Case: Dispersion surface tracking (2D scan)
This page demonstrates how to construct a dispersion surface by scanning a 2D parameter space (wave number k and propagation angle θ) and using branch tracking to follow a consistent mode across the scan.
We consider a typical electron–proton plasma and scan:
kin normalized unitsk/k_n, wherek_n = ω_pi/cθfrom 5° to 90°
using PlasmaBO
using PlasmaBO: c0
B0 = 5e-9 # [Tesla]
n = 5.0e6
T = 12.94
ion = Maxwellian(:p, n, T)
electron = Maxwellian(:e, n, T)
species = (ion, electron)
wn = abs(B0 * ion.q / ion.m)
wpi = plasma_frequency(ion.q, n, ion.m)
kn = wpi / c0
wci = wn
# Kinetic solver settings (adjust upward for accuracy)
N = 33Dispersion Curve Scan
julia> using CairoMakiejulia> θ = deg2rad(45);julia> k_ranges = (0.01:1:100) .* kn;julia> results = solve(species, B0, k_ranges, θ; N)Solving dispersion (k, θ)... 14%|███▎ | ETA: 0:00:06 Solving dispersion (k, θ)... 27%|██████▎ | ETA: 0:00:06 Solving dispersion (k, θ)... 39%|█████████ | ETA: 0:00:05 Solving dispersion (k, θ)... 50%|███████████▌ | ETA: 0:00:04 Solving dispersion (k, θ)... 60%|█████████████▊ | ETA: 0:00:04 Solving dispersion (k, θ)... 69%|███████████████▉ | ETA: 0:00:03 Solving dispersion (k, θ)... 78%|██████████████████ | ETA: 0:00:02 Solving dispersion (k, θ)... 86%|███████████████████▊ | ETA: 0:00:01 Solving dispersion (k, θ)... 94%|█████████████████████▋ | ETA: 0:00:01 Solving dispersion (k, θ)... 100%|███████████████████████| Time: 0:00:10 PlasmaBO.DispersionSolution{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, Float64, Matrix{Vector{ComplexF64}}}(9.819772774325405e-8:9.819772774325405e-6:0.0009722557023859583, 0.7853981633974483, Vector{ComplexF64}[[-126621.4672679358 + 3.828484288995915e-11im, -126181.2335617592 + 1.5791842202502766e-9im, -125742.53624571368 + 9.847263783716302e-11im, -2638.5615125116046 - 0.24087074983545376im, -2638.561512511591 - 0.24087074983393109im, -2638.5615125115755 - 0.2408707498342142im, -2638.4470794029603 - 0.2651185849232014im, -2638.4470794029558 - 0.2651185849246076im, -2638.4470794029535 - 0.26511858492351326im, -2638.354345226277 - 0.2802846465683645im … 2638.354345226296 - 0.28028464656731644im, 2638.4470794029653 - 0.2651185849247123im, 2638.4470794029676 - 0.26511858492450746im, 2638.4470794029867 - 0.26511858492194995im, 2638.5615125115687 - 0.24087074983569498im, 2638.561512511591 - 0.24087074983492585im, 2638.561512511606 - 0.2408707498358758im, 125742.53624571396 - 3.6524036462368963e-11im, 126181.23356175912 - 1.5268851946137258e-9im, 126621.46726793551 - 9.722567060424265e-11im]; [-126647.93288366735 + 2.9409363833110547e-11im, -126198.78017643755 - 1.7497786027715013e-10im, -125768.49399567676 + 1.2540599114707618e-10im, -2671.7111478460683 - 24.32794573333239im, -2671.7111478460606 - 24.327945733332342im, -2671.711147593096 - 24.327946381600125im, -2660.1534038743707 - 26.77697707736556im, -2660.153403874369 - 26.776977077366034im, -2660.1533937083386 - 26.77696518381656im, -2650.7873334594565 - 28.308737348809018im … 2650.787333459467 - 28.308737348808677im, 2660.1533937083555 - 26.77696518381754im, 2660.1534038743443 - 26.77697707736614im, 2660.153403874352 - 26.776977077366183im, 2671.7111475930897 - 24.327946381601173im, 2671.7111478460565 - 24.32794573333177im, 2671.7111478460574 - 24.327945733332776im, 125768.4939956772 - 1.3610743497535535e-11im, 126198.78017644341 + 9.50608229915713e-10im, 126647.93288366724 - 1.2909750098879196e-10im]; … ; [-313940.6425219099 - 4.5767317397199683e-8im, -313861.5463967749 - 4.98813864885042e-8im, -123618.23035965722 - 2.452496356421211e-5im, -5887.225775290701 - 2360.7742191325947im, -5887.056696170275 - 2360.953233797769im, -5774.271214401578 - 2344.2165982012616im, -5223.734490921901 - 2588.8106207791034im, -5109.542451117759 - 2214.2248373400066im, -5007.815769904562 - 2360.77421913258im, -5006.87270501756 - 2361.695774973426im … 5006.872705017591 - 2361.6957749734397im, 5007.815769904577 - 2360.7742191325806im, 5109.542451117768 - 2214.224837340033im, 5223.73449092193 - 2588.81062077915im, 5774.271214401537 - 2344.2165982012425im, 5887.056696170302 - 2360.953233797787im, 5887.225775290671 - 2360.7742191325797im, 123618.23035965688 - 2.4524236249676405e-5im, 313861.54639677383 - 4.979119694326073e-8im, 313940.64252190944 - 4.5657557734557486e-8im]; [-316611.46915751847 - 3.790530697871135e-8im, -316532.827505263 - 5.763574355945643e-8im, -123505.55495153666 - 2.591520506563352e-5im, -5920.375410625136 - 2384.861294116078im, -5920.198173682382 - 2385.048571954691im, -5802.211344564239 - 2366.0793519951685im, -5259.865771929476 - 2621.3752887317282im, -5147.171836575296 - 2234.949525835599im, -5040.96540523904 - 2384.8612941160754im, -5039.996687838439 - 2385.806051148127im … 5039.996687838509 - 2385.806051148149im, 5040.965405239044 - 2384.8612941160663im, 5147.171836575312 - 2234.9495258356374im, 5259.865771929589 - 2621.375288731788im, 5802.21134456414 - 2366.079351995137im, 5920.198173682412 - 2385.0485719547005im, 5920.375410625124 - 2384.861294116074im, 123505.55495153635 - 2.591481620457372e-5im, 316532.82750526344 - 5.7552485661227374e-8im, 316611.4691575189 - 3.7814108960056436e-8im];;])
plot(results, kn, wn)
using PlasmaBO: plot_branches
# k, ω pairs for initial branch points (see `BranchPoint` for more control over tracking)
initial_point = (50 * kn, -600 * wn *im)
branch = track(results, initial_point)
f, (ax1, ax2) = plot_branches((branch,), kn, wn)
ylims!(ax2, -3500,500)
f
Dispersion Surface Scan (2D)
julia> ks = (0.1:10:100.0) .* kn;julia> θs = deg2rad.(10.0:10.0:90.0);julia> res2d = solve(species, B0, ks, θs; N)Solving dispersion (k, θ)... 16%|███▋ | ETA: 0:00:06 Solving dispersion (k, θ)... 29%|██████▋ | ETA: 0:00:05 Solving dispersion (k, θ)... 42%|█████████▊ | ETA: 0:00:05 Solving dispersion (k, θ)... 53%|████████████▎ | ETA: 0:00:04 Solving dispersion (k, θ)... 64%|██████████████▉ | ETA: 0:00:03 Solving dispersion (k, θ)... 74%|█████████████████▏ | ETA: 0:00:02 Solving dispersion (k, θ)... 86%|███████████████████▋ | ETA: 0:00:01 Solving dispersion (k, θ)... 97%|██████████████████████▎| ETA: 0:00:00 Solving dispersion (k, θ)... 100%|███████████████████████| Time: 0:00:09 PlasmaBO.DispersionSolution{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, Vector{Float64}, Matrix{Vector{ComplexF64}}}(9.819772774325405e-7:9.819772774325404e-5:0.000884761526966719, [0.17453292519943295, 0.3490658503988659, 0.5235987755982988, 0.6981317007977318, 0.8726646259971648, 1.0471975511965976, 1.2217304763960306, 1.3962634015954636, 1.5707963267948966], Vector{ComplexF64}[[-126621.80061025688 + 1.3707465971481408e-10im, -126181.24222535486 + 1.1710538913166825e-9im, -125742.87425618316 + 5.846987748231946e-11im, -2642.8468602835806 - 3.354675534484284im, -2642.8468602835674 - 3.3546755344838677im, -2642.8468602835583 - 3.3546755344827615im, -2641.2531177883707 - 3.6923820397085603im, -2641.2531177883634 - 3.692382039713136im, -2641.253117788358 - 3.692382039713067im, -2639.9615824983184 - 3.9036040996082892im … 2639.961582498284 - 3.9036040996070027im, 2641.2531177883598 - 3.692382039711575im, 2641.25311778838 - 3.6923820397086375im, 2641.253117788384 - 3.6923820397128404im, 2642.846860283552 - 3.3546755344834764im, 2642.8468602835646 - 3.3546755344821992im, 2642.8468602835655 - 3.3546755344838157im, 125742.87425618309 - 1.3775428751476485e-10im, 126181.2422253549 - 1.1702149486582913e-9im, 126621.80061025695 - 5.806187423019334e-11im] [-126621.78541550916 + 1.1393387969182989e-11im, -126181.27204070387 + 4.409050115787424e-9im, -125742.85882737253 + 1.8187386726455134e-10im, -2642.635357548592 - 3.2009941384429936im, -2642.6353575485905 - 3.2009941384420117im, -2642.6353575485887 - 3.20099413844507im, -2641.1146261595877 - 3.523229935210295im, -2641.1146261595763 - 3.5232299352105345im, -2641.1146261595245 - 3.5232299351497507im, -2639.882257529749 - 3.7247756788961897im … 2639.8822575297627 - 3.7247756788950643im, 2641.11462615951 - 3.52322993514872im, 2641.114626159568 - 3.5232299352092444im, 2641.114626159589 - 3.5232299352113614im, 2642.6353575485773 - 3.2009941384415574im, 2642.635357548587 - 3.200994138446278im, 2642.635357548589 - 3.2009941384438694im, 125742.85882737218 - 1.120260879156892e-11im, 126181.27204070383 - 4.406439868809818e-9im, 126621.7854155089 - 1.8458618917769351e-10im] … [-126621.64001744706 - 1.005780415549995e-10im, -126181.56491331327 - 1.8513439913088687e-9im, -125742.71134676352 - 1.1086671931523871e-11im, -2639.0440903432673 - 0.5915198082506914im, -2639.04409034325 - 0.5915198082521512im, -2639.044090343189 - 0.5915198084834401im, -2638.7630705411 - 0.6510665766854982im, -2638.763070541073 - 0.6510665766835503im, -2638.763070537369 - 0.6510665724883152im, -2638.535338052059 - 0.688310720012476im … 2638.535338052059 - 0.6883107200132039im, 2638.7630705373654 - 0.6510665724885557im, 2638.763070541039 - 0.6510665766846514im, 2638.7630705410747 - 0.6510665766858601im, 2639.04409034318 - 0.5915198084831068im, 2639.044090343249 - 0.59151980825141im, 2639.0440903432695 - 0.5915198082509551im, 125742.71134676364 + 1.0234394501875162e-10im, 126181.56491331324 + 1.8497782618260056e-9im, 126621.64001744661 + 7.536233439882894e-12im] [-126621.6348741384 + 4.480657944360817e-11im, -126181.57526794306 + 1.4798307720002027e-9im, -125742.70613518421 + 1.8768330248387587e-12im, -2638.2300161582734 + 4.652112028935562e-13im, -2638.230016158267 - 3.170796958329447e-13im, -2638.2300161582634 - 3.126388037344441e-13im, -2638.2300161582616 - 2.525697828562826e-17im, -2638.230016158261 + 5.542233343748866e-13im, -2638.2300161582607 + 1.9895196601282805e-13im, -2638.2300161582575 + 1.2967404927621828e-12im … 2638.2300161582552 + 1.3278267374516872e-13im, 2638.230016158256 + 7.673861546209082e-13im, 2638.2300161582575 + 1.1191048088221578e-13im, 2638.230016158259 + 1.8474111129762605e-13im, 2638.230016158263 + 6.021849685566849e-13im, 2638.2300161582652 + 2.0592555438625482e-13im, 2638.230016158279 - 3.655686864334484e-13im, 125742.70613518418 - 4.0342364892168155e-11im, 126181.57526794236 - 1.4138924061216656e-9im, 126621.63487413849 + 1.441791943403444e-13im]; 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… ; [-267526.6896833227 - 1.3867035375129098e-8im, -267333.9463406511 - 1.4566527924190962e-8im, -126172.7721827667 - 8.131178452262035e-5im, -6336.322160536527 - 2687.0951031208115im, -6336.322158828375 - 2687.0951044719004im, -6336.238399615031 - 2687.144919137476im, -5456.912155150438 - 2687.0951031208297im, -5456.9118919419425 - 2687.0952726809214im, -5452.350753536302 - 2688.524627532267im, -5059.887961221407 - 2955.986624568311im … 5059.887961221375 - 2955.986624568299im, 5452.350753536297 - 2688.524627532258im, 5456.9118919419325 - 2687.0952726809237im, 5456.912155150461 - 2687.0951031208033im, 6336.238399615068 - 2687.144919137492im, 6336.322158828386 - 2687.0951044718913im, 6336.322160536536 - 2687.0951031208147im, 126172.77218276713 - 8.131450927870767e-5im, 267333.9463406517 - 1.4479155652225018e-8im, 267526.6896833233 - 1.3991666492074728e-8im] [-267503.4261346463 - 4.307133727370854e-8im, -267319.9842710019 - 4.4340176957773765e-8im, -126162.67962935817 - 5.874620238931358e-5im, -6166.908469825472 - 2563.996304892293im, -6166.908057921622 - 2563.9966480718776im, -6164.913344368277 - 2564.44976743686im, -5287.498464439411 - 2563.9963048922887im, -5287.483269726841 - 2564.008140447459im, -5251.723729394561 - 2561.2244723228005im, -4965.899650538577 - 2797.0124657559786im … 4965.899650538592 - 2797.012465755955im, 5251.723729394592 - 2561.224472322812im, 5287.48326972686 - 2564.00814044748im, 5287.498464439375 - 2563.9963048922928im, 6164.91334436826 - 2564.449767436855im, 6166.908057921591 - 2563.9966480718717im, 6166.908469825476 - 2563.996304892278im, 126162.6796293575 - 5.874393249095913e-5im, 267319.98427100206 - 4.412231646711007e-8im, 267503.42613464635 - 4.307003109715879e-8im] … [-266971.7162995503 - 3.724220405430934e-10im, -265749.96109124133 - 3.1226818729827694e-11im, -118413.0055746402 - 7.311455005152145e-10im, -3290.3034383581835 - 473.8073664097955im, -3290.1314368297753 - 474.11199820359855im, -3257.822805925344 - 512.8095690429591im, -3222.7561454448582 - 423.95071088852626im, -3065.206576794044 - 521.5043279243731im, -3061.797835173338 - 514.4723376860115im, -2972.366400505535 - 742.6364617585281im … 2972.3664005055157 - 742.6364617585447im, 3061.7978351733486 - 514.4723376860126im, 3065.2065767940426 - 521.5043279243706im, 3222.7561454448737 - 423.9507108885265im, 3257.8228059253547 - 512.8095690429702im, 3290.1314368297603 - 474.1119982035952im, 3290.303438358158 - 473.80736640979444im, 118413.00557464018 - 5.561473669732289e-10im, 265749.96109124226 + 5.044035653968433e-11im, 266971.7162995505 + 4.832862998682685e-10im] [-267036.43381136836 + 1.3664764966144298e-10im, -265686.80541522405 - 2.538051940690696e-12im, -117722.58317410506 - 5.054184844916654e-12im, -2638.2300161582607 + 2.1316282072803006e-13im, -2638.230016158253 - 8.705924869900628e-12im, -2638.2300161582516 - 6.3842264808045e-12im, -2638.2300161582516 + 1.2789769243681803e-13im, -2638.2300161582502 + 4.547473508864641e-13im, -2638.230016158249 - 2.640387262465721e-13im, -2638.230016158249 + 4.547473508864641e-13im … 2638.2300161582543 - 1.9539925233402755e-13im, 2638.230016158255 - 3.090860900556436e-13im, 2638.230016158257 - 9.39558369249945e-14im, 2638.230016158259 - 3.315906577094907e-13im, 2638.2300161582602 - 1.2967404927621828e-12im, 2638.230016158261 - 1.0653700144303002e-12im, 2638.230016158644 + 3.4876334432663327e-9im, 117722.58317410512 + 5.383422562120607e-12im, 265686.8054152236 - 1.1723955140041653e-12im, 267036.43381136813 - 3.659322734250219e-10im]; 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plot(res2d, kn, wn)
# Choose a reference angle and reference k for seeding the tracked mode
seed = (90.5 * kn, deg2rad(20), (1000 - 3000im) * wn)
branch = track(res2d, seed)
plot_branches((branch,), kn, wn)