JP1.17 TORNADOES, THOMSON, AND TURBULENCE . -

JP1.17TORNADOES, THOMSON, AND TURBULENCE:AN ANALOGOUS PERSPECTIVE ON TORNADOGENESIS ANDATMOSPHERIC COHERENT STRUCTUREMarcus L. Büker *Western Illinois University, Macomb, ILGregory J. TripoliUniversity of Wisconsin-Madison1. INTRODUCTIONTheoretical development regarding atmosphericturbulence has been one of the most challenging scientificproblems over the last century. Given the breadth ofapplication (e.g. wind energy, tornadogenesis, and climatechange), it is clear why there is such an interest in thesubject. However, while incremental advances have beenmade in recent years (with the help of exponentiallyimproving computational resources) there has not been aclear paradigm shift in some time. Even so, there is aslowly growing body of literature utilizing the potentiallysynergistic relationship between the fields of electromagnetism (EM) and hydrodynamics (HD). This synergyis based on the similarity between the equations thatgovern fluid dynamics and the equations that governelectro-magnetism. Thomson (1931) was one of the firstto employ this type of “analogous thinking” whendeveloping a visual framework for electromagnetism.A table of analogous variables has been compiled fromseveral sources (Belevich, 2008; Marmanis, 1998,Pinhiero, 2009) in Fig. 1. Exploring and exploiting thisanalogous relationship may be a catalyst for a radical shiftin how coherent structure in fluids is studied.2. VORTEX DYNAMICS: METHOD ANDDIAGNOSTICSOne of the main struggles in the numericalmodeling of tornadoes is the tendency for over-productionof turbulent diffusion in the vicinity of the vortex. Severalmethods have been developed to combat this numericalloss of kinetic energy, including “vortex confinement”, (e.g.Steinhoff and Underhill, 1994) and other antidiffusionmethods. However, it seems desirable to find physicalreasoning for why a tornado does not break down intoturbulence, given the extreme deformation fields near thevortex. Observations show that there can be coherentorganization, subsequent breakdown, and reorganizationof the parent vortex and surrounding vortex filaments (e.g.Rotunno, 1984).In order for vortex filaments to merge into theparent vortex, they must (1) be aligned with the parentvortex and (2) be advected toward (or into) the core of theparent vortex. Using the Lamb vector and some of the* Corresponding author address: Marcus L. Büker,Western Illinois University, Geography Department, Macomb, IL,61455; e-mail: ml-buker@wiu.edu.relationships well-known in electromagnetism, weexamined the behavior and self-organization potential ofvortex filaments surrounding a tornado in a high-resolutionsimulation.We use the University of WisconsinNonhydrostatic Modeling System (UWNMS: see Tripoli,1992), with up to 5 nested grids, obtaining horizontal andvertical resolution below 25 meters. An idealized verticalsounding, based on a tornadic supercell environment, isused to initialize the meteorological fields.One can imagine randomly oriented small-scalevortices (or angular moments) embedded within the largerscale rotation of the tornado. By decomposing the Lambvector into mean and perturbation components, we findonly one combination ( using the mean vorticity andperturbation velocity) will geometrically yield a torqueacross a segment of a perturbation vortex filamentembedded within a large-scale vorticity fieldutwhereuω ula(1a)ω is the meanl a is this component of the Lamb vector.is the perturbation velocity,vorticity, andThe magnitude of the curl of this term,la, is used todiagnose “gyroscopic alignment torque”, which is the HDanalog to the torque on a magnetic dipole momentimmersed in a large-scale magnetic induction:τ μ B(1b)where B is the large-scale magnetic induction, and μ isthe magnetic moment of the small dipole. Theseparameters are illustrated in Fig. 2.Regarding the alignment of vortex loops, StokesTheorem states that vortex lines cannot just „end‟ in themiddle of a fluid; they must either „loop back onthemselves, or terminate at a surface. That does notpreclude alignment of vortex segments, however. One canenvision a poloidal configuration of vortex lines aligningthemselves within the tornado.We investigate another EM-analogous parameterwhich is related to a mechanism in terms of hurricaneprediction (known as the “beta effect”). We term this new

parameter (which is a vector) the “vortex beta force”, β .Hurricanes (small vortices) are known to be drawnupgradient toward higher same-signed vorticity (as theCoriolis parameter increases with higher latitude) throughthe conventional “beta-effect”. This is absolutelyconsistent with this new β parameter, and it is directlyanalogous to the same mechanism responsible for“magnetic shielding” of charged particles: the force on theparticle in proportional to the gradient of the configurationof vorticity aligned with the particle‟s magnetic moment:„actual‟ radius ( 500m) we have confidence that the dataare meaningful.We put this type of interaction in the context ofvortex loops generated by a pulsing rear-flankingdowndraft near a mesocyclone. The gyroscopic torquepreferentially rotates the vortex loop into the „correct‟configuration, where this vorticity reconnects with themesocyclone and brings the funnel to the ground (Fig. 3,right side).(μ B)Once the surrounding vortices are aligned withthe parent vortex, another mechanism is needed to drawthe small-scale vortex upgradient. This “vortex beta”parameter is shown in Figure 4, where an isosurface of 3D vorticity magnitude (the main feature is the tornado) iscolored by the intensity of this vortex-beta parameter.Yellow values indicate high levels of the vortex-beta force.F(2a)Substituting in the analogous variables, oneobtains a formula for the force on a small local moment,which is generally aligned with the large-scale rotation:F(μ ω) β(2b)We investigate these two analogous formulationsof vortex interaction and discuss them in the followingsection.3. “EM-LIKE” TORQUING AND MERGING: RESULTSAND DISCUSSION3.1 Gyroscopic torqueTo simplify the gyroscopic torque formulation, welooked at the mathematical formulation for the electromagnetic torque, and applied this analogously to thevortex interaction problem. To calculate this newparameter, information was needed about the LOCALLYdefined angular momentum. Since the pressure gradientforce is the only force (besides friction) that acceleratesthe flow in a Lagrangian sense, the normal component ofpressure gradient acceleration to velocity was computed,and from this a local radius of curvature was obtained.Obviously, there is a strong correlation of local angularmomentum and the local curvature radius:μ r uWhereμis the angular moment,(3)r is the radius,isdensity, and u is velocity.Again, substituting in the analogous variables,one obtains a formula for this torque on the local moment:τμ ω(4)While there were reservations and questionsregarding the use of a variable point of reference for thelocal angular momentum calculation, there wasoverwhelming agreement with the Lamb vector formulation(Fig. 3). Given this information, plus given the calculatedcurvature radius near the tornado strongly agreed with the3.2 Merging parameterThis vortex-beta force competes against the socalled Magnus effect, which acts through the Lamb-vectorterm using the mean velocity and the perturbation vorticity:utωulb(5)This force will act to eject like-signed embeddedvortices away from the center of large-scale rotation.Thus, there will be a competition between the beta force(strongly dependent upon the gradient of like-signed largescale vorticity) and the Magnus effect in the final mergerprocess. These two forces can define a fundamentalbalance relationship, and formulated into a “net mergingforce” :α β lb(μ ω)ω u(6)This upgradient force on like-signed vorticalstructure is referenced in several places in the literature,even including motion in the Great Red Spot on Jupiter(Marcus, 2000), where it is argued that it is energeticallyfavorable for „prograde‟ (like-signed) vorticity to be drawnupgradient, while expelling „adverse‟ vortical structures.4. CONCLUSION AND FUTURE WORKWe have illustrated the concept of utilizing theEM-HD analogy in terms of vortex interaction duringtornadogenesis, highlighting two self-organizationalmechanisms of reorientation and upgradient advection oflocal angular moments embedded within a large-scalerotation. Self-alignment and merging of small-scalemoments in the presence of a large-scale field isubiquitous in both EM and HD. It is likely that the reasonis linked to the tendency of a physical system to seek thelowest energy state. We plan to investigate other vortexinteraction problems (e.g. hurricanes and boundary-layerturbulence) using this “analogous thinking” in upcomingwork.

5. REFERENCESBelevich, M., 2008: Non-relativistic abstract continuummechanics and its possible physical interpretations. J.Phys. A: Math. Theor. 41 (045401) doi: 10.1088/17518113/41/4/045401Rotunno, R., 1984: An investigation of a three-dimensionalasymmetric vortex. J. Atm. Sci., 41, 283-298.Marcus, P.S. and Kundu, T., 2000: Vortex dynamics andzonal flows. Phys. Plasmas, 7 (5) 1630-1640.Steinhoff and Underhill, 1994: Modification of the Eulerequations for vorticity confinement„‟: Application to thecomputation of interacting vortex rings, Phys. Fluids 6,2738; doi:10.1063/1.868164 [13]Marmanis, H., 1998: Analogy between the Navier-Stokesequations and Maxwell„s equations: Application toturbulence. Phys. Fluids, 10 (6), 1428-1437.Thomson, J. J., 1931: On the analogy between theelectromagnetic field and a fluid containing a large numberof vortex filaments. Phil. Mag. S. 7.12 (80) 1057-1063Martins, A. and Pinheiro, M. J., 2009: Fluidicelectrodynamics: Approach to electromagnetic propulsion.Phys. Fluids 21, 097103; doi:10.1063/1.3236802Tripoli, G. J.,1992: A nonhydrostatic numerical modeldesigned to simulate scale interaction, Mon. WeatherRev., 120, 1342–1359Figure 1. A table showing the analogous mathematical structure and variables between the equations of fluid dynamicsand electromagnetism. Variables are color-coded to highlight the analogies.

Figure 2. The left side of the diagram shows the geometric configuration of a segment of perturbation vorticity ( ω )embedded within a larger scale vortical field ( ω ). The curl of the second term in the Lamb vector decomposition (ω ula )describes the same type of “counter-torque” yielding alignment of a flywheel in a gimbal gyroscope.This quantity lies in the direction of the cross product of the local moment and the mean rotation, mathematically identicalin form to the torque on a current loop or magnetic dipole placed in an external magnetic field (right side of diagram).Figure 3. There is remarkable comparison of the EM-formulated gryoscopic torque (left) and the Lamb vector formulation(right). The bright yellow patches indicate strong torque. The area in the southeast part of the domain is the region ofstrong vorticity associated with the advancing gust front associated with the rear flanking downdraft. Also note thereduction of numerical noise in the EM-formulation. In the far right portion of the figure, a schematic of the tornadic vortexevolution is given. For example, a new downdraft pulse is also forming north of the mesocylone.

Figure 4. Isosurface of vorticity magnitude, colored by the magnitude of the “3-D vortex beta vector”. Orange arrowindicates direction of vortex beta force.

govern fluid dynamics and the equations that govern electro-magnetism. Thomson (1931) was one of the first to employ this type of “analogous thinking” when developing a visual framework for electromagnetism. A table of analogous variables has been compiled from several sources (Bele