know direction as well. This is not actually considered by Gurka in (i). while Moncrieff and Miller's study, (ii), is essentially two-dimensional, and presupposes knowledge of the direction of propagation. Plausibly, however, the respective directions of the wind squall and mean wind may be assumed in these two cases, so that the three estimates of the storm velocity are: (i) the maximum wind squall velocity;
(ii) the Moncrieff-Miller velocity, i.e. the mean wind velocity, with speed increased by an amount 0*32(APE)G;
(iii) the maximum mid-tropospheric wind velocity. These are tested against the data in Fig. 3. Figure 3(a) shows that, on average, the
storm moves faster and from a more northerly direction than the squall, maximum wind or mean wind. Figures 3(b) to (e) give scatter diagrams of the storm velocity relative to the ground, and to the three estimates for the storm velocity. The scatter is least in Fig. 3(b), suggesting that, in the particular situation at Minna, the mean storm velocity 079", 16.4 m/s is the best estimate of velocity of a SL. For a general prediction rule however, Figs. 3(c) to (e) must be considered, and in these, the magnitude of the relative velocity, (v,( = (4 +of)@, gives a measure of the error of the estimate; in each case, values of . .
AFRICAN SQUALL LINES 699
N
Storm t V
U
'J r
0 5 10 15 20mk -
Figure 3. (a) Means of storm velocity (16.4 m / s , 079"), squall velocity (14.5 m / s , 093"), maximum mid-tropospheric wind velocity (15.5 m / s , 089') and mean tropospheric wind velocity (6.9 m / s , 099'). (b) Scatter diagrams of storm velocity ( 4 s ) ; u and v are the westerly and southerly components, and each 5 m / s x 5 m / s square gives the number of times values of u and v occur in the relevant range. (c) to (e) As for (b), but with u and v replaced by relative velocity components ur and vr along and perpendicular to: (c) the maximum wind squall; (d) the Moncrieff-Miller velocity; (e) the maximum mid-level wind. All
diagrams are drawn to the same scale, as indicated at the top right.
Velocity statistics for squall lines affecting Minna in the period June 1974 to August 1976.
700 D. BOLTON
lvrl as great as the actual storm speed are not uncommon. In particular, for the squall and Moncrieff-Miller velocities, the error Iv,I actually exceeds the storm speed in 20% of all cases, while the magnitude of the vector mean value of vr is at least 5 m / s , indicating an unacceptably large systematic error, in addition to the enormous scatter. The sys- tematic error and scatter are somewhat less for the maximum velocity, which is thus suggested as the most satisfactory general estimate of SL velocity, thus confirming the results of Fernandez (1980). It does in fact give a slight under-estimate of speed, as seen in Fig. 3(a).
In conclusion, two results have emerged which will prove important to the SL model to be presented: (i) in the conditions for occurrence of SLs, a strong jet maximum at 650 mb is essential; (ii) as a general estimate of storm velocity, the maximum jet velocity is best, the storm
actually moving on average at a speed slightly greater than this maximum.
2. SQUALL LINE MODEL
The results of the Minna observations are consistent with Ludlam (1963), who concluded that, whereas mid-latitude SLs typically occur in conditions where the wind shear is more or less unidirectional, and move at a velocity typical of the wind in the middle troposphere, tropical SLs tend to occur in conditions where the wind speed has a maximum in the middle troposphere, and to move at a speed greater than this maximum wind speed. Another relevant observation is that in both types of SL the line may be seen to be composed of individual convective cells, usually triggered off by a single initial cell (Houze 1977); for the tropical SL in particular, the forward edge is often marked by a low-level arc-shaped roll cloud separated from the main cloud mass, with the main cloud mass usually having a sharp leading edge in which the active convective cells are embedded (Fortune 1980).
Moncrieff and collaborators (Moncrieff and Green 1972; Moncrieff and Miller 1976) have produced simple models of the airflows in Ludlam’s two categories of SL. By ignoring the cellular structure and representing the principal airflows as steady with two-dimensional inflow and outflow, they have carried out steady two-dimensional inflow/outflow analyses using equations similar to those of Dubreil-Jacotin (1935) and Long (1953), while Moncrieff (1978) has provided a two-dimensional model of the internal structure of a mid-latitude SL. Moncrieff‘s model, however, gives no idea as to why motion should develop in a two-dimensional, and not a three-dimensional configur- ation, while in all the models, dynamical considerations indicate the need for a three- dimensional internal structure with interlocking airstreams (thus agreeing with obser- vational evidence of occurrence of cells).
Short of a full numerical integration such as that of Moncrieff and Miller (1976), there is one type of model which does retain three-dimensionality and time dependence, but at the expense of omitting nonlinear terms, and this is the linear (gravity wave) model. Such a model will be seen to clarify some aspects of the initial development of a SL, and of its arc front; it does, however, over-simplify both the dynamics and thermodynamics, which leads to problems in the mature stage of SL growth. In a similar way the Simmons and Hoskins (1979) linear model of an unstable baroclinic wave was found to represent well the development of new depressions on the fringes of the region of motion, but to break down in the centre, where nonlinear effects were important.
Gravity wave models have been widely studied in connection with convection because they generate regions of low-level convergence which can trigger and sustain
AFRICAN SQUALL LINES 70 1
growth of convective elements. Most familiar are stable gravity waves in which gravity acts solely as a restoring force, as in Hamilton and Archbold’s (1945) SL model. These, however, lack an energy source, so would decay in an hour or two owing to viscosity and upwards energy propagation (Lilly 1979) and cannot therefore model either the explosive development or the persistence of a SL. Raymond (1975) and Fernandez and Thorpe (1979) attempted to remedy this deficiency by use of the forced gravity wave (wave-CISK) concept of Lindzen (1974), in which, within a stable environment, low- level convergence triggers off cumulus convection, which in turn acts as a driving mechanism for the wave. Mathematically, however, their model reduces to a linear gravity wave model with an unintended phase shift introduced between maximum forcing and maximum vertical motion (Bolton 1980b), and, while it obtained good predictions of propagation speed, it was unable to predict direction of propagation adequately.
Whilst the unstable gravity wave model does not answer the problem of modelling convection within a stable environment, in that it misses out altogether on gravity as a restoring force, it does avoid the phase shift of the wave-CISK model. Surprisingly it has received little attention in relation to SLs, the only previous work being single case studies by Marroquin and Raymond (1979, 1982) which concentrate on the prediction of storm velocity. This model will therefore be studied in some detail, setting out conclusions reached in Bolton (1981), where a fuller account may be found.
3. BASIC EQUATIONS
The basic equations which will be used are the linearized non-dimensional Boussinesq system
with boundary conditions
w = h = OatZ = +1.
Here (X, Y, Z) is position, non-dimensionalized so that the boundaries are at Z = +1, while T is non-dimensional time. The horizontal velocity (Su, Su) is taken as a small perturbation about a mean flow U(Z) = (U, V), while w is assumed small. Pressure is given by a small deviation Sp from a mean state po(Z), po(Z) being the mean density.
The variable h is defined as S@/& where d@ is the log(potentia1 temperature) deviation from a mean state &(Z), and Bo is the static stability, defined so as to incorporate the effect of latent heat release as the vertical gradient of the log(potentia1 temperature) excess, S&,, given by (1). As a fixed inflow level near the surface was taken, this had the effect of making the updraught buoyant over an unrealistically large altitude range, and of misrepresenting the downdraught thermodynamics, with conse- quences which will be noted in the concluding remarks. Buoyancy is incorporated via
702 D. BOLTON
the Richardson number
R = gB/S2 (9) where B is a representative (negative) static stability for the flow region, and S is a representative shear; the variation in buoyancy with height is given by the non-dimen- sional lapse rate L ( Z ) .
Diffusion is represented by the terms 4 ( - R / R u ) ’ w H . The 4(-R/Ru)’D is actually a reciprocal Reynolds number, but here it is convenient to work in terms of the Rayleigh number
Ra = -gBD4/K2 (10) where D is the depth of the convective layer, and K is the eddy diffusion coefficient (assumed constant). Only the horizontal derivatives V’, = d2/dX2 + d2/aY2 are used, these being essential, or otherwise the fastest growing convection would in general consist of infinitely short waves; the vertical derivatives a2/dZ2 can plausibly be omitted since attention will be confined to tall narrow cells.
4. NORMAL MODE ANALYSIS
Although this paper is concerned primarily with growth in time of an initially localized perturbation, it is necessary first of all to recall some results for individual normal modes. Thus solutions of (4) to (8) will be sought with an exponential dependence
exp[ik{Xcos 8 + Y sin 8 - cT} ]
c = cr + ici,
then cr is the phase speed, and k~ the growth rate, though plotted results will use growth rate non-dimensionalized with respect to buoyancy:
( 1 1 )
(12)
where k is the wavenumber, and 8 is the direction of propagation. Writing
= k s / ( - R ) @ . (13) For the normal mode analysis, attention will be restricted to the case of unidirectional
mean flow V = 0, with constant mean density po and lapse rate L ( ~ 1 ) . The linear system (4) to (7) with exponential dependence (11) thus reduces to
&U/dZ2 U ( Z ) - c’ - 4ik(-R’/Ra)* + ] i7= 0 (14)
a% R‘ az2 { U ( Z ) - c’ - 4ik(-R’/Ru)*}’
which is a form of the Taylor-Goldstein equation with added diffusion terms. Here Z is the Fourier transform of w while
R’ = R sec‘ 8 is the modified Richardson number appropriate to the particular direction of propagation 8, with correspondingly
c‘ = c sec 8 = c: + ic’ . With boundary conditions (8), this is an eigenvalue problem: to find values of c’ for which there is a non-zero solution for F.
One case is of particular importance in the analysis: the longitudinal roll 8 = in. Here R’ and d 2 both become infinite; their ratio, however, remains finite, and (14) has
(15)
(16)
703 AFRICAN SQUALL LINES
a solution
17 = constant x sin Inn(2 + 1)
where n is an integer and -R’ -R’ 112
c’ = i ( k2 + f n 2 d )” - 4ik (x) .
Growth rate a~ in (13) is then
aN = (1 + fn2d/k2)-* - 4k2Ra-’l2.
This is plotted in Fig. 4 against log k for different values of Ra for the first mode: n = 1; the log (to base 10) of wavenumber is used here and subsequently to show more clearly the short-wave behaviour.
Figure 4. Longitudinal mode, n = 1 : non-dimensional growth rate plotted against log(wavenumber) for different Rayleigh numbers.
In the absence of diffusion (Ra = a), growth rate increases with wavenumber, approaching unity in the short-wave limit k + a. If diffusion is included, growth rate is reduced, especially for short waves, which are damped (m < 0) for k greater than some critical wavenumber k,; maximum growth rate now occurs for some finite wave- number k, , given for Rail4 %- nn approximately by
k, = (nn)”/22-s/4Ra1/8 = 0.745n’/zRa’/S. (20)
These results show good qualitative agreement with those of Asai (1970a) who retained the a2/aZ2 diffusion terms in the Boussinesq system, and this gives some justification for their omission in this paper.
Apart from the longitudinal roll, no simple solutions of (14) exist for general mean flow profiles. Numerous studies of particular cases have, however, been made, and these are reviewed for example in Kelly (1978). Two particular results for unidirectional mean flows emerge from these studies which are of significance in the modelling of SLs. (i) Bolton (1981) has shown that it is very difficult for a mode to utilize both mechanical and thermal energy. For those modes which are thermally driven, the longitudinal roll (17) is the fastest growing, since “shear introduces a tilt into the convection cell which has a stabilizing influence by feeding perturbation energy into the mean flow by a Reynolds stress” (Kelly 1978; Asai 1970a, b). Squall lines are thermally driven (deriving
704 D. BOLTON
their buoyancy chiefly from latent heat release), as shown by the low-level up-gradient momentum transfer (Moncrieff and Miller 1976); yet they are transverse to the mean wind, not longitudinal. Modes with a preferred transverse orientation do occur for mean flows with a sufficiently pronounced inflexion point (as for instance in Eisler 1965), but only for very small buoyancies, and these are mechanically driven, transferring momen- tum down-gradient. (ii) Amplifying linear modes must, by the theorem of Miles (1961), have a steering level, i.e. a level at which phase velocity is equal to mean flow velocity. In contrast, tropical SLs usually travel faster than the mean flow at any level.
A tropical SL cannot therefore be modelled by a single amplifying linear mode. Comparison of eigenvalues for antisymmetric and symmetric flows did however emphasize the importance of group velocity, and this will be illustrated by the plane Couette and plane Poiseuille profiles:
U = Z (21)
u= 1 - 2 2 , (22)
and
previously studied by Taylor (1931), Eliassen etaf. (1953), Kuo (1963) and Asai (1970a), and by Lin (1955) and Gage and Reid (1968) respectively. The eigenvalues were computed by a combination of numerical methods, chiefly a Galerkin adaptation of the matrix method of Asai (1970a) to obtain a rough estimate, which was then improved by the shooting method. For stationary modes of the Couette profile, a graphical method similar to that of Eisler (1965) was used, while asymptotic behaviour for large wavenumber was given by short-wave expansions. Results (Figs. 5 and 6) show: (a) growth rates (b) two-dimensional phase speeds c: and group speeds
plotted for Ra = WJ (full line) and Ra = Id (dashed line);
C; = a(kc:)/ak; (23)
(c) the mean flow profiles, plotted on the same vertical scale as the phase and group speeds, so that it is immediately possible to read a particular phase/group speed back on to the mean flow. For the Couette profile, the growth rate graph branches for any value of -R’ greater
I6
I
0 log k -2 -1
(a)
Figure 5 . Plane Couette profile U = Z, R’ = -5. (a) Growth rate plotted against log(wavenumber) in the inviscid case (full line) and with Rn = 10s (dashed line). (b) Two-dimensional phase and group speeds plotted against log (wavenumber). (c) Mean flow profile. 0 marks the point of maximum growth rate for Rn = 1W.
AFRICAN SQUALL LINES 705
than 2 (illustrated, Fig. 5, for R’ = -9, and greatest growth rate occurs to the left of the branch point, i.e. for relatively long waves. For these long waves, phase speeds are zero, which shows that they move at the mid-level mean flow speed. For short waves, however, phase speed increases from 0 at the branch points to 5 1 in the short-wave limit, indicating a movement from the middle to the boundaries of the flow with increasing wavenumber. Like phase speed, group speed is zero for long waves, but at the branch points it becomes infinite, as in Kuo (1963), before approaching + 1 in the short-wave limit.
For the Poiseuille profile, the fastest growing mode never branches; for any value of -R’ less than about 2 it is in fact the only mode (illustrated, Fig. 6, for R’ = -1). This fastest growing mode always has its growth rate for Ra = steadily increasing to 1 in the short-wave limit, while phase and group speed both approach the mean flow speed at the velocity maximum, the approach for group speed being from above.
- . - - . -.
log k -2 -1 0
IRo = lo5
-2 L -1 0 log k I I I I
0 ’ log k -2 -1
( b)
-1 0 2 1
Figure 6. As Fig. 5 but with U = 1 - z2, R‘ = -1.
Thus the constraint that phase speed must be equal to some mean flow speed does not apply to group speed for either the Couette or the Poiseuille profile. So, provided the motion of tropical SLs is governed by group speeds, this overcomes the problem that they move faster than the mean flow at any level. There is, however, a difference between the two profiles. For Couette flow, greatest growth rate is for long waves, which move with the mid-level flow, while for Poiseuille flow, greatest growth rate is for short waves propagating at the maximum mean flow speed. At this point, the inclusion of diffusion becomes important, its effect being to reduce the growth rate of the short propagating waves (dashed lines in Figs. 5(a), 6(a)-shown for Ra = 105). For the Couette profile this does not affect the growth rate maximum, but for Poiseuille flow, maximum growth rate is now shifted to a finite wavenumber, at which group speed actually exceeds the maximum mean flow speed. (Maximum growth rates and corresponding group speeds are marked by 0 on the graphs.)
This result is highly significant in view of the observations of section 1: tropical SLs occur when there is a mid-level maximum to the mean flow, and this is precisely the situation in which group speeds of fastest growing waves may exceed the maximum mean flow speed; mid-latitude SLs on the other hand occur when the shear is almost uni- directional, and in this case fastest growing waves have group speeds equal to some mid-level mean flow speed. Bearing this in mind, it now becomes necessary to consider superposition of normal modes in order to discover what precisely is the role of group speed.
706 D. BOLTON
5. SUPERPOSITION OF NORMAL MODES
(a) Role of the group velocity Simmons and Hoskins (1979), following Ben (1975), have shown in two dimensions
that, when superposing amplifying exponential modes, the position of maximum ampli- tude moves at the group speed of the normal mode with the greatest amplification rate for a real wavenumber, provided the initial conditions do not give zero amplitude for that optimum wavenumber (in this the optimum wavenumber is assumed finite, which is assured here by inclusion of diffusion). Thus, if motion were restricted to be two- dimensional, i.e. allowing only transverse modes, the point of greatest instability for Poiseuille flow would indeed move faster than the mean flow at any level.
Unfortunately, however, the situation in three dimensions is not so simple, since it is not the transverse mode of Fig. 6, but the longitudinal mode of Fig. 4, which is fastest growing. Now, by a straightforward extension of the result of Simmons and Hoskins, motion of the point of greatest instability can be shown to be governed by the full three-dimensional group velocity of the mode with greatest amplification rate, which may be expressed in polar coordinates as
cg = (a(kc,)/ak)ek + (acr/aO)e, (24)
where ek, ee are unit vectors in the directions of increasing k and 8. in Eq.
(14), so for evaluation of the derivatives in (24) a large Richardson number expansion will be required. Such an expansion has been given by Banks et al. (1976) for stably stratified flows, and an entirely analogous expression can be written for unstably stratified flows in the form
The fastest growing longitudinal mode corresponds to the limit R' + -
i?ke(z) = C O S b d - ik(-Rf)-*1(z) - k2(-Rf)-'ti12(Z) + . . . (25)
(26)
(27)
c' = ik"R"/2{ao(k) - ik(-R')-v2co(k) - k2(-Rf)-'al(k) + ik3(-Rf)-3/2Cl(k) + . . .},
where
a0 = (1 + &rZ/k2)-'f2 - 4k2Ra-'n
is the growth rate of the longitudinal mode n = 1 in (19), co is the first approximation to the phase speed, and q, cl, . . . are successive (real) terms in the expansion. Hence, taking the real part of (26),
c: = c,(k) + O(k2/Rf)
C, = co(k) cos e + o ( k 2 ~ - J C O S ~ e).
(28)
(29)
so by (15) and (16)
Thus by (24), ignoring the remainder term (the contribution from which vanishes as e-, w,
cg = [{ a(kco)/ak} cos 01 ek - [CO sin Ole, (30)
Cg = -Coe@ (31)
so that the fastest growing mode, f3 = in, has
where ee is in the negative X direction; the group velocity is thus equal to the phase
AFRICAN SQUALL LINES 707
velocity of the transverse mode, or, more precisely, has magnitude equal to the large -R' limit of the phase speed, with propagation in the positive direction of the mean flow.
This rather surprising result may be illustrated by a simple physical argument. By (29), ignoring the remainder term, phase velocity vectors of modes for fixed k lie on a circle of diameter co as 8 varies (Fig. 7). Drawing the wavefronts of these modes (dashed lines), they are found to intersect at the forward edge of the diameter, so constructive interference of the modes will occur at a point moving with the phase velocity of the transverse mode.
I ' I ' 1 I
Figure 7. Unidirectional jet profile: phase velocity vectors (arrows) and wave fronts (dashed lines) showing constructive interference at the leading edge of the diameter of the circle.
For antisymmetric profiles, co = 0 for the fastest growing mode, so a disturbance will merely propagate at the mid-level mean flow speed. For jet profiles, however, short-wave phase speeds are close to the maximum mean flow speed; a patch of convection will thus propagate, though the point of greatest instability cannot move faster than the maximum mean flow speed because of the Miles (1%1) steering-level constraint. The actual speed of propagation depends on which k gives the greatest growth rate, and this in turn depends via (20) on the Rayleigh number which is chosen. Thus there is some uncertainty in the result, though a helpful first approximation is obtained in the limit Ra + 03, when phase and group velocity both approach the maximum mean flow velocity. This approximation, suggested observationally by Fernandez (1980), was the one found in section l(b) to give the best prediction of SL velocity. In fact, this proof only applies to unidirectional profiles, so caution is advocated when there is a significant transverse wind component.
Examining the jet profile case further, it is found that as 6' varies, group velocity vectors, by (30), lie on a second circle (Fig. 8), the rear edge of which (8 = in) moves at speed CO, while the forward edge (8 = 0) moves at the two-dimensional group speed, a ( k c ~ ) / d k , which can be greater than the maximum mean flow speed. A picture is thus suggested of an amplifying expanding ring of convection, the leading edge of which can,
708 D. BOLTON
Figure 8. Unidirectional jet profile: circle traced out by group velocity vectors.
in certain cases, move faster than the mean flow at any level, and this bears a clear resemblance to the observed structure of a tropical squall line. However, it should be noted that the growth rate at the forward edge is not as great as that at the rear, so that the analysis of Simmons and Hoskins (1979), which relates only to the position of maximum amplitude, can offer no information as to the role of the group velocity at this forward edge. An asymptotic analysis is now needed, therefore, in order to justify this suggested picture.
(b ) Asymptotic analysis To examine asymptotic behaviour of an initial perturbation, it is necessary to
superpose individual exponential modes of the form (11) using a Fourier integral which, for vertical velocity w , when expressed in polar coordinates, takes the form
w = [l,z”f(k, 8)Gke(Z) exp[ik{Xcos 8 + Y sin 8 - c(k, 8)T}] kd8dk (32)
wheref(k, 8) is the Fourier transform of the initial perturbation, and Eke(Z) and c(k, 6) are obtained by solving the normal mode eigenvalue problem, in this case making use of expansions (25) and (26) about the fastest growing longitudinal mode. Benjamin (1961), when dealing with neutrally stratified flow, made use of similar expansions, though about the transverse mode, which was the fastest growing in his case.
Remembering that the point of maximum instability moves with the phase speed (29) of the fastest growing longitudinal mode, it is convenient to set
X/T - co(k) = C cos @
Y/T = C sin @
(33)
(34) where
C = [{X/T - Co(k)}’ + (Y/T)’]@, (35)
CT being a measure of the distance of (X, Y) from the point moving with speed CO. Now
AFRICAN SQUALL LINES 709
making use of the method of steepest descent, it can be shown (Bolton 1981) by substitution in (32) that, provided the large Richardson number long-wave condition k(-R)-’/’ -G 1 holds, asymptotic behaviour for large time is given by
w 4J$(ko, in) COS~XZ { C ~ ~ ( ~ O ) T ) - ” ~ { C O S ( ~ ~ C - )n)},A (36)
where
Here ko is the wavenumber of maximum amplification for the longitudinal mode, i.e. that which maximizes Q in (27), C and @ are evaluated at ko, and
cg = a(kco)/ak. (38)
The cos(k0C - an) term, by (35), shows that the phase lines are circles centred on the point moving with speed CO. Assuming cg # CO, the exponential term A is greatest for fixed @ when
(39) c = (cg - co) cos @
(since &(ko) < 0 for ko to give a maximum of Q). So, bearing in mind that (C, $) are measured from the point moving with speed co, it follows that, as @ varies, the points of maximum amplitude lie on an expanding circle with points moving at speeds co and cg at opposite ends of the diameter (Fig. 9), thus confirming the picture deduced by consideration of group velocity. Drawing in phase lines also, the circle is split up, with the rear divided into cells, but with an unbroken arc at the front (Fig. 10(a)). Though the cells have the larger growth rate, the arc is also amplifying; it will be the first part of the ring of convection to arrive at an observer, and may even move faster than the mean flow at any level. This picture bears a strong resemblance to a tropical SL with its arc-shaped leading edge (roll cloud) arriving first, followed by distinct cumulonimbus cells.
If cg = co (as for the Couette profile), the condition (39) for maximizing of the exponential term reduces to C = 0: i.e. the previous circle of convection has collapsed to a point moving with speed co. Moving away from this point, the lines of equal
Figure 9. Circle traced out by points of maximum instability as @ vanes.
710 D. BOLTON
Y/T
Figure 10. Asymptotic form of perturbation in case (a) co # cg ; (b) co = cg (taken to be both zero). Concentric circles are phase lines, and hatching (heavier for larger amplitude) denotes positive w.
amplitude are ellipses elongated in the Y direction, so, with the phase lines, the picture which now emerges is Fig. 10(b). The arc-shaped leading edge has disappeared, but the line of cells transverse to the mean flow, moving at some mid-level mean flow speed, now has an obvious resemblance to the middle latitude SL.
6. NUMERICAL INTEGRATION
Some numerical integrations will now be presented which confirm the results of the previous section. Integration was carried out by means of a double Fourier transform in the horizontal using Simpson’s rule, with 16 strips in each direction, resolution being greater in the vicinity of the wavenumber k, of maximum amplification so as to resolve asymptotic behaviour; in the vertical, a 5-term Fourier series sufficed. The equations were linear in time, so could be integrated using a technique outlined by Froberg (1969, p. 285). Full details are given in Bolton (1981).
It was convenient, except in the simulation of section 6 ( 4 , to subtract off the average mean flow velocity, so that non-dimensional velocities have a = 7 = 0; the Poiseuille profile then becomes
u = 4 - zz. (40) Now a point moving at the average mean flow speed will remain at X = 0. Time was non-dimensionalized so that a point moving at the maximum mean flow speed will be at X = Tat non-dimensional time T. Initial conditions were chosen to correspond to the circularly symmetric updraught
w a exp{-ik8(X2 + Y 2 ) } sin h ( Z + 1) (41) h = 0.
AFRICAN SQUALL LINES 711
Except where otherwise stated: ko took the value 3; Richardson number R was taken to be -1 (a typical value for West African SLs-see Bolton 1981); lapse rate L was assumed constant ( ~ 1 ) ; variation of mean density with height was neglected; and Rayleigh number Ra was taken to be 16. This 16 was the smallest realistic value which could be taken for Ra in (lo), corresponding to B = -10-5m-’, D = 10 km and K = 3x 103m2s2 (appropriate for eddies with mixing length of order 1 km, speed 3 4 s ) . Referring to (20), this gives wavenumber k, of maximum amplification to be as large as 3, implying a spacing of adjacent cells approximately equal to the depth of the convective layer. Recent observations, e.g. those of Houze (1977), suggest a rather greater spacing, a discrepancy which may be due to exclusion of nonlinear effects from the model.
The results which will be presented are all of perturbation quantities. Since the model is linear, the amplitudes are arbitrary, so the results have in all cases been scaled so that the initial vertical velocity is unity. Because of the exponential growth in time, subsequent values had to be divided by some suitable scale factor before plotting contours of results; the scale factor was usually chosen as the value which gave the clearest contours, rather than basing it, say, on the maximum vertical velocity, so diagrams in which the scale factor was taken to be small will have a relatively large number of contours. In interpreting the results, it should be remembered that nonlinear terms will ultimately become important, and the model will break down; however, the time at which this breakdown occurs will depend on the actual value of the initial amplitudes. This question will be taken up in the final section.
( a ) Plane Poiseuille flow With the flow profile (40), results, as anticipated, show development of a ring of
convection. The plan sections, Figs. l l(a) to (d), show development of successive ‘arc front’ updraughts and downdraughts to the right of the main convection area; these are formed at X slightly greater than T for a given time T, so that the position of the successive arc fronts moves faster than the mean flow at any level. As new arc fronts form, the old ones amplify mainly to the sides, thus giving rise to the convective ring. Greatest amplitude, however, always remains at the rear of the ring with the original updraught, and this does not move as fast as the maximum mean flow velocity. By T = 4, motion is beginning to approximate to the longitudinal rolls which would be expected as having the greatest amplification rate.
The vertical sections at T = 2 are shown in Fig. l l(e) (a smaller scale factor had to be used than in Fig. ll(c) so as to show clearly the arc front). The section in the X-2 plane shows the extent to which the initial updraught dominates the arc front. As there is no Y-Z plane of symmetry, the transverse section is taken through the downdraught arc to the right of the initial updraught; it shows descent in the centre, weak ascent to the sides.
The study for this profile was repeated with various parameters changed, and the results (presented in full in Bolton 1981) may be summarized as follows: (i) Increasing the width of the initial updraught via use of a smaller value for ko in (41) gave a similar development, with cells initially spaced further apart, but tending to the spacing of Fig. 11 as the wavenumber 3 became dominant. (ii) Reducing diffusion by increasing Ra to lo7 meant that the wavenumber for maximum amplification was increased. As ko was taken to be 3, initial development was as in Fig. 11, but the effect of the shorter dominant wavelength soon became apparent in- a closer spacing of the cells.
712 D. BOLTON
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Figure 11. Plane Poiseuille flow, U = f - Z2, R = -1. (a) to (d) Horizontal sections of vertical velocity through Z = 0. Contours are plotted at f , 1, 2 and 4 times the scale factor, with thicker lines for positive values (upward motion), thinner lines for negative values. Velocities in excess of the scale factor are hatched with ///// upward motion \\\\\ downward motion. (e) Vertical sections in the planes Y = 0 and X = 2.4. Contours show potential temperature excess, plotted at f , 1 (thicker line), 2 and 4 times the scale factor. Arrows show perturbation velocities in the plane of the section, scaled so that the maximum vertical velocity is equal to the grid length; actual velocities can be
deduced by referring to the horizontal section.
(iii) When mean density variation was included with
r = -(l/po)(dfi/dZ) = 0.75,
it was found that growth of the perturbation proceeded at very much the same rate as for r = 0. Amplitudes, however, were much larger at upper levels, being proportional to Kfl , and kinetic energies at upper and lower levels thus remained comparable. (iv) When Richardson number R was taken to be - 5 instead of -1, the form of the
AFRICAN SQUALL LINES 713
perturbation was very similar to that of Fig. 11, but amplification was very much more rapid (scale factor =lo6 at T = 4). This is to be expected because the effect of increasing - R may be regarded as increasing the buoyancy while holding constant the shear which determines the non-dimensional time T. (v) Most important of all the changes was to allow variable lapse rate. In particular, with
L = -22, (43) a profile which is statically unstable only in the lower half of the flow region, the greatest amplitudes of vertical velocity were displaced from the middle towards the lower bound- ary, and apart from an initial weak downdraught arc, the perturbation moved not to the right, as in Fig. 11, but to the left. The reason for this change of behaviour can be deduced from consideration of normal modes. At the level of maximum mean flow, the fluid is now no longer buoyant, so the short-wave mode there (if any) cannot amplify rapidly. Instead, greatest amplification rates for short waves are likely to occur for modes travelling near the lower boundary, where the buoyancy is greatest and, with the form of mean flow adopted in (40), these modes will indeed propagate to the left.
Thus, in order to obtain a perturbation travelling faster than the mean flow at any level, it is necessary not only to have a mean flow maximum but that there should be appreciable buoyancy at this level. This would explain the fact that while in West Africa there are two jets, it is only the mid-level African easterly jet which is important for SL formation, the upper-level tropical easterly jet being too high for there to be adequate buoyancy (see section l(a)).
(b ) Plane Couette flow For the plane Couette profile (21) with Richardson number R = -1, the dominant
instability was stationary, as predicted in the asymptotic analysis. Two other propagating regions of instability did also develop, with maximum amplitudes near the lower and upper boundaries respectively. These are not however buoyancy induced, as similar results were obtained with neutrally stratified flow, R = 0, so are assumed not to be strictly relevant to the growth of SLs.
Results which are presented are therefore for R = - 5 , in which case the buoyancy is sufficient to dominate completely any other instability. The horizontal sections (Figs. 12(a) to (d)) show development of an elliptical patch of convection, with formation of successive cells in a line transverse to the mean flow, as predicted in Fig. 10(b). The gradual elongation of the cells into longitudinal rolls, already noted for Poiseuille. flow, is even more apparent here. Note that at T = 16, contours are open at the sides since vertical velocities were only calculated over a restricted range of X values in the neighbourhood of the maximum amplitude, the results then being plotted on the extended grid to show more clearly any motion.
The vertical section at T = 4 (Fig. 12(e)) shows the longitudinal roll structure, and also the tilting of the cells in the direction of the mean flow associated with feeding of perturbation energy into the mean flow ((i) in section 4).
( c ) Mixed Poheuille and Couette flow The simplest polynomial profile in which the shear vector changes direction with
height is a combination of Poiseuille and Couette flow which may be taken as
u = a($ - 22) V = bZ. (44)
714
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- 2 -
- 4 -
D. BOLTON
1 Y 4 -
2 -
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0-
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a. TIME I 0.0 SCRLE F A C T O R 1.00 EO b. TIME = 4.0 SCALE FRCTOR = 8 .75 € 1
I " " " " - 4 -
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-2 - - - 4 - -
- . . . . .
, , , . . .
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Figure 12. Plane Couette flow, U = Z, R = -5. (a) to (d) Horizontal sections of vertical velocity through Z = 0. (e) Vertical sections of potential temperature excess (contours) and perturbation velocity (arrows) through
Y = 0 and X = 0. For a full explanation see Fig. 11.
The component in a direction making angle 8 with the X axis is then
W = a($ - Z z ) cos 8 + bZ sin 8 = -a cos 8[{Z - (b/2a) tan 8)' - (b/2a)'tanZ 8 - 91. (45)
The component in the X direction, 8 = 0, is a symmetric parabolic profile; as 8 is increased, the parabola becomes asymmetric until for tan 8 = - t -h /b , the velocity maxi- mum has moved to one or other boundary; increasing 8 still further, the profile no longer has a maximum, but tends to the constant shear obtained with 8 = f ~ .
AFRICAN SQUALL LINES 715
Y 4 -
2 -
0 -
- 2 -
- 4 .
Normal mode results for parabolic profiles suggest that the greatest growth rate will occur for the short-wave mode at a velocity maximum for which IdZU/dp) is smallest, and here this occurs for modes with tan 8 = +2a/b, for which the velocity maximum is at a boundary. Two cases will be considered below: (i) a = 1, b = 4, (ii) a = 4, b = 1; the respective values of 8 for the fastest growing modes are *81" and 545".
(i) U = f - Z 2 , V = 4 Z , R = -1 Here the greatest shear is in the X direction, and results (Figs. 13) show initial
development of updraught and downdraught arc fronts at points moving faster than the maximum mean flow. These fronts display a tilt (Fig. 13(e)) in the Y-Z section owing
Y 4
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. . . . . . . . . . . . . . . . . . . . . - 1
I Y 2 - 2 - I 0
e. T I M E = 2.0 SCALE F A C T O R = 5 . 0 2 EO
Figure 13. (a) to (d) Horizontal sections of vertical velocity through Z = 0. (e) Vertical sections of potential temperature excess (contours) and perturbation velocity (arrows) through
Y = 0 and X = 2.4. For a full explanation see Fig. 11.
Mixed Couette-Poiseuille flow, U = $ - 2?, V = $ Z , R = -1.
716 D. BOLTON
to the influence of the two fastest growing short-wave modes for 8= +81" which propagate near the upper and lower boundaries. After a further time, the two modes become distinct, and there is eventual separation into upper- and lower-level pertur- bations propagating in different directions (though admittedly not inclined to the X axis by as much as the predicted 331").
(ii) U = f(f - l?), V = 2, R = -5 Here greatest shear was in the Y direction, so the larger Richardson number was
used as in section 6(b). Results (not shown) again indicate formation of tilted arc fronts, but now the two fastest growing short-wave modes for 8= 245" were propagating in sufficiently similar directions that no separation became apparent even by T = 8, rolls transverse to the constant shear component forming instead.
The splitting obtained with (i) could have relevance to the formation of severe right- and left-moving storms in mid-latitudes. The right-moving perturbation, having its maximum amplitude at a lower level, where in the atmosphere there is a greater energy supply, would be more likely to develop, which is consistent with the more frequent occurrence of severe right-moving storms (Ludlam 1980, p. 249). Raymond (1976), using an actual atmospheric sounding, obtained a similar splitting with his wave-CISK model, and he likewise attributed this to a double maximum in the growth rate/direction graph; his profile, however, bore more resemblance to (ii) than to (i) above, and his results do not show any evidence of a distinction between upper- and lower-level pertubations. Full numerical integrations, such as those of Klemp and Wilhemson (1978a, b), have obtained storm splitting even with unidirectional shear profiles. Rotunna and Klemp (1982) explained the split as an essentially nonlinear effect, due to dynamic forcing and rainwater loading; using a linearized pressure equation they were, however, able to show that once a split has occurred, there will be preferred development of the right cell if the environmental wind shear veers with height; in contrast, the wind shear in the present simulations actually backs with height.
(d ) Simulation of squall line at Minna A simulation was carried out using the 24-hour 'before' mean sounding of Figs. 2
as input, variables being incorporated by Fourier analysis. The tropopause was around 100 mb (16 km), with B = -3~10-~m- ' , S = 5 ~ 1 0 - ~ s - ' , so that the Richardson number was about -1. Other parameters were given their previous values, except that mean density variation was included with r = 0-75.
Results are shown in Figs. 14. The horizontal sections show development of suc- cessive downdraught and updraught arcs. The first updraught arc at T = 2 corresponds to a time of 15 minutes. By T = 8, corresponding to one hour, this arc has split into two cells, with formation of a new updraught at the front. The vertical sections, which show the structure of the cells, are in the W-E and N-S directions through the initial cell, which is always the most rapidly amplifying. They show a warm updraught, cold down- draught everywhere, with the downdraught originating from around 2 = 0.5, corre- sponding roughly to 200 mb.
Both arc front and disturbance as a whole moved from approximately 085", with the arc front oriented perpendicular to the motion. The speed of the system was 12 m / s , so that agreement with the observed mean propagation velocity, 16.4 m/s from 079", was quite good, though whereas the observed mean speed was greater than the average mean wind at any level, the model speed was slightly less than the maximum average wind. This may be due in part to underestimation of the maximum wind by the Fourier series which was used, but it is interesting to note that Thorpe (personal communication) was
AFRICAN SQUALL LINES 717
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d. T l H E I 8 . 0 SCALE FACTOR z 2 . 4 0 €10
I I
zoI - I
e. TIHE = 1.0 SCRLE F R C T O R = 9.30 E O
Figure 14. Simulation based on Minna mean sounding. (a) to (d) Horizontal sections of vertical velocity through Z = -0.6, corresponding approximately to 700 mb. T = 1 corresponds to a time of approximately 8 minutes; X = 1 corresponds to a distance of approximately 8 km. (e) Vertical sections of potential temperature excess (contours) and perturbation velocity (arrows) in the planes
Y = 0 and X = -1.2. For a full explanation see Fig. 11.
unable to generate a SL with the model of Moncrieff and Miller (1976) using a West African profile, apparently because the storm speed, as determined by the location of successive cells, did not match the speed of the downdraught outflow.
7. COMPARISON WITH OTHER MODELS
The most widely studied SL in the literature is the Venezuelan SL of 24 July 1972 (disturbance No. 47 of VIMHEX), previously modelled by Moncrieff and Miller (1976), Raymond (1975), and Marroquin and Raymond (1979). In order to permit a comparison with these studies, a simulation was carried out with the present model, using data from
718 D. BOLTON
this storm as input. Results (not shown) give a development very similar to that of Fig. 14, except that the motion of the model storm, at 18m/s from 115", was no longer perpendicular to its arc front orientation 040-220", and was significantly different from the motion of the observed storm-though agreeing quite well with results obtained from two of the other models. This may be because motion of the observed SL was determined by some synoptic-scale feature such as a trough, as suggested by Moncrieff and Miller (1976). If so, it may not be a very representative storm to study, which is unfortunate in view of the fact that it is the one most widely modelled.
(a) Storm velocity The speed of the model storm, 18 m / s , is slightly greater than the maximum environ-
mental wind speed, though the observed storm speed, 15.3 m / s , was rather less. Errors in orientation were greater: whereas the model storm moved from 115" with an arc front orientation 040-220", the observed storm moved from 065" with arc front orientation 155-335". The storm motion as estimated from the maximum environmental wind speed according to the Fernandez (1980) rule shows a similar error, since the strongest wind in the Venezuelan profile (that at 700 mb.) was 16-5 m / s from 108". It is significant that both the present model and the Fernandez rule gave very much better estimates for West African SLs (see sections l (b ) and 6(d)).
Marroquin and Raymond (1979) computed the group velocity vector for this storm and found that it varied considerably with wavenumber. The storm velocity, as found from the point of greatest instability, was 10.3 m/s from 071", which represents a good estimate of direction, but an error of 5 m / s in speed. A subsequent study, Marroquin and Raymond (1982), using another of the VIMHEX SLs, gave very much less variation in the group velocity vector, and a much better estimate for the storm velocity. Marroquin and Raymond (1979) also gave the storm velocity as estimated by the wave-CISK model of Raymond (1975): this was in error by nearly 1 5 4 s . For other SLs, however, the observed storm speed did usually lie close to the curve traced out by the computed group velocities, even though not near to the point of maximum amplification.
Moncrieff and Miller (1976) obtained an estimated storm speed of 13 m / s , which was slightly small; the estimated orientation, 155-335", was good, but there was con- siderable error in the estimated direction, 095", though not as great an error as in Fig. 14. The Moncrieff and Miller (1976) formula (2) gives a slight over-estimate of the speed, 17 m / s , but if the direction is assumed to be that of the mean wind, 070", it does represent the best estimate of storm velocity; for West African SLs, however, agreement was seen in section l (b) to be very much poorer.
(b) Storm structure Though Raymond has not actually published any initial value problem results for
SLs, his simulations of other storms, (1976), suggest certain shortcomings in common with the simulations of the present model, notably growth to unrealistically large amplitudes, failure of the initial cell to dissipate, cold downdraughts, rising again of downdraught air, and origin of the downdraught near the tropopause. These problems of linear systems will be taken up in the closing section, and are clearly avoided by a fully nonlinear numerical model such as that of Moncrieff and Miller (1976), which was able to give a much better picture of storm structure, particularly as regards the down- draught. One aspect in which the present model is superior, however, is the formation of the arc front (presumably made possible by the greater resolution achieved with the Fourier transform method); various other interesting deductions, especially on the role of the jet, have also arisen from the linear analysis.
AFRICAN SQUALL LINES 719
8. CONCLUDING REMARKS
Certain limitations of the linear system are evident in the case studies presented, notably: (a) Very large scale factors appear quite early in the integration (in excess of 10" by
T = 8 in Fig. 14). Even assuming initial perturbation speeds to be only those appro- priate to a cumulus, of order a few centimetres per second, these large scale-factors imply quite unrealistic updraught speeds. This growth to unrealistic amplitudes must be due partly to the omission of nonlinear terms, partly to the simplified thermo- dynamics which are likely to over-estimate the buoyancy, especially in the downdraught.
(b) The initial cell did not eventually dissipate, but always had the greatest amplitude. Again omission of nonlinear terms may be partly to blame, but the main cause must be thermodynamic, in that air is treated as buoyant, no matter what its past history, so that, for example, air rising in the arc front is then available to rise again in the main updraught.
(c) The unrealistic thermodynamics also affect the downdraught structure in that down- draught air is automatically negatively buoyant wherever the updraught air is buoyant. Thus the downdraught is a priori prescribed to be cold (whereas the mesoscale downdraught of a squall line is warm), and originates from a level near the tropopause, rather than from the middle troposphere as observed (Zipser 1969, 1977).
(d) As remarked in section 6, the spacing obtained for the cells is less than that normally observed even when Rayleigh number is given its smallest realistic value. Here it is likely that the main reason is omission of the nonlinear terms. In spite of these limitations, the straightforward unforced linear model can give
considerable insight into some aspects of SL generation, in particular: (a) In the tropics, the need for a jet to generate disturbances moving as fast as, or faster
than, the maximum environmental wind speed is explained (section 5) and a reason is given ((v) in section 6(a)) why this jet should be at the mid level rather than at an upper level; also a justification is provided (section 5(a)) for the Fernandez (1980) forecasting rule that the storm velocity should be close to the maximum mid-level environmental wind velocity. In addition, the nature of a tropical SL is seen (section 5(b) ) as an expanding ring of convection in which an arc front is followed by individual cells, rather than as a two-dimensional line, and this picture corresponds well with satellite images (e.g. Fortune 1980).
(b) In middle latitudes, on the other hand, where there is no jet, insight is provided (section 6(b)) as to why SLs should take the form of transverse lines of cells moving at some mid-level environmental wind speed. Furthermore, the influence of a trans- verse velocity component in which there is a maximum is suggested (section 6(c)) as a cause of storm splitting.
ACKNOWLEDGMENTS
I would like to thank Dr F. B. A. Giwa and Mr G. Dugdale who first interested me in this study, Dr J. S. A. Green for his stimulating supervision in it, Mr A. Seaton for advice on the use of the computer, and Mrs J. Ludlam and Mrs R. Assaf for help in typing my manuscript. Thanks are also due to University of Ibadan, Nigeria and Atmospheric Physics Group, Imperial College, London, for extensive use of computing and other facilities, and to Nigerian Meteorological Services for their kind provision of surface and upper air data.
720 D. BOLTON
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