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.