3-D Macroscopic Electromagnetic Equilibrium for Toroidal Confinement and
Resonance Shielding in Tokamaks
Resonance Shielding in Tokamaks
ComputationalProofofToroidalPlasma
confinement
IntegratingNavier-StokeswiththeLIFEFramework
IntroductionAchievingstablemagneticconfinementinnuclearfusionplasmas(suchasinaTokamak)remainsaprofoundtheoreticalandgeometricchallenge.TraditionalMagnetohydrodynamic(MHD)modelstreatmechanicalfluiddynamicsandelectrodynamicsascoupledbutdistinctsystems.Thisnotebookexploresaunified,fundamentallydifferentapproachbasedontheLocalizedIntrinsicFieldEquilibrium(LIFE)framework.ByintegratingthemechanicalfluiddynamicsoftheNavier-StokesequationswiththeLIFEtheory,wecanexpressallphysicalinteractions—mechanicalpressure,electromagneticfieldtensors,andspatialinertia—inidenticaldimensionsofforcedensity(N/m³).Thisexactdimensionalconsistencyeliminatesthemathematicalboundariesbetweenthematerialplasmaandtheenergeticconfinementfields.TheMathematicalChallenge:ToroidalGeometryToapplythisunifiedN/m³equationtoaTokamakreactor,wemustmaptheseforcedensitiesintoahighlycomplex,non-Euclideantoroidalcoordinatesystem.Withinthisgeometry,fieldconfinementisanisotropic.Thegoalofthiscomputationalnotebookistomathematicallyprovetheexistenceofzero-forceequilibriumstatesandtoidentifythepreciseboundaryconditionsrequiredforstabletoroidalconfinement.WhatThisNotebookDoesThefollowingMathematicacalculationsprovidetherigorousgeometricandtensorproofsforthisunifiedconfinementmodel.Inthisnotebook,wewill:Definethecomplexmetrictensorandspatialboundariesofthetoroidalcoordinatesystem.ApplytheunifiedLIFEforce-densityequations,seamlesslyintegratingNavier-Stokesmechanicalpressurewithelectromagneticfielddynamics.Computetheanisotropicpressurevectorstoverifythezero-forcespatialequilibrium.Visualizetheexacttopologicalboundariesandvelocitynodesrequiredforstableconfinement.IinvitetheWolframcommunitytorunthecode,manipulatethetoroidalboundaryparameters,andexplorethemathematicaleleganceofintrinsicfieldequilibrium!Author:WimVegtPhysicist&Researcher(Ret.),DepartmentofPhysics,EindhovenUniversityofTechnologyRelatedpreprintsregardingtheLIFEframeworkareavailableviaZenodo.In[]:=
email:wimvegt@quantumlight.sciencewebsite:
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The toroidal coordinate system Toroidal[r, φ , θ , R], parameterized by R (Radius of the Torus), is obtained by rotating bipolar coordinates about an axis perpendicular to the axis connecting the two foci. The coordinate θ specifies the rotation. The default value for parametera is1.
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R=.
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r=.
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θ=.
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φ=.
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t=.
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<<"VectorAnalysis`"
—
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InverseFunctions->True
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InverseFunctionsTrue
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<<Calculus`DSolve`
—
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$Failed
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Needs["DifferentialEquations`NDSolveProblems`"]
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Needs["DifferentialEquations`NDSolveUtilities`"]
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Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]
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<<Calculus`DSolveIntegrals`
—
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$Failed
Vektor Potentiaal Toroidal
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InverseFunctions->True
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InverseFunctionsTrue
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SetCoordinates[Toroidal[θ,r,φ,R]]
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Toroidal[θ,r,φ,R]
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{Coordinates[Toroidal],
CoordinateRanges[Toroidal]}
CoordinateRanges[Toroidal]}
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{{θ,r,φ},{-π<θ≤π,0≤r<∞,-π<φ≤π}}
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CoordinatesToCartesian[Coordinates[Toroidal],Toroidal]
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,,
RCos[φ]Sinh[r]
-Cos[θ]+Cosh[r]
RSin[φ]Sinh[r]
-Cos[θ]+Cosh[r]
RSin[θ]
-Cos[θ]+Cosh[r]
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R=.
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{Coordinates[Toroidal],
CoordinateRanges[Toroidal]}
CoordinateRanges[Toroidal]}
Uitkomsten