9_9:3
THE STRUCTURE OF VORTICITY IN
CUMULONIMBUS CONVECTION:
A NUMERICAL STUDY
by
Martin Johnson Atmospheric Physics Group Department of Physics Imperial College of Science and Technology
A thesis submitted for the Degree of Doctor of Philosophy in the University of London
August 1978 •
THE STRUCTURE OF VORTICITY IN CUMULONIMBUS CONVECTION :
A NUMERICAL STUDY
Martin Johnson
ABSTRACT
This thesis investigates the vorticity and potential vorticity fields generated by a three-dimensional primitive equation numerical model of deep convection.
Using the momentum and continuity of mass equations solved by Miller's deep. convection model, a vorticity equation is derived in both continuous and finite-difference forms. The latter is derived in such a way as to be consistent with both its continuous analogue and the primitive equations.. It is shown that the primitive equation model time integration scheme introduces second-order time errors into the horizontal components of the vorticity equation, and a change in the scheme is suggested which would remove these errors.
Using the finite-difference form of the vorticity equation, the fields of the three components of (i) the vorticity vector and (ii) the diagnostic terms in the vorticity equation are computed for three different simulations: a squall line, a stationary storm, and a splitting storm system. In all cases the vorticity field is surprisingly well organised. It is found that the horizontal components of vorticity reach values of order of magnitude 10-2 s-1 in all the storms around the edges of the updraught cores and downdraught 'cold pools'. The vertical vorticity attains this magnitude over a relatively large area in the first and third cases, and over a very restricted area in the second. This corresponds to a significantly larger amount of (horizontal) vorticity in the mean flow of the first and third cases. In all the storms, the vertical compohent appears as a 'vortex doublet', the positive (cyclonic) value being about twice the negative; the area of positive vertical vorticity lies just inside the updraught on the side nearest to the downdraught, while the negative area lies on the opposite side of the updraught. Both are contained mainly in the 500-800 mb layer.
A simplified model of the vorticity field is described in terms of vortex lines, and using the results of the diagnostic calculations, a theory is suggested for their development.
Comparisons are made between the results of the above, and dual- Doppler radar observations of severe storms of the United States and computer models of tornadoes.
Equations are obtained for potential vorticity, liquid water potential vorticity, and equivalent potential vorticity, and it is shown that due to baroclinity, diffusion and diabatic effects, these quantities are not conserved. The results are presented of the computation of the first of these quantities. CONTENTS
ABSTRACT 1
NOTATION 5
CHAPTER ONE - INTRODUCTION 6
1.1 Observations of cumulonimbus convection 7 1.2 Analytical models of cumulonimbus convection 10 1.3 Numerical models of convection 13 1.4 Tornadoes 17
CHAPTER TWO - DEFINITIONS OF TERMS AND DERIVATION OF VORTICITY AND CIRCULATION EQUATIONS IN
CONTINUOUS FORM 20
2.1 Definitions 20 2.2 Derivation of the vorticity equation 21 2.2.1 Primitive equations 22 2.2.2 Vorticity equation 23 2.3 Circulation equation 27 2.3.1 Derivation of circulation equation 27 2.3.2 Consequences of circulation equation 27
CHAPTER THREE - DERIVATION OF VORTICITY EQUATION - FINITE DIFFERENCE FORM 29
3.1 Primitive equation model of cumulonimbus convection 29 3.1.1 Definition of symbols 29 3.1.2 Configuration of model grid 30 3.1.3 Primitive equations in finite difference form; time integration scheme 32 3.2 Derivation of vorticity equation 34 3.2.1 Choice of finite-difference scheme and points in grid for calculations 34 3.2.2 The vorticity vector in finite difference form; further simplification of the vertical momentum equation 37 3.2.3 Derivation of the vorticity equation: x-component 38 3.2.4 Derivation of the vorticity equation: y-component 44 3.
3.2.5 Derivation of the vorticity equation: vertical component 45
CHAPTER FOUR - THE VORTICAL STRUCTURE OF SIMULATED CUMULONIMBUS 48
4.1 Method of presentation of results 48 4.2 Case A : Tropical cumulonimbus 49 4.3 Case B : The Hampstead storm 51 -1 4.3.1 • Results for 32-44 minute means 52 4.3.2 Results for 64-76 minute means 53 4.4 Case C : Splitting storm system 53 Key to diagrams 55 Diagrams for case A 56 Diagrams for case B (i) 68 Diagrams for case B (ii) 80 Diagrams for case C 92
CHAPTER FIVE - INTERPRETATION OF RESULTS 104
5.1 Model of vorticity field in cumulonimbus 104 5.2 Physical interpretation of results 108 5.3 Comparison of model and simulation results 116 5.4 Generation of vertical component; comparison with dual-Doppler observations 121 5.5 Tornadoes 123
CHAPTER SIX - POTENTIAL VORTICITY 125
6.1 Equation of general potential vorticity of deep convection 125 6.2 Special cases 126 6.2.1 'Ordinary' potential vorticity 126 6.2.2 Equivalent potential vorticity 128 6.2.3 Liquid water potential vorticity 129 6.2.4 Potential vorticity of deep convection 129 6.3 Potential vorticity equations_insimplified models of cumulonimbus 131 6.3.1 Moist inviscid cumulonimbus model 131 6.3.2 Dry inviscid cumulonimbus model 132 6.4 Potential vorticity fields in simulated cumulonimbus 133 4.
CHAPTER SEVEN - SUMMARY AND CONCLUDING REMARKS 143
7.1 Summary 143
7.2 Suggestions for future research 145
APPENDIX - PRIMITIVE EQUATION MODEL TIME INTEGRATION
SCHEME 147
REFERENCES 152
ACKNOWLEDGEMENTS 156 5.
NOTATION
Cp Specific heat of air at constant pressure F Initiating heat source F F F Acceleration components due to subgrid scale oe 3 P momentum transfer Vector (Fe Fy,- ygps) g Acceleration due to gravity g(1 + Z ) g* h Height of isobaric surfaces K K8' q Diffusion coefficients Total mass of liquid water per unit mass of air Z Mass of cloudwater per unit mass of air c Mass of rainwater per unit mass of air Latent heat of condensation of water p Pressure q Specific humidity Q Sink of specific humidity due to condensation T Temperature u E (uw,w) Velocity vector
✓ Terminal velocity of raindrops Circulation Vorticity vector Potential temperature 0 Virtual potential temperature eE Equivalent potential temperature W Liquid water potential temperature Potential vorticity of cumulonimbus convection TICB Equivalent potential vorticity HE LW Liquid water potential vorticity P Air density In 6
$* Zn 64 Zn e (I)E E cLW in OLW 0 Flux of vorticity along vortex tube; strength of vortex tube X (e*ye s ) - gh' + 1/2 (u2 v2) Dp/Dt SZ Instantaneous angular velocity of fluid element CORRIGENDA AND ADDENDA
Page 21 line 6 Delete 'It can be shown,' .... 'that'.
Page 37 line 14 For 'differerence' read 'difference'
Page 38 line 15 For 'AE ' read 'AE 1/2,1/2,1/2 0,0,1/2 Page 38 line 16 For '6Y.F s ' read I SYPX ' P - Page 49 line 25 It should be made clear that the horizontal dimensions of the vorticity model were restricted to 20km x 20 km due to limits on computer storage area, Although slight problems do exist near the primitive equation model boundaries, the fact that these are avoided in the vorticity model is merely a co-incidental advantage _ of the restriction.
Page 50 line 15 For 'Fig. 4.7(a)' read 'Fig. 4.9(a)'
Page 104 line 15 Delete 'i.e. the dominant wavenumber of the vorticity field is the same as that of the storm system'.
Pages 106-7, Figs. 5.1-5.3 Vortex lines are assumed to be moving from left to right in these figures.
Pages 108, 137, Figs. 5.4, 6.4 The trajectory shown in these figures are computed according to the method presented in Miller and Betts (1977).
Pages 108-110 In addition to the information presented on Fig.5.4 and in the text, the data in the following Table may be found useful. The Table illustrates in more detail how fl and the terms in the equation for ri vary-along the trajectory in the immediate vicinity of the updraught.
Position on n Dn/dt Stret- Tilt- Comp- Brclnty Diff- trajectory ching ing ress. usion relative to 29kmx29km grid 10-7s-2 I J P 10-3s-1 10-7s-2 10-7s-2 10-7s- 10-7s-2 10-7s-2
11.6 8.6 939 -6.57 -99 -16 2 3 -510 390 12.4 8.6 904 -2.21 221 24 -6 5 159 39 13.0 9.2 816 3.30 53 34 -14 -35 76 -8 13.2 9.3 606 7..57 322 77 7 -12 763 -513 14.6 9.2 460 5.75 262 108 17 -76 399 -186 15.6 9.5 447 6.31 450 100 -9 -36 270 125 Corrigenda and Addenda (continued)
Page 110 Line 2 For '-88' read
Page 155 The following reference should be added:
Miller, M.J. and Betts, A.K. 1977 Traveling Convective Storms over Venezuela, Mon.Wea.Rev., 105, pp.833-848. 6.
CHAPTER ONE
INTRODUCTION
'Convection' is the name given to any motion arising directly
from the conversion of available potential energy and which is involved
in the exchange of heat, moisture and momentum. This type of motion
occurs on all scales, and there is a close interaction between them;
thus precise knowledge of the features of one particular scale of motion
is required in order to understand fully the other scales of motion with
which it interacts.
For this reason, development of models of global, synoptic and meso-
scale atmospheric motions demands knowledge of the motions and processes
of 'sub-grid-scale', i.e. smaller than meso-scale, phenomena in order that
they may be parameterised in terms of the larger-scale variables.
One of these sub-grid-scale regimes is that of cumulonimbus, or deep
convection; it is the most common type of weather in the tropical regions
and is important for the transfer of heat, mass, moisture and momentum
from low to high levels in the troposphere. In mid-latitudes (the mid-
Western United States, for example) cumulonimbus of the 'supercell' type
are renowned for their longevity, propagation speed and accompanying
- severe weather such as hail and tornadoes.
A knowledge of the development of vorticity in deep convection is
important for the parameterisation problem, since many of the large-scale
models are formulated in terms of vorticity, and also for a better
understanding of the mechanism of formation of the tornado, which is a
local but intense manifestation of atmospheric vorticity.
Study of cumulonimbus convection has been carried out using various
approaches; the remainder of this chapter will be devoted to the
presentation of some of the more recent work. 7.
1.1 OBSERVATIONS OF CUMULONIMBUS CONVECTION
The first intense study of cumulonimbus using aircraft and radar
techniques was carried out in the Thunderstorm Project by Byers and
Braham (1949), who measured updraught and downdraught speeds and areas,
cloud top height, propagation speed and time-development of the storms.
However, the storms studied by them were less severe than the typical
mid-Western U.S.A. severe storms.
Newton and Newton (1959), using rainfall data, tracked the move-
ments of individual storms over hundreds of kilometres for times of about
twelve hours, thereby deducing that storm tracks were on average about 20°
to the right of the mean 850 mb - 500 mb wind direction. They discuss
the differing effects of wind shear on deep and shallow convection, and
put forward the possibility that the cumulonimbus might act as a barrier
to the mid4Tlevel mean flow.
After a study of the Wokingham storm of 9 July 1959, Browning and
Ludlam (1962) noted that vertical windshear is important for the maintenance
of convective overturning. All three of these papers stressed the
importance of organised downdraughts in determining storm structure.
Fankhauser (1971), using observations of two Great Plain thunderstorms
obtained by aircraft, radar and rawinsonde soundings, determined the
kinematic properties of the near-cloud air-flow, moisture budget, and the
dynamical interactions between the clouds and their surroundings. The • results tended to confirm the theory of Newton and Newton (1959) that a
mature cumulonimbus acts as an obstacle to the mid-level flow, although
mid-level air was found not to be entirely insulated from the internal
cloud circulation. Hydrodynamic drag and deflection forces (e.g. the
Magnus effect) were thought to have a strong influence on the propagation
velocity. A three-dimensional graphical model of a severe right-moving
storm was presented. 8.
Chisholm (1973) used data gathered by the same methods as those of
Fankhauser (1971), together with hail reports, to analyse four hailstorms occurring during the Alberta Hail Studies Project. A one-dimensional
'loaded moist adiabatic' model was used to estimate the vertical velocity,
temperature and water content in the updraught core. Airflow models were deduced for each case, and these were shown to be dependent on the environmental wind and its interaction with the updraught. The vertical velocity and water content of the updraught core were found to be related to the convective available potential energy.
Ruprecht and Gray (1974) combined satellite and rawinsonde observations of tropical weather systems in a large statistical survey of individual cloud organisations. Regional differences between Western Pacific and West
Indies cloud clusters and their dynamics and structure were investigated.
Direct association of low-level vorticity and divergence was found to be possible if these quantities were interpreted in terms of their deviations from a regional mean state; more agreement was then found with the wave-
CISK idea of Lindzen (1974). The mean cloud-cluster vertical vorticity budget was calculated; it was found that the clusters gained large amounts of vertical vorticity in the lower troposphere due to convergence
(horizontal transport). The cumulonimbus was then thought to selectively
transport vertical vorticity upwards, whence much was lost by divergence in the upper levels. Significant differences were found to exist between
the Western Pacific and West Indies systems, but variations within a system were typically much larger than the mean parameter differences between them.
Detailed examination of the synoptic situation associated with a severe
tornado outbreak was carried out by Hoxit and Chappell (1975) following the
'jumbo' outbreak of 3-4 April 1974. They concluded that a surface cyclone
with its resultant convergence fields, together with low-level heating, was
responsible for the intense convection over such a wide area.
r 9.
This outbreak was also studied by Agee et al (1975) who considered
the synoptic aspects more briefly than Hoxit and Chappell (1975), but
went on to analyse in more detail three particular storm cells, and
using a motion film of one tornado family, they were able to present a
suction vortex model based on that of Fujita (1972). Again, strong
wind shear, both magnitudinal and directional, between 850 mb and 500 mb
was an important feature.
Use was made of a single-Doppler radar measuring the radial
component of velocity by Burgess et al (1975) to study a splitting severe
storm which occurred on 24 May 1973. The whole life-cycle of the storm
was simultaneously photographed by a NSSL intercept team, with the result
that a radar signature directly related to a tornado was discovered, and
the evolution and relationship to the parent storm were found.
Brown et al (1975) presented the results of the first dual-Doppler
measurements of the vector winds in a tornadic storm, which tended to
confirm the conclusions made after single-Doppler measurements, such as
that of Burgess et al (1975), concerning the association of mesoscale
cyclonic circulation with a definite single-Doppler reflectivity pattern.
Kropfli and Miller (1976) used the dual-Doppler technique to determine
the three-dimensional structure of a convective storm, which grew in an
environment of weak shear, during its decaying stage. Some features of
the storm were found to be similar to the Wokingham Storm studied in s Browning and Ludlam (1962), being an upshear-tilted updraught, surface gust
front, downdraught consisting of middle-level cool dry air carrying precipi-
tation, and a vortex pattern similar to that of obstacle flow. Flow
vectors relative to the storm velocity were plotted, and fluxes of mass,
water vapour, energy, momentum and the vertical component of vorticity were
calculated using the dual-Doppler soundings.
Ray (1976), also using dual-Doppler data, calculated the vorticity
and horizontal divergence fields observed during two tornadic storms which 10.
occurred on 20 April and 8 June, 1974. The results are described in
more detail in Chapter Five of this thesis.
Betts, Grover and Moncrieff (1976) describe some of the results of
the Second Venezuelan International Meteorological and Hydrological
Experiment (VIMHEX II) conducted in Northern Venezuela in 1972. The
tracks, life-cycles and storm-top heights of tropical thunderstorms and
squall-lines were observed by radar, and pre-calibrated radiosondes were
used to measure the state of the atmosphere before, during and after the
storms. It was found that the squall-lines were well-organised,
propagated in a distinctive manner and significantly modified the dynamic
and thermodynamic structure of the synoptic-scale flow.
The first part of Grove (1977) deals with the influence of the
large-scale flow on cumulonimbus convection, the second part dealing with
stationary and slow-moving cumulonimbus. A detailed observational study
of the Hampstead Storm of 14 August 1975 is presented.
Heymsfield (1978) carried out an investigation into the three-
dimensional kinematic and dynamic structure of the severe storm of 8 June,
1974 (the second of the storms studied by Ray (1976)). Using the dual
.Doppler data, the horizontal convergence, vertical component of vorticity,
and the rate of change of the latter due to stretching and tilting of vortex
lines were calculated. The results are, as with the work of Ray (1976),
presented in more detail in Chapter Five.
1.2 ANALYTICAL MODELS OF CUMULONIMBUS CONVECTION
Green and Pearce (1962) first introduced the concept of Richardson
number into the study of deep convection. A steady, two-dimensional,
inviscid flow was considered. Green found that the steering-level of
the incompressible flow is an eigenvalue of an eigenvalue problem,
determined by the extreme outflow height and the value of a non-dimensional
quantity of the form of a Richardson number. With continuity of pressure 11. as boundary condition at the updraught/downdraught interface, Green found that this interface must be orientated with the updraught under- neath the downdraught.
Pearce assumes that the flow is consistent with the descriptive model of Browning and Ludlam (1962); from the vorticity distribution, by integration, the distribution of EP-0'0 was calculated (where e. is the deviation of potential temperature from the undisturbed state, and
is the value of 0' at the interface), thus yielding the distribution eto of heat sources and sinks necessary for the maintenance of the flow. The two individual parts of the study in this paper therefore gave conflicting results for the slope of the interface.
Moncrieff and Green (1972) obtained two conservative quantities by analytical integration of the equation of motion and thermodynamics for a
Boussinesq fluid. The first, a vorticity constraint, is valid only for two-dimensional flow, and was applied to the problem of steady two- dimensional convective overturning in shear. The second was a kind of
Bernoulli quantity for a Boussinesq fluid, valid for three-dimensional flow.
A Richardson number for steady overturning in shear was defined, and the steering-level and propagation speed determined in terms of this and a density scaling height. Comparison with several observed storms was favourable, especially for conditions exhibiting a Richardson number of order unity. Heat and momentum transfer were expressed in terms of the large-scale parameters, and it was found that these could not be represented by a Fickian-type diffusion.
In Moncrieff and Miller (1976), Moncrieff uses the three-dimensional conservative quantity obtained in Moncrieff and Green (1972) to model tropical cumulonimbus. Here, the system propagates in the same sense relative to the flow at all levels, and thus inflow is at the front, outflow behind, the storm; full three-dimensionality is therefore necessary in the storm interior in order to be topologically possible. It was assumed that 12. part of the downdraught air originated from the upper troposphere, which although unrealistic, still provided useful results. The
propagation speed was found to depend on the convective available
potential energy and, weakly, on the windshear of the large-scale
flow. The modification to the atmosphere by the storm was calcu-
lated, resulting in an enhancement of the mid-level jet by a large-
amplitude momentum transfer.
Moncrieff (1978) tackled the interface slope problem and the
dynamical feasibility of two-dimensional models. Constant shear
was assumed in the ambient flow, and with the asymptotic solution
for the steering-level and remote flow found in Moncrieff and Green
(1972) used as lateral boundary conditions, the non-linear momentum
and thermodynamic equations were solved to determine the internal
streamline structure and updraught/downdraught boundary configuration,
the latter being a free boundary. A dimensionless parameter (the
ratio of convective available potential energy to the square of the
cloud-layer shear) was found which determined the flow orientation;
for all realistic ranges of this parameter, the interface sloped down-
shear, being little affected by compressibility, by vorticity generated
by the interfacial boundary layer, or by the parameter taking different
values in the updraught and downdraught branches. Upshear orientation
could only be induced by providing thermodynamic forcing not consistent
with wet-adiabatic processes, but the flow was not steady. Steady, two-
dimensional severe storms are thus impossible in constant vertical shear,
and although not examined, flows with other ambient velocity profiles
were expected to exhibit broadly similar features. This work therefore
resolved the paradox of the interface configuration in Green and Pearce
(1962). Further, while two-dimensional modelling was acknowledged as
being valuable for the study of certain aspects of deep convection, the
results of this paper suggest that three-dimensional numerical simula-
tions are a basic necessity for the theoretical development of the severe 13. local storm.
1.3 NUMERICAL MODELS OF CONVECTION
Recent rapid increases in the capabilities of computers have led to considerable advances in the field of convection simulation. Two- dimensional models have become fairly sophisticated. Orville and Sloane
(1970) modelled mountain cumuli in two dimensions, assuming incompressible flow. Features of the model include non-zero ambient winds, heating and evaporation at slopes and valley surfaces, cloud shadow effects and a warm rain process. A storm evolved, lasting for about one hour, and passed through the cumulus stage with updraughts, the mature stage with co-existing up- and downdraughts, and the dissipation stage with predominant but weakening downdraughts. The authors concluded that for a severe travelling storm, a three-dimensional model would be required so that the inflow could be 'protected' from the rain. However, the results were claimed to resemble past observed storms.
Takeda (1971) presented a two-dimensional model which incorporated a size-distribution of water drops. Upper and lower boundaries were solid, and on the lateral boundaries variables were kept constant and equal to their initial values. The simulations were terminated after 60 minutes; clouds completing or nearly completing their life-cycle in this time were called 'short-lived', while those showing no signs of dissipating after
60 minutes were called 'long-lived'. Thus the complete life-cycle of
'long-lived' clouds was not achieved. The formulation of the latter cloud- type necessitated a jet in the lower layer of the initial wind-field, the height of which was constrained to lie within certain limits. Cloud-top height was found to depend on the instability and cloud physics, but little correlation was found between duration and intensity of the modelled storms.
There have been several two-dimensional studies of squall-lines.
Hane (1973) concluded that the kinetic energy for the motion was derived 14. from the convective available potential energy, and that a third dimension would be desirable since flow around the storm at about 500 mb was expected.
Rather than reach a steady state, the storm underwent three or four develop- ments in a hundred-minute period, but each development, at its most intense, exhibited similar structure. Schlesinger (1973) deduced a relation between the initial shear and moisture supply, and the duration and intensity of the storm and updraught slope:
Large moisture supply + weak shear : intense short-lived updraught.
Rainfall and downdraught downshear of updraught, which
isolated latter from its source.
Moisture + moderate-strong shear moderate-strong long-lived updraught.
Rainfall and downdraught upshear of updraught, perpetuating
latter.
Less moisture + strong shear : no strong or long-lived updraught.
Rainfall and downdraught upshear of updraught, but lowest
air flowed underneath it, limiting buoyancy.
Again, little correlation was found between duration and intensity.
Takahashi,(1974) modelled tropical showers in two dimensions with cyclic lateral boundary conditions, rigid lower boundary, but free upper boundary.
Water droplets were classified into 61 groups according to size. The rainfall pattern was seen to be determined by (i) the re-entry of drops from the cloud boundary into the cloud, (ii) a strong downdraught,
(iii) inflow of warm, dry air from outside the cloud, which suppressed the updraught and triggered a sudden downdraught with fall of the accumulated raindrops. The model exhibited strong inflow near cloudbase at the developing stage, and formation of a new airflow cycle near the ground at the dissipating stage. Rainfall intensity increased as cloud-height increased. No comparison was made with earlier two- dimensional models due to lack of water in those models. However, a comparison was made between this and an axisymmetrical model of tropical convection in Takahashi (1975), in which the cylindrical model region had 15. rigid smooth boundaries and the drop-size spectrum was modelled as in
Takashi (1974). The disturbance area of the 1975 model was more localised than that of the 1974 model. When the boundary conditions were modified to give the same total rainfall as the axisymmetric model, the resultant maximum inflow height in the former was found to be twice that in the latter; this also altered the raindrop size distribution.
Two main limitations of two-dimensional (slab) models were acknow- ledged by the above authors. Firstly, compensating downdraught had to occur in the same plane as the updraught and in a limited area. It can thus only spread out in the plane of the model on reaching the ground, and is more intense. Secondly, any relative middle- and high-level flow 'cannot pass around the updraught, which is therefore tilted down- shear and some rain falls into the updraught. Furthermore, the theoret- ical work of Moncrieff (1978) shows that the assumption of two- dimensionality severely restricts the storm dynamics development. Three- dimensional models were therefore desirable.
Pastushkov (1973)describes a three-dimensional model of moist deep convection. This was a primitive equation model using height as the vertical
co-ordinate. The flow was assumed symmetrical about the central X - plane, with solid no-slip lower boundary, solid upper boundary on which
variables were held at their initial values, and zero disturbance on the
lateral boundaries. Two types of convection were identified : (i) weak
convection, for which there existed a critical shear below which convection
was retarded; this critical shear was determined by the initial conditions,
and decreased as the available potential energy decreased. (ii) strong
convection, intensified by vertical shear and for which there existed a
finite shear giving maximum intensification.
Wilhelmson (1974) presented both a two-dimensional model, and a three-
dimensional model which assumes symmetry about the x -z plane, using the
primitive equations in height co-ordinates, and compares the results of 16. these models. Points in common were initial propagation (with the wind at - 2 km), an updraught core which never leans upshear, and a pressure force of which the vertical component generally opposed cloud growth at all times. However, the mature cloud in the three- dimensional model grew - lkin higher, developed faster, lasted longer, travelled faster and further than that of the two-dimensional model, and had up- and downdraught intensity maxima at twice the height.
Schlesinger (1975) presents the preliminary results of a three- dimensional primitive equation model in height co-ordinates in which the horizontal gridlength is 3.2 km and vertical gridlength 0.7 km. Lateral boundaries were open, lower and upper boundaries smooth and rigid. No turbulence or liquid precipitation was allowed. If there was shear in the ambient flow, a vortex doublet was found at mid-levels; updraught development was less than in the no-shear case. In the case of ambient wind veering with height, a mid-level pressure gradient resulted which was directed to the right of the cloud motion. The strongest vertical forces were pressure and buoyancy - these were in opposition.
All of the above numerical models use height as a vertical co-ordinate, forming a Cartesian system; however, for the modelling of deep convection, if the perturbation of pressure from its ambient value is not taken into account, the computation of the condensation rate of vapour to liquid water leads to inconsistencies. This problem does not occur if pressure is used as a vertical co-ordinate.
A system of primitive equations expressed in pressure co-ordinates and relevant to deep convection is derived in Miller (1974). This equation set describes a non-hydrostatic system while, providing suitable boundary conditions are chosen, the acoustic wave is eliminated completely.
Results of the numerical model based on these equations are presented in
Miller and Pearce (1974), Moncrieff and Miller (1976), Miller (1978) , and 17.
Thorpe and Miller (1978). The results of three integrations, one with vertical shear and two without, are discussed in the first of these four papers, while in the remaining.three•the initial conditions for the simulations are derived from soundings taken before observed storms.
A description of the analytical results of Moncrieff and Miller
(1976) has appeared in (1.2) above; it is found that this steady state analysis can represent the basic features of the quasi-steady-state reached by the numerical simulation. Generally the two approaches have been found to be complementary and comparison between them encouraging.
The predictions of Moncrieff and Miller (1976) are compared to the characteristics of a squall line observed in the VIMHEX II experiment by Betts, Grover and Moncrieff (1976). Despite significant differences between analytically-predicted and observed outflow velocity profiles and overall vertical mass transport, considerable agreement was found between the analytical model and observation. However, the observed squall-line more nearly resembled the numerical simulation.
It is with Miller's numerical model and its results that this thesis is concerned; the actual equations solved by the model, and the finite- difference scheme used by the model, are presented in Chapters Two and Three where the equations forming the basis of this thesis are also derived. Some of the results of the simulations are presented in later chapters where rele-
vant.
1.4 TORNADOES
Although this thesis is concerned with the development of vorticity on
the cumulonimbus scale, and is therefore not directly concerned with tornadoes
(which have a typical horizontal dimension of about 100 m), a brief
description follows of some of the later research into them. 18.
A tornado is defined in Fujita (1973) as a rotating column of air
which possesses a parent cloud, may or may not possess a funnel cloud, and
which occurs over land. Direct measurement of tornado parameters is -1 almost impossible due to the extreme wind speeds (- 100 m s ) and pressure
drops (- 100mb) associated with their passage. However, maximum wind
speed, path width and path length can be estimated from photographs of
damage tracks, and propagation velocity from visual observation or radar
'tracking of the parent storm. A system of classification of tornadoes
according to maximum wind speed, path length and path width, the, FPP
Tornado Scale, is described in Fujita (1973); this is designed to be
coarse enough for the easy classification of all tornadoes. The results
of classifying the 1971 and 1972 tornadoes in the United States according
to the FPP system are presented in Fujita and Pearson (1973). Fujita
(1976) contains the results of a study of the tornado 'super-outbreak' of
3-4 April 1974. Photographs depicting the life histories of three of
these tornadoes are shown together with maps of their tracks. Maps of
significant tornado paths for each decade since 1930 are also presented.
Turner (1966) produced vortices in a laboratory experiment in a
cylindrical can of water, rotating with its axis vertical, by releasing a
stream of bubbles from a fine tube along the upper part of the axis.
Leslie (1971) and Bode et al (1975) described numerical simulations of
Turner's experiment, the latter concentrating on an investigation of the
effect of surface friction on such vortices. It was found that an increase
in drag caused the vortex width to increase, the frictionally-induced inflow
strength and therefore vertical flow-strength to increase, but the vortex
strength (as measured by the maximum swirling velocity attained) to decrease.
A sudden increase in drag during the life of a given vortex also induced
these changes, with the opposite occurring if the drag was suddenly reduced.
These results were found to agree with the laboratory experiments of
Dessens (1972), and the observation by Golden (1971) in which the circulation 19. of a waterspout decreased rapidly during a 1 km traverse over land, and then reformed on moving back over water. Fujita (1976), however, includes two photographs of a tornado which widened considerably as it moved over water; he attributed this to inflow of relatively warm, moist air from the river.
In all the above computer and laboratory experiments, vorticity is provided at all levels in the initial flow field, and all concentrated vortices thus formed touch the lower boundary. However, these two features are not observed naturally in all tornado-producing thunderstorms.
Smith and Leslie (1978) therefore investigated the effect of confining the vorticity of the initial flow field above a given level by computer model.
Several experiments were performed to determine the effect of varying the minimum height of the vorticity-bearing layer; the side boundary conditions were also varied. Flows with suspended vortices and with vortices touching the ground were successfully simulated, the former occurring when the imposed swirl is concentrated in higher levels, above
'cloud base' (defined here as the lower end of the imposed body force,
1.5 km above the lower boundary in this model). Furthermore, vortices could only be generated if the body force was confined within certain limits depending on the strength of the initial vorticity. This extra constraint could explain why only a few of the cumulonimbus which appear to be suitable candidates for spawning tornadoes actually do. A mechanism of tornado formation consistent with these results is also described. 20. •
CHAPTER TWO
DEFINITIONS OF TERMS AND DERIVATION OF VORTICITY AND CIRCULATION EQUATIONS IN CONTINUOUS FORM
2.1 DEFINITIONS
(a) Vorticity : The motion of an infinitesimal element of a fluid can be
expressed as the superposition of
(i) a uniform translation
(ii) a pure straining motion
(iii) a rigid-body rotation.
It can be shown that the angular velocity CI of this solid-body
rotation is given by
= 2 VX where is the velocity vector. The vector 'tiNg, is called the
vorticity vector, denoted by 3 with Cartesian components (i ) / , 3 ).
Mathematically,
(2.1) ) rI , x ut. (b) Vortex line and vortex tube : A vortex line is a line whose tangent
is everywhere parallel to the vorticity vector; thus in a Cartesian
co-ordinate system the family of vortex lines at a particular time t are
solutions of dx — dz ,-b) — ,t) S(s ,t) (2.2) A vortex tube is the surface in the fluid formed by drawing all the
vortex lines through a reducible closed curve wholly in the fluid.
(c) Flux of vorticity and strength of a vortex tube : Let C1 be a
closed curve on a vortex tube, and let. S1 be the open surface bounded by
The flux of vorticity, , along the vortex tube through Si is C1 . defined by
= (2.3) 21. •••
where d.$ is an element of area of S1 with the direction of the local normal to S1.
Let C2 , S2 and 4,2 be similarly defined at a different location on the same vortex tube; then
12 =fs2 .3..'
It can be shown, since V.3 is identically zero, that 4/2 = 451
i.e., the flux of vorticity along a vortex tube is independent of the
open surface used to measure it; this quantity is known as the strength of
the vortex tube.
It therefore follows that a'vortex tube cannot begin or end in the
interior of the fluid.
(d) Circulation : The circulation r around a closed material curve C
wholly in the fluid is defined as (t) = cts, (2.4) where ds is a line element of the curve C. Application of Stoke's Theorem then gives r(t)-= $63, as (2.5)
where S is the open surface bounded by C and ca is an element of
that surface. Thus the circulation around any reducible closed curve is
equal to the flux of vorticity across the open surface bounded by that
curve, and also to the strength of the vortex tube formed by all the
vortex lines passing through C •
2.2 DERIVATION OF THE VORTICITY EQUATION
Miller (1974) obtained a consistent set of primitive equations in
pressure co-ordinates valid for the description of a non-hydrostatic system,
and showed that this equation set filters acoustic waves completely. 22.
Miller and Pearce (1974) used this equation set in a slightly modified form as the basis of a numerical model of deep convection.
A vorticity equation is required for the study of the development of the vorticity field generated by Miller's model; this vorticity equation must be consistent with the primitive equation set used in the model, and therefore it is derived directly from them and is expressed in pressure co-ordinates:
2.2.1 Primitive equations
The primitive equations used in the model are as follows : (from
Miller and Pearce (1974))
Momentum: + v3 — cot +Fx
— u3 -- E0 01-1- Fu trG a9 ap
gisstrhS) 1" 9s ? to +1 + Eifs 0
Continuity of mass: au. ?Ai (2.9) - + + Bx 61) y is defined by Vi = + (2.10) Miller (1974) showed also that W can be approximated by —4?4,s, and that this approximation is valid when differentiated. Also, ri)z. can be approximated by --9A Z Using these approximations arid the 15 //af) definition of 3 (equation 2.1), 3 is given approximately by r ÷ /16 .6‘j (2.11) ges a Bp Z u, Zo) (2.12) 9 = •P'sF( and (2.13) ax z9 23.
Some manipulation of equation (2.8) is necessary to facilitate the derivation of the vorticity equation. Multiplying (2.8) by -9, and
neglecting terms including products of h' , ei , and L (exactly the
approximation made in Miller and Pearce (1974)).gives, after some
rearrangement, 92esZ +91-0-ftt Substituting for he using equation (2.10) : —co esfip —sekap vap)-14a7H- ges —vati—H ges —cul-Hiap ges -9 Substituting equations (2.11) and (2.12) yields finally : Fp atl9, — 9esy,; + LIN es +9(91*-1)—es* Ws •(2.147 This is the form of the vertical component of the momentum equation
required for the analysis which follows.
2.2.2 Vorticity equation The three components of the momentum equation (equations (2.6), (2.7),
(2.14)), together with the mass continuity equation (2.9), are now used to
derive the vorticity equation; they are here presented together for
convenience
+Fx (2.6)
av = _ ziy% _ _ co +F (2.7) at 0 9
(2.14) kVei=gesrp+.19—v —w-Sz4i ;Ves) gketz re-s 4t. + ?IV + bu) _ 0 (2.9) ax zp
The vorticity equation is found by taking the curl of both sides of
the momentum equation. a The X -component is thus obtained by adding % of equation (2.14) dolJ to 9% of equation (2.7), giving L to 6F4 444-1 ( (04-4114-n -1) ?±? at eszpk gsztp ap/ gsrp aj v / k kilo) 3a0s* ges 24.
It should be noted that the terms containing NV have cancelled out
exactly.
On expanding the differentiation of products and rearranging
considerably, the following equation can be obtained
oi_op ILL al) B._coll +9N —39e /ii as:10211_,VA_A+n.A.F.9_—i*P (2.15) at ----- by ,,sap —vay ap ay sap ay —9e*ge,Jrg ess 3rS ge -aY .r au. For a reason to be explained below, vs-,-jc is added then
subtracted from the right-hand side of equation (2.15). The required
form of the X-component of the vorticity equation is = -- Rat) —v11-4,_)+It.7t-39est _icizet +p_gesiii351)
(2.16) 9 a 61A-4 ) se5 P s Since 5 is the curl of another vector, and therefore its divergence
is zero, the first term on the right-hand side can be replaced by —1.1.40(4 .
The 4-component of the vorticity equation is derived by adding
a of equation (2.6) to —si of equation (2.14), giving
—gesTp"( v ÷ (c1)N—9e sarl—ap tx(v-Oft+thogt1-93?-4-1)÷getstx Again, the terms in IP have cancelled exactly.
By an exactly similar method as that applied to the X -component
(except that frii is. added and subtracted from the right-hand side instead
of 'Sr, the following equation is obtained „( al „ „ as _03M +-sh+,,i ?Iy. _3 k —Max bx Mbp/ bp bx 1 by ges bp bx by 3Ksapl.ges.1) ( 9_1 _ ÷ a r-,c (2.17) —9 67(VA 9e5 ax —9es ap Sirice the divergence of S is zero, the second term can be replaced
by —VIuy .
The vertical component of the vorticity equation is derived by
subtracting 1 of equation (2.6) from of equation (2.7); this ay 3_dX 25.
gives
be bX ZX 4 ax NJ ap 9 The terms containing If have cancelled out for this component also.
By expanding the differentiation of products, rearranging, and
adding and subtracting Sgesal)(gWes) from the right-hand side, the
following equation for 3 can be obtained _ a3 ZS ZS acoke. Zu)20/ at — 11:57( V + 11 — a17 + esigal —3(t+z—ito[z+-e—c-; (2.18) It can easily be shown that the fourth and fifth terms are equal
to
Equations (2.16), (2.17), (2.18) are the three components of the
vorticity equation. On making use of the observations noted after each
equation, they can be written as a single vector equation thus
—6ANY§, 4- (S:V).4!:1 34)lh 9144)1- AUK +17XE (2.19)
A B C t< D >} E
An explanation follows of the significance of each of the terms
A-E.
(A) Rate of change of S due to advection.
(B)(3. 10011. can be rewritten as 131 Um g;- Poo P61
where P and Q are two neighbouring points on a vortex line and Skt;
is the velocity of the fluid at Q relative to that at P. Then the
fractional rate of change of vorticity due to this term, i.e. —1 d'S 1 I --6E , is identical to the fractional rate of change of the
material line element vector Thus in the absence of
other effects, the local vorticity vector behaves like a material 26.
line element coinciding instantaneously with the local vortex
line; part of the change comes from an extension or contraction
of the line element (component of SIL parallel to 3), called
'stretching', and part of the change comes from a rotation of the
line element (component of &M perpendicular to 3 ), called 'tilting'.
(C) This term and the advection term can be moved to the left-hand side
of the equation and combined such that the left-hand side becomes
ej_e_) ; thus in the absence of other effects lk is
conserved following an element of the fluid; if the element expands,
the same amount of vorticity has to be shared by a greater volume of
air and thus 3 is reduced, with the opposite true in the case of
a contraction. • Since, if VAIL=2° , i.e. the fluid is incom-
pressible, term C is zero, this term is due to the compressibility
of the fluid. Using the mass continuity equation (2.9) valid for
this motion, term C reduces to =. 19_ b co f.ip nessibitit9 ges bPkges) (D) Rate of change of 3; due to horizontal gradients of fa, or I i.e. baroclinity. There is no baroclinic term in the vertical
component due to this particular formulation of the primitive
equations in pressure co-ordinates.
(E) Rate of change of 3 due to diffusion of vorticity. aV The reason for introducing ) ) 3geepTes) into the X) vertical components respectively is now evident; the partial cancellation of the stretching and compressibility terms is purely mathematical, and these terms must be present in order to describe fully the stretching and compressibility processes. 27.
2.3 CIRCULATION EQUATION
2.3.1 Derivation of the circulation equation
The circulation r around a closed material curve C contained
wholly in the fluid is defined by equation (2.4) as r(t) = It can be shown that differentiating both sides of this equation
with respect to time gives
(2.20) dt =cis + tid(14-.4 Substituting the momentum equation relevant to deep convection
(equations (2.6), (2.7), (2.14)) into equation (2.20) gives, after some
rearrangement at(t)r-- IV14.4,+ gtaii -91:s.dviE.4 +paw). cts, =gfeC1-91:s.ds +1E.ds,
Using Stoke's Theorem then yields
ft(t) = TAI: (2.21)
where S is the open surface bounded by ,C, and d..S is a directed element of that surface. Equation (2.21) is the circulation equation valid for cumulonimbus convection.
2.3.2 Consequences of the circulation equation
Consider the special case of equation (2.21), viz. c111(t) = 0 (2.22) dt This can only be true if the sum of all the body forces in the fluid can be written as the gradient of a single-valued scalar function of position, i.e. the total body force is conservative. 28.
Consider a material tube coinciding at an initial instant with a
vortex tube of arbitrary cross-section; by definition, no vortex line
initially passes through the tube surface, and the circulation round
any closed curve lying on the tube and passing round it n times is
n multiplied by the vortex-tube strength. By equation (2.22), the
circulation round each of the curves is constant, and in particular
the circulation round any closed curve lying on the tube but without
passing round it remains zero. Therefore the flux of vorticity
through the open surface bounded by such a curve is zero, and this
- is only possible if this surface continues to lie on the vortex tube.
Also, equation (2.22) shows that the strength of the vortex tube
remains constant. Therefore, in a fluid whose total body force is
conservative, any vortex tube moves with the fluid and its strength
remains constant.
However, in the model of convection used here there are two terms
(baroclinity and diffusion) which are non-conservative, both singly and
in sum. Equation (2.22) is not true, and therefore the above argument
,:breaks down. In fact, in the case of deep convection, while the
pattern of vortex lines can be drawn for a given instant, and while this
is a useful and instructive way of representing the instantaneous
vorticity field, there is no way in which a particular vortex line can
be identified at different instants. 29.
CHAPTER THREE
DERIVATION OF THE VORTICITY EQUATION - FINITE DIFFERENCE FORM
In order to facilitate the discussion of the development of vorticity in a numerical model of cumulonimbus convection, it is necessary to formulate the vorticity equation using finite differences; each of the terms on the right-hand side of the vorticity equation can then be calcu- lated by computer using the velocity and thermodynamic fields generated by the primitive equation numerical model. This thesis will be concerned with the results of the Miller three-dimensional primitive equation model of deep convection, the derivation and description of which can be found in
Miller and Pearce (1974) with some amendments in Moncrieff and Miller
(1976). However, for convenience, a brief description is given of the configuration of the model grid before the vorticity equation is derived.
3.1 PRIMITIVE EQUATION MODEL OF CUMULONIMBUS CONVECTION
3.1.1 Definition of symbols
If 4) is any property of the atmosphere contained within the bounds of the model region, then s 0(tAx, j dy , Up) with respect to the origin of the model co-ordinates. Note that increasing j a decreasing y.
Using standard notation, means of (I) in the x, y, p directions are respectively defined as: frA [0114-113,lik 00-1jijk)
[ OIL jiklk E 2 (kiik
N/ jk+y2 + Li jk+i) 30.
and [021)31, j, ,k 01,j , k4-2) •
Note that the subscript on the left-hand side of each definition refers to the position of the new variable thus formed. Any combination of these symbols is also possible, e.g.:
[041314j+1>k E z 931(.4 k+[01144 tj+i)k — 2 1,4j47i )
Again, using standard notation, differences of in the x,y,p directions are respectively defined as: 6X, ] CIAA Ow*
cs403i,j2)1( s coithk -
re3L,j,k+1
and
Ce4195ii,,j k+I = C5i,j,k+2 01,j,k
Note that due to the fact that increasing j E decreasing y, the Sy operator is defined in the opposite sense to the others. As with the means of (I), any combination of 6-operators is permissible.
Furthermore, any combination of 6- and mean-operators can be defined, and since the operators are linear and all variables assumed to be continuous in space, the order of the operators may be changed, with no effect on the value of the result.
3.1.2 Configuration of model grid
The model uses a staggered grid as shown in figure (3.1). The table below shows the positions on the grid at which the various quantities is, 31.
Fig.3.1 Configuration of model grid 32. are computed. The symbols correspond to those on figure (3.1).
POSITION ON GRID VARIABLE COMPUTED BY MODEL =x, O i,j,k u, v, yFx, Fy =x u =x u O i+1, j+i, k h', C, IP; u V 3- =X y,p •i+1, j+1, k+1 w, et*, (38*, p33 Z, U 3 5X3Y3P, Fp =x,y X i,j, k+1
Hencefortli(i,j3k) will be subtracted from the subscripts of the variables; thus,
(1)7:+1, 7c4.71 will be written 41,m,n
IP are defined in the model in terms of u, v, h' as follows:
--WN7- cxx vz,u2,,o9] [Prelv/2,0 (3.1) 69
fr_„ „12 f=xx. = veo (3.2)
3.1.3 Primitive equations in finite difference form;
time integration scheme
The primitive equation set solved by the model is as follows
(Miller and Pearce, 1974):
.x,31p[82Pubp,„ R Momentum: 1X110,0,0 —{8gV/10,010 +Voloi030)0fi (3.3) At 6.X °Ivo 2 Ap
c. 2P I" EE.1 Avo,o,o =_- —[8911o,o)o-110Ao so,op 11*[ 4,o,0)0+P (3.4) At py 33.
40.4.1 _444E-Avir[sx - x.19 =Au [SSW 11 [S2t(t)11 14 kit ii)1i 24 )2
_9e51[10 — g2e,,3 (9..444— tti,11,1 (3.5) 61) 2 02 .2 016 + F
Continuity: CCXTiti 4- [S1791. n DPC414- (3.6) 6tX Ay Ap
The time dimension has deliberately been omitted from the subscripts of the variables; it is assumed for the purposes of the following analysis that all the variables are held at the same timestep. However, examination of the description of the time integration scheme presented in Miller and
Pearce (1974) reveals that this assumption implies that second-order time error terms arise in the vertical momentum equation, and this is discussed in detail in the Appendix. Fortunately, the numerical errors introduced by making the above assumption have been found to be negligible, and except for the discussion in the Appendix, the problem is henceforth ignored.
As in the derivation of the continuous form of the vorticity equation, the vertical momentum equation (3.5) is simplified by eliminating h' in favour-of u and V. Substituting equation (3.2) into equation (3.5) and rearranging yields:
,x4r1 rs47.X,X,91 rs2p 03 Aaji [X,91 reral 117. At - - k,24 44-1. — iL 1'2'2 A9 24'5 24 (SPAi I i +9217-7(b9,t, rcPrpien 4711 i [SPV'9],,, I i& Air p balt—aTr 2-
—92(3.(=/t)ii— L. 1 ÷ FP 2 Wig 2. /Z12
s Multiplying throughout by _w) , and using the fact that p is 2 independent of x, y and t :
34.
r 2p co 1 _...— nos[S PA Nes)14.0.6 ies)}1,vi Ai 90 4 2Ap
_rx.536 47- .41 t[sx( 9 A re:VIA Ax Ap [ ,..„(73.(tx,51, • [-sx'3, I?" —ger-llivil- 9es(se "11, — v 11 nil .64 i---ip
+ g ilapi,14., —LA.,-1 — . --Fi'llovit1 (3.7) el lri ge li
This equation will be further simplified in a later section.
3.2 DERIVATION OF VORTICITY EQUATION
3.2.1 Choice of finite difference scheme and points in
grid for calculations
The vorticity equation is found by taking the curl of the
momentum equation. In finite difference form, there are very many
ways in which the operator 'V' may be defined, each one:
(a) producing slightly different expressions for the vorticity vector
and its equation, and (b) calculating the vorticity and its equation
at different points on the grid.. There are, however, two constraints
on the method which can be used in order to minimise numerical errors:
(i) the resulting equation must be consistent with the
primitive equation set,
(ii) the resulting equation should be consistent with its
35.
continuous analogue.
The first condition is satisfied automatically by deriving the
vorticity equation directly from the finite-difference primitive
equations (as opposed to deriving a vorticity equation in continuous
form then rewriting it using some finite-difference scheme).
The effect of the second condition is now discussed by means of
two examples.
Consider the simplified momentum equations:
Au0,0,0 _ — [Sx (3.8) At 46,x
tiv0,0,0 Cg9 1110,0,0 (3.9) b,y
gesis- = gel (3.10) At
The x-component of the vorticity equation is formed using
equations (3.9) and (3.10). Centred space differences will be used
in both examples, since they were used throughout the development of
Miller's primitive-equation model, and also their use results in
smaller numerical errors than either forward or backward differences.
Example 1: Differentiating (3.9) with respect to p, averaging in the x-direction, and multiplying both sides by ws„...
A s SPTil s [S116974)1 (94 ti,m) = ,d 1i304 At AP 9t1p Differentiating (3.10) with respect to y:
ge Co40,1 at. Log 36.
Adding these two equations:
.c.gesSevlio t ( ifDit,5,11= ges Pt 7 Pp J
The expression inside the curly bracket on the left-hand
side of this equation is a possible finite-difference
expression for 1.40)11k however, inspection shows that the terms on the right-hand side do not cancel
completely; this equation for 1,0A. is, therefore,
not consistent with the continuous form, since in Chapter
2 it was shown that terms containing i should cancel exactly.
Example 2 Differentiating (3.9) with respect to p and multiplying
both sides by ge,
ge,s DM°" - - 9e.s DEV 10,0 a Sp /2. AH AP Differentiating (3.10) with respect to y and averaging
with respect to x :
A0 ( opA) sPV9 ISA _ At k ge dy Adding these two equations, and using the fact that the
order of operators can be changed: A f9eNNa0,4 41P6K)Lil
The contents of the curly bracket form this time a possible
finite-difference expression for i0,01A , and the terms
in * have cancelled exactly. Condition (ii) above is
therefore satisfied by this method.
It can be seen therefore that Condition (ii) determines the finite-
difference scheme to be used when taking the curl of the momentum equation, 37. and hence the expressions for vorticity and its equation, and the position on the model grid at which these are calculated. Similar simple experiments to those given in these examples can be performed in order to find the finite-difference scheme for the y- and vertical components of the vorticity equation. The rules thus obtained are as follows:
x-Component : ge Z15 of the y-momentum equation, plus Al
of the x-mean of the vertical momentum equation (RULE A) y-Component : —9e11 2(,1 Ap of the x-momentum equation, minus V of the y-mean of the vertical momentum eauation (RULE B)
Vertical Component : afaX of the y-mean of the y-momentum equation,
minus of the x-mean of the x-momentum
equation (RULE C)
3.2.2 The vorticity vector in finite differerence form; further simplification of the vertical' momentum equation
Use Of Rules A, B, C to calculate the left-hand side of the three components of the vorticity equation gives the following finite-difference formulations for t , 3 :
E _CS9(0)30A. + 9e Ci)vjt)Ah (3.11)
g,ns14is:Lpoom csgA10?)),,2. (3.12) to(
-- Le701A)0 ,Jk)0
The definition for is seen to be identical to that 38. given in equation (3.1).
Equations (3.11) and (3.12) can be used to further simplify the vertical momentum equation (3.7). Forming trIvzsi.a/.2. and according to the rules given in section (3.1.1), and substituting into equation (3.7):
1—w1e5„/2)101 1) [eV] Fir@ s) Pit [s2r1 At 167p 9ek 2JSp
-1 viz + 9,71-:_ FP 1,2,9di (3.13) [°* gesti The three components of the momentum equation (equations 3.3, 3.4,
3.13) and the continuity of mass equation (3.6) are now in the required form for the direct derivation of the vorticity equation according to
Rules A, B, C given above.
3.2.3 Derivation of the Vorticity Equation: x-Component
Applying Rule A to equations (3.4) and (3.13) yields p = _9s[8PN1314he2 x) 91S2PV 30)0A dt -9e7Li s A (p)2
[SPF of [64 942. —Tsp-•LO) 905.F X/ 1
re e3p,p.s2pit40,1 [0(,240,P r9A.1041 +L k 29esii 4,69 (34 x _ Mr3q")10,04 g [6‘ 4)]040,ii (3.14) 63 t4 39.
Terms containing the differentiation of a product are now expanded as
follows:
gek[EP AP - ge[uPer90,0,i _gestilx:18:100,0,1 (3.15) Lip
[89 ( 07" 62f 2 top,1 = [ PIP'984(S2fiAe0,0,1 2904 apay 2gek 6.0.9
[erg. s9 (yreS PiA 0,0A (3.16) 2gelt A1)48'
[Sia xig)P=,imx =Yam sY7, om.x 9 Am0A rxisYS9P91110)0A (3.17) Ay
X _p(x,y,p TAN x_ 4 1- r=149.4 pi -xm,t9strI x LI S V ' (3.18) 9)13ri •••• 0340 . 6,9 Dy Dy
The second term on the right-hand side of equation (3.14) is expanded
and rewritten thus: r gfrisa-)4 9,1 rsPlrig'PP s coUzi 9 1P,P 6P( 2Plig 41 — gds 40,0,i = opyli gel/. ti L 2k (Apr geli 2 0101 2 (Apy.
• anS ce(0,=424 V411) t-1734MR-seePaoA4 -6"1"(52PViSe(gelLa- .—azti 2(A0- 2(4)1 2(A1)
P SP c=7-1- XIS 11 —804.[%2Pv °04. (3.19) moz
40.
As in the derivation of the x-component in continuous form, giklk X must be added to and subtracted from the right-hand side of -
equation (3.14). The finite difference form may be chosen at will
(since it is both added and subtracted); as will be seen below, the
form most suitable here is
(3.20)
Equations (3.15), (3.16), (3.17), (3.18), (3.19) are now substituted
into equation (3.14), and expression (3.20) is added to and subtracted
from the right-hand side. On arranging the terms into groups according
to their physical effect, as in the continuous case, the resulting
compressibility term is found to be cornmssibility _[T(.90(sxrtxly,piV fix'Msyr49,P3X0,0 — 4 (6°P)i) Ay
co (S2Pv S(306%%1. 20,02 x 24, P p7191z- _gest6 V ,E 0) [ 211(15.1. e(co--TP)9] it0, (3.21) 2(0' 2gesti Api^9
The first two terms on the right-hand side of (3.21) are each of
the form A.B. It can easily be shown that they can be replaced by
expressions of the form 71.3+ (31468. Thus
p v_Ex,Vccx=_474341 TM.(%1X13exgi 49'1;41 u. = ° -1°Pit [SgrX.64rellopA (3.22) 46X AX
reg sy timorix (t)93. 6,3rciy,p,x, (6X g xty1.S 0iO4(3.23) Ay Ay 41.
Adding together equations (3.22) and (3.23) gives, for the first
two parts of equation (3.21)
rxig,x1 9 x tx,y,P,9 s9 4 ) E (b tix ey 040li
X' 9A rotte6s. sxp7x,u,p,i -- Is g g .53(&xf2- 4P)10,04 __ ...10,01.1 (3.24) AK Ali The continuity of mass equation (3.6) is
WE'14,14 o + [89 + (Sew) At J14,0 = 0 x
Averaging this equation in the x, y and p directions and recalling
that the order of operators can be changed gives
Esqx4.69-1 rs,P(*vAl , (3.25) JoAia L V -10)0,11 = 0 Ax p
Substituting (3.25) into expression (3.24), substituting the
result into (3.21), and noting that
Exi,tp] • 0.05,1 = [e2' ta' Lip 24
the compressibility term now becomes 4 yd49 ,2p=w41 s is2P P efxisi tAgool =[7 4 0, 00 _ v . 00 at res5miti, 2 kap (hoik 2 002 X
.043"(62PV .SP(geSajo,li [S2PCce S9(a)=—")}6.A.I. 2 (A02 29e4
s4(sx axq ,PN csx sx AliPCMIN ' v hop (3.26) Dx Dy
The fourth term on the right-hand side of (3.26) is now rewritten 42.
x x [eV. S9( ct)''s )9100,1 =IN .82e6p.vccoesne,0,1÷[-1.2es2PwwPA,oit 2 9 esvAA, 2 9es„4 ApAti 2964 Ap
=ffsIfs2P159.VPi-i'940,1- --CroPfAs.s9(5IelbA-s x (3.27) 2gesti NIA9 29 4 e%eli Apt9
Substituting (3.27) into (3.26) gives
2tes oMP)P4051 ) • =. [C7962 V .V(9010,04. [c,70 116 .S4 2 (AP? 2.9 e.,0,0,Ap 2p2p +{.174349- etc-73491 _go .eirt9,01, WV( '11 JoAt 4-0,0a 12 2 gel tip Ay
=Am jOrAsnaGsi 4 _ sleroeavk (3.28) Ax
Careful examination shows that this equation consists of the following: Wa s (a) The first two terms are a finite difference form of 53eszpl9el
i.e. the true compressibility term (on comparison with the
continuous case).
(b) the next three terms consist of two finite difference forms of
the same quantity, but of opposite sign and thus
partially cancelling. These three terms together therefore
constitute a second-order quantity.
(c) the final two terms are immediately seen to be second-order.
Therefore only the first two terms contribute to the true compressi-
bility term, the remainder being the second-order numerical error terms arising due to the use of finite differences. In the actual computation 43. of the diagnostics, the compressibility term is limited to the first two terms of equation (3.28), the other five terms being computed as a separate 'spatial error' term.
The six terms of the x-component of the vorticity equation, plus
the 'spatial error', can now be written down
1?-7,41 p [FLX49aPdS15 e,o 21 -Jo .2 1—Pu. sP= "30 ,4. At adtectig, _ . /19 Op
•
+ [07"V reitalso rrlm,PY 6ellePS2P010,0,1i (3.29a) 29esti 6P AY — 2 OW
(AgOA51 JoAt (3.29b) At ::::11X191X&exl"IP19
x [roe (Aso,biti) %ME L 17 0 u- -10,0,3 — sesti 0 '4044 (3.29c) ‘Lit kath,9 Ay
100,4 —GT43'"(S2P4(96)P16,0,ii—LVP.62Pes.V(MP40,12. (3.29d) tat &rye:guts 2 (4)2 29E4. esk 4
g1t0,0A (3.29e) kAt 2)
ges[sPF,}0,0, _ 64F: lc , )2. (3.29f) ,Ap ges4 x 2P l OE 0,04 __ IlX1/x96%49 _ gestrA2PrO4 10„,,i N GlIt000e 0.30,11 kX sepa „Al 402 29 i
Wrvb9(Sxr13')]0,04 — [E14538Vi- xto)] (3.29g) Ax 119 44.
3.2.4 Derivation of the Vorticity Equation : y-component
Application of Rule B to equations (3.3) and (3.13) gives the following equation for 90,0,1
— _ ges[eNT1,1 + gigl"PeeetOlopit — ge0F0,,A1 tie a 2 (6p)2 Ap
—(SrePS2Peid041— [Sxr09T19)1:4 4E8)(0" Pr 40.1 29es &ix) Ax Ax
14 ret 9 vpvio3031 - - (3.30) g fr) .10,0,2 +[6x QX Wit Ax
By expanding the differentiation of products, rearranging using
a method exactly similar to that of the x-component, and adding and
subtracting a finite difference form of , the final form
of the y-component of the vorticity equation is found to be
ru. 1, P,0 a 4c="x `91 r SEV ere tS] At7 4 9 J0,04 +1..r9iPq ]0) 041,-9fi lt aAa At abWitm Ax Ap
..CIMIP'XA(841Cial9 tr"Pegg et41030, (3.31a) 20,i 6P Ax 2(4)z
x (46■90A5) = (7 r) 1919 SY vEx'8 'n,004 (3.31b) kAL /*doling by
iriPsX (A■vispi )04 [rIPSN] (3.31c) 96 4 op,2 littifig
P r 1 1.=Y0) 2P c2P-e Sn4 bithboi) 0 . SX(COe 0,0,1 (3.31d) 6"- cuRretsivin§ 2 OW 29 es3t4 ax
45.
H )1— q = — 9 Cif , opi (3.31e) Ax
[MLA_ gereF,Im (3.31f) At laftAs10f1 gE%Lix 112741T-
oph ) Tmls2p-N41 pt /vial error L 264) -10,04 ti (Mt 2gesti 6,p&
rgriji9 SX(S979 rOiRMIX OWFX190Ni + Lt3 1J0A-1 (3.31g) ny Lac
3.2.5 Derivation of the vorticity equation : vertical component
Rule C is applied to equations (3.3) and (3.4) yielding
3 Ube = — [SUisx sli . 2i, 0 _[Sx0x1'1182PvA246, I Stx F91 14,0 Lit
X —r.X 1 4 ra ji I — [SX V? )] -1-18TOL"871,014 0 rx 2. 2/ 1/2/0 (3.32) 261)6 Ay
Expanding the differentiation of products and rearranging
■yiH L1314,0 — _11-cxsxF {7/-989?r9i,-- to_.[EraieicgliKesa.,,,o,r9."moa 0 -- ax" iho 212,1 46,,y 2.84Ax " 244
X estp x elx0yr y i [84Z-394'11)1191 + 7.1 21v— Ni(211PgieV11,110 24pAy MP 6( Ax Ay
[S9rn 4 1 0 (3.33) Ax Ay
46.
A finite difference form of ISeqp(0) now has to be
added to and subtracted from the right hand side of equation (3.33)
in order to complete the stretching and compressibility terms. The
form which allows the greatest simplification of the equation is [rvt‘j 81N 9 es .1Ap
The resulting compressibility term is then seen to be
....11 lio rmt-rlpi'' _Ertul..92. 410 P-91vo CATE"t+°)amvonsui: — 3 AK Ay
The final two terms on the right-hand side of this equation are
both of the form A.B; replacing them by terms of the form T.FA-oA6B,
and expanding the differentiation of the quotient in the first term,
gives after some algebra
= VI.601) +[ }i ir."(Ax Zig11 la kint3 . lx" -)11 S(964 14o s)13+: 22
yr-: Xilt4, rAi [S414'4815XLIAisttO 2 2 -YfW%Iljt iii° AX L1 y
Using the continuity of mass equation (3.6) this reduces to
tontz4retS$XIEF rri"STS9VA. 101,0 (3.34) (MA4.1112°) i AX dg 2.11
Inspection reveals that the first term on the right-hand side of
(3.34) is a finite-difference form of the true compressibility term,
the remaining two being second-order; the latter are therefore written
together in a separate 'spatial error' term, as in the two horizontal 47. components.
The full equation for the vertical component of vorticity is therefore as follows
3 212,0 ur mrs901, . 011,4 — 2 1 VY- (3-. 35a) At 46AtiOrl 2Aptx 2444
- r x13. xl -1 s p(6) (3.35b) the z L AP -11-4o
X ES9 (731691P 62t91 I.n CgrojCIIPS2PVIL lip (3.35c) 11 2 — 212. —611121 titling 2 ApA9 Zap Ax
rivisx 43 re 8%61 (.3..35d) wpfessimbi 1.5 tf-5s
it i go — 21.111.0[V WM (3.35e)
2 1 clifficko Ax a 1
4% xis id XIga sysxto4 1.0
ETNA it 11110L. k 1° 5 -(3.35f) alerfor Qx ay
A finite-difference version of the vorticity equation has thus been obtained in sections (3.2.3), (3.2.4) and (3.2.5) which can be seen to be consistent both with the continuous analogue of the vorticity equation (described in Chapter 2) and with the finite-difference primitive equation set; this vorticity equation is therefore in the form required for computation, some results of which are presented in the following chapters. 48. CHAPTER FOUR
THE VORTICAL STRUCTURE OF SIMULATED CUMULONIMBUS
In Chapter Three, a vorticity equation was derived in finite- difference form which is consistent with both the continuous analogue and the primitive equation set used by Miller's model. The expressions for the three components of the vorticity vector thus derived are now used to calculate the fully three-dimensional vorticity field generated by Miller's model of cumulonimbus convection. An attempt to interpret these results using the diagnostic vorticity equation will be made in
Chapter Five.
4.1 METHOD OF PRESENTATION OF RESULTS
The results of the calculations performed here are presented in two basic ways:
(a)Contours of an individual component of the vorticity vector;
vertical and horizontal sections are taken through the model area
and contours shown of the component of vorticity normal to this
section, i.e. the contours represent the magnitude and sense of an
instantaneous local rotation in the plane of the section. Contours
of the modulus of the vorticity vector are shown on horizontal sections.
In order to assist comparison and interpretation, the sections of
airflow corresponding to those of vorticity are also shown.
(b)Plan views of some vortex lines. The pressure is written alongside
certain points on the vortex lines to. give some indication of the
vertical development of them.
It should be emphasized that, since the vorticity model is confined to a region of horizontal dimensions 20 km x 20 km, whereas the primitive equation model has horizontal dimensions 29km x 29 km, the lateral boun- daries of the vorticity model are far enough away from those of the
I 49.
primitive equation model for the effects of the latter to be negligible.
The vertical dimension of the vorticity model is equal to that of the
primitive equation model, i.e. 800 mb in the first case study, and 900 mb
in the others.
Three particular cases run on the primitive equation model are
studied, being (A) a tropical cumulonimbus, (B) the Hampstead Storm,
and (C) a splitting storm system.
4.2 CASE A : TROPICAL CUMULONIMBUS
Results of the simulation of a tropical cumulonimbus are described
in Moncrieff and Miller (1976). The initial conditions for the simu-
lation were obtained from a radiosonde sounding taken prior to a squall
line observed during the Venezuelan International Meteorological and
Hydrological Experiment (VIMHEX II, Observation No.147).
The perturbation used to initiate the convection grew first into a
cumulus and then into a cumulonimbus, the decay of which produced a
strong downdraught and thus a density current propagating near the ground;
this density current moved relative to the wind at all levels and the
resulting boundary-layer convergence initiated the 'second-generation'
cumulonimbus. By the 40 minute stage of the simulation, two such cells
were observed. These rained out and decayed, new cells growing in the
period 60-80 minutes.
The results of the vorticity calculation are presented for a time-
average of the primitive variables over the period 84-92 minutes
(Figures 4.1-4.11) during which time two cells were identifiable together
with a coherent cold pool and gust front. (The calculation was also
carried out for the instantaneous fields at 88 minutes with very similar
results; the effect of the time-averaging has been to smooth out some
of the less permanent and important features, although all magnitudes are 50. very slightly reduced. The results of this calculation are not presented here.)
The vertical sections of the horizontal vorticity components
(Figs. 4.1, 4.3) show a slight phase tilt with height, corresponding to that of the updraught core (Figs.4.2,4.4)- Largest magnitude -2 -1 (1-2 x 10 s ) occur in regions of maximum streamline curvature, where the air is entering and leaving the updraught, and where the downdraught is forced to spread as it nears the ground. The total and perturbation fields of the two horizontal components have similar structure.
The horizontal sections of the vertical component of vorticity
(equal to the perturbation field, since the initial vorticity field was horizontal) for 300 mb, 600 mb and 1000 mb (Figs. 4.5, 4.9(b)) -3 -1 show a maximum value of less than 5 x 10 s . At 800 mb, however, -2 -1 (Fig.4.\ (a)) there exists a region of strong (> 10 s ) positive vertical vorticity near the interface between the updraught and down- -3 -1 draught of the more active cell, with a slightly weaker (- 5 x 10 s ) negative region to the south-west of this updraught. This is also the case at 700 mb (not shown), these two levels being within the region where updraught and downdraught exist strongly side by side (Fig. 4.10(a)).
The section of airflow for 1000 mb (Fig. 4.8(b)) shows contours of horizontal divergence on this level, which apart from a multiplicative constant, is numerically equal to w at the 950 mb level; comparison of these contours with those of ICI for the same level shows some correlation between horizontal gradients of w, and the value of M.
Since horizontal gradients of w contribute to the horizontal components of and since the vertical component of is relatively small at this level, it can be concluded that the horizontal variations of w dominate the vorticity field here.
The figures of plans of vortex lines (Fig. 4.11) show that the vortex lines have a similar configuration at most levels in the storm, the main 51.
difference being the orientation with respect to the storm. Those
regions where the horizontal components of vorticity have large
magnitudes correspond to regions where the vortex lines are relatively
close together, around the updraught core and downdraught gust front.
Maximum vertical development occurs in the updraught near the updraught/
downdraught interface.
4.3 CASE B : THE HAMPSTEAD STORM
Miller (1978) presents the results of the simulation of a storm
which occurred over North London on 14 August 1975, known as the
Hampstead Storm. Observations of this storm, the characteristics and
environment of which are described in Grove (1977), reveal that it lasted
for about three hours, with heavy rain and hail affecting a very small 2 'area (approximately 100 km ). The storm was therefore considered
stationary, and it had multicellular structure, several cumulonimbi
contributing to the rainfall.
The initial conditions for the simulation were taken from a Crawley
sounding, modified to take account of surface observations in London.
Since the ice phase is important in mid-latitude convection relative to
tropical convection, the precipitation fall-speed for the Hampstead
Storm model was made to be almost double that of the tropical cumulo-
nimbus model described above, in an attempt to model the presence of
falling hail.
In the simulated storm the perturbation used to initiate convection,
located south-south-east of the centre of the model region, moved slowly
towards the north-west then grew rapidly into a precipitating cumulo-
nimbus. By 30 minutes a downdraught had begun to spread out at the
surface, the strongest boundary-layer convergence being on the south-east
edge of the 'cold pool'; new cells were thus generated which followed a
similar lifecycle to the primary cell, moving from the south while growing, 52. then moving towards the north-east when mature.
The calculation of the vorticity field was carried out for
(i)time-average of the fields over the period 32-44 minutes, and
(ii)time average of the fields over the period 64-76 minutes. As in the case of the tropical cumulonimbus, means were selected in order to smooth out less important transitory features whilst preserving the important ones.
4.3.1 Results for 32-44 minute means
During this period the first cell was very active, with both up- and down-draughts in existence.
The vertical sections of the horizontal vorticity components (Figs.
4.12, 4.14) reveal a level in the atmosphere near which the horizontal -3 -1 components are very small; at about 850 mb the values are < 10 s , compared to the values above (1-- -6 x 10-3 s-1. ), around the updraught -1 core, and below (± - 10-2 s ), in the region where the downdraught approaches the ground and is forced to spread out. The horizontal slices of the vertical component at 300 mb, 600 mb (Figs. 4.16(a) and
(b)) show that the updraught is divided into two parts, one part in an area of cyclonic vorticity, the remainder in an area of anticyclonic vorticity. This applies also to the downdraught outflow on the lower surface (Fig. 4.20(b)). Fig. 4.18 shows that the largest values of
1 0 correspond to large horizontal gradients of w, as in the tropical cumulonimbus case, at 950 mb.
The vortex lines are seen in Fig. 4.22 to pass around the updraught
core and downdraught outflow; again, the configuration is similar through-
out the storm region, alignment with respect to the storm being the major
difference between the configuration at different levels. 53.
4.3.2 Results for 64-76 minute means
During the period 64-76 minutes three cells were identifiable; one growing, one mature and one decaying. Figs. 4.23-4.33 reveal that the same basic features are present for each of the cells as were present for the cell active in the 32-44 minute calculation; magnitudes are, however, lower, and the continued spreading out of the cold downdraught is shown by the greater separation of the regions of maximum values of horizontal vorticity below 850 mb in Figs. 4.23(b) and 4.25(b).
4.4 CASE C : SPLITTING STORM SYSTEM
A paper by Thorpe and Miller (1978) describes the results of two simulations. The second of these used as initial conditions soundings based on the routine 123 radio.onde ascent at Crawley for 9 July 1959i this is considered representative of the atmospheric conditions in the
Wokingham area where a severe storm (described in Browning and Ludlam
(1962)) occurred a few hours later. No attempt was made, however, to compare the simulation results with the observed storm in Thorpe and
Miller (1978).
During the first 44 minutes of the simulation a cumulonimbus cell grew to maturity and produced a wedge-shaped cold air outflow aligned along the axis of the upper and lower level winds. This cell split into three updraught cores in the period 44-56 minutes, and by 60,minutes only the two dominant updraught cores existed; these both had similar velocities to the parent cell, both propagating almost due west.
Results of the vorticity calculation are presented for the means over
32-36 minutes (in Figs. 4.34-4.44) during which time the parent cell was in its mature stage.
Figs. 4.34 and 4.36 show a strong phase tilt with height of the horizontal components of vorticity, corresponding to that of the updraught 54. core (Figs. 4.35, 4.37). Maximum values are found at the top of the -2 updraught and in the downdraught outflow, both being - 1-2 x 10 s-1.
Horizontal sections at 300 mb, 600 mb (Fig. 4.38) show a division of the area in and downwind of the updraught into two parts, one with cyclonic vorticity, one with anticyclonic. Vertical vorticity -2 reaches values of 10 on the 600 mb level, the cyclonic vorticity being more intense than the anticyclonic. Again, vortex lines are seen, in
Fig. 4.44, to pass around updraught and downdraught cores. 55.
KEY TO DIAGRAMS
-1 Vorticity : Contour interval 0.005 s Red > 0 Green = 0 Blue < 0
Vertical Velocity ! Contour interval 1.5 m s-1 Continuous > 0 Dash-Dot = 0 Dash < 0
Horizontal Divergence : Contour interval 0.0015 ' 1 Continuous > 0 Dash-Dot = 0 Dash < 0
Wind vector arrows : Length proportional to wind -1 speed 4 mm =10 m s 56.
CASE A
TROPICAL CUMULONIMBUS 84 - 92 MIN MEANS
FIGS. 4.1 - 4.11 57.
Fig.4.1(a) Vertical section through storm along line AA' marked on Figs. 4.6, 4.8, 4.10, showing contours of the component of vorticity perpendicular to that plane.
Fig.4.1(b) As for Fig.4.1(a), but vertical section along line BB' marked on Figs. 4.6, 4.8, 4.10. 57.
Fig.4.1(a) Vertical section through storm along line AA' marked on Figs. 4.6, 4.8, 4.10, showing contours of the component of vorticity perpendicular to that plane.
Fig.4.1(b) As for Fig.4.1(a), but vertical section along line BB' marked on Figs. 4.6, 4.8, 4.10. 58.
—) —4 —4 . —) —4 —4 —) • —4
A' •
• • • • • •
•
• • • SA■ Fig.4.2(a) Vertical section through storm along line AA' marked on Figs. 4.6, 4.8., 4.10, showing cell-relative flow in that plane,, 0.1 g kg-1 cloudwater contour, and vertical hatching of rainwater >0.1g kg 1.
iiti • ' I !I •••0 ■I■ ••■• •••■ . •••• •4 -4. -4 !.1 i t; ; •••11 —4 ••10 •••• •-■ —4 1i 1;st r!
• • • •
••• 4.4 .me % • • • .MD
-) -0 -3 • ••• ..—.), —4 —4 Fig.4.2(b) As 4.2(a), but vertical section along line BB' marked on Figs. 4.6, 4.8, 4.10. 59-
F g.4..fla) As for Fig.4.1(a), but showing contours of perturbation vorticity.
0
Fig.4.3(b) As for Fig.4.1(b), but showing contours of perturbation vorticity. 59.
Fig. 4. 3 (a) As for Fig.4.1(a), but showing contours of perturbation vorticity.
B
Fig.4.3(b) As for Fig.4.1(b), but showing contours of perturbation vorticity.
60.
—
--) --) —) • —) —4
—• ••-0. "—) •••-0 —0 —0 •••4
A A' On. ••• - ••• . •••
• %
••■ ' •••-.) —4 —) -4 •• •ID Fig. 4. 4 (a) As for Fig. 4.2 (a)
---) ---) ---) ---> ---) ---) ---) —4--3--4--3--3-3-3 -4-3
. ti til
—0 —0 ..4 ■10
•
. - .
.4 4 .4 -. __. s % • • . —4 —9 _4 _4 —4 ■■•) —4 -0 Fig. 4.4 (b) As for Fig. 4. 2 (b) 61.
Fig.4.5(a) Horizontal section through storm at p = 300 mb showing contours of the vertical component of vorticity.
Fig.4.5(b) As for Fig.4.5(a), but horizontal section at p = 600 mb. 61.
NNNNNNNNNN NNNNNNN_ NNNNNN NNNNN . NNNN_NNN. N N NNN NNNNN NNNNNNN: .NNNNN NNNNNNNNNNN. \NNNNNNNNNNNNNNNNN \IN\NINN\1%NN NNN NN'c\I"\ .NN N.NNNNN 1\\I \NINNN- .N\NN NN ANN‘'N \N\NNN NNIN
•NINN\INNNN -N N N NNN'
N_NNN\INN\INNNN%NNNANSI
N
Fig.4.5(a) Horizontal section through storm at p = 300 mb showing contours of the vertical component of vorticity.
Fig.4.5(b) As for Fig.4.5(a), but horizontal section at p = 600 mb. 62. --NNNNN\INNNINN\INN J INN\INN \IN INN • NNN\INNNeNN\INN \INN\IN\I 1\NAN \INNN\N\I NNNN • \NN\IN\IN\NNNN NNNNN\NNN\NN!NN \NN\I\NNN
\N-N\N\II\N\t\rNj
NNN\INNN INNN\INNN 1\1\1\NN\I- NN
N\N\INNN INN\I (a) p = 300 mb A' Fig.4.6 Horizontal section through storm showing cell-relative horizontal air flow and contours of vertical velocity. Lines AA' and 135' show positions of vertical sections in Figs. 4.1-4.4. 63.
Fig.4.7 Horizontal section through storm at p = 950 mb showing contours of the modulus of the vorticity vector. -
63.
1 •-• • N . B -
(a) p = 800 mb A'
Fig. 4.8 As for Fig. 4.6 but in (b) contours are r_Tf horizontal divergence
A
--• -A --A /1- __9 --4 -4 -4
--A ----A ---A "---A
■
0. 0
. --4 --A --A N -4 --A --)
• -4 --4
--1 --A ----A
• ----A ----A --A
---A --A --A ---A ------A - -•
Fig. 4.7 Horizontal section through storm at p = 950 mb showing contours of the modulus of the vorticity vector.
Fig. 4. 8 As for Fig.4.6 but in (b) contours are of horizontal divergence
A
----) :44 ""`-> ----) ---) --/ 3 i/ I I 01 LH , -"--.)---A -"--) ---3 ---A4f-e„,-1 ---5----1 ---> ---* --9 -4 ‘ • / —3-9-9 i I 1 •-•-••)•••••• •••••■ vr-•-> • • I I ‘i• •
•J N:j9 B-4 i' 1 4- a 1:"'"•) ""*". .'"•3 -."41 %.*`‘ NI:
\ N \N, NI, Nu
I. tr.) -4-) --* %%'9 -9 ---; N--) --) - N.:hi N.); • s."tA
■ ---) ---0 -."13 ---) ---) ---0 "-- ",4 5-3 5-3 -3-3-3-3 ---3 ---3 ---3 ---3 ---0 ---3 ---* ---3 "-) ---0
(b) p = 1000 mb 65.
Fig.4.9(a) As for Fig.4.5(a), but horizontal section at p - 800 mb.
Fig.4.9(b) As for Fig.4.5(b), but horizontal section at p = 1000 mb.
65.
N
"IN Fig. 4.9 (a) As for Fig. 4.5 (a) , but horizontal section at p = 800 mb.
--4 --4 ,..-- { La( LN --4 I --4
--4-4-4 —4 —4 —4 —b B
‘•AI
-4 —Ai ---1 kz.s. —9 \Sr \\ \ N 1 -4 --* """) """1. .".11■ N / \
--4 --9 --4H --4 -.9 --4 --4
--4 --4 —4 --4
—4 —3 —9
--4 --4 --4
4 --y
--4 --4 - --9 ---9 ---4 ---9 -4 —9 ---9 --, -.4 --4 —4 --4 rr. Fig. 4.9 (b) As for Fig. 4.5 (b) , but horizontal section at p = 1000 mb. ITTTUTT1'117777.7171 ITTTUTV71777771 7i71 IITTTU1t7777717trr II TITI711777/;17t7T rittitt 777,171'7T AT ITI77 7 7 7 7.11' trrti TT 7.I/7//,117IIITT /11,etl'i'1 T Tr Is
it I 1. T•I .T rIttrTTTIT t tti IrtrITITTT T 0 TrIT7PIITTITTTJTITTT 0 trittir - rtrtrTT.TITTT rrti,T—Hrtrrt,i-17,TITT 1::T..71.:1.'_fr I I r•t . T :1 r'tr T Li_ r- J T :P TT TT T 5.3 a.
(a) (b) ( c )
Fig.4.11 Plan view of vortex lines. Pressure co-ordinates are marked at selected points on these lines. In (a) stipple represents area of horizontal divergence < -1.5 x 10-3's-1, hatching that of horizontal divergence > +1.5 x In (b) and (c), stipple represents area of w > 1.5ms-1, hatching that of W < 68.
CASE B (i)
HAMPSTEAD STORM 32 - 44 MIN MEANS
FIGS. 4.12 - 4.22 69.
Fig.4.12(a) Vertical section through storm along line AA' marked on Figs. 4.17, 4.19, 4.21, showing contours of the component of vorticity perpendicular to that plane.
Fig.4.12(b) As for Fig.4.12(a), but vertical section along line BB' marked on Figs. 4.17, 4.19, 4.21.
69.
•
Fig.4.12(a) Vertical section through storm along line AA' marked on Figs. 4.17, 4.19, 4.21, showing contours of the component of vorticity perpendicular to that plane.
0 0 0
4 -4
...... ,
Fig.4.12(b) As for Fig.4.12(a), but vertical section along line BB' marked on Figs. 4.17, 4.19, 4.21.
70.
0 006
A
Fig.4.13(a) Vertical section through storm along line AA' marked on Figs. 4.17, 4.19, 4.21, showing cell-relative flow in that plane, 0.1g kg71 cloudwater contour, and vertical hatching of rainwater >0.1g kg 1.
40 •
Fig.4.13(b) As for Fig.4.13(a), but vertical section along line BB' marked on Figs. 4.17, 4.19, 4.21. 71.
Fig.4.14(a) As for Fig.4.12(a), but showing contours of perturbation vorticity.
Fig.4.14(b) As for Fig.4.12(b) but showing contours of perturbation vorticity. 71.
_ Fig.4.14(a) As for Fig.4.12(a), but showing contours of perturbation vorticity.
--•
( Fig.4.14(b) As for Fig.4.12(b) but showing contours of perturbation vorticity. 72.
A
Fig. 4. 15 (a) As for Fig. 4.13 (a)
-) --► -► -4 ..41. L
•■• ■I■ ' .. . . .
4 ▪ •
•
4.4 qm, 1(.■ oel■ 41 T Fig. 4.15 (b) As for Fig. 4.13 (b) 73.
Fig.4.16(a) Horizontal section through storm at p = 300 mb showing contours of the vertical component of vorticity.
Fig.4.16(b) As for Fig.4.16(a) but horizontal section at p = 600 mb.
73.
7/ 7 r 7 7 7 7,7 7 7 i 7 7 l /7,7 7 r 7 7 7 7 717 7 7l l v l r r 1 7 7 7 7 7 7 r r r,); 7 1 7 7 ? 7 T r r ///' r r rNi 7 7 7 ie It 7 7 7'‘ 7 7 7 r 't I i 7 7'1
7 \ •7 7 7' 7 7'
7 7 ■ 4 24. -,11 2
I 1 1 A 21 7 7 I I +rn' 7 / 7 7 7 ^l 7 7 7 7 7/ /
7 7 l 7 7 7 7 7 7 7 7 / / /
7 7 7 7 7 , 7
7 7 7 / 7 7 7 7
7 l r 7 7 7 7 7
Fig.4.16(a) Horizontal section through storm at p = 300 mb showing contours of the vertical component of vorticity.
Jertical velocity. n'T
t ■
'I 1 1 1 • 1
1 1 1
1 1
'T \
Fig.4.16(b) As for Fig.4.16(a) but horizontal section at p = 600 mb.
74. . I' 7 7 7 7 '7 7 7 7 7 1 (r r 1 7 / 7 7 7 7 1' 7 7 7 7 7 7 7 r r 7 r 7 r 7 7 7 7 7/ r r 7 7/ Y. 7 Ay r 7 7 7 7 r, 777 77 P' 777771, 7 7 7 7 7 7 J 7 !, r. 7 T 1 7 .1 iv 7 7 7 r,'1 I I r 71 r 1 1 7`77 7 1 r A' I I t /r r r 7 7 7 7 7•7ii - 7 i r /I" I it 7 7 7 777'7' 1 7 r /A /A /A /A /A 7 l 7 7 r r 7 2' 7. 177Itrt. , va 7 • 1‘ 1' 7 7 l I ,7T TT I 1 1 1 r ik r r r Y ./1 7 7 7 7 i• : I' it 11 7 7 7 1 7 / 1 7 7 / / / / / 7 7 7 7 7 7 7 1 1 7 l 7 7 / 7 ___7 7/ 7 7 7 / 7 7 / 1 1 7 7 7 7 7 7 7 7 7 7 7 7 7 / 1 7 7 7 / 7 7 / 7 1 7 7 7 1, 7 7 / 1 T 1 1 7 1 7 1 7 / 7 7 7 7 7 1 7 7
Bt (a) p = 300 mb
Fig. 4.17 Horizontal section through storm showing cell-relative horizontal air flow and contours of vertical velocity. Lines AA' and BB' .show positions of vertical sections in Figs. 4.12-4.15.
(b) p = 600 mb 75.
Fig.4.18 Horizontal section through storm at p = 950 mb showing contours of the modulus of the vorticity vector.
(a) n = ROO mb
As for e. horizontal diverger.
A
Fig.4.18 Horizontal section through storm at p = 950 mb showing contours of the modulus of the vorticity vector. • % • Fr
A A A A % 5 1 \ 1 1 1 1 A % % A A t 55%11115155 itt‘ISSItltS15%155% A A A % 't I% N sk I I t I S I 1 % % A A A A 1 A 1155115555555115151 , , , % \ '‘ a\-11_IIIrti t 1 , N A 1 t 't 1 I t . f N 1 t t 1 A I 1 % \ \ 15\. 4\ 1 I'M t t • A A 1\ N.. IC.< i ' Y‘t t 7 1 t % % 1 t t 1 1/ ' 5. 5 '1/4 N N N g , 1/4 .1/4 Nsr Isi I ': t t r i % 1 1 5 1 ./t • .1 A. 4 N ▪ t-h • • 7 1 1 5 N 0 , 11/4 f, f. 1'•7 1 O 11 IT) 1 t 1• t r • • N st,ti •••• ."\ 1 t 1 1 1 • • • i ‘& / 7 t 1 1 rt / "- 5 5 N tT (-1-- 1 5 r • t .st• 1... 1Ys.- N. •'5 5 5 r• 'A* 1 5 I•4 (•-:t t 1 5 5. / •/, . 1 7 • N 5 I % • .. fa• .1 I I I c , 1. ■ / 1 1 I-•• • I I I r-.V.-- k...! N 1 ''. • . (D Co 5 t t I t 1.t 1 t % I A k ‘4-144..., ie4 .1Z • k: I I 0 Qtr o 1% %t %1 %1 II \ N (D 0 (-1- 5 5 5 :5 .. . , C s ... .0, I • . • • (D 1 A I ...fp A A ", % 1 1 % I '1 \515 % 1 t 1 A 1 1 % % I I I I I A 11115 55151111551 5 1% 03 1 1 1 1 5 1 1 A 5 % 1 5 1 1 1% 8 (-1" 1151555150A 'A I t t I I A t % A A 5 t 1 A' t t 1 A A 1 % 0 1 1 i t t t t %IASI% Al..% 1111 ' 1 1 1 5 1 5 A 1 t % A A A % % Alt t It t As ACS A t A\ A s I A A % 1 1 1 A 1 1 % 1 % '1., A I A A It %I AlttAilt I 1s- 111 0 77.
Fig. 4.20 (a) As for Fig. 4.16 (a) , but horizontal section at p = 800 mb.
Fig. 4.20 (b) As for Fig. 4.16 (a) , but horizontal section at p = 1000 mb. •
7 7 .
/ e
V V
/ < ✓ I 1 or•
I 1 or' sr"/ sr" or" or' / LA
1 4,-*. \ • 4 4r-
Or — 4,/ 4--
•-• r orr r 41.■
Fig. 4.20 (a) As for Fig. 4.16 (a) , but horizontal section at p = 800 mb.
Fig. 4.20 (b) As for Fig. 4.16 (a) , but horizontal section at p = 1000 mb.
78. IB'
ol• ••\
..... ✓ es/ er ✓/ ...... / . Ay .0 ‹, ' i e• i-- .-- .- 1 ...... N / / ■ _ er• ..• .-• 1 1 se / / e 0. ••••• *-- .•••• I...-- ...... II• ..fr gr.' /I/ • ...- e. e• e 1 1 .< 1 1/ b.' ..-• r- .-- I.- .- fr. ek le' fr I 1 / or ..,.. .. i I 1 11 ■./ te a." ■■•• ie'l a! ...* •••• •••• 4/ fe• •••• 1 1 / de ... e e e 1 I le!' le 44- te'/ 4 - " ,,,ire ••• • ...-i "r- ie ..- .-, 1 sh 0. . • e• e ■ e e I rtext••• fe• k" fee' ve fr- vs// fr. \de /I' / I I i • .0 fr. .0 / 1 i I . .1' of • sr"' 4••"" fe') It•I 60' ir•• 41 .••• el ‘ % . I / i a, / I •/ • • 1 e • 4.-- --4-..- 4••••:' se• a••• •••• or, %. ✓ • ' % v. •■•• ... 1 .• •• •• v. 4.1 4... 4.... 4.... 4...• ■-••• 4•:' *•• ao o. ■••• . • .. • 1 ■•••• s•• ••• N40` .. - .• •• ...• •II••-• • 4— / o••• .••• •••• ...‘ . . •••• ...... jo...- . .••• . de .•••• or e••• •-• ••••••• •■•• ••• 41,/ 4.- *•••• s...... •
B7' (a) p = 800 mb
Fig.4.21 As for Fig.4.19
IB
... ..• .9 e• ■•• ... ..• e e
... . A.' ...,.. • .0 .. V•..' e ..•, Ay a. N. N.. / N. • • %, • • Veer. .1 1 1 de •• •••• •••■•• .1.- II, g.„ % ■ . •
I I AI •••• ••■ IF. Os. C••■ IC\ >/ 11/4* . 4 % .... e-••• 4--- lIs, f.... .ISC,, . IC, IrN, •• S I I e e e e • fp. 41-- 4--- f--- ir---.. 5,-. *--.1..e yr ee . . I I ._ 4•-• *--- ft-- (---,.. t I...•••, I 14 / . 1 1 6-.• 4.-- 4—, 4.-- rsh 11 I' a' e a/ se e i ' I i 0' le' E."' 4--- 1 'NI % / s, I 4/ ... e, I , / i s' ,e• ir-. t..,--- • Vit •i • . ..- elk- N, h. 4:0 ., I ., / e e 1/ V 1,. l, • e • e I/ 4-- • V.11,, • e A, •••
B7 (b) p.= 1000 mb r
292
366
(a) (b) (c)
Fig.4.22 Plan view of vortex lines. Pressure co-ordinates are marked at selected points on these lines. In (a) stipple represents area of horizontal divergence < -1.5 x 10 3 s-1, hatching that of horizontal divergence > +1.5 x 10-3 g-1 . In (b) and (c), stipple represents area of w > 1.5ms-1, hatching that of w <-1.5m s-1 . 80.
CASE B (ii)
HAMPSTEAD STORM 64 - 76 MIN MEANS
FIGS. 4.23 - 4.33 81.
Fig.4.23(a) Vertical section through storm along line AA' marked on Figs. 4.28, 4.30, 4.32, showing contours of the component of vorticity perpendicular to that plane.
Fig.4.23(b) As for Fig.4.23(a), but vertical section along line BB' marked on Figs. 4.28, 4.30, 4.32. 81.
Fig.4.23(a) Vertical section through storm along line AA' marked on Figs. 4.28, 4.30, 4.32, showing contours of the component of vorticity perpendicular to that plane.
Fig.4.23(b) As for Fig.4.23(a), but vertical section along line BB' marked on Figs. 4.28, 4.30, 4.32. 82.
••• ale
A 40 00 • • •
•
04 4.• 4E. 414. 44.
•••• — 11.•• • • •••• -4 7-1. • Fig.4.24(a) Vertical section through storm along line AA' marked on Figs.4.28, 4.30, 4.32, showing cell-relative flow in that plane, 0.1g kg-1 cloudwater contour, and vertical hatching of rainwater >0.1g kg 1.
r • •
---) ---) ---> ---) ---)
B ■Is
.- - - w - ..- 4-7- 4– 4-- k".... 4". 4"..."
4= 4.. h". r• 1 ■••• 4.--* le" 4.""" 4...'' 4..- li.. 4.. 4-- 4- 4- 4-'''' 4.--- 4-*-- 4-•-- 4--- '. 4- (--- 4- (---- (-- 4-- 4- 4- 4- 4- 4- 4- 4- i- 4— 4- .- 0. . . Fig.4.24(b) As for Fig.4.24(a), but vertical section along line BB' marked on Figs. 4.28, 4.30, 4.32. 83.
Fig.4.25(a) As for Fig.4.23(a), but showing contours of perturbation vorticity.
J 1)
Fig.4.25(b) As for Fig.4.23(b), but showing contours of perturbation vorticity. 83. I
A'
Fig.4.25(a) As for Fig.4.23(a), but showing contours of perturbation vorticity.
Fig.4.25(b) As for Fig.4.23(b), but showing contours of perturbation vorticity.
• - • ▪ ▪
04( 84.
ft.
•
A •••
•••• 41.•
di•
• .••• 111. • . •
••• 1111M 1.1•1• 11■1•• •■•• ■•■111. Fig. 4.26 (a) As for Fig. 4.24 (a)
•
—4 —) —4 —4 —4 —5 --3 —4 —4 --> --->
B
■.• •.• el* •••
•••
41.• 4.""
•-• r ••• N ■••• 4... 4.- •4.- 4- ie.. k
4- 4(,-- 4- 4- 4- 4- 4-.. 4.... 4.. Fig. 4.26 (b) As for Fig. 4.24 (b 85.
Fig.4.27(a) Horizontal section through storm at p = 300 mb showing contours of the vertical component of vorticity.
Fig.4.27(b) As for Fig.4.27(a) but horizontal section at p = 600 mb. 85.
Fig.4.27(a) Horizontal section through storm at p = 300 mb showing contours of the vertical component of vorticity.
Fig.4.27(b) As for Fig.4.27(a) but horizontal section at p = 600 mb.
▪ • 86.
. 1/ /1/ / / 7 l ) 1 11171/ / / 7 /' / 1 r 1 t r 1 %/ 1' r 7 7' 1 )' l t r 1 r t, .1- 7 7 .1 7 V r \ 7 7\ 7 7 r r r r 1 i / / 1 i ! r 4 , 1, , 1 1 7 7 7 1 7 I 7 r, 1 ' .r, r i / ,,, 1 7 1 7 1 7 1 7 T )0 1 r r , 1 / ' / 1 1 / 1 7 ,7 7 .1 / 1- /- r. r r A'' 7 \ - 1 11 1 1 1 I' 7 7 11/4 7 7 7 7 7 >•,7 7 -7_ ■ , • N _ Ay 1 1 1 .4. /,./ .1\ 7 / 'if' 7 p 7 7 7 7 7 _ . 1 1 7 1 1 .1 7 1 r r r r.7 .7 / / 7 ; 7 / .1 / / 7 7 I i? ' / I . .7 / 1 t/ / / ...... / / / 7 7 7 7 7 1 1 ra . ....* / 41 I' N. / / / l / / • / / l 7 1 i l i ....* •••1 /1 / ••• / / 7 / /1 74 1 1 / 7 / ) :-• .., r.7 7 .7 7 7 7
/ l \ 1 / 7 •/ /\. .. - ...... / .... ./ / 1. / / / / \ 1 / / i / / / / / , \r , ? a , , , /I ....* -11 . II -S • • •••° ' • ‘• .... • .". , It 7 .7 7 .7 7 7 .7 • J" .7 .7 .7 1 .7 7 .. 7 • / 7 • "f? . •.... . -- • / 7 / / 7 7 7 7 7 7 7 7 7 7 7 7 7 7 r 7 r r r t r r l 7 7 / 7 l / 7
B7 (a) p = 300 mb
Fig.4.28 Horizontal section through storm showing cell-relative horizontal air flow and contours of vertical velocity. Lines AA' and BB' show positions of vertical sections in Figs. 4.23-4.26.
al3"
I • •• •
•••• • am. .the• • C• •
•••• ••• / • • N. • • • • • • .. . •••• • • . I. • . ... • . • 5 • \ • • •C • • a ... . o• .., •• •:\ •- ••• • \ A • • • 1, • • a .I. . • ••• 4.* ••• •• • • '1/4 • • l • • ‘ o • • • • • • • • .— . ••• .. \ ' 1 ■ • ‘..; • • • ‘• • S •. 4 5- f • 1 • S. • • ohN....• • •
I ••• •
J ..- • C. ft • • • i I
•
'"••• 41 • • • • ••••
B7 (b) p = 600 mb 87.
Fig.4.29 Horizontal section through storm at p = 950 mb showing contours of the modulus of the vorticity vector. 87. dB • / 1' 1 v 1 11 e" / 1 / / / k" e, / v" V e1 r; -,/ 1 / 1 e e" .• ,i/ 1" 1 / i /- 7 te'" tej e- be' e' 1 / ur‘ r r if r r ./.. r 1 i i i V„, / V brie ... „) 1 e• r r rd f r / / / 1 1 pl .)r r /N le V kr" V V , i r r r • r r r r r i e .-- r / .4 / - 42,1",k-' ir . k---- i I 4i, r r 'r'' e" lele t-'''' / e t / / 1 / 1 I/ of' r ir% 4.--- k'''' 4.--, -.- e e % i i i I i . 1 e I or' r 4.'". el''` 4**"... 4e..... 4.' l--- "P . le" I r r e '' r i r /r I e, t ,V ■•••• *3'. t.-- 4r- 4---- 4--- I4.---
r r r ) .• r r‘i i r /;,/ Je V or''' r'''' *--- t----t-- 4,--6 e" I. i r r r r r .1:r_.-- .--1 .--. 4.--- sr' e Li‘ I t r e e 1 e■ 1 .1 Nee' r''' tr." " • 4,-4 4-- Ie' V e \ \ \ % • i , oe ef d I 4, i ... 4KN 4(...... le". 4r 4-- - (--- 4-^', l---- 4 1 ig/ el \ W 4.• 4.• I 4-• so • 4••• / te" le‹.• 4•••; 4.-*" 4.--- ic--- *--"' 1e,"" or' e. el te' e or r 4.• .r.• .... p• -,b..0. 4.... 4.--- or'' or' ..-- or" x ..., 4 ... el" I/ ar• •••• .'" ."- ■ e .-- .-- ..-- .."
4••• 4.••• ••••••. . .11...... 4" ''. 4. k.."..' o--- or' e" r'" be" or- or- e
e.'" e, r ..-- .- 4-- 4-- 4:1-. t.".' e.' V V 41"' V V V. V
• e' er- te-
,e• ee L _ 87 (a) p = 800 mb
4, 30 As for Fig.4.28 but in (b) contours are of horizontal Jivergence
'4-- 4- 4-- --•
6-- 4-- 4-- 4--
4r--•" 4--- 4— 4--
4--
J
I/ ' 1 1 / 1 .
Fig.4.29 Horizontal section through storm at p = 950 mb showing contours of the modulus of the vorticity vector. •• •
88. dB' V e`e_e e e e eieeie r • e e r e e e 11 e e • e le "4.1 e •e e e e /%1( e I e A••• / e -e / 1 1 e tre 4,/ kr"' I 1 t 1 1 1 1 r. 10.- 'we ore try' Jr r %! 1 1 e I e e k-- 4-- r r r e Ie e ez■,, 4-- 4-- 114. Ay r r 2. r ✓if e fee je k-/ 4••••••*•"' 4••"4. 4••° r r r e re! are se" r r r I %to' tr-• 4.--e ire' re' re• • • • 40' 1 r 40 se. 40 6e% e..4-•• 4;-• 4— Ems'4-r) 44"" 4.-°' fee we
be . r i r r 14. • se• 4■•••
r 4.• lee lee ire te se'
or" re ••• 14.-• 4-•"' eeeee • "••• • 4••• 40 v...4- .••• 4— 41-- de" le' Ne re
r" re. re 0- •-• *re toe re re ae. e r"
e• tee 40" ree re• •-• - • • :4••••• 40"' ts," DeeeDeeeee
te• re we 4r• •e-• eeeeeeee•ee; BP (a) p = 800 mb
Fig. 4.30 As for Fig.4.28 but in (b) contours are of horizontal divergence
VB.
4•••••* 4••••• 4••••• 4•••• 14— • -e 4-- 4-. 4-- .4.-c • Is. 0.• 0•••
lee it' Or."' 4--
le•*•• 4g, - 4--* 4-- 4-- 4•-• ...411. ti .„) ti
Ay
e
e e
ma. • r ▪ e
B7 (b) p = 1000 mb 89.
Fig.4.31(a) As for Fig.4.27(a), but horizontal section at p = 800 mb.
Fig.4.31(b) As for Fig.4.27(a), but horizontal section at p = 1000 mb.
▪ • - 89.
1 ."'../. . e .' 1 I / / / 31(---li •1- el ,,,,," ar""
\ 1 / V ker ILI. 1 3" ur'. V .' 1 e- izzArZ` kr IV I e / e it/ t
/ r , r ! 1 1 1 I / 1 r- ..-- .1-- *-- 1 *.- e e le r I r/ e e el ,,,, . k--- 4r." 1,l 1 -... i , k k ie e ) e ✓1 i t e e re* ),./ ..... „..--„,, -- , I k e e e 4 •%. 4, we% oe or-- 4.-- ft".
ee e e 4 to' Er"' I •■••• e e at" ft,' V' ."
••••••• 0.." ow- tr V 1 r' • or_ kr- r r r r'
V e-
e". ✓ te/ se' te". w" le' V 1 .4-- Fig. 4.31 (a) As for Fig. 4.27 (a) , but horizontal section at p = 800 mb.
4-""" ■-•""
- 41-
k-- k--- A kr" t ND. / V kr"kV 1 \si. 1;
'LT
.11P /I- 1
- 4/ V ‘,/ 1 r ./
4r 1 r r r e' ar" •
Fig.4.31(b) As for Fig. 4.27 (a) , but horizontal section at p = 1000 mb.
, - , / i. ;I .-A -4 -::: .N. 1, 1, 1. 1, 1, 1, ti \ \ \ \, \„.._ N, \ 1 1 1 1 1 1 1 1 , . 1, ti \\ ‘• N, N, N, 1. >, >, 1, 1, t I t : 7. "--, ..../.1 ,..-4 N--, •-• N4 11 1, 1 \ 1, ti ti ti ‘, • 1 \ N Co / I PI \, .4..,---)N-.1 ,--• Nt 1. 1, \ 1, ti Sb Sb NT Ns).\,•\,'"S: Sb k• ,i t.3, \. \ \ 1, 1 'N. 9.•,t .1...\-;"16-`31 1 %, \ 1, 1, \, 1, i , ---/ .1---)--)\-4$.•yo, 1, 1, ti 1, ti A. \, ‘, ‘,,A, x . I / 7. 3!--, -4 -*„ •-• -... N„ N, \ N\I \ \, 1. l• 11* \, 1. 1, \,, 1, Ns \• 1, . . 1, ti •IA • • . -4 -' "'"*N. Sb `1, NI \ V•14,.-k. -'i 1 1 ..1) I. N, N, . \ 0 0 O N. ..11 ,-.c,C,,, \ \, 1 1 N, 11, L 1 1 coO 0 11 •• \, 1;•1... L....V.\ \ 1 11 1 N, \ --) *•-■■■ I I \ 1, N, AS -V-s), \ A N, sal c.* „ • •I••• \, 1 1, ‘• •- • N0 \ .. it. try \ "a, \ N. N. A-\ N\. -1-• 30 .j.1
\I 1, 1, 1, Ns- NI., N, 1, 1, 1 \I • 1 i • 1 1 NA 4. . / • '11% N,. Na 1, -NI 1, N. 1, \, )/ 1, •-z•l; I 1 ig. - --.9 1, N.' 'S, N NN11 111 1, 1 1, I. I 1 t F • r
1 \ 1, 11 18 1 1 11 1.ei 0. 1 1 1/. fo
'1 1 1 • V I I 1 As A co I 1 *\.• \\1\i\\CNOI .N. N. 1, N. ..N.• ..-N; N, • I 1 1 _ • • / \■ \ \ 17 • 1 1 1 1 1 1, 1 1 1,• 1 1 V A■ \15).\I S 11 1,- 1, • - ), N, N, 1 1 1, N, • N, N, \, \ 7'
• •
(a) (b)
Fig.4.33 Plan view of vortex lines. Pressure co-ordinates are marked at selected points on these lines. In (a) stipple represents area of horizontal divergence < -1.5 x 10 3 s 1 , hatching that of horizontal divergence > +1.5 x 10 3 s 1 . In (b) and (c), stipple represents area of w > 1.5m s-1 , hatching that of w <-1.5m s-1. 92.
CASE C
SPLITTING STORM SYSTEM 32 - 36 MIN MEANS
FIGS. 4.34 - 4.44 93.
Fig.4.34(a) Vertical section through storm along line AA' marked on Figs. 4.39, 4.41, 4.43, showing contours of the component of vorticity perpendicular to that plane.
Fig.4.34(b) As for Fig.4.34(a), but vertical section along line BB' marked on Figs. 4.39, 4.41- 4.43.
93. 00
r - - • - • Fig.4.34(a) Vertical section through storm along line AA' marked on Figs. 4.39, 4.41, 4.43, showing contours of the component of vorticity perpendicular to that plane.
Fig.4.34(b) As for Fig.4.34(a), but vertical section along line BB' marked on Figs. 4.39, 4.41- 4.43. • 94.
.1■
A
Fig.4.35(a) Vertical section through storm along line AA' marked on Figs.4.39, 4.41, .4.43, showing cell-relative flow in that plane, 0:1g kg-1 cloudwater contour, and vertical hatching of rainwater >0.1g kg-1,
•
-3 -3 -4 -3 -3 -3 -3 -3 -3 -3 -4
B -4-3-4-4-3 -4--4 -3-9-4 -4 -3
-4 -4 -4 -4- -4 -.4 -4 -9 -4 -4 -4
4.0
• • .
4- 4- 4-- 4- 4- E-- 4- 4-- (-I- 4- 44- 4- 4- 4- E- 4-- 4-- 4-- 4-- E- 4-- 4- (--t- +--- 4- .4- 4- Fig.4.35(b) As for Fig.4.35(a), but vertical section along line BB' marked on Figs. 4.39, 4.41, 4.43. 95.
Fig.4.36(a) As for Fig.4.34(a), but showing contours of perturbation vorticity.
Fig.4.36(b) As for Fig.4.34(b), but showing contours of perturbation vorticity. 9 5 .
(
Fig.4.36(a) As for Fig.4.34(a), but showing contours of perturbation vorticity.
o
4-- 4-- t- r - 4- 4- 1-- 4-- 4- 4-- 4-- .- r- E-- k-- E-- ,__. 4-- '- 4— 4— 4 Fig.4.36(b) As for Fig.4.34(b), but showing contours of perturbation vorticity.
▪
96.
•••
•
A - • •
Fig. 4.37 (a) As for Fig. 4.35 (a)
• •••
---> ---> ---> ---> —3 —4 —9
---)-->--> - B --> —3 —3 —3 --> --) —3 —3
I I I; , I •, ,r
! ▪ • ■•• -
4.••• 4•••• ■(•■ 4- E- 4- 4- 4- 4-- 4--;- . 47• - 4-- <--- <-- <--- <.-- <, 4- <- 4- 4- 4-- Fig. 4.37 (b) As for Fig. 4.35 (b) 97.
Fig.4.38(a) Horizontal section through storm at p = 300 mb showing contours of the vertical component of vorticity.
Fig.4.38(b) As for Fig.4.38(a) but horizontal section at p = 600 mb.
97.
Fig.4.38(a) Horizontal section through storm at p = 300 mb showing contours of the vertical component of vorticity.
••■
A A A A A A A , . ./. ././, "A A /I A i
, A A I ' /6 , I . p21 , I A A A , A AA
* A A A A , A 1 A ,A A' .."` A' At
' / / 1 A 7 7 7 7 A A A* r.1
' / / l't /? /AA-'--
///// 1 / / .7 A ...A .--.* -.■ A / ,.. t. _.
le / / / / 1 / / / ? ...... 4 .-.. -• 1 1 /
.. 2. / / , / / 7 ' is' — I\ I 1 1 A A t . 7 7 .-- .. r , , i ...... 0
'. / / / 2. / 2 / , 1 7 7 7
/ / / ./ / / / / C 7 7 ,- , r r l...... kiii!Il li1)17 7
A A A A A •r A A A A A A A A AA ", A A A' " A A A ," A A A A r A A' A A A A • A A A A A A A A A A A -' A A A A A AA A AAAA Fig.4.38(b) As for Fig.4.38(a) but horizontal section at p = 600 mb.
Ay 98. 11/7/777/7/7//////// 77/7/77/////7/7/77/ //77777//7/7/////7/ B. 77/777/77/7/7/77 e 7/77/////////77 777/7/77/'777 ykii /77(777ifivy7 777777777/// il/i1 7/17////11 / 4.1,1"7/1 77/)177i77/7,7\71-//1/1// /77)V7/77/ 777/z/i777
77/77/1/77///77/7/7/ 7777/7/77/77/7/Y/// 77771 ///t7177/7/7/P777/Not
(a) p = 300 mb Fig.4.39 Horizontal section through storm showing cell-relative horizontal air flow and contours of vertical velocity. Lines AA' and BB' show positions of vertical sections in Figs. 4.34-4.37. Ay
11 is .11 11 la 1. 14 14 /111 14 "4 /4 .01 .01 de •
11 :it 1. la la "la la la 0'4 0.4 1' l la 11 T 11 A. T T l ,■• A 11
1' 11 is 11 11 T A T 14 '1 ./1 ■e4 •," .sa .4 .
. Ali l T T T 11 1. I. ."1 T.
I. 1. IT • •
.11 711/
ej. oP
i■
11 r r .0% ••-t •-•••A •-.."t"" 11 it
All AN All II A• )1 • ...A' .0.21 ..!4/ "II •
1' .•11 ea 11 • .11 B (b) p = 600 mb 99.
Fig.4.40 Horizontal section through storm at p = 950 mb showing contours of the modulus of the vorticity vector.
99. A
, 4 •
• • ,, ,
I td.;
4 4 i i 1 l•• 1 1.0 -
I ..../ ■ .. .4. •••• ■ .
• • 4 1 so...) ■ ■ ••• "P.. ■ ow ..— ••••""" e",..: I - N. i ... ■ ■ am . / . 's
, . \ , . % ... , ‘..,:. i .„. / . • • it •• /
\ • I • • ;. • 1 % ... — — em .
o • • / % . ... SO ■ 4. 4. . / 1 /' • • .• •• S ••• ... . I • *.
• ,
. • do, - • 4. -
(a) p = 800 mb
Fig.4.41 As for Fig.4.39 but in (b) contour (-)1- horizontal divergence
A
Fig.4.40 Horizontal section through storm at p = 950 mb showing contours of the modulus of the vorticity vector. ▪ •
100. A
' y. •
44 vEY
ef
f I I . • ••• • • I / ••• •• ••• 4.0 i••••
4 • eel ••• • • ••• fr. 4•••. • '11 / / .." 4 I j ill••• 4.•... WI •• I.'''. 11.• 40. 4...... 1 4 4 • 4. 1 *Ls. e...... 1 "..: . ,s .r 4.. .. % / • . ..• • • N. . % .0, •••• •••• 4. ••• .- Os •• I • , ' ••• . • . • •• . 4.• . • 1 • . P - -r . .. ..>• .... ••• / i • I • a . ••• 0.• ••• . I ...... - .
. 00 ft
• •• ► •••• ••
••• • •• e• • •
% •••• • • • • I 40 IND / •
(a) p = 800 mb
Fig.4.41 As-for Fig.4.39 but in (b) contours are of . horizontal divergence
(b) p = 1000 mb 101.
Fig.4.42(a) As for Fig.4.38(a), but horizontal section at p = 800 mb.
Fig.4.42(b) As for Fig.4.38(a), but horizontal section at p = 1000 mb. 101.
Fig.4.42(a) As for Fig.4.38(a) but horizontal section at p = 800 mb.
7
Fig.4.42(b) As for Fig.4.38(a), but horizontal section at p = 1000 mb. ▪ ▪
102.
vv 4.
• •• Rs. /
• • 4' I 0..0° • 4 I i I I I 0 v.;
• I 4 .1•P
• I • ./ 0. -- , $, I I i Ito' 61, tri ••• ....•••• fr. ••• / I N I I I ' 4•:-% .- v. / 4 i •■'■ ..• .../ . • • . . / • • S . A ;'...... 1.. •- «7. 4, .1 4 • • . . , • ... . 1 • . . • ..;* ' 1
• • ▪ •ft. er.
I. 4. • • • • *a lb. ft.
• . • • • Mk
▪ dis
•
B7r (a) p = 800 mb
Fig.4.43 As for Fig.4.41
(b) p = 1000 mb
• •
900
925 900.
(c) (a) (b) Fig.4.44 Plan view of vortex lines. Pressure co-ordinates are marked at selected points on these lines. In (a) stipple represents, area of horizontal divergence < -1.5 x 10-3 s-1 , hatching that of horizontal divergence > +1.5 x 10-3 s-1. In (b) and (c), stipple represents area of w > 1.5ms-1, hatching that of W < - 1.5m s-1. 104.
CHAPTER FIVE
INTERPRETATION OF RESULTS
In Chapter Four, the vorticity field generated by the deep convection model was described for several case studies. The aim of this chapter is to compare and contrast these results, to offer some physical explanation for the observed structure of the vorticity field using the computed vorticity diagnostics, and to compare some of the results with those obtained by dual-Doppler observation of tornadic storms, and with those obtained by computer simulations of tornadoes.
)
5.1 MODEL OF VORTICITY FIELD IN CUMULONIMBUS
Examination of the figures presented in Chapter Four reveals that the vorticity fields generated by the three different storms differ from each other in only a few basic features. One of the most interesting characteristics of all three simulations is the fact that the vorticity field has the appearance of being very 'well-organised', i.e. the dominant wave number of the vorticity field is the same as that of the storm system.
That this should be so was by no means obvious from the appearance of the velocity fields.
The main differences between the vorticity fields of the three storms are the phase change with height and the strength of the vertical component generated. Features exhibited by all the case studies include:
(a) a strong horizontal component of vorticity around the edge of the
updraught, especially near the upper and lower extremities, and (of
opposite sign) around the edge of the downdraught, especially near
the gust front,
(b) any positive (cyclonic) vertical component of vorticity generated is
mostly found at and just above levels where updraught and downdraught
exist side by side, i.e. between 500 mb and 800 mb approximately, near 105.
the interface between them,
(c) a region of negative (anticyclonic) vorticity of lesser strength
than that of the positive is also found between about 500 mb and
800 mb; these doubletS of vorticity are found with varying
strength in all the storms.
The highly-organised nature of the vorticity field and the fact that
there exist only a few differences between the vortical structure of the
three case-studies permit the description of a simplified vortex-line
model of cumulonimbus.
The cell region can be divided into three horizontal slabs. The
uppermost, above about 700 mb, consists of a strong updraught core.
The middle slab, 700 mb-800 mb or 900 mb, contains updraught and down-
draught existing side by side, and the lower slab, 800 mb or 900 mb to
1000 mb, consists mainly of the downdraught 'cold pool' spreading out near
the lower boundary.
It should now be noted that figures 5.1-5.3 which follow are only
schematic diagrams intended to show the main features which are common to
all the case studies; both the alignment of these configurations and the
- vertical vorticity generated will be ignored. They will vary between
the types of storm studied, and in a given cell they will vary with
height. However, these simplified pictures of the vorticity field will
be found most useful for the discussion in the next section of how the
vorticity fields are physically generated.
In the upper slab, the vorticity field takes the form of almost
horizontal, large-magnitude, vortex rings with cyclonic sense near the
edge of the updraught core, while the initially straight environmental
vortex lines which are in the vicinity of the updraught are stretched
around them. The two configurations which appear in the results are
shown schematically in figs. 5.1(a) and (b). 106.
(a) (b) Fig.5.1 Schematic diagram of vortex-line configurations in 'upper slab'. Continuous lines are vortex-lines; hatched area is updraught core.
Fig.5.1(a) shows the case in which the approaching environmental vortex-line has the opposite sense to the leading edge of the updraught vortex ring, 5.1(b) the case when the senses of the environmental vortex line and vortex ring leading edge are the same.'
The 'middle slab' consists mainly of almost horizontal vortex rings around both updraught and downdraught cores, the former having cyclonic sense, the latter anticyclonic. Again, the environmental vortex lines are stretched around these rings. Many configurations are topologically possible here, but the two which actually appear in the results are illustrated in figs. 5.2(a) and (b). In (a), the environmental vortex line is in the opposite sense to the updraught vortex ring, in (b) they are in the same sense. 107.
(a) (b)
Fig.5.2 Schematic diagram of vortex-line configurations in 'middle slab'. Continuous lines are vortex-lines; area inside dot-dash line is downdraught.
The lower slab consists almost entirely of a 'cold pool' spreading out as it approaches the lower boundary; vortex rings with anticyclonic sense are generated around the edge of the cold
pool, and have a large magnitude near the gust front. Environmental
vortex lines are again stretched around the rings, and, like the upper
slab, there are two possible configurations, illustrated in figs.5.3(a)
and (b).
(a) (b)
Fig.5.3 Schematic diagram of vortex-line configurations in 'lower- slab'. Continuous lines are vortex-lines; area inside dot-dash line is downdraught.
Fig.5.3 is identical to fig.5.1, except that the sense of all the
vortex lines is reversed.
108.
5.2 PHYSICAL INTERPRETATION OF RESULTS
By using partical trajectories computed by the primitive equation
model, and the vorticity and diagnostic fields computed by the
vorticity model, an attempt is now made to explain physically the
structure of the vorticity field.
Fig.5.4 shows a typical computed updraught trajectory taken from
the simulation of the tropical storm; the view shown is. the projection
of the trajectory on to an x-p plane. This example will be used to
discuss the change of the y- and vertical components of vorticity along
the trajectory. However, the reasoning given for this case can also
be applied to the change of vorticity along trajectories in the other
two cases studied.
200
300
400 (6.31,-1.Vx103 p (rriti 500 (1.49, -1151)(10-3
600 (757, -0.581x10-3 C CR 700 800 (330, 2.03)x10-3 C-6.57, 0.45)x10-3 B 900 .00x10-3 (-220.86)x10 1000 A west )1kffx east
Fig.5.4 Computed trajectory of air in tropical storm; values of fl, are shown at various points thus: (fl,
Consider first the y-component of vorticity, fl, given by
gys + r. 109.
At region A on fig.5.4, a long way from the storm, the air
parcel will possess the vorticity of the undisturbed flow. However,
as it approaChes region B near the updraught, it will pass through
an intensely baroclinic region and the value of 1-1 of the parcel
will become large and negative there as the parcel is deviated from
the horizontal path. The equation for n, together with the value
of each term at region B shown immediately below, is (equation 2.17)
— 7~. t-'1\ hf.41+)-.-ao C _ L) ÷ (L. p_oe Dt by -)Ms Zip/ -Ekb -a'sapL9PsY geak s4$ _ 7 X10 16 2_ 3 -510 390 s- 2 As the air-parcel approaches region CL or CR its vorticity will
change due to the gradient of vertical velocity across the updraught
core induced by baroclinity; parcels near the core centre rise more
quickly than those at the edges. Parcels passing near CL will have
strong negative n, those near CR strong positive n. Diffusion
cancels out some of the vorticity near the centre of the updraught
core where the vorticities of opposite sign are close together. The
trajectory in this example passes through region CR. The terms of
the equation for n have the following values here: g=1/4-fne/-3.54)-ricat+t-2estp-ro-4.1-0 (-1-apLifP-wzs-F/ „, 7 X10 77 7 -12 763 -513 s-2 As the parcel, now approaches region D, it is forced by the
baroclinic term to move horizontally again; baroclinity thus imparts
a large amount of positive n to the air parcels as they enter the
anvil. The jet-like profile of the anvil also produces stretching
of the vortex lines: Parcels which have passed near CR therefore
a smaller negative have an even larger value of fl, those from CL or even slightly positive fl. The terms of the equation for n in
region D are, for the trajectory in this example 110.
P ÷(e_ 3a)_i —ge (2c — o t= sap g sx k% s P) X10 7 s-2 100 -88 -36 -270 -125 It can be seen that the tilting term is removing some of this
component; it will be seen later that in this region 11 is being
tilted into the vertical by the horizontal gradient of w.
As the air parcel moves away from the storm through region E
diffusion of vorticity restores n to the undisturbed value of the
level at which it is now situated.
The vorticity of the downdraught air can be similarly explained;
in this case, the baroclinic term is at its strongest close to the
lower boundary, for which reason the high values of vorticity extend
much further away from the storm centre than is the case in the anvil,
where the baroclinic term declines relative to the other terms.
Figs. 5.5(a) and 5.5(b) show vertical sections of the baroclinic
and diffusion terms of the equation for n for the tropical storm;
these correspond in time and location to the vertical sections of n
and n' shown in figs. 4.1(b) and 4.3(b).
One feature common to all the storms, but not part of the cell
system, and which has hereto been ignored, is a wave-like pattern in
the upper hundred or so millibars, upstream of the cell, with a wave-
length of two to three kilometres. This is a numerical, and not a
physical, feature. It is thought that it is part of the same problem
as that which introduces second-order time-error terms into the
horizontal components of the vorticity equation, i.e. the time
integration scheme employed in the primitive equation model, as
discussed in the Appendix. Attempts are being made at the present time
to amend the primitive equation model using the second of the solutions
suggested in the Appendix. •
I B •
, • op 1 ■ •
- ) 7 ------71 Fig.5.5(a) Vertical section through tropical storm along line BB' marked on Figs. 4.6, 4.8, 4.10 showing contours of baroclinic term in equation for n. Contour interval 0.00002 s-2.
Fig.5.5(b) As for Fig.5.5(a), but showing contours of diffusion term.
112.1
Consider next the vertical component of vorticity, given by
3 = ay The air-parcel in the region of A in fig.5.4, as stated above,
possesses the undisturbed value of vorticity, and thus has no vertical
component. However, as it moves through region B it obtains some
vertical component through the tilting term, due to the horizontal
gradient of w acting on the horizontal component of vorticity. By
the time this particular air parcel has reached region C, it has moved
from an area of positive (cyclonic) vertical component to one of
negative (anticyclonic) component, the latter being derived also through
the tilting term. The parcel continues to receive negative vertical
component near region D (where, as noted above, the horizontal component
was being reduced by tilting), but 'negative stretching' or compression
tends to reduce its magnitude (negative aW/az). Diffusion then
gradually removes the vertical component as the parcel approaches region
E. The terms of the equation for at the various regions have the
following values: tg Iges51,CP÷ ((&V+ ?kW]) -1—>(1--sesV ])+ (ax —6a9) Region B x 10 7s 2 4 44 0 -16 •
Region C : x 10-7S-2 10 115 -7 -60
Region D : x10-7s 2 51 -163 2 19
Although not much in evidence in the description of the vertical
component of vorticity along this particular trajectory, the stretching
term can have a larger effect than it does here in the region 600-400 mb
on the vertical component of vorticity of parcels which pass through
regions of large positive Waz.
This description of the contributions to the substantial derivative
of vorticity of the various terms of the vorticity equation can also be
used to describe the vortical structure of the other case studies. One 113. ar
difference in particular which will be found between the case-studies
is the amount of horizontal vorticity in the undisturbed state; this
results in there being a large difference in the amount of vorticity
available for tilting into the vertical. It should be stressed that
in all the case-studies of this thesis, there was no vertical component
in the ambient flow.
This study has shown that (a) the vorticity field in all the case
studies is well-organised, with only a few well-defined differences
between them, permitting a simplified model to be described, and
(b) that as far as the horizontal components are concerned, only the
baroclinic and diffusion forces are important, and then only in the
immediate region of the updraught/downdraught system. Stretching,
tilting and diffusion are important for the vertical component, again
only in the region of the storm cell.
These facts make it possible to use the simple vortex-line model
of cumulonimbus described in the previous section to explain the
structure of the vorticity field, as opposed to the trajectory
explanation given above. It was stated in Chapter Two that due to
the two non-conservative forces operating (baroclinity and diffusion)
a given vortex line could not necessarily be identified at two different
instants. However, as shown above, these forces are only important in
the immediate vicinity of the storm, and in the description which
follows, they are neglected except in this region. This means that
mean flow vortex lines are advected with the flow except near the storm
circulation.
Morton (verbal communication) proposed a description of vortex-line
development for the case of an updraught situated in a mean flow in which
the vortex lines are parallel straight lines; this is adapted for the
study in this thesis, in which the mean flow possesses directional shear
as well as magnitudinal shear, and a downdraught as well as an updraught. 114.
Consider a slab of air approaching an updraught core; the vortex lines in this slab are assumed to be parallel horizontal straight lines. The updraught core consists of approximately horizontal vortex rings with anti-clockwise sense. As seen in the previous section, there are two possible configurations, both of which occur in the results in the previous chapter: (a) vortex lines in the slab of air in the opposite sense to the leading edge of the updraught vortex rings, (b) the vortex lines in the slab of air in the same sense as the leading edge of the updraught vortex rings, illustrated in figs. 5.1(a) and (b). Consider the first case; in the early stages, its appearance would be as in fig.5.6(a). As the vortex line approaches the updraught, that section of it which is near the updraught, AB in fig.5.6(b), will come very close to the portion CD of the vortex ring which has opposite sense.
These two portions will therefore tend to self-annihilate by diffusion, thus instantaneously 'joining' A to D and B to C, resulting in the configuration of fig.5.6(c). This new vortex line ADECB can then
E
A
(b) (c) (d) (e) Fig.5.6 Time-evolution of vortex-line configuration: mean flow vortex lines in opposite sense to that of vortex rings. move away with the mean flow, diffusion of vorticity restoring it to a straight line, (figs.5.6(d) and (e)). Since this sequence of events occurs near the updraught, where there is a horizontal gradient of vertical velocity, there is some vertical development, region E on fig.5.6(c) being higher than regions AD and CB. This vertical 115. component is removed by diffusion as the vortex line moves away.
Consider now the second case in which the approaching vortex line and the leading edge of the updraught vortex rings have the same sense, as in fig.5.7(a). As the vortex line approaches, it will be retarded by the presence of the updraught and stretched around the vortex rings, at
the same time being lifted by the updraught in its central region, shown in fig.5.7(b). Again, two regions of the vortex line of opposite sense,
AB and CD, come into close contact and self-annihilate, leading to
fig.5.7(c). The vortex line can now move away with the mean flow, while
a new vortex ring is deposited around the updraught core (fig.5.7(d)).
This would not, however, lead to a build-up of vortex rings around the
updraught since diffusion is constantly annihilating vorticity of
opposite sense near the centre of the updraught core.
-->
(a) (b) (c) (d) Fig. 5.7 Time-evolution of vortex-line configuration: mean flow vortex lines in same sense as that of vortex rings.
This reasoning allows the vorticity field in the outflow to return
to the mean-state vorticity field away from the storm, as is observed in
the numerical simulations. Further, this explanation does not contradict
the statement of Chapter Two referred to above, that in the presence of
non-conservative forces, vortex lines cannot be identified at different
times.
Using the above argument, the vorticity field generated when a vortex
line approaches a downdraught core can easily be deduced, the only
difference being the sense of the vortex rings. A more interesting region
is the one where the updraught and downdraught exist side by side, as in 116. 4
fig.5.8(a); according to the above, the vorticity field after some
time will be as in fig.5.8(b).
(a) (b) Fig.5.8 Time evolution of vortex line configuration: vortex rings of both senses present at same level.
There will, however, be some vertical development due to gradients of
W. The section AB of the line shown in fig.5.8(b) will be tilted down-
wards, section BC will be approximately horizontal. Section CD will slope
fairly sharply upwards, due to the proximity of updraught and downdraught,
producing a relatively large positive vertical component of vorticity.
Section DE will be approximately horizontal, while section EF slopes
downwards. Note that the slope of section EF will be steeper than that
of AB, since stronger gradients of vertical vorticity are normally
associated with updraughts than with downdraughts. Section CD, once
tilted into the vertical by horizontal gradients of vertical velocity, can
then also be stretched by vertical gradients of vertical velocity. This
will result in an area of positive vertical vorticity at or above about
800 mb in the region just inside the updraught near the interface between
updraught and downdraught, and a region of weaker negative vertical
vorticity on the opposite side of the updraught.
5.3 COMPARISON OF MODEL AND SIMULATION RESULTS
This section tests the arguments of the previous section by using
them to attempt a prediction of the vortex-line configurations given in
the previous section. Figs. 5.9(a) to (c) show the initial fields of • 117.
• ■ A\ •
` 350
Az.50
4ftl*, (a)Tropical storm system
(b) Hampstead storm
.1350 A( K450
7:k?c) ..0 4i..1495° 650 - 850 750Ae
Fig.5.9 Fields of cell-relative initial velocity (dash-dot line, 2.5mm = lm s-1 ), and initial vorticity (solid line, 2.5mm = 10-3s-1). •
(c) Splitting storm system
• / )4,2 5 0 of / •
4350 • •
• :14(450
/950. •
550 850
750
Fig. 5.9 (c) As for (a) and (b) a 119.
vorticity and cell-relative velocity for the case-studies presented
in this thesis. The alignment of updraughts and downdraughts in the
various storms can be obtained from the horizontal sections of airflow
presented in Chapter Four.
Consider, for example, case (C), the splitting storm system.
Fig.4.43(b) shows the position of the downdraught core at 950 mb, and
the initial wind and vorticity vectors are obtained from fig.5.9(c),
thus as a vortex line approaches the downdraught the situation is as
in fig.5.10(a). According to the theory described in the previous
section, the resulting vorticity field should have the appearance of
fig.5.10(b).
(a) (b) Fig.5.10 Predicted vortex-line configuration in splitting storm system : 900 mb.
Comparison with fig.4.44(a), the vortex line configuration for
the region around 900 mb, shows that the basic pattern is very similar;
there is a slight difference in orientation due to the two figures being
at different levels.
The updraught/downdraught configuration for the 750 mb level is
shown in fig.4.43(a), and fig.5.11(a) shows the appearance of the
vorticity field as the vortex line approaches the cell. Fig.5.11(b)
illustrates how this is expected to develop. 120.
\\ ()1\\
(a) (b) Fig.5.11 Predicted vortex-line configuration in splitting storm system : 750 mb.
Since on this level updraught and downdraught exist side by side, an appreciable amount of positive vertical component of vorticity is expected to develop in the region AB, with a weaker negative component in the region CD. That this is indeed the case can be seen from figs.
4.42(a) and 4.44(b). Inspection of fig.4.38(b) shows that the vertical vorticity is still in evidence at 600 mb, and the diagnostic calculation
confirms that this is due to advection and stretching of the vorticity produced at lower levels, as suspected earlier.
Finally for this example, the position of the updraught near 300 mb
is obtained from fig.4.39(a), resulting in the appearance of fig.5.12(a)
of the vorticity field as the vortex line approaches the cell. This is
expected to develop as in fig.5.12(b).
Fig.5.12 Predicted vortex-line configuration in splitting storm system : 350 mb.
The general form of fig.5.12(b) concurs fairly well with that of
fig.4.44(c). Similarly, good agreement is found between this theory 121. , w
and the results for the other case studies.
5.4 GENERATION OF VERTICAL COMPONENT: COMPARISON
WITH DUAL-DOPPLER OBSERVATIONS
The magnitude of the vertical component of vorticity generated by
the deep convection model varied greatly between the case-studies, from
6 x 10 3 s 1 over a very small area in the Hampstead storm to -2 x 10 2 s-1 over a larger area in the squall line and splitting storm systems, even
fairly early on in the life cycle of the latter. It is notable that
around the levels where most of the vertical vorticity component is
generated by tilting, where updraught and downdraught co-exist, the
environmental (horizontal) vorticity varies in a similar fashion. The
initial vorticity fields at 850 mb for the three different case studies
were:
STORM SYSTEM 11 II x 10-3s 1 x 10-3s 1 x 10 3 s 1
SQUALL LINE -1.2 -6.0 6.12
HAMPSTEAD STORM -1.1 0.6 1.25
SPLITTING STORM -8.0 4.2 9.04
Thus the squall-line and splitting storm systems had much more horizontal
vorticity available for tilting into the vertical than did the Hampstead
storm system. This tilting occurs in all the case studies at and above cloud
base, just inside the updraught; a strong positive component is generated
in the region where updraught and downdraught are close together, and a weaker
negative component on the other side of the updraught core. Both negative
and positive vorticities can be enhanced by stretching between cloud-base and
about 500-600 mb.
A method has therefore been found whereby a vertical component of
vorticity can be generated in an environment in which there was none
initially, and concentrated mainly above cloud-base. Its strength depends 122. largely on the magnitude of the (horizontal) vorticity present in the undisturbed flow.
Ray (1976) presents the results of the computation of the vertical component of vorticity and horizontal divergence fields using data gathered by dual-Doppler observation of two tornadic storms which occurred on 20 April and 8 June 1974. The main feature observed in both storms was a vortex doublet with positive component stronger than negative, with an area of high horizontal convergence between the areas of oppoging sign and, around the
1-3 km region, an area of divergence on the opposite side of the positive vorticity region to the convergence. The strong conver- gence is interpreted as updraught, divergence as downdraught; thus
the strong positive vorticity is located just inside the updraught on that side of it which is nearest to the downdraught. The negative
vorticity is on the opposite side of the updraught. The vertical
vorticity has its maximum value (-10 2 s 1) on the level where the
divergence is zero, i.e. where the vertical velocity is highest, and is concentrated mainly above cloudbase. The horizontal grid-length
used in this study was 1.5 km.
Heymsfield (1978) also analysed the kinematic and dynamic
structure of the storm of 8 June 1974 using dual-Doppler data.
However, due to the elimination of small-scale phenomena by filtering,
only features with a horizontal length scale of larger than 4 km are
claimed to be represented. Despite this, very similar
results were obtained for the structure of the vertical vorticity
field to those of Ray (1976). Heymsfield.went on to calcu- late the stretching and tilting terms of the vertical component of the
vorticity equation. It was found that the largest production due to
tilting was 4 x 10-5 s-2 above 3 km, the largest negative production 123.
-2 x 10-5 s-2 in axeas consistent with the observed vortex doublet.
Mid-level stretching reached magnitudes of 2 x 10-5 S-5 and
-2 x 10 5 s-2 in the corresponding regions. The largest stretching, however, was found to occur at low levels beneath the updraught with
a magnitude of 6 x 10 5 s 2.
The location and size of the vertical component of vorticity and the stretching and tilting terms of the equation for observed by dual-Doppler radar techniques thus compare favourably with the fields produced by the primitive equation model, and are consistent with the notion that vertical vorticity is generated by tilting and stretching of horizontal environmental vortex lines above cloud-base.
5.5 TORNADOES
While the 1 km grid-length of the primitive equation model prevents the detection of tornadoes, which have a length-scale of
10-100 m, there are some observations of the storm-scale vertical
vorticity field which can be made and which are relevant to the
tornado problem.
'It has been noted above that vertical vorticity generated in
the model has been confined mainly to the region above cloud-base,
and that the region of cyclonic vorticity, two or three times as
large as the anticyclonic, is found in the updraught near the inter-
face of the latter with the downdraught. This should be compared
with the fact that most tornadoes are observed to occur near the
gust front (but inside the updraught), and in fact the cyclonic
low-level vertical vorticity found by Heymsfield (1978) was concen-
trated there. This suggests that the link between the vorticity
above cloud-base and tornadoes is the large stretching term found in
low-levels beneath the updraught by Heymsfield (1978). 124.
Smith and Leslie (1978), as mentioned in Chapter One, investi-
gated numerically the effect of confining the region of the atmosphere
possessing vertical vorticity above a given level, which they varied,
in a model of a tornado. They found that if the vertical vorticity
is confined to a region above cloud-base, a pendant vortex could be
formed providing the initial values of vertical vorticity and thermo-
dynamic forcing were suitably matched. If vertical vorticity was
allowed to exist to just below cloud-base, tornadoes could be
generated, again subject to the constraint that the initial vorticity
and thermodynamic fields were suitably matched. Such a pool of
vertical vorticity is present in all the case-studies of this thesis,
and also in the observations of Ray (1976) and Heymsfield (1978);
the initial conditions assumed in the model of Smith and Leslie would,
therefore, appear to be possible in real storms.
The initial conditions for the primitive equation model simula-
tions were all abstracted from soundings taken just before actual storms; only in the last of the case-studies (the splitting storm
system) were tornadoes actually observed in the real storm. While
it is not known whether the vertical vorticity and thermodynamic
forcing (measured here by the convective available potential energy) were suitably matched in the simulations, it is perhaps significant
that the simulated splitting storm system generated more vertical
vorticity over a larger area than the others.
Thus the results and conclusions of Smith and Leslie (1978) are
certainly not contradicted by either the case-studies presented here,
or the observations of Ray (1976) and Heymsfield (1978).
125. • CHAPTER SIX
POTENTIAL VORTICITY
In this chapter an equation is derived in continuous form for a general potential vorticity quantity of the form
.17R e 1 where R is some scalar property of the atmosphere, using the equations of the primitive equation model. Special cases are then considered;
these are (i) R = 4) (log potential temperature), (ii) R = E (log
equivalent potential temperature), (iii) R = (Pal (log liquid water
potential temperature), and (iv) R = (e ,,/es,)- z, the baroclinic term in the vertical momentum equation.
6.1 EQUATION OF GENERAL POTENTIAL VORTICITY OF DEEP CONVECTION
The vorticity equation relevant to deep convection, derived in
Chapter Two, can be written
= — 3(Q. u) + g k )07X + (6. 1) where % = 69/diSM-
The FULL continuity equation is De — Dt P\71:1; • (6.2) Substituting for V.0 in (6.1), using (6.2),
= ((.0)u I + gkx1i7X VxE VC Dt — +
(6. 3) 00)-7- CLv)141. g x v96 + VxF
Let R be some scalar property of the fluid; then on forming the scalar product of VR with both sides of equation (6.3),
OR.f(i)=17R.0,..00k,t1 -I- 9 VR.(1sxVX) i-OR.(Vx
Using the cyclic permutation property of the triple scalar product,
and rearranging,
• 126.
D_ (US .v(DR) + k . (MVO + (NO (6.4) Dt e \Dti e e
This is the required equation for the general potential vorticity (4. VR) /p.
6.2 SPECIAL CASES
Equations are now derived for the special cases mentioned above.
The following quantities are defined:
'Ordinary' potential vorticity (6.5)
"Equivalent' potential vorticity (6.6)
'Liquid water' potential vorticity (6.7)
'Potential vorticity of deep convection' (6.8)
The equations of thermodynamics and cloud physics which will be used
in this section are (Miller and Pearce (1974))
12.11 - 0 (6.9) eDt cTrLQ - + eeK i0 2 F--
(6.10) Dat = + Kirq,
((+v = (etr'4 + Kcipzci+ 'etc) (6.11)
6.2.1 'Ordinary' Potential Vorticity
Substituting (I) for R in equation (6.4) gives 127.
at = -# 47 ts.(vx x is* v4). (ox F) (6.12)
D can be replaced by the right-hand side of equation (6.9). The Dt quantity k . (V% x can be expanded as follows:
k . (VX x VC1)) = k .(71,X x 14))
-- 1ce Vhex -- Vh since =
gg V 0.4 — oh I, (6.13) 4 h By definition, e* = 0(1+aq) where a = 0.6077 + in (I +(ILO
= Oh(1) + +41,) (6.14) Thus, using equations (6.13) and (6.14),
1<.(vx x \RN = . 0 (VA x;44) (ViAx Vb,L) (VhI, x ;LI))
Substituting this and equation''(6.9) into (6.12) gives the required
equation for II : on _ v K Lvzo +L) Db. —T k CpT X t 3FtS it 076 (7:cq(oilq, XVI),t) +OA a +.11.-:WVhi Vi4 (Vx E) (6.15) The first term on the right hand side of this equation represents
the change in E due to diabatic changes in (P, the third term
representing diffusion of vorticity. The second term is present since
128.
the vector is not parallel to the vector V Vhf hx.
6.2.2 Equivalent Potential Vorticity
Substituting cbE, for R in equation (6.4) gives
=(gt9— IS. (V% X VIS) + VotPE (Vx f) (6.16)
The equation defining cpE is
GUDE = dq) cIrr The equation for (1)E, is obtained by eliminating Q between equations
(6.9) and (6.10), giving
= K Lv28 +1_- (6.17) Dt CpT v\ Since F— is only an initiating heat source in the numerical model, it can be seen that in an inviscid model of cumulonimbus convection, (P E is a conserved quantity. Substituting (6.17) into (6.16), and expanding the term k.(7xx VSE) using the definitions of X and (I)E, gives, after
: some algebra, the required equation for HE
I:ttE =evae +til7Kt/Vagrii/ +fqetccravAxvhohLxviNI---o=drophix4c
ii-tvos ÷ca_1--p0 ql.(oxE) (6.18)
The terms of (6.18) have the same meaning as those of (6.15). It can clearly be seen that in an inviscid model of deep convection, only the second term would remain. 129.
6.2.3 LIQUID WATER POTENTIAL VORTICITY
Substituting (P for in equation (6.4) gives EW R Ig = e• qui) + x %Lei) + i-VOug . (VX (6.19)
The equation defining (I) is EW d.0
The equation for (Pa, is obtained by eliminating Q between
(6.9) and (6.10), then eliminating Dq/Dt using (6.11), giving 110 L 2 .E_ tit = gefT ap eirq+ Weli/72.°—CTri Thus even in an inviscid model of deep convection, the value of (PLW of an element of air can be altered by a change in its rainwater concentration. Substituting (6.20) into ;(6.19), and expanding the term k.(17)(X VfLW) using the definitions of X and (.5w gives, after some algebra, the required equation for HEW: Dn = 3. Dt e ocp1.3--rap (e1 rVt) Ke-h172-0-1.T.KieViteii +1.t91"(vq,xv4),0:574- h)+(1 e1p.---N)(9„OTet xxvhq+A(vhixvv4 + IVO* ex j="' (6.21) 4 Vt CPTV 9. (VX The terms of (6.21) have the same meaning as those in (6.15) and (6.18). 6.2.4 Potential Vorticity of Deep Convection Equations (6.15)1 (6.18) and (6.21) for II, R and II respec- E LW tively show not only that these three quantities are non-conservative in the numerical model, but that also one of the terms contributing to 130. voi changes inthese quantities is not attributable to diabatic effects or vorticity diffusion, viz. the non-zero angle between V71(1), or Vh4E and Vh Vh(j)LW X. This can only be eliminated by choosing a quantity of the form ax+ b (a,b constants) for the variable R. Further, the vectors VIDE, V Lw contain a large contribution due to the initial fields, which are functions of pressure only, and which thus make no direct contribution to the change of vertical velocity or horizontal vorticity. Only horizontal gradients of x, which is a perturbation quantity, can contribute to the baroclinic term in the vorticity equation. This suggests that 1I will be smaller than II, RE or. 11.w, but that it may have more dynamical significance. Substituting X for R in equation (6.4), Dncg =3. .V(11) + -1--VX.(9XE) (6.22) rTt e The equation for •x is now obtained. By definition, e(1+4%) — 1-1 011+44s) . DX _ D .1 00+41 • • DE — DELe50+Gut5)j Dt 0( i44) 1 De 4. a.8 Pa, 40+04) W_ 8(1+ctip _pl. es(1+4) 9 Dt 66(livis) It 00401 Dt gs(1+igts) Dt Dt Using equations (6.9), (6.10), (6.11) then gives 131. DX — —a+KiC/20—n(1., N7q,s —16(4,V)e Dt e v—Aw sj + g -4,(etrVt)—Q Kqyak (6.23) Substitution of (6.23) into (6.22) yields the required equation for nCB Dr142 't [LA f i,1129+E EQ+Kfralft)*12—+:014100t1,5— 64j .S)01 CpT 0 T 0+01) g (e LA) Ks V141 + If V%. (7X E) (6.24) The terms of this equation are respectively diabetic effects and diffusion of vorticity. This equation is 'neater' than those for H, H and 11 in that the terms can be divided into two physical effects; E LW as stated above,. RC may also have dynamical significance for the B cumulonimbus problem, but this question has not been fully investigated. 6.3 POTENTIAL VORTICITY EQUATIONS IN SIMPLIFIED MODELS OF CUMULONIMBUS Brief mention has already been made above about the way that the potential vorticity equations (6.18, 6.21) would simplify under certain assumptions. This section examines in slightly more detail how all of them can be reduced. 6.3.1 Moist Inviscid Cumulonimbus Model The potential vorticity equations for a moist, inviscid model are obtained from equations (6.15), (6.18), (6.21), (6.24) by putting K = = 0 and F = 0 , but keeping all other terms containing 0 4 q, 4r' l c' The term F/T is also omitted, since it is merely an initiating heat source for the numerical simulation. The equations for H, H H and then become E LW CB 132. Dil = .vo +.4.{p6Akclixv„ci4vv,0,xv,i0+ovi,,,xvhs1 Raj= Q isiticT,_T-04)(vi,(0* AA) -(giLxVin-41-0„,0)(vv,tx vhcii Lw=i-..14(g-i;Tft[etryt3)+hipa°:---4)(VIAMI)+0-egOvitiloiliii)+A(VillxV64 "21,18 Dt P • (A ISIT 0 +xi) 04-citiCC t ) (LI NW —1es61 V)01 Thus in a moist inviscid model. II and H are subject to both LW diabatic effects and the effect of Ohl , VOLw not being parallel to V , while He is subject to diabatic effects only. IIE , however, hx B . not being parallel to V X. A is subject only to the result of V41)h E h quick 'order of magnitude' check suggests that this term may be much smaller than the diabatic terms of the other equations, and so HE is probably nearly conserved in this case. 6.3.2 Dry Inviscid Cumulonimbus Model The potential vorticity equations of a dry inviscid model are found from the last four equations by setting ic = Zr = q = Q = 0. In this case H and n are identically equal to II, and H reduces to E LW CB n. = . The equations for H and H are then CB = t_nto3 - v. as (4, MI 133. • s If it is assumed that (I) is horizontally stratified, then this further reduces to e es where g= ad2s Thus in this case, H is conserved exactly, while Tic is not. B 6.4 POTENTIAL VORTICITY FIELDS IN SIMULATED CUMULONIMBUS Computation of the quantities II and R show both to be CB well-organised, except near the 600 mb level, where the wave-length of the disturbance appears to be only about two gridlengths. Potential vorticity doublets appear in the updraught and in the cold pool in both Ti and fields. Magnitudes are greatest at lower levels, II CB CB reaching -10-8 m2 kg-1 S-1 in the Tropical Storm system, for example. II is two or three times as great as HcB, confirming earlier expecta- tions. Figs. 6.1-6.3 show the appearance of the HcB field for the Tropical Storm System. Fig.6.4 illustrates the variation of n and H along a trajectory CB in the Tropical Storm - the same trajectory as that used to show the variation of fl and C in Fig.5.4. No attempt has yet been made to or HCB but it calculate the terms of the equations for IF, nE' na7 H are conserved in can be seen from Fig.6.4 that neither II nor CB simulated cumulonimbus. Figs. 6.5-6.8 show the values of II and H averaged over a whole CB' — H , H ) for horizontal plane of the model, versus height (denoted by H CB the case-studies of this thesis. In the Tropical and Hampstead storms, — the HH and n fields most nearly resemble each other in the lower CB 300-400 mb, while above about 600 mb the e fields are very different -H —H from the CB fields. In the Splitting storm, however, the H and 134. t A Fig.6.1(a) Vertical section through tropical storm along line AA' marked on Figs.6.2, 6.3 showing contours of II . CB' Contour interval 10-8 m2 kg-1 s-I. Fig.6.1(b) As for Fig.6.1(a), but vertical section along line BB' marked on Figs.6.2, 6.3. 135. B Fig.6.2 Horizontal section through tropical storm showing contours of naB. Lines AA' and BB' show positions.of vertical tections in Fig.6.1. Contour interval 10-13 m2 kg-Is-1. A B B' (b) p = 600 mb A 136. • - e • .• • B' • I (a) p = 800 mb Fig.6.3 Horizontal section through tropical storm showing contours of 110. Lines AA' and BB' show positions of vertical sections in Fig.6.1. Contour interval 10-8' m2 kg-, s-a . A B' 200 300 400 (-440, -74) -300,77) (63,85) 500 (-311,-44) 600 19, 270) 700 800 580, 366) (150, 52) 900 (00) (11) (- —31 (-141, -163) 1000 • I 1 a I IL west )114n( east Fig.6.4 Computed trajectory of air in tropical storm; values of II are shown at various points - CB thus: (II, ). Units: 10-10 m2 B kg-ls-1. 138. • -4 0 1 2 3 4 5 6 7 x10-1° Fig.6.5 Vertical profiles of TrI and Tril for the tropical storm, CB 84-92 minute means. Units: m2 -kg1 sS.- 1 139. a —H -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 I Ice x10-1° Fig.6.6 Vertical profiles of rll and 711 for the Hampstead storm, CB 32-44 minute means. Units: m2 kg-i s-1. 140. e 100- 200. 300- (mb) 400. 500- 600- 700- 900. • 1000 -7 -6 -5 -4 :3 -2 -u1 0 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 x10-1° Fig. 6.7 Vertical profiles of and II for the Hampstead storm, e CB 64-76 minute means. Units : m2 kg-1 s-1. 141. 700. 800. goo- I . • 1- . . , 1-1H -7 -6 -5 -4 -3 -2 -1 3 4 5 6 7 I I x 10-1° 100/ 00- 600 700- 800., 900- 1000 —H -7 -6 -1 1 2 3 4 5 6 7 11 x10-1° --H Fig.6.8 Vertical profiles of IT and lICB for the splitting storm, 32-36 minute means. Units: m2 kg-1 s-1. 142. fields are basically similar throughout. Also, in both Tropical eCB n—H and Hampstead storms, CB is large and positive in the region above about 600 mb, while below the variation of e is on a much smaller space-scale. In the Splitting storm, e is large and positive below 600 mb, mainly negative above. This suggests that the Tropical and Hampstead Storms have some'characteristic in common which is not possessed by the Splitting storm. One such characteristic is the abrupt change in direction of the initial wind-shear in low-levels which is apparent in both of the first two cases but not the third. 143. • CHAPTER SEVEN SUMMARY AND CONCLUDING REMARKS 7.1 SUMMARY This thesis has described in detail the formulation and some results of a fully three-dimensional pressure-coordinate diagnostic model of the vorticity and potential vorticity of deep convection. Using the velocity, thermodynamic and cloud physics fields generated by the primitive equation model of Miller, this model calculates the three components of (i) the vorticity field and (ii) the terms of the vorticity equation, the 'ordinary' potential vorticity field, and the field of the potential vorticity of cumulonimbus convection. Vortex lines are also calculated. The results of three simulations have been investigated, these being a tropical squall-line, a stationary storm, and a splitting storm system. The vorticity fields of all these storms are surpri- singly well-organised and differ from each other in only a few well- defined features, e.g. the amount of vertical vorticity generated. The horizontal components of vorticity reach magnitudes of - 10-2 s-1 around the edges of updraught cores and downdraught 'cold pools'. The vertical component also reaches this order of magnitude, appearing as a vortex doublet inside the updraught. The cyclonic region is found near the boundary with the downdraught, the anticyclonic being of slightly smaller magnitude on the opposite side of the updraught. The actual size of the vertical vorticity generated, as stated above, varies from case to case. The organised nature of the vorticity field has permitted a simple model of cumulonimbus vorticity to be described in terms of vortex lines. The storm is schematically divided into three horizontal slabs, the upper consisting of updraught, the middle consisting of updraught and 144. and downdraught side by side, the lower consisting of downdraught outflow. The upper slab then contains vortex rings with cyclonic sense around the updraught, while the environmental vortex lines pass around them. The lower slab is identical, except that the vortex rings have opposite sense. The middle slab contains vortex rings of both senses side by side, the environmental vortex lines passing around both. The development of the vortex-line configuration has been discussed using the diagnostic calculations. It is found that in the case of the horizontal components, baroclinity and diffusion are the most important terms, the former generating the vortex rings, the latter being the agent which allows the environmental vortex lines to apparently 'pass through' the storm. The vertical component is generated by tilting of these vortex lines in the middle slab near the updraught/downdraught boundary by horizontal gradients of w, and then stretched in regions of positive Waz. Tilting on the opposite side of the updraught is responsible for the weaker anticyclonic component. The amount of horizontal vorticity available for tilting, and therefore the strength of the vertical vorticity generated, depends on the vorticity of the mean flow at around the 800 mb level. Thus the stationary storm system, having much less mean flow vorticity at this level, generated much less vertical vorticity than the other two cases. The introduction of dual-Doppler radar in the observation of severe storms in the United States has made possible the calculation of the vorticity fields within real storms. Several papers containing a description of the vertical vorticity field and some of its diagnostics have been published, and comparison with the results of the vorticity model described here is encouraging. Four types of potential vorticity have been defined being 'ordinary', equivalent, liquid water, and that of cumulonimbus convection. Equations 145. or have been derived for these quantities using the vorticity, thermo- dynamic and cloud physics equations of the numerical model. Special cases of these equations have been examined, and it has been shown that in the case of a moist, inviscid cumulonimbus model, equivalent potential temperature is conserved and equivalent potential vorticity is almost conserved. In a dry, inviscid model, po:tential temperature and 'ordinary' potential vorticity are exactly conserved. The results of calculating the fieldd for the tropical storm are presented. The horizontally averaged II and IlcB are plotted against height for all the cases, and it is shown that certain simila- rities exist between the tropical and Hampstead storms, while the -HH --H and fields of the splitting storm had a different structure; it has been suggested that this might correspond with the fact that the first two cases have a sharp change in the direction of the low-level shear, while the third does not. 7.2 SUGGESTIONS FOR FUTURE RESEARCH The model here presented is capable of providing the complete vorticity field, in both component and vortex-line form, the diagnostics, and the potential vorticity fields using previously-generated velocity and thermodynamic fields. It will thus be possible to examine the vorticity of various types of simulated storm in order to determine in greater detail the relation between the initial conditions and the resulting vorticity and potential vorticity fields, and to investigate the role of vorticity in individual storm features, such as storm splitting. It would be instructive also to calculate the equivalent and liquid potential vorticity fields, and to formulate a diagnostic model of potential vorticity. The transport and budgets of vorticity could be computed. 146. The results of these studies can be used to provide a basis for theoretical research of the type carried out by Moncrieff and Green (1972) and Moncrieff and Miller (1976), the latter then giving ideas for further simulations. One important objective is the parameteri- sation problem. Another is the tornado problem; some comparison can be made between the vorticity fields generated by the model, and those used as 'input' to tornado models to gain a greater understanding of the conditions required for tornadoes. This list of suggestions contains only some of the possibilities, and thus future research using the model developed.here should prove interesting and profitable. 147. APPENDIX PRIMITIVE EQUATION MODEL TIME INTEGRATION SCHEME It was stated in Chapter Three that due to the time integration scheme used in the primitive equation model, difficulties arise in the vertical momentum equation. The scheme is here examined in detail in order to determine the precise nature of the difficulties, and it is shown that by making a change in the scheme, the difficulties can be avoided. The scheme, described in the Appendix of Miller and Pearce (1974), consists of alternate Euler-backward and Euler timesteps. Since the even timestep is a simple forward difference, and is therefore much less complicated than the odd timestep, it only will be considered in detail. The equation set solved by the model is of the form: _ a, 9 Fc + (1) at — +gfg (g) W rit) = —9 {92e a +9est—LA 9fp(W au +av --e= 0 ax a! ap whence the diagnostic equation for h' is a2 )111 4. 92 _( t32 = [ (flit a_Fit + LEA a.-F.P (AS) (-632. - S ap p s es4 ax ap • Let (I) = 4(nLt) for any non-negative integer n. n Assume um, wm, 1301,m, Zm, h;n . are known where m is odd; the calculation of these variables at time (77+1)At is then an even timestep, i.e. a forward time difference without temporary values. The order in which the variables are computed is now described. 148. Firstly, u are computed from finite-difference versions m+1 , of equations (Al) and (A2) giving uitio = wry) + (11fri gfx„,) (A6) (A7) vrico = v + (—g ay1*" + Then, using equation (A4), W is given by m+1 h)Alft (A8) bP ox by Next, er are computed in terms of V* *m+1" m+1 mm" m+1'+1' m+13 wm+1 using the thermodynamic and cloud physics equations. Finally, h'7724.1 is computed from equation (A5) using the values of ec, 1, fp at time (m+1)At, and the values of fx, fy at time mAt. It is thus seen that, although equation (A3) is used to form the diagnostic equation for h', equation (A5), it is not itself integrated forward in time as are equations (Al), (A2), and the thermodynamics and cloud physics equations. It is now shown that, due to the method of integration just described,'the right-hand side of the vertical momentum equation which is consistent with this method of integration will contain second-order time integration errors. Substituting equations (A6) and (A7) into (A8), 30011-i = ZtLin — + ?Alm At + Bp Bx By axe g a'irnzy2 Atax — 9 -24 " At — 9-yr de Substituting equation (A8) computed at time mAt and rearranging, — con) 9gn1 Acm 3119m (A9) = + -- 9 aP )(1 Z92- ax ay The quantity Mtn will have been computed at the final stage of the previous timestep using the equation e af-xm-f+ gl4 tn-i 4-?..Ep in (Am) xa2. +at )111m +g a (e5a -Fifi9"-: ap {e5rim -44] T 4p x a 4 149. Substituting (A10) into (A9) — a/I'M-WM) - I 3 1*14\ e./* ( At k 9 es Ti )-41.gles{(-64+ 9tm afrn tm-)— g(tm t*1 = -ip(93eln) [Alm —14] as? —n&A.fjyTA J at ( ax } Integrating with respect to p by the method used to integrate the continuity equation for w, cDrn+i —corn = — 93 e2 grin iten —6) + s ap S `e —g if* k 17 .kin lyni-1)4 (A11) ati k Zx y This is the form of the vertical momentum equation which is consistent with the method of integration in the primitive equation model, and as can be seen the fourth term is a time-differencing error. This term was neglected when computing the vorticity diagnostics. By computing the values of 3E/at, an/at using the right-hand side of the horizontal components of the vorticity equation derived from equation (A11) neglecting the time-differencing error, and comparing with the true values of (c1.1-Enl)/At, (flm+i-nm)/At for an even timestep, it was confirmed that this term is in fact negligible. Obviously, no such problem arises in the vertical component of the vorticity equation, which is obtained wholly from the horizontal momentum equations and continuity of mass. 150. This problem can be averted in two ways. Solve for um M+ using equations ( A6), (A7) respectively; (1) +1 V 1. obtain wir174.1 by integrating forward in time the finite-difference form of equation (A3) with all variables on the right-hand side held at time mkt, analagous to the treatment of (A6) and (A7). can then be computed in terms of ecle Z *m+1'm+1 m' um+1' h' is computed using the values of -.)7,114_1, 971+1. m+1 ALL variables at time (m+1)Lt. The continuity equation is still satisfied, since it is implicit in the diagnostic equation for h'. (2)Solve for U V w 6 exactly as at present m+1' m+1' m+1' *m+1.,m+1 from the in the primitive equation model, then compute h'm+1 balance equation with ALL the variables held at time (m+1)At. An analysis of this method as performed on the present primitive equation model scheme yields identical results up to and including equation (A9). However, in equation (A10), the fxm_i, fym-1 are replaced here by fy . Thus" using this scheme, h' is given by m' m m (Al2) 412)hirn# 92f,p(eg- 411= —4[drte m —111.11÷4+ trA+ teffi Substituting (Al2) into (A9), =__L(- 930211, 1fit_.:11-91e (flm _61}1 gAr.pm (A13) 41, k dP j Sxfh: ap Integrating (A13) with respect to p, 0440-0)04 - 93eZ. m g2es (gni- to) gfprvi (A14) lit p Equation (A14) contains no new time-error terms; it is exactly the equation used for the derivation of the horizontal components of the vorticity equation. Methods (1) and (2) can be seen on examination to yield identical results and are thus equivalent; both differ from the integration scheme actually used in the model in that Am, hm are used in place of Am-1, fym...1 to calculate him. Both methods satisfy continuity of 151. • mass and conserve vorticity; the primitive equation model satisfies continuity of mass but does riot conserve the horizontal components of vorticity. The adoption of method (1) would involve large-scale changes to the present primitive equation model; the adoption of method (2), however, would merely require that fxre fym are computed BEFORE instead of after; no extra computing time or storage area is m thus involved in this change. The stability of the model if either method (1) or method (2) were adopted in place of the present scheme is not known, but is likely to be improved since methods (1) and (2) increase the implicitness of the scheme. Application of the above analysis to the odd (Euler-backward) timestep is not carried out here, as mentioned previously, due to the added complication of the 'temporary values'. However, it can be stated in retrospect that although larger than those of the even time- step, the errors in the odd timestep are still small enough to be neglected. • 152. REFERENCES Agee, E., Church, C., 1975 Some synoptic aspects and dynamic Morris, C. and Snow, J. features of vortices associated with the tornado outbreak of 3 April 1974. Mon.Wea.Rev., 103, pp.318-333 Betts, A.K., Grover, R.W. 1976 Structure and motion of tropical squall- and Moncrieff, M.W. lines over Venezuela. Quart.J.R.Met.Soc., 102, pp.395-404 Bode, L., Leslie, L.M. 1975 A numerical study of boundary effects and Smith, R.K. on concentrated vortices with application to tornadoes and waterspouts. Quart.J.R.Met.Soc., 101, pp.313-324 Brown, R.A., Burgess, D.W., 1975 NSSL dual-Doppler radar measurements in Carter, J.K., Lemon, L.R. tornadic storms: a preview. and Sirmans, D. Bull.Amer.Met.Soc., 56, pp.524-526 Browning, K.A. 1962 Airflow in convective storms. and Ludlam, F.H. 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(Accepted for publication) Turner, J.S. 1966 The constraints imposed on tornado-like vortices by the top and bottom boundary conditions. J.Fluid Mech., 25, pp.377-400 0 155. Wilhelmson, R. 1974 The life cycle of a thunderstorm in three dimensions. J.Atmos.Sci., 31, pp.1629-1651 • 156. Nr ACKNOWLEDGEMENTS The author wishes to express his most sincere and grateful 'thanks to his supervisor, Dr. M. W. Moncrieff, for continual help, guidance and constructive criticism given throughout the period of this work. Thanks are also conveyed to Dr. M. J. Miller for a great deal of advice, and for permission to use and reproduce some of the results of his storm simulations, to Mr. A..G. Seaton for valuable programming assistance, and to Mrs. J. Ludlam for so care- fully and patiently typing this thesis. The author is also most indebted to his wife for tracing by hand all copies of the coloured illustrations of Chapter Four, and for being a constant source of help and encouragement. The receipt of a N.E.R.C. Research Studentship is gratefully acknowledged. 4