This is the defence presentation for the dissertation Dynamics of Helical Flow Structures

1. This is the defence presentation for the dissertation Dynamics of Helical Flow Structures A Description of Vortex Formation in Turbulent Fluids by Nikolaj Nawri given on 2 April 2003 at the Department of Meteorology of the University of Maryland at College Park. It is set up for full screen viewing. This can be changed under Slide Show > Set Up Show. In full screen view the speaker notes can be displayed by right-clicking on the running presentation.

2. “You don't seriously believe that a theory must restrict itselfto observables? Perhaps I did use this sort of philosophy, but it's nonsense. Only the theory decides what one can observe.” Albert Einstein to Werner Heisenberg (1926)

3. Dynamics of Helical Flow Structures A Description of Vortex Formation in Turbulent Fluids Nikolaj Nawri Department of Meteorology University of Maryland College Park, MD, USA

4. Tornadolike Vortices A tornadolike vortex is any atmospheric vortex with a diameterof a few hundred metresassociated with a storm system or otherwise driven by larger-scale flowfeatures such as fronts or squall lines. Dynamics of Helical Flow Structures

5. Tornadolike Vortices • Tornadolike vortices, by definition, are associated with severe thunderstorms. • This associationof the vortex to the larger-scale circulation of the storm system leads to a strongcoupling of a wide range of spatial and temporal scales. • The complete description of tornadogenesistherefore requires a comprehensive theory of fluid flow including both the intrinsic, universalproperties of small scales and a consideration of the energy input on large scales by specific external forcing. Dynamics of Helical Flow Structures

6. Storm Research • A common approach employed inobservational tornado research is to calculate certain forecast parameters from data ofthe storm system (obtained prior to tornadogenesis) and to compare their values between tornadic andnontornadic cases. • Despite unreliable and sparse storm data it appears asif the currently employed parameters are not adequate for reliable predictions of tornadogenesis without high false alarm rates. Dynamics of Helical Flow Structures

7. Storm Research • Climatological study of severe thunderstorms by Rasmussen and Blanchard [1998] • Analyses all of the 0000 UTC soundings from the United States made during the year 1992 that have nonzero CAPE. • Storms are classified as ordinary thunderstorms, nontornadic supercell thunderstorms, and tornadic supercell thunderstorms. • Forecast parameters characterising vertical wind shear, static instability, and various combinations thereof are calculated. Dynamics of Helical Flow Structures

8. Operational Forecast Parameters

9. Operational Forecast Parameters

10. Operational Forecast Parameters

11. Environments of Vortex Formation • Prior to storm formation atmospheric motion is predominantly horizontal, where over even terrain variability primarily is in the vertical stratification. • The most favourable large-scale vertical wind profile for the formation of storm rotation is horizontally homogeneous and vertically veering. • Tornadolike vortices on the other hand are embedded in a storm flow with intense vertical motion and strong horizontal gradients in velocity. • They commonly form on or near outflow boundaries and shear zones with intense horizontal variability on the vortex scale. Dynamics of Helical Flow Structures

12. Storm and Vortex Flow • Sincethe instabilities of the storm flow are associated with spatial variability on the vortex scale, a separation of the flow into large and small scales is not possible. • Instead the velocity field is separated into a slowly evolving “background” flow representing the storm motion, and a rapidly evolving perturbation flowrepresenting the tornadolike vortex. Dynamics of Helical Flow Structures

13. Dynamical Systems Analysis • From the Fourier transformed equations of motion for storm and vortex flow low-dimensional dynamical systems are derived through spectral truncation. • In the dynamical system for the rapidly evolving expansioncoefficients the slowly evolving expansion coefficients, over short periodsof time,are considered to be constant parameters. • Since the“tornadic” and “nontornadic” parameter regions and the corresponding background flows follow from a bifurcation analysis, an explicit solution of theequations of motion is not necessary to determine the qualitative evolution of the fastflow. Dynamics of Helical Flow Structures

14. The Quest for New Forecast Parameters • The dynamically derived parameters are defined as various combinations of the background flow expansion coefficients. • They are uniquely determined by a given background flow, but given the set of parameters the background flow is not completely specified. • They are very abstract and unlike the currently employed forecastparameters cannot easily be interpreted in terms of physical concepts such as “static instability” or “vertically veering wind profile.” Dynamics of Helical Flow Structures

15. Overview Flow in physical space • Basic kinematical variables and their importance for the motion of the fluid • Definition of eddies and flow structures • Spectral and helical decomposition of storm and vortex flow • Scales of the storm system Dynamics of Helical Flow Structures

16. Overview Flow in phase space • Three-dimensional dynamical systems • Equilibria of these systems • Stability of equilibria and transitions between them • Interpretation of bifurcations in terms of flow instabilities Implications for tornado forecasting Outlook Dynamics of Helical Flow Structures

17. Flow Properties - Energy For a given 3D velocity field v = v(t,x) with vorticityw = r£ v, the following energy related kinematical variables are defined: Kinetic energy: Enstrophy: Intensity: Dynamics of Helical Flow Structures

18. Flow Properties - Helicity To express the relative orientation of velocity and vorticity by scalar fields define: Alignment: where Helicity: Dynamics of Helical Flow Structures

19. Eddies • Eddies are defined as positive kinetic energy perturbations from the normal flow state. • More specifically, helical eddies are defined as positively correlated positive perturbations in intensity and alignment. • The normal flow state in this study is defined as the relatively slowly evolving velocity field associated with the thunderstorm. Dynamics of Helical Flow Structures

20. Flow Instability • The inertial forcing term v £ wis responsible for basically allinteresting kinematical phenomena in fluid dynamics, in particular for the generation of turbulentvelocity perturbations and energy transfer between spectral components of the velocity field. • The magnitude of inertial forcing decreases with increasing alignment. Dynamics of Helical Flow Structures

21. Flow Stability • While theinertial forcing and shear instabilities are responsible for the onset and maintenance ofturbulence, these mechanisms are damaging for an increase in the intensity of large eddies. • To survive longer as a coherent kinetic energy perturbation, an eddy must acquirekinematical properties that minimise the very forcing term that created the initial disturbance inthe normal flow from which it grew. • Intense velocity perturbations with strong alignment can therefore be expected to be morepersistent than velocity perturbations with weak alignment. Dynamics of Helical Flow Structures

22. Flow Structures • Since dynamically significant eddiesare intensethe meaning of helicity for eddiescan loosely be expressed as • Intense and persistent eddies thatpreserve their qualitative kinematical properties over sufficiently long periods of time to be ableto interact with each other are referred to as flow structures. Dynamics of Helical Flow Structures

23. Helicity Extremisation For a given intensity helicity is extremised in a finite volume if velocity and vorticity are perfectly aligned: with a positive constant c. These maximally helical flows are called Beltrami flows. Dynamics of Helical Flow Structures

25. Beltrami Vortex Flow • In simplified form,if the periodic vortex pattern is taken as a representative of the full spectrum, helical vortices can be described by two entangled waves with a fixedamplitude and phase relationship. • As they intensify or weaken the amplitude and phase relationshipbetween the vortex waves is maintained. The corresponding expansion coefficients therefore musthave a very similar time-dependence. • If this entanglement is destroyed the vortexdisintegrates. Dynamics of Helical Flow Structures

26. Truncation • Spectral truncation: only the two vortex waves and one catalyst wave for each dynamical system are considered (one-triad-interactions). • Helical truncation: for the two vortex waves only the positively helical component is considered, and for the catalyst wave only the nonhelical component. • For the vortex waves the real part of the expansion coefficients is set identically zero, and for the catalyst wave the imaginary part. Dynamics of Helical Flow Structures

27. Dynamical Systems For both triads qualitatively the same dynamical system is obtained with Dynamics of Helical Flow Structures

28. System Equilibria • The dynamical system has two coexisting equilibria: • The phase space origin (0,0,0) exists for all parameter values. • The nontrivial equilibrium exists for p¹ 0. Dynamics of Helical Flow Structures

30. Linear Stability The linear stability of the two equilibria of each system is analysed by calculating theeigenvalues of the Jacobian matrix evaluated at the stationary solutions. Dynamics of Helical Flow Structures

31. Linear Stability • The linear stability of the system near any of the fixed points depends on the sign of the realparts of the three eigenvalues li. • For asymptotic stability Re[li] < 0 8 i = 1,2,3. • Conversely, for instability Re[li] > 0 for at least one eigenvalue. • To characterise the combinedstability or instability of all three eigendirections of the linearised flow around any equilibrium, attractor and repellor strengths are defined. Dynamics of Helical Flow Structures

32. Attractor Strength The attractor strength of an equilibrium X is defined as for all Re[li] negative, else A(X) ´ 0. Dynamics of Helical Flow Structures

33. Repellor Strength The repellor strengthof equilibrium X is defined as the magnitudeof the vector containing all its unstable eigenvalues Dynamics of Helical Flow Structures

36. System States • Without asymptotically stable equilibria there can only betransient fluctuations and the momentary development of a vortex flow would not be a convincingargument for the formation of tornadolike vortices in a turbulent fluid. • Generally, a meaningful definition of “state of a system” requires at leastsome degree of persistence. • In the following, “state” will therefore always be referring to anequilibrium of the system. Dynamics of Helical Flow Structures

37. Initial and Final States • In practicethe slow velocity field is calculated fromobservations of the storm system prior to tornadogenesis. • Relevant initial conditions for phase spacetrajectories are therefore small perturbations from the phase space origin. • To find criteria for vortex formation it is necessary to find the background flow conditions that lead to a transition from the ground to the vortex state. Dynamics of Helical Flow Structures

38. Transitions Between Equilibria • Since stable (orunstable) manifolds of two distinct hyperbolic fixed points cannot intersect, a point arbitrarilyclose to the origin, which lies within the basin of attraction of the vortex fixed point, mustnecessarily also lie on the unstable manifold of the origin. • Then vortex formation from smallinitial perturbations, i.e., atransition from a state close to the origin to the vortexequilibrium, takes place. • To characterise the transition probability between states of the system, acombined measure of the instability of the initial state and the stability of the final state mustbe introduced. Dynamics of Helical Flow Structures

39. Transition Probability Based on the definition of repellor and attractor strength, the transition probabilityfrom equilibrium state X1 to equilibrium state X2 is defined as the product of the repellor strength of the initial state with the attractor strength of thefinal state: Dynamics of Helical Flow Structures

45. System Parameters • After eliminating some of the slow expansion coefficients, simple interpretations of the system parameters can be found. • Parametersp andr determine the vertical background velocity field, and parameters qand s determine the horizontal velocity field. • Parameter q describes hyperbolic, directional shear, and parameter s describes straight-line convergence and divergence. Dynamics of Helical Flow Structures

47. Bifurcations • Inside the region of stability the real parts of all three eigenvalues of the respective equilibrium must be negative. • By continuity, at the boundaries at least one eigenvalue must have a vanishing real part. • These transition regions in parameter space are called bifurcation points. • There are two types of bifurcations: • Transcritical bifurcations: exchange of stability • Hopf bifurcations: creation or destruction of equilibria Dynamics of Helical Flow Structures

48. Dynamical Regimes The groundstate is stable for parameter values satisfying Since in that region arbitrary, small perturbations of the fast flow are damped out, this part of parameter space is referred to as the “laminar” region. Dynamics of Helical Flow Structures

49. Dynamical Regimes Similarly, the part of parameter spacesatisfying in which the vortex state is stable, is referred to as the “tornadic” region. Dynamics of Helical Flow Structures

50. Bifurcation Chart Dynamics of Helical Flow Structures