Functional validation of the HDOV framework at the heliopause crossings of Voyager 1 and Voyager 2

Valida ion o he HDOV Model a he Heliopause
C ossing by Voyage 1 and Voyage 2
A noldo Wal e Fe nández
[email p o ec ed]
PREPRINT — No embe 22, 2025
Abs ac
This wo k p esen s a alida ion o he Vib a ional Wa e Dispe sion (HDOV)
model applied o he heliopause c ossing by he Voyage 1 and Voyage 2 p obes.
Based on eal da a om he MAG and PWS ins umen s, a dynamic p o ile o he
κlocal
(
) unc ion is de ined, in e p e ed as an e ec i e medium impedance o e ec i e
mesoscale cohe ence damping a e, whose e olu ion allows iden i ying anomalous
e en s ha exhibi s a is ical an icipa ion wi h espec o changes obse ed by
con en ional magne ohyd odynamic models.
The ab up ansi ion in
κlocal
p ecedes discon inui ies eco ded in plasma in en-
si y and magne ic ield magni ude, indica ing an ope a ional p edic i e capabili y o
he heliopause ansi ion in he local en i onmen o he p obes. The compa ison
be ween HDOV simula ions and MHD models shows a s uc u al co ela ion be ween
he magne ic ield, plasma p essu e, and he
(
)cohe ence unc ion o he mesoscale
medium.
C oss- alida ion wi h Voyage 2 con i ms he obus ness and ans e abili y o
he indica o wi hin obse a ional unce ain ies and he limi ed numbe o a ailable
e en s. In his wo k,
κlocal
is p esen ed as a phenomenological index o egime change
in u bulen plasmas; a b oade in e p e a ion wi hin he HDOV amewo k, which
connec s hese ansi ions wi h no ions o g a i a ional and quan um cohe ence, is
discussed in complemen a y wo ks and exceeds he s ic ly obse a ional scope o
his p ep in .
1
Con en s
1 In oduc ion and Objec i es 5
1.1 Mo i a ion ................................... 5
1.2 Objec i es ................................... 5
1.3 Con ex and Concep ual F amewo k ..................... 6
1.4 S uc u e o he Wo k ............................ 6
2 Da a Acquisi ion and P ep ocessing 7
2.1 Da a Sou ces and Fo ma s .......................... 7
2.2 Ope a ional De ini ion o κlocal( )...................... 7
2.3 Uni ica ion and Expo ............................ 9
3 De ini ion o κlocal( )9
3.1 Linea Fo mula ion o κlocal ......................... 9
3.2 Dimensional Valida ion ............................ 10
3.3 Calib a ion and Pa ame e Unce ain y ................... 10
3.3.1 Physical Ancho ing o Coe icien s β3and β4............ 11
3.4 Robus ness Analysis wi h Con inuous Wa ele T ans o m ......... 12
3.5 Physical P o iles and Slow Damping Me ic on a 1-hou G id ....... 13
3.6 Quan i a i e Compa ison wi h MHD Models and O he App oaches . . . 15
3.6.1
Di ec Compa ison wi h MHD P oxies on a Common Tempo al G id
16
3.6.2 Mul i-scale Robus ness Analysis ................... 17
3.6.3 Value-Added Analysis and Ope a ional Ad an ages ........ 17
3.6.4 Implica ions o Ope a ional Moni o ing .............. 18
3.7 Ex ension o Voyage 2: Uni e sal Cons uc ion o κlocal .......... 18
4 Physical In e p e a ion o he Mechanism Behind κlocal( )19
4.1 Physical In e p e a ion o κlocal ....................... 19
4.1.1 Ope a ional De ini ion o Phase Cohe ence ............. 19
4.1.2 Connec ion wi h Es ablished Physical Models ........... 20
4.1.3 Physical De i a ion o Coe icien s βi................ 20
4.1.4 Physical Basis o he Choice o Scale L.............. 21
4.1.5 Validi y o he G adien P oxy and Taylo Regime ......... 21
4.1.6 Link o Di ec Physical Obse ables ................. 22
4.1.7 Physical Mechanisms o κlocal P edic i i y ............. 22
4.1.8 Componen s: The mal and Magne ic ................ 23
4.1.9 Cohe ence Analogy wi h Phase Diag ams .............. 23
4.1.10 Eme gence as Cohe ence B eakdown ................ 24
4.2 Limi a ions and Conside a ions ....................... 24
4.2.1 Theo e ical Limi a ions ........................ 24
4.2.2 Obse a ional Limi a ions ...................... 25
4.2.3 S a is ical Limi a ions ........................ 25
4.2.4 Gene aliza ion Limi a ions ...................... 26
4.2.5 Con ex and Pe spec i e ....................... 26
5 Syn hesis o Main Resul s 26
2
6 Valida ion o κlocal( )wi h Real Da a 27
6.1 Co ela ion wi h Real Ab up E en s .................... 27
6.2 In e p e a ion o Modes Co ela ions in Complex Sys ems ........ 28
6.2.1 Compa ison wi h Th esholds in Plasma Physics .......... 29
6.3 Ex ended Valida ion wi h Pa icle P oxies ................. 29
6.4 Limi a ions and Conside a ions ....................... 30
6.5 Limi a ions o he P esen S udy ...................... 31
7 Cohe ence Mechanism o κlocal om Fi s P inciples 31
7.1 Func ional In e p e a ion in he Gene alized HDOV F amewo k ..... 31
7.2 Connec ion wi h Gene alized Theo y .................... 32
7.2.1 Mas e Equa ion De i ed om E ec i e Ac ion .......... 32
7.3 P ope ies o κlocal .............................. 33
7.3.1 Uni s and No maliza ion ....................... 33
7.4 Quali y Con ol and Ve i ica ion ....................... 33
8 Implica ions o Plasma Physics 33
8.1 Plasma Tu bulence Theo y .......................... 33
8.2 As ophysical Plasma Diagnos ics ...................... 33
8.3 Connec ion wi h Fundamen als o Plasma Physics ............. 34
8.4 Pe spec i es o Fu u e Resea ch ....................... 34
8.5 Gene aliza ion o O he Plasma En i onmen s ............... 35
8.5.1 Te es ial Magne opause ....................... 35
8.5.2 Tokamaks (T anspo Region) .................... 35
8.5.3 Sola Co ona ............................. 35
9 Ope a ional Fo ecas ( 2) 36
9.1 ROC/PR Cu es and Pe o mance Me ics ................. 36
9.2 Fea u e Abla ion S udy ............................ 36
10 Conclusions and Pe spec i es 39
A De i a ion om Fi s P inciples o κlocal 39
A.1 Wa e Equa ion in Inhomogeneous Plasmas ................. 39
A.2 WKB Ansa z and Scale Sepa a ion ..................... 40
A.3 Mesoscale A e aging ............................. 40
A.3.1 Tempo al Te m ............................ 40
A.3.2 Spa ial Te m ............................. 40
A.3.3 Damping Te m ............................ 40
A.4 De ini ion o κlocal ............................... 41
A.5 Connec ion wi h µ(z)and n(p)....................... 41
A.6 Dimensional Valida ion (De ails) ...................... 42
A.7 Appendix Conclusion ............................. 42
B Rep oducibili y and Robus ness Valida ion 42
B.1 Rep oducibili y ................................ 42
B.1.1 Da a and Sou ces ........................... 42
B.1.2 P ocessing P ocedu e ......................... 43
B.1.3 Code and Ve i ica ion ......................... 43
3
B.2 Mul i-scale Robus ness Analysis ....................... 43
B.2.1 Mul i-LSweep ............................ 43
B.2.2 Sensi i i y o Pa ame e s ....................... 44
B.3 C oss- alida ion wi h Voyage 2 ....................... 44
B.3.1 Me hodology and P ocessing ..................... 44
B.3.2 Main Resul s ............................. 44
B.3.3 Analysis o Th ea s o Validi y ................... 44
B.4 Ins umen al Unce ain y Analysis ..................... 45
B.5 Appendix Conclusions ............................ 45
C O de s o Magni ude and Physical Bounds 45
C.1 Re e ence Pa ame e s in he Heliopause En i onmen ........... 45
C.2 Rele an Tempo al and Spa ial Scales .................... 46
C.3 Expec ed Bounds o κlocal .......................... 46
C.4 Consis ency wi h he HDOV In e p e a ion ................. 46
4
1 In oduc ion and Objec i es
The alida ion o he HDOV (Hypo hesis o Vib a ional Wa e Dispe sion) model in eal
con ex s cons i u es an essen ial s ep o di e en ia e i om o he heo e ical app oaches,
such as magne ohyd odynamic (MHD) models o s anda d geome ic in e p e a ions o
he in e s ella medium (Fe nandez,2025b). In his wo k, we p opose o apply HDOV
o he analysis o he heliopause c ossing by he Voyage 1 p obe (pe iod 2012-2013). In
addi ion o eplica ing he inding in Voyage 1, we applied HDOV o Voyage 2 and
ound a CUSUM an icipa ion o he c ossing om No embe 5, 2018, o Sep embe 25,
2018, 18:00 UTC (+40.2 d), wi h he same sign and o de o magni ude as in V1 unde
iden ical h eshold and me ic con en ions.
1.1 Mo i a ion
The HDOV model posi s ha ce ain egime ansi ions in he s a es o he medium can be
an icipa ed h ough a local pa ame e
κlocal
de i ed om magne ic and he mal p ope ies
o he en i onmen (Fe nandez,2025b). In his sense, he heliopause, unde s ood as he
bounda y be ween he sola wind and he in e s ella medium, ep esen s a pa adigma ic
case whe e condi ions change ab up ly and a cohe en cohe ence scales.
Expe imen al e idence om Gu ne e al. (2013a) and Bu laga e al. (2013a) sugges s
discon inui ies ha do no exac ly ma ch MHD p edic ions. This opens he possibili y o
ein e p e ing he da a om a cohe ence pe spec i e based on HDOV, in which he loss
o cohe ence is an icipa ed by he dynamics o κlocal( ).
1.2 Objec i es
The main objec i es o his s udy a e:
1.
To de ine and calcula e he unc ion
κlocal
(
) om eal da a o Voyage 1 (MAG, PWS,
and CRS), building a slow cohe ence me ic o he local heliosphe ic en i onmen .
2.
To compa e he e olu ion o
κlocal
(
)wi h ab up e en s de ec ed in MAG, PWS,
and CRS, cha ac e izing i s ela ionship wi h discon inui ies in plasma cohe ence
and he heliopause s uc u e.
3.
To e alua e he p edic i e capabili y o he HDOV model ega ding discon inui ies
in en i onmen al cohe ence, quan i ying empo al leads (∆
lead
) a he heliopause
c ossing and compa ing wi h classical MHD p oxies.
4.
To ex end he cons uc ion and calib a ion o
κlocal
(
) o Voyage 2 and e i y he
ep oducibili y o he esul s in a second heliopause ajec o y, including V1–V2
c oss- alida ion o leads and i me ics.
5.
To con as he p edic ions o he HDOV app oach wi h MHD models and al e na i e
empi ical pa ame e iza ions, iden i ying s eng hs and limi a ions o he unc ional
amewo k p oposed o he desc ip ion o he heliopause.
5

1.3 Con ex and Concep ual F amewo k
In p e ious wo ks (Fe nandez,2025b), i has been p oposed ha wa e dispe sion in plasma
allows de ining a space ime cohe ence s uc u e based on he cohe ence o he local plasma
en i onmen . This idea is implemen ed in HDOV h ough he unc ion
κ
(
z
)o
κ
(
), which
ep esen s an “e ec i e wa e densi y” o “medium impedance” o he collapse o s a es.
In as ophysical con ex s, his unc ion can acqui e a decisi e ole in dis inguishing
cohe en phases om non-cohe en ones. Pa icula ly, when a local maximum o discon-
inui y in
κlocal
is obse ed, he HDOV model p edic s he eme gence o a new spa ial
phase, e en i mac oscopic pa ame e s such as p essu e o densi y do no show isible
discon inui ies (Fe nandez,2025b).
1.4 S uc u e o he Wo k
This a icle is o ganized as ollows. Sec ion 2 desc ibes he acquisi ion and p ep ocessing
o magne ic ield (MAG) and elec on densi y (PWS) da a om Voyage 1, along wi h he
adop ed empo al g id, he no maliza ions used, and he c i e ia o cleaning and quali y
con ol.
Sec ion 3 in oduces he ope a ional de ini ion o
κlocal
(
), discusses i s in e p e a ion
as a slow damping me ic, and p esen s he linea model in e ms o local obse ables. I
de ails he policy o adjus ing he coe icien s
βi
, alida es he dimensional closu e, and
shows he physical and
κlocal
p o iles on a 1-hou g id, including quan i a i e compa ison
wi h MHD p oxies and he ex ension o he analysis o Voyage 2.
Sec ion 4 de elops he physical in e p e a ion o he mechanism behind
κlocal
(
),
analyzing he combined ole o weak collisions, ield g adien s, and u bulence in phase
decohe ence, as well as he main limi a ions o he model in his pa ame e domain and
i s con ex wi hin plasma heo y.
Sec ion 5 p esen s a s uc u ed syn hesis o he main esul s and hei concep ual
ele ance as an indica o o unc ional accessibili y o he local heliosphe ic en i onmen .
Sec ion 6 quan i ies he co ela ion be ween
κlocal
(
)and ab up e en s obse ed by
MAG, PWS, and pa icle p oxies, discussing he physical meaning o modes bu obus
co ela ions in phase lag and hei compa ison wi h usual h esholds in plasma physics.
Sec ion 7 de i es a o mula ion o
κlocal
(
) om i s p inciples wi hin he gene alized
HDOV amewo k and analyzes he unc ional in e p e a ion o each e m o he model,
emphasizing he cohe ence mechanism and i s connec ion wi h gene al heo y.
Sec ion 8 discusses he implica ions o he app oach o as ophysical and labo a o y
plasmas, including i s mul iscale na u e and possible ex ensions o o he en i onmen s
( e es ial magne opause, okamaks, sola co ona).
Sec ion 9 explo es he po en ial o
κlocal
(
)as an ope a ional o ecas indica o , p e-
sen ing ROC/PR cu es and abla ion expe imen s on he model componen s in Voyage 2.
Finally, Sec ion 10 p esen s he gene al conclusions, he main limi a ions o he s udy,
and pe spec i es o u u e wo k. Appendix A de elops he de i a ion o
κlocal
om
i s p inciples, Appendix B documen s he ep oducibili y and obus ness aspec s o he
analysis pipeline, and Appendix C collec s he dimensional and physical annex, wi h he
uni s, no maliza ions, o de s o magni ude, and bounds o he model coe icien s ha
gua an ee he physical consis ency o he o mula ion.
6
No e. In his wo k we ha e ied o be as ca e ul as possible wi h da a handling, s a is ics
and documen a ion, bu he HDOV amewo k applied o he heliopause is s ill
an explo a o y p oposal and, a he ime o his e sion, has no ye unde gone
pee e iew in specialised jou nals.
We elease he da a, sc ip s and igu es so ha anyone can:
•check each s ep,
•c i icise he me hodology,
• ep oduce and, i needed, e u e he esul s.
2 Da a Acquisi ion and P ep ocessing
To alida e he HDOV model in he immedia e in e s ella en i onmen , magne ic ield
(MAG) and plasma wa e (PWS) da a p o ided by he Voyage 1 p obe we e used,
co esponding o he pe iod be ween July 2012 and Feb ua y 2013. This in e al includes
he heliopause c ossing, iden i ied in he li e a u e be ween Augus and Sep embe 2012
(Gu ne e al.,2013a;Bu laga e al.,2013a).
2.1 Da a Sou ces and Fo ma s
The da a we e downloaded om he NASA CDAWeb po al
1
, using he
g1_p ep ocess_-
om_cdaweb.py
sc ip , de eloped speci ically o his wo k. The ollowing p oduc s we e
p ocessed:
•
MAG: Magne ic ield da a wi h 48-second esolu ion, con e ed o hou ly ime
se ies by esampling.
•
PWS-LR: Plasma densi ies de i ed om he spec al peak equency, wi h 1-second
esolu ion, also esampled o 1 hou .
2.2 Ope a ional De ini ion o κlocal( )
The cen al me ic o his wo k,
κlocal
(
), is ope a ionally de ined as he e ec i e damping
a e o loss o phase cohe ence in he plasma. I s cons uc ion is based on he concep o
phase deco ela ion ime
τc
and he cohe ence leng h
Lc
o Al én wa e modes, ollowing
he o mula ion o he HDOV model (Fe nandez,2025a).
The undamen al de ini ion is:
κlocal ≡1
τc
= A
Lc
,(1)
whe e he deco ela ion ime
τc
is de ined as he in e al in which he phase au oco ela ion
Cϕ(τ)decays o 1/e o i s ini ial alue:
Cϕ(τc) = 1
e.(2)
1h ps://cdaweb.gs c.nasa.go /
7
Fo mally, o a na owband il e ed magne ic ield
B
(
) ha con ains a dominan
Al én mode, he ins an aneous phase
ϕ
(
)can be de ined by he Hilbe ans o m
H
(
·
):
ϕ( ) = a g [H(B ( ))] .(3)
This cons uc ion es ablishes a concep ual b idge be ween phase cohe ence and deco -
ela ion ime
τc
. Howe e , in he b oadband u bulen egime o he heliopause, we do
no use his Hilbe phase as a di ec obse able on eal Voyage da a, as i s physical
in e p e a ion deg ades in he p esence o widely dis ibu ed equency spec a.
In his wo k,
τc
and
κlocal
a e ope a ionally es ima ed om a e ages o
|B|
,
ne
,
|∇B|/B
and Γ
o al
o e mesoscale windows
L
, acco ding o Equa ion
(5)
, which ac as obus
p oxies o phase cohe ence loss in he plasma.
The Al én eloci y Ais calcula ed om he local plasma pa ame e s:
A=B
√µ0ρ,(4)
whe e
B
is he magne ic ield magni ude,
µ0
is he acuum pe meabili y, and
ρ
is he
plasma mass densi y.
Ope a ionally,
κlocal
(
)is cons uc ed as a linea combina ion o he empo al a e ages
o he main physical a iables ha modula e plasma cohe ence:
κlocal =β1|B|τ+β2neτ+β3|∇B|
Bτ
+β4Γ o alτ.(5)
Each e m in Equa ion (5) has a clea physical meaning:
•β1⟨|B|⟩τ
: magne ic ield con ibu ion. A mo e in ense ield ends o s uc u e he
plasma, bu can also inc ease aniso opy and educe cohe ence a ce ain scales.
•β2⟨ne⟩τ
: cap u es he mal and densi y e ec s. Highe elec on densi ies inc ease he
plasma equency and modula e he medium’s abili y o sus ain cohe en modes.
•β3|∇B|/Bτ
: quan i ies he in luence o magne ic g adien s and u bulence. S eep
g adien s indica e inhomogenei ies ha dispe se wa es and educe cohe ence.
•β4⟨
Γ
o al⟩τ
: inco po a es o he damping sou ces, such as collisional and kine ic
e ec s.
The coe icien s
βi
we e empi ically calib a ed by minimizing he oo mean squa e
e o (RMSE) be ween
κlocal
(
)and a e e ence p o ile, using boo s ap sampling wi h
N= 104 esamples o es ima e 95% con idence in e als (Fe nandez,2025a). The alues
ob ained a e:
β1= (1.02 ±0.05) ×10−6s−1nT−1,
β2= (19.8±0.9) ×10−6s−1cm3,
L= 22.4±1.1days,
whe e
L
is he empo al a e aging scale. In physical uni s, his implies ha he e ec i e
coe icien s a e in he o de o magni ude
β1∼
10
−6s−1nT−1
and
β2∼
2
×
10
−5s−1cm3
,
consis en wi h he o de -o -magni ude es ima es discussed in he physical in e p e a ion
sec ions and appendices. The na owness o he con idence in e als sugges s ha hese
8
pa ame e s a e obus p ope ies o he local en i onmen . A igo ous de i a ion om
i s p inciples ha jus i ies he o m o Equa ion
(5)
and p o ides physical ancho s o
he coe icien s βiis p esen ed in Appendix A.
The
κlocal
(
)me ic hus de ined ac s as a slow en elope ha cap u es he e olu ion o
plasma phase cohe ence. I s uni s a e a e o equency (
s−1
), and in igu es i is epo ed
in milliHe z (mHz) o easie eading. A high alue o
κlocal
indica es a apid loss o
cohe ence, while a low alue sugges s a mo e o de ed and cohe en medium.
2.3 Uni ica ion and Expo
The p ep ocessed da a we e in eg a ed in o a single uni ied able wi h he ollowing
columns:
ime UTC Da e and Time
Bmag Magne ic ield magni ude (|B|, nT)
ne Elec on densi y (ne, cm−3) de i ed om PWS
Mass densi y closu e. We use
ρ
=
µ mpne
wi h
µ≃
1
.
2(H wi h small ac ion o He).
This closu e ensu es ep oducibili y in he calcula ion o
A
and main ains cohe ence wi h
he adop ed con en ion o
ne
. The esul was sa ed in CSV o ma and used as inpu o
he calcula ion o κlocal( )in he nex sec ion.
3 De ini ion o κlocal( )
The Vib a ional Wa e Dispe sion (HDOV) Hypo hesis p oposes ha e e y scala ield
ψ
in a dynamic space ime backg ound is go e ned by a damping unc ion ha modula es
i s e olu ion. This unc ion is exp essed locally as an e ec i e damping a e
κlocal
(
),
dependen on he physical en i onmen and no as an in a ian pa ame e (Fe nandez,
2025b).
In his con ex , he heliopause ansi ion zone is ideal o e alua ing he a ia ion
in he cohe ence o
κlocal
, as i is cha ac e ized by an ab up change in he medium’s
opology, plasma densi y, and magne ic ield in ensi y.
3.1 Linea Fo mula ion o κlocal
In his e sion o he model, we ede ine
κlocal
(
)as a linea combina ion o he ele an
physical a iables
Xi
(
), wi h coe icien s
βi
ep esen ing he sensi i i y o he damping
a e o each ac o . This linea o mula ion is consis en wi h obse ed phenomenology
and simpli ies physical in e p e a ion:
κlocal( ) =
N
X
i=1
βiXi( ),(6)
donde
Xi
(
)a e he inpu a iables (e.g., magne ic ield in ensi y, plasma densi y) and
βia e he i ing coe icien s.
In his wo k, we conside
N
= 4 main channels, de ined as a e ages o e a empo al
window o leng h L:
X1( ) = |B|L( ), X2( ) = neL( ), X3( ) = |∇B|/BL( ), X4( ) = Γ o alL( ),
(7)
9
•
Lead Time: An icipa ion ime ela i e o he ac ual e en . Posi i e alues indica e
p edic i i y, nega i e alues indica e delay. HDOV o e s he highes an icipa ion
(+62 h).
•
Complexi y: Quali a i e e alua ion o compu a ional and heo e ical equi emen s.
•Da a Requi emen s: Va iables necessa y o ope a ional implemen a ion.
3.6.1 Di ec Compa ison wi h MHD P oxies on a Common Tempo al G id
To quan i y he p edic i e pe o mance o
κlocal
agains con en ional magne ohyd odynamic
indica o s, we pe o med a sys ema ic compa ison unde s ic ly equi able condi ions.
Bo h ime se ies—
κlocal
and he MHD p oxies—we e aligned on a common hou ly g id,
o cing a common empo al suppo ha elimina es disc epancies due o i egula sampling.
Table 4: Compa ison o
κlocal
(Voyage 1) agains MHD p oxies on a 1-hou g id (common
suppo ). Lag in hou s (posi i e = delayed p oxy).
P oxy RMSE (s−1)a(scale) ∆AIC Lag (h) / Co
σ24h(|B|) 2.462 ×10−13 1.567 ×10−11 — 62 / 0.17
δB/B 2.756 ×10−13 4.682 ×10−12 −582.23 −44 / 0.09
In e p e a ion o Table 4:
•
RMSE (Roo Mean Squa e E o ): Roo mean squa e e o be ween he se ies.
Lowe alues indica e be e i .
•a(scale): Slope in he linea eg ession, indica es he sensi i i y o he p oxy.
•
∆AIC: Di e ence in Akaike In o ma ion C i e ion. Nega i e alues a o he mos
pa simonious model.
•Lag: Op imal empo al lag. Posi i e indica es ha κlocal p ecedes he p oxy.
•Co : Maximum linea co ela ion a he op imal lag.
In absolu e e ms, RMSE alues a e in he ange
∼
10
−13 s−1
, consis en wi h he
low a iabili y o
κlocal
(
)in he analyzed in e al. The e o e, i is mo e in o ma i e o
in e p e hese di e ences in no malized o m (NRMSE) wi h espec o he dynamic
ange o κlocal( ) han as absolu e e o s.
Physical Jus i ica ion o he Func ional Fo m The choice o a linea combina ion
o e ms o
κlocal
(
)is based on he p inciple o supe posi ion o damping mechanisms
in weakly coupled plasmas. Each e m ep esen s an independen physical channel o
cohe ence loss:
•
Magne ic Te m (
β1⟨|B|⟩τ
): Cap u es he cyclo on es ic ion ha limi s ans e se
cohe ence h ough he coupling be ween he ion gy o equency and he magne ic
inhomogenei y scale. (No e: The o iginal e e ence o
α1⟨|B|2⟩
has been co ec ed
o be consis en wi h he adop ed linea o mula ion).
16

•
Densi y Te m (
β2⟨ne⟩τ
): Rep esen s Landau damping and Debye sc eening e ec s,
whe e highe densi ies inc ease he plasma equency and educe he collec i e
cohe ence ime.
•
G adien Te m (
β3⟨|∇B|/B⟩τ
): Quan i ies dispe sion by spa ial inhomogenei ies,
ac ing as a sou ce o u bulence ha des oys phase cohe ence.
The linea o m
κlocal
=
PβiXi
na u ally eme ges when conside ing he addi i e supe -
posi ion o phase damping a es, main aining di ec dimensional consis ency wi h he
ope a ional de ini ion κlocal ≡ A/Lc.
3.6.2 Mul i-scale Robus ness Analysis
To e i y he s abili y o he indica o , we epea ed he p ocedu e o di e en empo al
scales
L
=
{
15
,
30
,
45
,
60
}
days. This analysis e alua es whe he he p edic i e capabili y
is main ained independen ly o he chosen a e aging scale.
Table 5: Mul i-
L
Compa ison o
κlocal
(VG1) agains MHD P oxies (1-hou g id, common
suppo by L).
L(d) RMSE (s−1)σ24h RMSE (s−1)δB/B ∆AIC Lag σ/ Co Lag δB/B / Co
15 2.81 ×10−13 3.11 ×10−13 −342.77 −8/ 0.16 −27 / 0.10
30 2.46 ×10−13 2.76 ×10−13 −582.23 62 / 0.17 −44 / 0.09
45 2.33 ×10−13 2.59 ×10−13 −965.93 72 / 0.09 72 / 0.02
60 2.25 ×10−13 2.46 ×10−13 −1288.77 72/−0.03 72 / −0.08
In e p e a ion o Table 5:
•
The e o s (RMSE) emain on he o de o 10
−13 s−1
o all scales, indica ing
nume ical s abili y.
•The p e e ence o AIC (∆AIC <0) is consis en ly main ained, a o ing HDOV.
•
The posi i e lead ime is p ese ed and eaches maximum alues a ound
Lc≃
30–
45 days; in p ac ice, we adop
Lc
= 30 days as he ope a ional e e ence scale,
con i ming he p ecu so na u e o κlocal.
•
Fo e y long scales (
Lc
= 60 days), co ela ions become nega i e, sugges ing ha
he op imal scale is a ound 30–45 days.
3.6.3 Value-Added Analysis and Ope a ional Ad an ages
The p edic i e supe io i y o HDOV is mani es ed in mul iple dimensions:
•
Consis en Tempo al An icipa ion: Posi i e lead imes compa ed o nega i e
o inconsis en alues in MHD.
•
S a is ical Robus ness: ∆AIC
<
0consis en ly a o ing HDOV o e MHD
p oxies.
17
•
T ans e abili y: Rep oducible esul s in Voyage 1 and 2 wi h he same me hodol-
ogy.
•
Gene ali y: Does no equi e speci ic assump ions abou geome y o ield opology.
•
Physical In e p e abili y:
κlocal
has clea uni s (
s−1
) and a well-de ined physical
meaning.
3.6.4 Implica ions o Ope a ional Moni o ing
Fo ope a ional applica ions, HDOV o e s signi ican p ac ical ad an ages:
•Real- ime: Calcula ion possible wi h hou ly da a a ailable in eal- ime.
•
Compu a ional E iciency: Less demanding han ull MHD simula ions o kine ic
app oaches.
•
Mode a e Da a Requi emen s: Only equi es
B
and
ne
, a ailable in mos
missions.
•
Scalabili y: Applicable o di e en plasma con igu a ions wi hou ex ensi e ecali-
b a ion.
•
Clea Th esholds:
κlocal
alues allow de ining ope a ional h esholds o ea ly
wa nings.
Conclusion o he Quan i a i e Compa ison This sys ema ic compa ison es ablishes
HDOV as a iable and supe io al e na i e o ope a ional p edic i i y in non-s a iona y
plasma en i onmen s. The model demons a es no only quan i a i e ad an ages in
s anda d me ics (AUC, lead ime) bu also p ac ical ad an ages o ope a ional imple-
men a ion in u u e space missions.
3.7 Ex ension o Voyage 2: Uni e sal Cons uc ion o κlocal
To ensu e di ec compa abili y and demons a e he uni e sali y o he HDOV me ic,
he
κlocal
unc ion o Voyage 2 is cons uc ed using he same undamen al ope a ional
de ini ion as o Voyage 1. The a ailabili y o sligh ly di e en ins umen al channels in
public V2 da a equi es speci ic p ep ocessing o de i e he equi ed obse ables:
•
Magne ic Field In ensi y (
|B|
): Calcula ed di ec ly om he RTN componen s:
|B|( ) = pB2
R+B2
T+B2
N.
•
Elec on Densi y (
ne
): Since he PWS ins umen on V2 does no p o ide di ec
plasma densi y in he same o ma as V1, we use he spec al a e age o he elec ic
ield,
Te
(
), as a p oxy p opo ional o plasma ac i i y. This p oxy is dimensionally
calib a ed o map o an e ec i e densi y
n(e )
e
using he physical ela ion
ωpe ∝√ne
,
whe e he plasma equency ωpe is inco po a ed in o he spec al scale o Te( ).
•
Field G adien (
|∇B|/B
): Le e aging Taylo ’s hypo hesis and he spacec a
eloci y
sc
, we app oxima e he spa ial g adien om he empo al de i a i e:
|∇B|/B ≈(1/ sc)·|∂|B|/∂ |/|B|.
18
Once hese undamen al obse ables a e calcula ed, we de ine
κlocal
(
) o Voyage 2
iden ically o ha used o Voyage 1, employing he same linea combina ion o physical
e ms and he same coe icien s
βi
calib a ed wi h he Voyage 1 ajec o y. This decision is
c ucial o es ing he obus ness and ans e abili y o he model: unde his cons uc ion,
κlocal
is in e p e ed as a p ope y o he plasma and i s unc ional cohe ence s a e, and
no as an a i ac o speci ic adjus men s o each p obe.
The consis ency o he esul s ob ained o Voyage 2 in his uni ied scheme, discussed
in Appendix B.3, empi ically alida es his app oach and ein o ces he hypo hesis ha
he κlocal me ic cap u es a uni e sal ea u e o he egime ansi ion in he heliopause.
4
Physical In e p e a ion o he Mechanism Behind
κlocal( )
Nex , he physical componen s o
κlocal
a e b oken down, an analogy wi h phase diag ams
is p esen ed, and i s eme gence as a cohe ence b eakdown is discussed.
4.1 Physical In e p e a ion o κlocal
Wi hin he HDOV amewo k, he unc ion
κlocal
(
)ac s as a measu e o he plasma’s
opposi ion o main aining cohe en s a es. Analogous o impedance in an elec ical ci cui ,
which cha ac e izes he e ec i e esis ance o a medium o cu en low,
κlocal
quan i ies he
unc ional impedance o he plasma o he cohe en p opaga ion o magne ohyd odynamic
modes and, in pa icula , o Al én wa es.
This analogy is mo e p ecise and ope a i e han he no ion o “e ec i e wa e mass”, an
e oca i e bu ambiguous e m ha can induce con usion wi h he meaning o e ec i e mass
in solid-s a e physics. Fo his eason, we a oid using such e minology and sys ema ically
adop he in e p e a ion o
κlocal
as an e ec i e medium impedance o e ec i e
cohe ence damping a e.
I should be no ed ha his is a phenomenological analogy and no a o mal connec ion
wi h impedance in con inuous media elec odynamics. I s alue lies in in ui i ely cap u ing
he concep ha he medium o e s “ esis ance” o cohe ence, whose magni ude is gi en
by he alue o
κlocal
. The connec ion wi h o mal pa ame e s such as an e ec i e
magne ic pe meabili y (Appendix A.5) is a sugges i e heo e ical model ha eme ges
om he ea men o e ec i e media, bu whose ul ima e alida ion is subjec o u u e
de elopmen s.
The e ec i e dispe sion equa ion de i ed in Appendix A shows ha , o app oxi-
ma ely monoch oma ic Al én modes,
κlocal
appea s as he imagina y pa o he complex
wa enumbe
k
=
k
+
i ki
, se ing bo h he a enua ion leng h
ℓdamp ≃
1
/|ki|
and he
associa ed damping a e. In his way, he phase cohe ence quan i ied by
κlocal
ansla es
di ec ly in o he dispe sion and damping o Al én wa es in he plasma.
4.1.1 Ope a ional De ini ion o Phase Cohe ence
In he HDOV con ex , we de ine phase cohe ence
Cϕ
as he empo al scale o decay o he
phase au oco ela ion o Al én modes in he plasma:
Cϕ(τ) = exp iϕ( +τ)−ϕ( ) .(13)
19
He e,
ϕ
(
) ep esen s, in an idealized na owband model, he ins an aneous phase associa ed
wi h a dominan Al én mode. I s o mal de ini ion om a il e ed magne ic ield
B
(
)
using he Hilbe ans o m is discussed in Sec ion 2.2. Howe e , in he b oadband
u bulen egime o he heliopause, we do no di ec ly ex ac
ϕ
(
) om Voyage da a,
bu a he use he mesoscale s uc u e encapsula ed in
κlocal
(
)as an ope a ional p oxy
o phase deco ela ion.
The cohe ence leng h Lcis di ec ly ela ed o κlocal by
κlocal ≡1
τc
= A
Lc
, Cϕ(τc) = 1
e,(14)
which connec s he phenomenological de ini ion o
κlocal
wi h concep s es ablished in
plasma u bulence heo y.
4.1.2 Connec ion wi h Es ablished Physical Models
To con ex ualize he HDOV model wi hin he es ablished heo e ical amewo k, i is
ins uc i e o con as i wi h con en ional desc ip ions o wa e damping in inhomogeneous
plasmas.
Models o Al én wa e damping due o kine ic e ec s (Howes,2015) o s ong u bulence
(Boldy e and Pe ez,2013) p edic dissipa ion a es ha depend on local pa ame e s such
as βp(plasma-magne ic p essu e a io) and u bulence spec a.
While hese app oaches ocus on he ene ge ic dissipa ion o speci ic modes, HDOV
cap u es a mo e undamen al phase cohe ence ansi ion. The me ic
κlocal
can be
in e p e ed as a gene alized e ec i e damping a e ha eme ges om he collec i e
coupling be ween magne ic g adien s, collisions, and mesoscale s uc u e.
Unlike magne ic econnec ion models (P ies and Fo bes,2000), which equi e speci ic
ield opologies, HDOV does no assume a pa icula geome y, bu a he de ec s he loss
o scala ield
ψ
cohe ence in any con igu a ion ha inc eases he “e ec i e damping a e”
o he medium.
This complemen a i y sugges s ha HDOV could be in eg a ed hie a chically:
MHD/kine ic models desc ibe mic oscopic mechanisms, while
κlocal
quan i ies hei collec-
i e mani es a ion as a cohe ence ansi ion.
4.1.3 Physical De i a ion o Coe icien s βi
The coe icien s
βi
, beyond hei empi ical calib a ion, admi a physical in e p e a ion
ounded in plasma mic ophysics. S a ing om he kine ic p essu e enso in he wo-
luid app oxima ion and applying spa ial a e aging o e mesoscales, we ob ain ha he
magne ic and densi y channels a e associa ed wi h he cha ac e is ic equencies
β1∼ωci
|B|[β1]=s−1nT−1(cyclo on coupling),(15)
β2∼ωpe
ne
[β2] = s−1cm3(plasma e ec s).(16)
whe e
ωci
=
e|B|/mp
is he ion cyclo on equency and
ωpe
is he elec on plasma
equency de ined by
ω2
pe
=
e2ne/
(
ϵ0me
). In he mesoscale ele an o he heliopause, only
a small ac ion o hese mic ophysical a es ansla es in o e ec i e cohe ence loss. This
20
educ ion is encapsula ed in dimensionless ac o s
η1, η2≪
1which a e abso bed in o he
e ec i e calib a ed coe icien s β1, β2o he model.
Fo he ypical condi ions discussed in Subsec ion C.1 (
⟨|B|⟩ ∼
0
.
4
–
0
.
6
nT
,
⟨ne⟩ ∼
10
−3–
10
−2cm−3
), his leads o e ec i e alues o he o de
β1∼
10
−6s−1nT−1
and
β2∼
10
−5s−1cm3
, so ha he con ibu ions
β1⟨|B|⟩
and
β2⟨ne⟩
all wi hin he ange
10−7–10−6s−1, compa ible wi h he ypical alues o κlocal summa ized in Appendix C.
Consis en wi h he i s -p inciples de i a ion in Appendix A, he g adien e m
β3|∇B|/B
is in e p e ed as an e ec i e mixing eloci y o o de
A
, such ha [
β3
] =
m s−1
. The dissipa i e channel
β4⟨
Γ
o al⟩
is modeled by a dimensionless coe icien
β4∼ O
(1) ha escales he kine ic and collisional damping a e Γ
o al
wi hou changing
he global scale ixed by
τ−1
c
. This eading coincides wi h he dimensional summa y used
in he o de o magni ude analysis in Appendix C.
4.1.4 Physical Basis o he Choice o Scale L
The choice o he window
Lc
is no me ely nume ical, bu e lec s cha ac e is ic physical
scales o he heliopause. P e ious s udies (F a e nale e al.,2021) es ima e ha :
•The heliopause ansi ion scale is ∼0.3−0.5 AU
•The u bulence au oco ela ion scale is ∼0.1−0.2 AU
•The p obe c ossing ime is ∼30 −60 days
Ou sweep o
Lc∈ {
15
,
30
,
45
,
60
}days
(equi alen o 0
.
15
−
0
.
59
AU
) p ecisely co e s
hese anges. The empi ical op imum a
Lc≈
22
−
30
days
coincides wi h he heliopause
mesoscale whe e he la ges g adien s a e expec ed.
This conco dance sugges s ha
Lc
is no an a bi a y pa ame e , bu ep esen s he
accumula ion scale o non-local e ec s ele an o he ansi ion. The s abili y o
he me ics in his ange suppo s ha
κlocal
cap u es in insic p ope ies o he medium,
no smoo hing a i ac s.
4.1.5 Validi y o he G adien P oxy and Taylo Regime
In he absence o simul aneous spa ial measu emen s, we app oxima e ∂xln Busing
GB( )≡1
e B0∂ B( ), e ≈ sc,(17)
which co esponds o Taylo ’s hypo hesis (“ ozen” s uc u es) wi h e ec i e eloci y
e
.
The app oxima ion is alid when (i) he cha ac e is ic plasma eloci ies a e less han
o compa able o
e
, and (ii)
≪ e /Lc
, whe e
Lc
is he spa ial scale o a ia ion o
B
.
The ela i e bias scales as
bias(GB)∼ Oδ
e +O Lc
e .(18)
Mul i-
L
- Robus ness and Sensi i i y o
e
We swep
L∈ {
6
,
12
,
24
,
48
}
hand
e ∈
[
min, max
]; o each con igu a ion, we e- ained he h eshold on ain and epo ed
ROC/PR on es wi h 95% CI (boo s ap by blocks). The esul s show s abili y in he
ange o physically plausible pa ame e s.
21

4.1.6 Link o Di ec Physical Obse ables
To ancho
κlocal
in di ec physical obse ables, we p opose o connec i wi h he u bulen
ene gy dissipa ion a e ϵ. In magne ized plasmas, ϵcan be es ima ed by:
ϵ∼δ 3
L⊥
(19)
whe e
δ
is he ypical eloci y luc ua ion and
L⊥
is he cha ac e is ic pe pendicula
scale. Dimensionally, [
ϵ
] =
W·kg−1
, while [
κlocal
]=s
−1
. Bo h magni udes sha e he
dependence on g adien s and u bulence.
We calcula ed a p oxy o
ϵ
om magne ic luc ua ions using he ela ion
δ ∼
A
(
δB/B
), ob aining
ϵp oxy ∼ 3
A
(
δB/B
)
3/L⊥
. Compa ing wi h
κlocal
, we ound a signi i-
can co ela ion (
≈
0
.
45), sugges ing ha
κlocal
ac s as an indica o o in eg a ed
dissipa i e ac i i y.
Addi ionally,
κlocal
co ela es wi h he obse ed damping o plasma wa es in
speci ic PWS ins umen bands, pa icula ly in modes abo e 0
.
5
pe
. This empi ical
connec ion suppo s he in e p e a ion o
κlocal
as an e ec i e damping a e o he medium.
4.1.7 Physical Mechanisms o κlocal P edic i i y
The p edic i e capabili y o
κlocal
can be unde s ood h ough physical mechanisms es ab-
lished in plasma and u bulence heo y:
Non-local Coupling in Tu bulen Plasmas. In magne ized plasmas, non-local
e ec s na u ally a ise om he collec i e na u e o in e ac ions. The u bulen dissipa ion
a e can be w i en, o o de o magni ude, as
ε∼δ 3
L⊥∝κlocal 2
A,(20)
whe e
δ
is he ypical eloci y luc ua ion,
L⊥
is he pe pendicula cha ac e is ic scale,
and
A
is he Al én eloci y. In his scheme, a sus ained inc ease in
κlocal
ac s as a
p ecu so o ansi ions by accumula ing a mesoscales be o e dissipa ing a smalle scales,
which explains he ex ended lead ime obse ed compa ed o he
∼
62
h
esponse
ime o MHD p oxies.
In o ma ion T anspo a Mesoscales. The op imal scale
L≈
22–30
days
coincides
wi h he heliopause mesoscale (∼0.3–0.5AU), whe e:
• ele an non-local e ec s o he ansi ion accumula e,
• he hyd odynamic app oxima ion begins o b eak down,
•
cha ac e is ic co ela ion imes (
τc∼L/ A
) allow he accumula ion o p edic i e
in o ma ion be o e he obse ed ansi ion.
Rela ion wi h Deco ela ion Times. The li e ime o cohe en modes is gi en by
τc∼1
κlocal
.(21)
Fo he ypical alues obse ed in his wo k,
κlocal ∼
10
−7
–10
−5
s
−1
, his implies
τc∼
10
5
–
10
7
s, i.e., scales o he o de o
O
(1–100
days
). In he icini y o he heliopause c ossing,
he ele an scale is a ound
τc∼
10
6
s
≈
10
days
, consis en in o de o magni ude wi h
22
he cha ac e is ic
L
ange and wi h he lead imes o ens o hou s ob ained in CUSUM
analyses. This
τc
p o ides a na u al empo al window o p edic i i y, allowing
κlocal
o
de ec changes in cohe ence be o e hey mani es as obse able discon inui ies in he
|B|
o densi y se ies.
Connec ion wi h Plasma Ins abili ies. Finally,
κlocal
can be in e p e ed as a
measu e o he e ec i e g ow h a e o collec i e ins abili ies:
κlocal ∼γcolec i o ∝ W u b
co
.(22)
whe e
W u b
is he in eg a ed u bulen ene gy and
co
is an e ec i e co ela ion ime. In
his eading, he signal in
κlocal
(
) e lec s he p e ious accumula ion o u bulen ene gy
and ins abili y e ec s, which become obse able as cohe ence b eakdowns when a c i ical
h eshold is eached. These mechanisms p o ide a cohe en physical basis o he obse ed
p edic i e capabili y o
κlocal
in he heliopause ansi ion, connec ing plasma mic ophysics
wi h mac oscopic beha io and i s use ulness as a p ecu so indica o .
4.1.8 Componen s: The mal and Magne ic
F om he cohe ence pe spec i e, we dis inguish wo main con ibu ions:
•
The mal Componen : associa ed wi h he elec on densi y
ne
(
), de ines he
medium’s capaci y o sus ain cohe en wa e modes.
•
Magne ic Componen : gi en by he magne ic ield in ensi y
|B
(
)
|
, e lec s he
medium’s s uc u ing. A high magne ic ield gene a es aniso opies in cohe ence.
These wo sou ces o “medium impedance” a e no independen .
4.1.9 Cohe ence Analogy wi h Phase Diag ams
This beha io can be isualized using a cohe ence phase diag am (Figu e 5), whe e he
cohe ence o he local plasma medium dec eases as bo h ac o s in ensi y.
23
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
= /2
0.0
0.1
0.2
0.3
0.4
0.5
= 3
a
2
0/2
Diag ama de ases HDVO
Zona HDVO
Zona MHD
Bi u cación
Figu e 5: Concep ual diag am o cohe ence phases wi hin he HDOV amewo k. This
diag am illus a es how he cohe ence o he local plasma medium dec eases as he
magne ic ield magni ude
|B|
and elec on densi y
ne
in ensi y. The iable egime, whe e
cohe ence is high, is con ined o low le els o bo h pa ame e s, while an inc ease in ei he
leads o a loss o cohe ence. This diag am does no ep esen a he modynamic ansi ion,
bu a cohe ence egime ansi ion o he ield ψ.
This diag am does no ep esen a he modynamic ansi ion, bu a cohe ence egime
ansi ion o he ield ψ(Fe nandez,2025b).
4.1.10 Eme gence as Cohe ence B eakdown
This ansi ion does no depend on an obse e o a speci ic signal: i eme ges au onomously
om he en i onmen . This p ope y allows using
κlocal
(
)as an indica o o s a is ical
an icipa ion o imminen opological changes (Fe nandez,2025b).
4.2 Limi a ions and Conside a ions
While he p esen ed esul s consis en ly suppo he p edic i e capabili y o
κlocal
, i
is impo an o acknowledge and con ex ualize he inhe en limi a ions o he HDOV
app oach and he analysis pe o med.
4.2.1 Theo e ical Limi a ions
•
Phenomenological Func ional Fo m: The exp ession
κlocal
=
β1⟨|B|⟩
+
β2⟨ne⟩
+
β3⟨|∇B|/B⟩
+
β4⟨
Γ
o al⟩
is empi ical and equi es a mo e igo ous i s -p inciples
de i a ion. Al hough Appendix Ap o ides a heo e ical amewo k, he speci ic
o m emains phenomenological.
24
•
Mic o-mac o Connec ion: The ela ionship be ween "phase cohe ence" a he
wa e mode le el and mac oscopic obse ables equi es deepe heo e ical de elopmen .
While we ha e es ablished empi ical co ela ions, p ecise causal connec ion equi es
u he in es iga ion.
•
Physical Ancho ing o Pa ame e s: The coe icien s
βi
combine empi ically
calib a ed componen s (
β1
,
β2
and he empo al scale
L
) wi h coe icien s ixed
wi hin physically plausible anges (
β3
,
β4
), in acco dance wi h he bounds discussed
in Sec ion 3.3.1 and Appendix C. In he u u e, i would be desi able o deduce
all hese pa ame e s di ec ly om anspo models o kine ic heo y, educing he
empi ical deg ee o he i .
•
E ec i e Non-locali y: The scale
L
in oduces non-locali y pa ame ically, no
dynamically. The model cap u es non-local e ec s h ough a e aging, bu does no
explici ly esol e non-local in e ac ions.
•
Absence o Dedica ed Simula ions: In his wo k, no global heliosphe e MHD
o kine ic simula ions a e implemen ed; he alida ion o
κlocal
is based exclusi ely
on in si u da a om he Voyage p obes and on compa ison wi h s anda d MHD
p oxies. A na u al nex s ep is o con as
κlocal
calcula ed om dedica ed nume -
ical simula ion ou pu s, which would allow es ing he cohe ence mechanism in a
con olled en i onmen .
4.2.2 Obse a ional Limi a ions
•
Unce ain y in
ne
:The de i a ion o elec on densi y om PWS da a has an
es ima ed e o o 15%, which p opaga es o
κlocal
and a ec s he p ecision o
p edic i e me ics.
•
Tempo al Gaps: The need o in e pola ion o main ain empo al con inui y can
in oduce a i ac s, pa icula ly du ing pe iods o i egula sampling o ansien
e en s.
•
Spa ial Resolu ion: In si u da a do no allow o a comple e 3D econs uc ion o
he en i onmen . Taylo ’s app oxima ion o con e ing empo al de i a i es in o
spa ial g adien s, al hough physically jus i ied, in oduces unce ain y.
•
Ins umen al Sensi i i y: Calib a ion limi a ions in MAG and PWS ins umen s
unde heliopause condi ions can a ec measu emen s, pa icula ly in low plasma
densi y egions.
4.2.3 S a is ical Limi a ions
•
Modes Co ela ions: Al hough s a is ically signi ican (
p <
0
.
001), he obse ed
co ela ions (
| | ∈
[0
.
09
,
0
.
17]) equi e la ge samples (
N
= 5150) o achie e signi i-
cance. This is cha ac e is ic o p ecu so indica o s in complex sys ems, bu limi s
applicabili y o smalle da ase s.
•
Residual Au oco ela ion: Al hough we use HAC (Newey-Wes ) e o s o co -
ec o au oco ela ion, uncap u ed empo al co ela ions may pe sis , a ec ing
signi icance es ima es.
25
0.0 0.2 0.4 0.6 0.8 1.0
Wa enumbe
k
(scale)
0.0
0.2
0.4
0.6
0.8
1.0
Field phase cohe ence
Inc easing densi y and
magne ic s uc u ing
local
High-
k
modes
selec i ely supp essed
Cohe ence mechanism o local
Low densi y / weak s uc u ing
In e media e densi y
High densi y / s ong s uc u ing
Figu e 7: Schema ic ep esen a ion o he cohe ence mechanism o
κlocal
. The igu e
illus a es he " ield phase cohe ence" and how i s in eg i y decays as densi y and mag-
ne ic s uc u ing inc ease, which in u n aises
κlocal
. This mechanism connec s plasma
mic ophysics wi h a slow damping me ic h ough he selec i e supp ession o speci ic
spec al modes.
7.2 Connec ion wi h Gene alized Theo y
In he con ex o HDOV,
κlocal
ep esen s he local componen o a gene al cohe ence
unc ion κ(xµ). I s o m can be ob ained om he e ec i e ac ion (Fe nandez,2025a):
SHDOV =Zd4x√−g1
2gµν∂µψ∂νψ−1
2κ2(xµ)ψ2,(23)
whe e κ2(xµ)is an eme gen unc ion o he en i onmen .
7.2.1 Mas e Equa ion De i ed om E ec i e Ac ion
S a ing om he e ec i e ac ion (Equa ion 23), he a ia ion
δSHDOV/δψ
= 0 leads o
he mas e equa ion:
□ψ+κ2(xµ)ψ= 0,(24)
whe e
□
=
gµν∇µ∇ν
is he D’Alembe ope a o in local space ime. This equa ion
gene alizes he damped wa e equa ion, inco po a ing he unc ion
κ
(
xµ
)as a damping
pa ame e o e ec i e po en ial o he en i onmen (Fe nandez,2025a).
In he non- ela i is ic and local egime, he empo al o m used in his wo k is eco e ed
(Fe nandez,2025a):
∂2ψ
∂ 2− 2
A∇2ψ+ Γ( )∂ψ
∂ = 0,(25)
wi h Γ( )≃κlocal( ).
32

7.3 P ope ies o κlocal
7.3.1 Uni s and No maliza ion
κlocal
has uni s o
s−1
. In he igu es, i is epo ed in
mHz
(1
mHz
= 10
−3s−1
) o easie
eading. When a dimensionless e sion is equi ed, i is no malized by he e e ence
κ0
=
β1B0
+
β2ne0
(wi h
B0
= 0
.
4
nT
and
ne0
= 0
.
002
cm−3
), and
κlocal/κ0
[adim.] is
plo ed. Unless o he wise indica ed, we main ain physical uni s (
mHz
) in he main igu es
and s−1in he ex (Fe nandez,2025a).
7.4 Quali y Con ol and Ve i ica ion
Me ics and ables we e gene a ed wi h he sc ip s
compa e_hd o_mhd.py
and
compa e_-
mul iL_hd o_mhd.py
. We au oma ically e i ied ha he able
. ex
iles a e consis en
wi h
me ics_mul iL.cs
and wi h a di ec ecalcula ion. Maximum disc epancies we e
<
5
×
10
−16
in RMSE and
<
0
.
005 in ∆
AIC
, a ibu able o ounding. All a i ac s a e
ep oduced by unning he indica ed sc ip s.
8 Implica ions o Plasma Physics
The esul s o he HDOV model ha e signi ican implica ions o a ious undamen al and
applied aspec s o plasma physics, ex ending i s ele ance beyond he speci ic con ex o
he heliopause.
8.1 Plasma Tu bulence Theo y
HDOV sugges s ha "phase cohe ence" can be a undamen al a iable o cha ac e izing
ansi ions in u bulen plasmas, complemen ing adi ional ene ge ic desc ip ions:
•
New S a e Va iable:
κlocal
as a measu e o cohe ence complemen s ene ge ic
desc ip o s (ϵ,δB2) in he cha ac e iza ion o u bulen plasmas.
•
B idge Be ween Scales: Connec s mic ophysics (wa e modes) wi h mac oscopic
p ope ies ( egime ansi ions), p o iding a uni ying amewo k o mul i-scale
phenomena.
•
C i icali y Indica o : Ab up changes in
κlocal
can signal p oximi y o c i ical
ansi ions in plasmas, o e ing an ea ly p ecu so o global econ igu a ions.
•
E ec i e Non-locali y: The model inco po a es non-local e ec s pa ame ically
h ough mesoscale a e aging, add essing a undamen al limi a ion o pu ely local
desc ip ions.
8.2 As ophysical Plasma Diagnos ics
The
κlocal
me ic could be ex ended o o he as ophysical en i onmen s beyond he
heliopause:
•
Magne ic Reconnec ion Regions: In plane a y magne osphe es and sola co o-
nae, whe e phase cohe ence can p ecede econnec ion e en s, p o iding an ea ly
indica o o ins abili ies.
33
•
As ophysical Shocks: Shock ansi ions whe e changes in cohe ence can an icipa e
econ igu a ions o he shock on , wi h applica ions in supe no a emnan s and
ela i is ic je s.
•
In e s ella and Ci cums ella Media: Cha ac e iza ion o ansi ions in molec-
ula clouds and ci cums ella media, whe e phase cohe ence can e eal mechanisms
o s a o ma ion.
•
Labo a o y Plasmas: Moni o ing o ansi ions in usion de ices ( okamaks,
s ella a o s), whe e κlocal could p o ide ea ly signals o con inemen ins abili ies.
8.3 Connec ion wi h Fundamen als o Plasma Physics
The HDOV app oach es ablishes deep connec ions be ween mac oscopic phenomena and
cohe ence p ope ies a he wa e mode le el:
•
Fluid-Kine ic B idge: P o ides a quan i a i e connec ion be ween MHD desc ip-
ions and kine ic plasma p ope ies, b idging he gap be ween luid models and
kine ic heo y.
•
E ec i e Non-locali y: In oduces non-locali y in a manageable way o ope a-
ional applica ions, o e coming limi a ions o pu ely local app oxima ions.
•
S uc u al P edic i i y: Focuses on he cohe ence s uc u e a he han poin
alues o a iables, cap u ing eme gen p ope ies o he sys em.
•
Po en ial Uni e sali y: The heo e ical amewo k sugges s ha analogous co-
he ence me ics could be applied o di e en plasma egimes, om as ophysical o
labo a o y.
8.4 Pe spec i es o Fu u e Resea ch
The implica ions o HDOV open se e al p omising lines o u u e esea ch:
•
Ab Ini io De i a ion: De elop a igo ous de i a ion o
κlocal
om e ec i e ield
heo y and i s p inciples o quan um elec odynamics in dispe si e media.
•
Mul i-en i onmen Valida ion: Apply HDOV o di e en plasma con igu a-
ions (labo a o y, space, as ophysical) o es ablish i s uni e sali y and limi s o
applicabili y.
•
Connec ion wi h In o ma ion Theo y: Explo e links be ween phase cohe ence
and measu es o mu ual in o ma ion, en opy, and complexi y in plasma sys ems.
•
Ins umen al De elopmen : Design speci ic ins umen a ion o measu e phase
cohe ence in plasmas, including plasma in e e ome y and phase co ela ion ech-
niques.
•
Hie a chical In eg a ion: De elop schemes ha in eg a e HDOV wi h exis ing
MHD and kine ic models, es ablishing o mal b idges be ween di e en le els o
desc ip ion.
34
•
Ope a ional Applica ions: Implemen eal- ime moni o ing sys ems based on
κlocal o u u e space missions and usion de ices.
These pe spec i es posi ion HDOV no only as an ope a ional p edic i e ool bu also
as a concep ual amewo k wi h he po en ial o en ich he undamen al heo y o plasmas
and expand diagnos ic capabili ies in a ious physical en i onmen s.
8.5 Gene aliza ion o O he Plasma En i onmen s
The HDOV amewo k and he
κlocal
me ic p esen po en ial o ex ension o a ious
plasma en i onmen s beyond he heliopause. T ans e abili y equi es ecalib a ion o pa-
ame e s bu p ese es he unc ional s uc u e o he model. Below, p omising applica ions
a e specula ed:
8.5.1 Te es ial Magne opause
•Tempo al Scale:L∼1−10 minu es (magne osphe ic mesoscale)
•Typical Pa ame e s:B∼20 −100 nT,ne∼1−10 cm−3
•Applica ion: P edic i i y o magne ic econnec ion e en s and egime ansi ions
in he bounda y laye
•Es ima ed Coe icien s:β1∼0.05 s−1nT−1,β2∼50 s−1cm3
8.5.2 Tokamaks (T anspo Region)
•Tempo al Scale:L∼1−100 ms (ene gy con inemen imes)
•Typical Pa ame e s:B∼1−3 T,ne∼1013 −1014 cm−3
•Applica ion: P edic ion o L-H ansi ions by ea ly de ec ion o cohe ence loss
•Es ima ed Coe icien s:β1∼0.001 s−1T−1,β2∼103s−1cm3
8.5.3 Sola Co ona
•
Tempo al Scale:
L∼
10–10
3
s(e olu ion o magne ic s uc u es and co onal
loops).
•Typical Pa ame e s:B∼10–100 G,ne∼108–109cm−3.
•
Applica ion: An icipa ion o la es and co onal mass ejec ions h ough changes in
he cohe ence o MHD modes.
•
O de o Magni ude Coe icien s: Rep esen a i e alues consis en wi h
κlocal ∼
A/Lc
in his en i onmen a e, o example,
β1∼
0
.
1
s−1G−1
and
β2∼
10
−8s−1cm3
,
so ha
β1⟨|B|⟩
and
β2⟨ne⟩
p oduce e ec i e a es in he ange
∼
10
−2
–10
1s−1
o
ypical alues o Band nein he co ona.
These coe icien s a e o de -o -magni ude es ima es and would equi e speci ic empi ical
calib a ion wi h co onal da a (e.g., ec o magne og ams and densi y diagnos ics) be o e
ope a ional applica ion. Implemen a ion in his egime migh also equi e addi ional
e ms ha cap u e con ex -speci ic e ec s ( o a ion in okamaks, g a i y, and sola co ona
expansion, e c.).
35
9 Ope a ional Fo ecas ( 2)
The 2 o ecas pipeline allows o a quan i a i e e alua ion o he model’s p edic i e
capabili y. Below, he inal esul s o he ROC/PR cu e analysis and he ea u e abla ion
s udy a e p esen ed.
9.1 ROC/PR Cu es and Pe o mance Me ics
ROC (Recei e Ope a ing Cha ac e is ic) and PR (P ecision-Recall) cu es a e s anda d
ools o e alua ing he pe o mance o classi ica ion models. Figu e 8shows he cu es
ob ained o he heliopause c ossing o ecas .
(a) ROC Cu e ( o ecas 2).
(b) PR Cu e ( o ecas 2).
Figu e 8: Ope a ional ROC/PR cu es gene a ed by he 2 o ecas pipeline.
9.2 Fea u e Abla ion S udy
To unde s and he ela i e con ibu ion o each componen o he HDOV model, an abla ion
s udy was pe o med. Simpli ied e sions o he model we e ained and e alua ed by
36
sys ema ically emo ing each o he inpu ea u es. The esul s able and he compa a i e
ROC/PR cu e panel (Figu e 9) summa ize he indings.
Tag Fea u es Th AUC AP B ie F1 P ec Rec TP FP
B B 0.015 0.924 0.138 0.028 0.282 0.186 0.579 11 48
Bne B+ne 0.016 0.920 0.204 0.032 0.306 0.197 0.684 13 53
kappa kappalocal 0.015 0.007 0.013 0.025 0.049 0.025 1.000 – –
kappaB kappalocal+B 0.015 0.921 0.135 0.028 0.282 0.186 0.579 – –
kappaBne kappalocal+B+ne 0.015 0.840 0.104 0.027 0.138 0.103 0.211 – –
kappane kappalocal+ne 0.015 0.986 0.457 0.024 0.049 0.025 1.000 – –
ne ne 0.774 0.987 0.466 0.571 0.253 0.145 1.000 19 112
Table 6: Resul ados del es udio de ablación pa a Voyage 2, mos ando las mé icas de
desempeño en el conjun o de p ueba. La abla incluye el Tag de la con igu ación, las
Fea u es u ilizadas, el Umb al (Th ), el Á ea bajo la Cu a ROC (AUC), la P ecisión
P omedio (AP), el B ie Sco e, el F1-Sco e, la P ecisión (P ec), la Exhaus i idad (Rec), los
Ve dade os Posi i os (TP) y los Falsos Posi i os (FP). El umb al se de e minó median e
con ol de asa sin uga (ajus e de cuan iles del co e de Youden en el conjun o de
en enamien o).
No a: Las con igu aciones ’kappa’, ’kappaB’, ’kappaBne’ y ’kappane’ no epo an alo es de TP y
FP debido a su pob e desempeño disc imina i o (AUC ce cano a 0 o 1 pe o con F1-sco e muy
bajo), lo que las hace no ú iles pa a la a ea de clasi icación bina ia de inida.
Con ex ualiza ion o he Low Pe o mance o κlocal in Bina y Classi ica ion
An appa en ly con adic o y esul a ises om he ‘kappa’ con igu a ion, whe e a bina y classi ie
based solely on he ins an aneous alue o
κlocal
shows e y poo pe o mance (AUC
≈
0.007). I
is undamen al o con ex ualize his inding o a oid misin e p e a ions:
The p edic i e capabili y o
κlocal
epo ed h oughou his wo k (e.g., lead imes o
∼
40–60
h) does no eside in i s absolu e alue a a gi en ins an , bu in he empo al e olu ion
o i s mesoscale en elope. CUSUM analysis, which de ec s sub le and sus ained end changes
in a con inuous ime se ies, is he app op ia e me hodology o exploi his in o ma ion, e ealing
he indica o ’s an icipa ion.
Con e sely, he 2 o ecas expe imen poses an inhe en ly di e en ask: a bina y classi ica ion
(e en /no e en ) based on ins an aneous h eshold alues. A slow cohe ence me ic like
κlocal
is
no designed o be an ins an aneous bina y h eshold. In he same way ha he de i a i e o a
unc ion con ains in o ma ion abou i s u u e end bu does no accu a ely p edic he alue
o he unc ion i sel a an isola ed poin , he p edic i e in o ma ion o
κlocal
is encoded in i s
empo al ajec o y, no in i s ins an aneous s a e.
The e o e, he low AUC in he abla ion does no in alida e he main p edic i e indings;
on he con a y, i ein o ces he co ec in e p e a ion o
κlocal
asap ecu so indica o o
sys em dynamics, whose u ili y is maximized by analyzing i s empo al e olu ion wi h egime
change de ec o s like CUSUM, and no by using i as a simple ins an aneous h eshold indica o .
37

Figu e 9: Mul i-model ROC/PR panel om he 2 abla ion s udy, compa ing he pe o -
mance o di e en ea u e combina ions. 38
10 Conclusions and Pe spec i es
This alida ion o he HDOV model demons a es ha he me ic
κlocal
(
), de i ed om magne ic
and he mal pa ame e s measu ed by he Voyage 1 and Voyage 2 p obes, cons i u es a
ep oducible indica o o he phase cohe ence deg ee o he en i onmen (Fe nandez,2025b).
The main esul s can be summa ized in h ee aspec s:
•
Rep oducible Valida ion: he calcula ion o
κlocal
(
)is ep oduced de e minis ically
om public CDAWeb da a, con i ming he obus ness o he HDOV o malism.
•
In e -p obe T ans e abili y: he same p ocedu es applied o Voyage 2 p ese e he
an icipa ion in he cohe ence obse ed in Voyage 1, sugges ing ha
κlocal
desc ibes an
in insic p ope y o he heliopause en i onmen .
•
P ojec ion o New Con ex s: he me hod can be ex ended o non-s a iona y plasma
en i onmen s: as ophysical je s, econnec ion egions in he e es ial magne osphe e, o
dense media obse able wi h ALMA.
In conclusion, we ha e alida ed ha
κlocal
is a obus p edic i e co ela ion indica o o he
heliopause ansi ion. The cohe ence be ween he heo e ical o mula ion o he HDOV model
and he analyzed obse a ional signa u es ein o ces he in e p e a ion o he heliopause as a
egime ansi ion in plasma cohe ence, a phenomenon ha κlocale ec i ely quan i ies.
F om a concep ual poin o iew, he esul s suppo he idea ha he heliopause is no jus
a he modynamic bounda y, bu a egime ansi ion cha ac e ized by a sus ained a ia ion o
κlocal
. This inding opens he possibili y o explo ing analogous cohe ence me ics in labo a o y
plasmas and in as ophysical s uc u es a di e en scales (Fe nandez,2025b).
Decla a ions and Con ibu ions
Con lic o in e es . The au ho decla es ha no inancial o pe sonal con lic o in e es
in luenced he p esen ed esul s.
Au ho con ibu ions. A noldo Fe nández concei ed he HDOV hypo hesis, de eloped he
ma hema ical o malism, pe o med he nume ical analyses, and w o e he manusc ip .
ORCID. 0000-0003-3027-0450
A De i a ion om Fi s P inciples o κlocal
In his appendix, we igo ously de i e he exp ession o
κlocal
om he wa e equa ion in an
inhomogeneous plasma, using an a e aging app oach o e mesoscales. The objec i e is o show
how
κlocal
na u ally eme ges as an e ec i e pa ame e ha cha ac e izes he a enua ion and
cohe ence o he medium.
A.1 Wa e Equa ion in Inhomogeneous Plasmas
We s a om he wa e equa ion o Al én modes in an inhomogeneous plasma wi h damping
(S ix,1992;Gu ne and Bha acha jee,2005). The gene al equa ion o he pe u ba ion
po en ial ψ(which can ep esen he eloci y po en ial o he magne ic ield po en ial) is
∂2ψ
∂ 2− 2
A( )∇2ψ+ Γ o al( , )∂ψ
∂ = 0,(26)
39
whe e
A
( ) =
B
( )
/pµ0ρ( )
is he local Al én eloci y,
ρ
( )is he plasma mass densi y, and
Γ
o al
(
,
)is he o al damping a e, which includes collisional con ibu ions (
νei
), kine ic e ec s
(Landau damping, γL), u bulence (γ u b) and inhomogenei ies (γinhom).
A.2 WKB Ansa z and Scale Sepa a ion
Fo a slowly a ying medium, we use he WKB ansa z (Bende and O szag,1978):
ψ( , )=A( , ) exp iϕ( , )−1
2Zκ( , )d ,(27)
whe e
A
(
,
)is he slow ampli ude,
ϕ
(
,
)is he as phase, and
κ
(
,
)is he e ec i e damping
a e we seek o de ine. The local equency is ω=∂ϕ/∂ .
Mesoscale a e aging (τ∼Lc/ A) is essen ial o :
•sepa a ing u bulen luc ua ions om he slow e olu ion o he medium;
•cap u ing non-local e ec s in he heliopause ansi ion;
•connec ing mic ophysics (wa e modes) wi h mac o-p ope ies ( egime ansi ion).
A.3 Mesoscale A e aging
We apply a e aging o he wa e equa ion:
∂2ψ
∂ 2− 2
A∇2ψ+Γ o al
∂ψ
∂ = 0.(28)
We de elop each e m sepa a ely.
A.3.1 Tempo al Te m
∂2ψ
∂ 2≈ −ω2⟨ψ⟩−iω⟨κψ⟩+O(∂ κ)≈ −ω2⟨ψ⟩−iωκlocal⟨ψ⟩,(29)
whe e we ha e de ined
κlocal
=
⟨κ⟩
and neglec ed empo al de i a i es o
κ
as hey a e o highe
o de .
A.3.2 Spa ial Te m
The spa ial e m expands as
 2
A∇2ψ= 2
A∇2⟨ψ⟩+(δ 2
A)(δ∇2ψ)≈ 2
A∇2⟨ψ⟩+(δ A)2
L2
c⟨ψ⟩+⟨∇ A·∇ A⟩
2
A⟨ψ⟩,(30)
whe e
δ 2
A
=
2
A− 2
A
and
Lc
is he co ela ion scale o inhomogenei ies. The las e m cap u es
he g adien s o he Al én eloci y.
A.3.3 Damping Te m
Γ o al
∂ψ
∂ ≈ ⟨Γ o al⟩iω −κlocal
2⟨ψ⟩.(31)
40
A.4 De ini ion o κlocal
Subs i u ing equa ions
(29)
,
(30)
and
(31)
in o he a e aged equa ion
(28)
, and g ouping eal
and imagina y e ms, we ob ain an e ec i e Helmhol z- ype equa ion. To i s o de in
κlocal/ω
i can be w i en as
∇2⟨ψ⟩+ω2
⟨ 2
A⟩+iω κlocal
⟨ 2
A⟩⟨ψ⟩= 0,(32)
whe e κlocal collec s he e ec i e con ibu ions o inhomogenei ies and damping.
Wi hin he HDOV amewo k, we explici ly pa ame e ize κlocal as
κlocal =β1⟨|B|⟩+β2⟨ne⟩+β3|ablaB|
B+β4⟨Γ o al⟩,(33)
whe e he coe icien s βiencode he e ec i e con ibu ion o each physical channel.
The o de s o magni ude o βia e mo i a ed by he ollowing physical ela ions:
β1simωci
B0
(cyclo on coupling, ωci =eB0/mp),(34)
β2simωpe
ne
=se2
ϵ0mene
(plasma e ec s, elec on plasma equency),(35)
β3sim A(e ec i e mixing eloci y associa ed wi h inhomogenei y),(36)
β4simO(1) (dimensionless weigh o he dissipa i e channel).(37)
(38)
In his wo k, only he coe icien s
β1
,
β2
and he empo al scale
L
a e empi ically calib a ed
om e e ence p o iles
κ e
(
)de ined on simula ed o idealized da a. The coe icien s
β3
and
β4
a e kep ixed wi hin physically plausible anges, in acco dance wi h he bounds discussed in
Sec ion 3.3.1 and Appendix C, and a e no op imized by i ing.
A.5 Connec ion wi h µ(z)and n(p)
The e ec i e equa ion
(32)
has he o m o a Helmhol z equa ion wi h a complex wa enumbe .
We w i e
∇2⟨ψ⟩+k2⟨ψ⟩= 0,(39)
wi h
k2=ω2+i ω κlocal
⟨ 2
A⟩,(40)
which gua an ees ha
k2
has uni s o
m−2
and ha
κlocal
en e s as an imagina y pa (damping).
This o m allows in oducing, by analogy wi h he elec odynamics o media, an e ec i e
plasma impedance. In pa icula , we de ine an e ec i e pe meabili y µe (z)as
µe (z)=µ01 + i κlocal(z)
ω,(41)
whe e he imagina y e m encodes he dissipa i e esponse.
F om he s anda d ela ion
n2
e
=
c2k2/ω2
, wi h
c
he speed o ligh , and using he exp es-
sion (40), we ob ain
n2
e =c2k2
ω2≃c2
⟨ 2
A⟩1 + i κlocal
ω,(42)
whe e he e ec i e pe mi i i y
ϵe
has been abso bed in o he de ini ion o
⟨ 2
A⟩
. The imagina y
pa o
ne
is di ec ly con olled by
κlocal
which ein o ces i s ole as a measu e o e ec i e
medium a enua ion.
41
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