text: − | + E[Y (a¯ , A (a¯ )) Y (a , A (a ), A (a , A (a ))) H (a¯ ) = h ] (cid:2) 4 2 3 2 4 1 2 1 3 1 2 1 (cid:3)2 1 2 | − | + E[Y (a , A (a ), A (a , A (a ))) H (a¯ ) = h ] E[Y (a , A (a ), A (a , A (a ))) H = h ] (cid:2) 4 1 2 1 3 (cid:3) 1 2(cid:2) 1 (cid:3) 2 1 2 4 1 2 1 3 1 2 1 1 1 − | ¯ + E Y (a , A (a ), A (a , A (a ))) Y (A ) H = h (cid:2) 4 1 (cid:3) 2 1 3 1 2 1 4 3 1 1 | − ¯ ¯ + E Y (A ) H = h E Y (A ) 4 3 1 1 4 3 ¯ + E Y (A ) . 4 3 Denote A (a¯ − , A (a¯ − )) by Aa¯u−1, A (a¯ − , A (a¯ − ), Aa¯u−1) by Aa¯u−1 and so on with u+1 u 1 u u 1 u+1 u+2 u 1 u u 1 u+1 u+2 A (a¯ − , A (a¯ − ), Aau−1, . . . , Aa¯−u−1) by Aa¯u−1. Using this compact notation the treatment t u 1 u u 1 u+1 t 1 t blips in (8) can be(cid:2)rewritten as (cid:3) − | µ (h , a¯ ) = E Y (a¯ , Aa¯u , . . . , Aa¯u) Y (a¯ − , Aa¯u−1, . . . , Aa¯u−1) H¯ (a¯ − ) = h . u,t+1 u u t+1 u u+1 t t+1 u 1 u t u u 1 u The telescoping sum for general using this compact notation is t (cid:2) (cid:3) | E[Y (a¯ ) H (a¯ − ) = h ] t+1 t t t 1 t − | = E Y (a¯ ) Y (a¯ − , Aa¯t−1) H (a¯ − ) = h t+1 t t+1 t 1 t t t 1 t | − | + E[Y (a¯ − , Aa¯t−1) H (a¯ − ) = h ] E[Y (a¯ − , Aa¯t−1) H − (a¯ − ) = h − ] t+1 t 1 t t t 1 t t+1 t 1 t t 1 t 2 t 1 − | + E[Y (a¯ − , Aa¯t−1) Y (a¯ − , Aa¯−t−2, Aa¯t−2) H − (a¯ − ) = h − ] t+1 t 1 t t+1 t 2 t 1 t t 1 t 2 t 1 | − | + E[Y (a¯ − , Aa¯−t−2, Aa¯t−2) H − (a¯ − ) = h − ] E[Y (a¯ − , Aa¯−t−2, Aa¯t−2) H − (a¯ − ) = h − ] t+1 t 2 t 1 t t 1 t 2 t 1 t+1 t 2 t 1 t t 2 t 3 t 2 · · · · · · − | + E[Y (a , Aa1, , Aa1) Y (A¯ ) H = h ] t+1 1 2 t t+1 t 1 1 | − ¯ ¯ + E[Y (A ) H = h ] E[Y (A )] t+1 t 1 1 t+1 t (cid:88) (cid:88) ¯ + E[Y (A )] t+1 t t t = E[Y (A¯ )] + µ (h , a¯ ) + (cid:15) (h , a¯ − ), (11) t+1 t u,t+1 u u u,t+1 u u 1 u=1 u=1 where | (cid:15) (h , a¯ − ) = E[Y (a¯ − , Aa¯u−1, . . . , Aa¯u−1) H (a¯ − ) = h ] u,t+1 u u 1 t+1 u 1 u t u u 1 u − | E[Y (a¯ − , Aa¯u−1, . . . , Aa¯u−1) H − (a¯ − ) = h − ], t+1 u 1 u t u 1 u 2 u 1 | are nuisance functions that satisfy the constraint E[(cid:15) (H (a¯ − ), a¯ − ) H − (a¯ − )] = 0, u,t+1 u u 1 u 1 u 1 u 2 A3

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text: Process rATLAS Analysis rCMS Analysis → max max pp χχ 0.37 atlas conf 2013 035 0.08 cms 1303 2985 → pp t˜ t˜(cid:63) 0.15 atlas conf 2013 047 0.44 cms 1502 06031 → 1 1→ pp G gg 0.65 atlas 1308 1841 0.00 cms 1303 2985 → 1× pp g˜g˜ 1 3.33 atlas conf 2013 089 2.89 cms 1502 06031 → × pp g˜g˜ 1 1.31 atlas 1308 1841 0.77 cms 1502 06031 → × 16 pp g˜g˜ 1 0.62 atlas 1308 1841 0.38 cms 1502 06031 32 Table 2: Exclusion r by ATLAS and CMS analyses, obtained with CheckMATE. max √ LAS card with pileup suitable for s = 13 TeV. The ATLAS diphoton search selection is reproduced as far as this is possible with fast simulation. Photons are required to | | | | satisfy η < 2.37, excluding the region 1.37 < η < 1.52. Photon isolation requires γ γ E0.4/E < 0.022 where E0.4 is the calorimeter energy in a cone of radius ∆R = 0.4 around cal γ cal (but excluding) the photon. The analysis requirements are these: • at least two isolated photons with Eγ > 30 GeV • T at least one isolated photon with Eγ > 40 GeV • T leading photon satisﬁes Eγ/m > 0.4 • T γγ subleading photon satisﬁes Eγ/m > 0.3 T γγ where mγγ is the diphoton mass. After full signal selecti(cid:82)on, the signal eﬃciency is approx- imately 40% for both signal samples. See Figure 1 for the signal dip√hoton spectrum after full signal selection, assuming × → L − σ → BR(G γγ) = 8 fb, s = 13 TeV and dt = 3.2 fb 1. The m dis- pp G1 1 γγ tributions for the G , the radial excitations G , and their sum are ﬁt with a double sided 1 n Crystal Ball (DSCB) as deﬁned in the ATLAS search [6]. The ﬁtted full width at half maximum is approximately 40 GeV. Also shown in Figure 1 is the diphoton spectrum of the Pythia8 signal added to the → → → MG5 aMC@NLO background processes qq γγ, qg qγγ and gg γγ together with a background ﬁt. The ﬁt employs the probabi√lity density function used by ATLAS, namely − f(x; b, a0) = (1 x1/3)bxa0 where x = mγγ/ s and a0, b are free parameters [6]. . 5 Conclusion ≈ We have identiﬁed pseudoscalar gluinonia, bound states of gluinos with m 380 GeV, g˜ as an explanation for the diphoton excesses reported by ATLAS and CMS. A light gluino below the threshold for decay to stop or sbottom has a small width because it must decay via virtual squarks, allowing two gluinos to form bound states with widths much larger than the gluino width. A scan is performed in NMSSM parameter space with NMSSMTools4 to identify a ≈ benchmark point g˜ consistent with the benchmark h [4] featuring m 380 GeV which 380 60 g˜ survives experimental constraints. In order to identify a viable point with a gluino mass small enough to lie below the threshold for two-body decay, the gaugino mass uniﬁcation ≈ constraint M 3M is ignored. We verify that g˜ survives LHC8 search constraints with 3 2 380 CheckMATE. An analysis based on the ATLAS diphoton search is carried out with events generated by Pythia8 with Delphes3 d√etector simulation which reproduces the important features of the diphoton excess in the s = 13 TeV data. We ﬁnd that to reproduce the 750 GeV diphoton excess requires an enhancement of the nominal gluinonium to diphoton signal × strength of 40, and assume that some of this enhancement is due to the underestimate | | of the ground state wavefunction at the origin ψ(0) 2 in the literature. An explanation for the remaining enhancement is undetermined. Acknowledgements 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: [15] Yevgeny Kats and Matthew J. Strassler. Resonances from QCD bound states and the 750 GeV diphoton excess. JHEP, 05:092, 2016, 1602.08819. [16] Ulrich Ellwanger, John F. Gunion, and Cyril Hugonie. NMHDECAY: A Fortran code for the Higgs masses, couplings and decay widths in the NMSSM. JHEP, 0502:066, 2005, hep-ph/0406215. [17] Ulrich Ellwanger and Cyril Hugonie. NMHDECAY 2.0: An Updated program for sparticle masses, Higgs masses, couplings and decay widths in the NMSSM. Com- put.Phys.Commun., 175:290–303, 2006, hep-ph/0508022. [18] G. Belanger, F. Boudjema, C. Hugonie, A. Pukhov, and A. Semenov. Relic density of dark matter in the NMSSM. JCAP, 0509:001, 2005, hep-ph/0505142. [19] Ulrich Ellwanger and Cyril Hugonie. NMSPEC: A Fortran code for the sparti- cle and Higgs masses in the NMSSM with GUT scale boundary conditions. Com- put.Phys.Commun., 177:399–407, 2007, hep-ph/0612134. [20] Debottam Das, Ulrich Ellwanger, and Ana M. Teixeira. NMSDECAY: A Fortran Code for Supersymmetric Particle Decays in the Next-to-Minimal Supersymmetric Standard Model. Comput.Phys.Commun., 183:774–779, 2012, 1106.5633. [21] M. Muhlleitner, A. Djouadi, and Y. Mambrini. SDECAY: A Fortran code for the decays of the supersymmetric particles in the MSSM. Comput.Phys.Commun., 168:46–70, 2005, hep-ph/0311167. [22] Vardan Khachatryan et al. Search for Physics Beyond the Standard Model in Events with Two Leptons, Jets, and Missing Transverse Momentum in pp Collisions at sqrt(s) = 8 TeV. JHEP, 04:124, 2015, 1502.06031. [23] Manuel Drees, Herbi Dreiner, Daniel Schmeier, Jamie Tattersall, and Jong Soo Kim. CheckMATE: Confronting your Favourite New Physics Model with LHC Data. Com- put. Phys. Commun., 187:227–265, 2014, 1312.2591. [24] Torbjorn Sjostrand, Stephen Mrenna, and Peter Z. Skands. A Brief Introduction to PYTHIA 8.1. Comput.Phys.Commun., 178:852–867, 2008, 0710.3820. [25] Torbjorn Sjostrand, Stephen Mrenna, and Peter Z. Skands. PYTHIA 6.4 Physics and Manual. JHEP, 0605:026, 2006, hep-ph/0603175. [26] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro. The automated computation of tree-level and next-to-leading order diﬀerential cross sections, and their matching to parton shower simulations. JHEP, 07:079, 2014, 1405.0301. [27] J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaˆıtre, A. Mertens, and M. Selvaggi. DELPHES 3, A modular framework for fast simulation of a generic collider experiment. JHEP, 02:057, 2014, 1307.6346. 6

label: reference

text: Yevgeny

text: Kats

text: and

text: Matthew

text: J.

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text: Resonances

text: from

text: QCD

text: bound

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text: and

text: the

text: GeV

text: diphoton

text: excess.

text: JHEP,

text: Ulrich

text: Ellwanger,

text: John

text: F.

text: Gunion,

text: and

text: Cyril

text: Hugonie.

text: NMHDECAY:

text: A

text: Fortran

text: code

text: for

text: the

text: Higgs

text: masses,

text: couplings

text: and

text: decay

text: widths

text: in

text: the

text: NMSSM.

text: JHEP,

text: Ulrich

text: Ellwanger

text: and

text: Cyril

text: Hugonie.

text: NMHDECAY

text: An

text: Updated

text: program

text: for

text: sparticle

text: masses,

text: Higgs

text: masses,

text: couplings

text: and

text: decay

text: widths

text: in

text: the

text: NMSSM.

text: Com-

text: put.Phys.Commun.,

text: G.

text: Belanger,

text: F.

text: Boudjema,

text: C.

text: Hugonie,

text: A.

text: Pukhov,

text: and

text: A.

text: Semenov.

text: Relic

text: density

text: of

text: dark

text: matter

text: in

text: the

text: NMSSM.

text: JCAP,

text: Ulrich

text: Ellwanger

text: and

text: Cyril

text: Hugonie.

text: NMSPEC:

text: A

text: Fortran

text: code

text: for

text: the

text: sparti-

text: cle

text: and

text: Higgs

text: masses

text: in

text: the

text: NMSSM

text: with

text: GUT

text: scale

text: boundary

text: conditions.

text: Com-

text: put.Phys.Commun.,

text: Debottam

text: Das,

text: Ulrich

text: Ellwanger,

text: and

text: Ana

text: M.

text: Teixeira.

text: NMSDECAY:

text: A

text: Fortran

text: Code

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text: Supersymmetric

text: Particle

text: Decays

text: in

text: the

text: Next-to-Minimal

text: Supersymmetric

text: Standard

text: Model.

text: Comput.Phys.Commun.,

text: M.

text: Muhlleitner,

text: A.

text: Djouadi,

text: and

text: Y.

text: Mambrini.

text: SDECAY:

text: A

text: Fortran

text: code

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text: the

text: decays

text: of

text: the

text: supersymmetric

text: particles

text: in

text: the

text: MSSM.

text: Comput.Phys.Commun.,

text: Vardan

text: Khachatryan

text: et

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text: Search

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text: Standard

text: Model

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text: Herbi

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text: Schmeier,

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text: and

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text: Kim.

text: CheckMATE:

text: Confronting

text: your

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text: Com-

text: put.

text: Phys.

text: Commun.,

text: Torbjorn

text: Sjostrand,

text: Stephen

text: Mrenna,

text: and

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text: Z.

text: Skands.

text: A

text: Brief

text: Introduction

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text: PYTHIA

text: Comput.Phys.Commun.,

text: Torbjorn

text: Sjostrand,

text: Stephen

text: Mrenna,

text: and

text: Peter

text: Z.

text: Skands.

text: PYTHIA

text: Physics

text: and

text: Manual.

text: JHEP,

text: J.

text: Alwall,

text: R.

text: Frederix,

text: S.

text: Frixione,

text: V.

text: Hirschi,

text: F.

text: Maltoni,

text: O.

text: Mattelaer,

text: H.

text: S.

text: Shao,

text: T.

text: Stelzer,

text: P.

text: Torrielli,

text: and

text: M.

text: Zaro.

text: The

text: automated

text: computation

text: of

text: tree-level

text: and

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text: cross

text: sections,

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text: simulations.

text: JHEP,

text: J.

text: de

text: Favereau,

text: C.

text: Delaere,

text: P.

text: Demin,

text: A.

text: Giammanco,

text: V.

text: Lemaˆıtre,

text: A.

text: Mertens,

text: and

text: M.

text: Selvaggi.

text: DELPHES

text: A

text: modular

text: framework

text: for

text: fast

text: simulation

text: of

text: a

text: generic

text: collider

text: experiment.

text: JHEP,


text: Bifurcations in a Q2D Kolmogorov-Like Flow 17 0.3 Exp Exp NPS (ν: +7%) 5.5 NPS (ν: +7%) NPS (α: +22%) NPS (α: +22%) 0.2 NPS (β: −6%) 5 NPS (β: +6%) hu2yi λ¯ 4.5 x 0.1 4 3.5 0 10 12 14 16 18 0 1 2 3 4 5 6 Re Re Re − c (a) (b) Figure 8: The eﬀect of the variation in model parameters. (a) A bifurcation diagram and (b) the average wavelength of the pattern in the modulated ﬂow regime. The numerical results correspond to either a 7% increase in ν, a 22% increase in α, or a 6% decrease in β − compared with the depth-averaged values for the straight ﬂow (α = 0.064 s 1, β = 0.83, × − and ν = 3.26 10 6 m2/s). The uncertainty bars in (b) are only shown for every other data point for clarity. what underestimates the critical Reynolds number. To resolve this discrepancy, we tested the sensitivity of the transition in the NPS to changes in the forcing proﬁle as well as variations in parameters β, ν, and α. In particular, we found that Re is fairly insensitive c to spatial variation in the strength of the magnets in the array. Consequently, we turned our attention to studying the sensitivity of Re to the values of the parameters β, ν, c and α. In order to match Re in the NPS and experiment, we had to either decrease c β by 6%, increase ν by 7%, or increase α by 22%. Figures 8 (a) and (b) show that the variation of parameters has a fairly weak eﬀect on both the amplitude of the modulation and the wavelength of the pattern, which suggests that the disagreement between the simulation and experiment is primarily due to the deviation of the ﬂow and/or forcing from quasi-two-dimensionality, which is discussed in Appendix . Note that the Reynolds number Re = Uw/ν is deﬁned using the measured rms velocity U and parameter ν which cannot be measured, but has to be computed. While in the simulation the value of ν is well-deﬁned (it is one of the parameters of the model), in the experiment it is not, so the corresponding Re depends on the choice of ν. Hence, to enable a proper comparison of experiment with numerics, we deﬁned Re in both cases × − using the analytically computed depth-averaged value ν = 3.26 10 6 m2/s (Suri et al. 2014), regardless of the actual value of ν used in the simulation. Matching Re using this convention is eﬀectively equivalent to matching the rms velocity U. 4.4. Secondary Instability As we increase the forcing further, the modulated ﬂow in the experiment becomes ± ± unstable giving way to a time-periodic ﬂow with a period T = 42.8 0.4 (120 1 s in p± dimensional units) at onset, which corresponds to Re = 17.6 0.1. The modulated state −p × − in the NPS, with the depth-averaged (α = 0.064 s 1, β = 0.83, and ν = 3.26 10 6 m2/s) as well as the adjusted parameters, undergoes a Hopf bifurcation as the forcing is increased. Table 1 compares Re and T in the experiment with those from the NPS p p for the diﬀerent parameter sets. A video comparing the time-periodic ﬂow in the NPS ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: Bifurcations

text: in

text: a

text: Kolmogorov-Like

text: Flow

font: default

text: Exp

text: Exp

text: NPS

text: (ν:

text: NPS

text: (ν:

text: NPS

text: (α:

text: NPS

text: (α:

text: NPS

text: (β:

text: NPS

text: (β:

text: λ¯

text: x

text: Re

text: Re

text: Re

text: c

text: (a)

text: (b)

text: Figure

text: The

text: eﬀect

text: of

text: the

text: variation

text: in

text: model

text: parameters.

text: (a)

text: A

text: bifurcation

text: diagram

text: and

text: (b)

text: the

text: average

text: wavelength

text: of

text: the

text: pattern

text: in

text: the

text: modulated

text: ﬂow

text: regime.

text: The

text: numerical

text: results

text: correspond

text: to

text: either

text: a

text: increase

text: in

text: ν,

text: a

text: increase

text: in

text: α,

text: or

text: a

text: decrease

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text: β

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text: the

text: depth-averaged

text: values

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text: the

text: straight

text: ﬂow

text: (α

text: s

text: β

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text: ν

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text: uncertainty

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text: (b)

text: are

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text: shown

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text: every

text: other

text: data

text: point

text: for

text: clarity.

text: what

text: underestimates

text: the

text: critical

text: Reynolds

text: number.

text: To

text: resolve

text: this

text: discrepancy,

text: we

text: tested

text: the

text: sensitivity

text: of

text: the

text: transition

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text: we

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text: that

text: Re

text: is

text: fairly

text: insensitive

text: c

text: to

text: spatial

text: variation

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text: the

text: strength

text: of

text: the

text: magnets

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text: the

text: array.

text: Consequently,

text: we

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text: sensitivity

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text: the

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text: Re

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text: the

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text: we

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text: increase

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text: or

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text: and

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text: show

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text: the

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text: a

text: fairly

text: weak

text: eﬀect

text: on

text: both

text: the

text: amplitude

text: of

text: the

text: modulation

text: and

text: the

text: wavelength

text: of

text: the

text: pattern,

text: which

text: suggests

text: that

text: the

text: disagreement

text: between

text: the

text: simulation

text: and

text: experiment

text: is

text: primarily

text: due

text: to

text: the

text: deviation

text: of

text: the

text: ﬂow

text: and/or

text: forcing

text: from

text: quasi-two-dimensionality,

text: which

text: is

text: discussed

text: in

text: Appendix

text: Note

text: that

text: the

text: Reynolds

text: number

text: Re

text: Uw/ν

text: is

text: deﬁned

text: using

text: the

text: measured

text: rms

text: velocity

text: U

text: and

text: parameter

text: ν

text: which

text: cannot

text: be

text: measured,

text: but

text: has

text: to

text: be

text: computed.

text: While

text: in

text: the

text: simulation

text: the

text: value

text: of

text: ν

text: is

text: well-deﬁned

text: (it

text: is

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text: the

text: corresponding

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text: the

text: choice

text: of

text: ν.

text: Hence,

text: to

text: enable

text: a

text: proper

text: comparison

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text: experiment

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text: numerics,

text: we

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text: Re

text: in

text: both

text: cases

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text: analytically

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text: depth-averaged

text: value

text: ν

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text: et

text: al.

text: regardless

text: of

text: the

text: actual

text: value

text: of

text: ν

text: used

text: in

text: the

text: simulation.

text: Matching

text: Re

text: using

text: this

text: convention

text: is

text: eﬀectively

text: equivalent

text: to

text: matching

text: the

text: rms

text: velocity

text: U.

text: Secondary

text: Instability

text: As

text: we

text: increase

text: the

text: forcing

text: further,

text: the

text: modulated

text: ﬂow

text: in

text: the

text: experiment

text: becomes

text: unstable

text: giving

text: way

text: to

text: a

text: time-periodic

text: ﬂow

text: with

text: a

text: period

text: T

text: s

text: in

text: p±

text: dimensional

text: units)

text: at

text: onset,

text: which

text: corresponds

text: to

text: Re

text: The

text: modulated

text: state

text: −p

text: in

text: the

text: NPS,

text: with

text: the

text: depth-averaged

text: (α

text: s

text: β

text: and

text: ν

text: as

text: well

text: as

text: the

text: adjusted

text: parameters,

text: undergoes

text: a

text: Hopf

text: bifurcation

text: as

text: the

text: forcing

text: is

text: increased.

text: Table

text: compares

text: Re

text: and

text: T

text: in

text: the

text: experiment

text: with

text: those

text: from

text: the

text: NPS

text: p

text: p

text: for

text: the

text: diﬀerent

text: parameter

text: sets.

text: A

text: video

text: comparing

text: the

text: time-periodic

text: ﬂow

text: in

text: the

text: NPS

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text: 18 J. Tithof, B. Suri, R. K. Pallantla, R. O. Grigoriev, M. F. Schatz Re T p p ± ± Experiment 17.6 0.1 42.8 (120 1 s) NPS (Depth-Averaged) 15.6 43.2 (137 s) NPS (ν: +7%) 16.4 43.2 (130 s) NPS (α: +22%) 17.1 48.1 (139 s) − NPS (β: 6%) 16.5 45.8 (137 s) Table 1: Critical transition parameters characterizing the stable periodic regime for the experiment and the NPS with diﬀerent sets of parameters. (with depth-averaged parameters) and experiment is included as online supplementary material. Re and T in the simulation are within 15% of the experimental measurements for p p the depth-averaged values of parameters computed using the vertical proﬁle P(z) that corresponds to the straight ﬂow. However, the values of ν, α, and β should vary slowly with Re, since P(z) is weakly dependent on the horizontal ﬂow proﬁle. Hence, a diﬀerent set of parameters is required to describe the two instabilities and, more generally, there is no universal set of parameters β, ν, and α that correctly describes the experimental ﬂow at all Re. From table 1 we see that Re and T show very diﬀerent sensitivity to p p changes in each of the parameters. Hence, while separately modifying ν, α, and β shows some improvement in matching either Re or T , it should be possible to obtain even p p better agreement by modifying all the model parameters simultaneously, each by only a few percent. (cid:112) The necessity for modifying parameters across diﬀerent dynamical regimes also raises the question of how robust α, β, and ν are to changes in the (local) wavenumber of the ≈ ﬂow. To test this, we have recomputed the parameters using the wavenumber k 5/4κ associated with the modulated ﬂow. We found that β and ν change by less than 1%, and α by about 3.5%, compared to those computed using k = κ. This robustness suggests that, once adjusted to match the experiment, the 2D model should provide a reasonably accurate description of the dynamics even in the weakly turbulent regime where the wavenumber may vary in space and time (Suri et al. 2017). 5. Nature of the Primary Instability An important consequence of conﬁning the ﬂow in the longitudinal or transverse direc- tions is that we restrict the set of coordinate transformations (symmetries) that leave the governing equation (1.4) equivariant. The symmetries of the governing equation, in turn, determine the number of, and the relation between, distinct modulated ﬂow solutions created as a result of the primary bifurcation. Below we discuss each of the diﬀerent ﬂow domains, in the order of decreasing symmetry. 5.1. DPS On an unbounded or a doubly-periodic domain, equation (1.4) is equivariant under the following symmetry operations (Chandler & Kerswell 2013): T → (a) Continuous shift by δx in x: δx(x, y) (x + δx, y). x R T → (b) Reﬂection in x combined with a discrete shift of half a period in y: w(x, y) − x y ( x, y + w). R R → − − (c) Reﬂections in both x and y: (x, y) ( x, y). R Rx y Note that the double reﬂection is equivalent to a rotation by angle π about the x y R T z-axis, while the square of the symmetry operation w corresponds to a discrete T R T T x y ≤ shift 2w in the y-direction, i.e., ( w)2 = 2w. For the DPS, 0 δx < L , so the y x y y x ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: J.

text: Tithof,

text: B.

text: Suri,

text: R.

text: K.

text: Pallantla,

text: R.

text: O.

text: Grigoriev,

text: M.

text: F.

text: Schatz

text: Re

text: T

text: p

text: p

text: Experiment

text: s)

text: NPS

text: (Depth-Averaged)

text: s)

text: NPS

text: (ν:

text: s)

text: NPS

text: (α:

text: s)

text: NPS

text: (β:

text: s)

text: Table

text: Critical

text: transition

text: parameters

text: characterizing

text: the

text: stable

text: periodic

text: regime

text: for

text: the

text: experiment

text: and

text: the

text: NPS

text: with

text: diﬀerent

text: sets

text: of

text: parameters.

text: (with

text: depth-averaged

text: parameters)

text: and

text: experiment

text: is

text: included

text: as

text: online

text: supplementary

text: material.

text: Re

text: and

text: T

text: in

text: the

text: simulation

text: are

text: within

text: of

text: the

text: experimental

text: measurements

text: for

text: p

text: p

text: the

text: depth-averaged

text: values

text: of

text: parameters

text: computed

text: using

text: the

text: vertical

text: proﬁle

text: P(z)

text: that

text: corresponds

text: to

text: the

text: straight

text: ﬂow.

text: However,

text: the

text: values

text: of

text: ν,

text: α,

text: and

text: β

text: should

text: vary

text: slowly

text: with

text: Re,

text: since

text: P(z)

text: is

text: weakly

text: dependent

text: on

text: the

text: horizontal

text: ﬂow

text: proﬁle.

text: Hence,

text: a

text: diﬀerent

text: set

text: of

text: parameters

text: is

text: required

text: to

text: describe

text: the

text: two

text: instabilities

text: and,

text: more

text: generally,

text: there

text: is

text: no

text: universal

text: set

text: of

text: parameters

text: β,

text: ν,

text: and

text: α

text: that

text: correctly

text: describes

text: the

text: experimental

text: ﬂow

text: at

text: all

text: Re.

text: From

text: table

text: we

text: see

text: that

text: Re

text: and

text: T

text: show

text: very

text: diﬀerent

text: sensitivity

text: to

text: p

text: p

text: changes

text: in

text: each

text: of

text: the

text: parameters.

text: Hence,

text: while

text: separately

text: modifying

text: ν,

text: α,

text: and

text: β

text: shows

text: some

text: improvement

text: in

text: matching

text: either

text: Re

text: or

text: T

text: it

text: should

text: be

text: possible

text: to

text: obtain

text: even

text: p

text: p

text: better

text: agreement

text: by

text: modifying

text: all

text: the

text: model

text: parameters

text: simultaneously,

text: each

text: by

text: only

text: a

text: few

text: percent.

text: The

text: necessity

text: for

text: modifying

text: parameters

text: across

text: diﬀerent

text: dynamical

text: regimes

text: also

text: raises

text: the

text: question

text: of

text: how

text: robust

text: α,

text: β,

text: and

text: ν

text: are

text: to

text: changes

text: in

text: the

text: (local)

text: wavenumber

text: of

text: the

text: ﬂow.

text: To

text: test

text: this,

text: we

text: have

text: recomputed

text: the

text: parameters

text: using

text: the

text: wavenumber

text: k

text: associated

text: with

text: the

text: modulated

text: ﬂow.

text: We

text: found

text: that

text: β

text: and

text: ν

text: change

text: by

text: less

text: than

text: and

text: α

text: by

text: about

text: compared

text: to

text: those

text: computed

text: using

text: k

text: κ.

text: This

text: robustness

text: suggests

text: that,

text: once

text: adjusted

text: to

text: match

text: the

text: experiment,

text: the

text: model

text: should

text: provide

text: a

text: reasonably

text: accurate

text: description

text: of

text: the

text: dynamics

text: even

text: in

text: the

text: weakly

text: turbulent

text: regime

text: where

text: the

text: wavenumber

text: may

text: vary

text: in

text: space

text: and

text: time

text: (Suri

text: et

text: al.

text: Nature

text: of

text: the

text: Primary

text: Instability

text: An

text: important

text: consequence

text: of

text: conﬁning

text: the

text: ﬂow

text: in

text: the

text: longitudinal

text: or

text: transverse

text: direc-

text: tions

text: is

text: that

text: we

text: restrict

text: the

text: set

text: of

text: coordinate

text: transformations

text: (symmetries)

text: that

text: leave

text: the

text: governing

text: equation

text: equivariant.

text: The

text: symmetries

text: of

text: the

text: governing

text: equation,

text: in

text: turn,

text: determine

text: the

text: number

text: of,

text: and

text: the

text: relation

text: between,

text: distinct

text: modulated

text: ﬂow

text: solutions

text: created

text: as

text: a

text: result

text: of

text: the

text: primary

text: bifurcation.

text: Below

text: we

text: discuss

text: each

text: of

text: the

text: diﬀerent

text: ﬂow

text: domains,

text: in

text: the

text: order

text: of

text: decreasing

text: symmetry.

text: DPS

text: On

text: an

text: unbounded

text: or

text: a

text: doubly-periodic

text: domain,

text: equation

text: is

text: equivariant

text: under

text: the

text: following

text: symmetry

text: operations

text: (Chandler

text: Kerswell

text: T

text: (a)

text: Continuous

text: shift

text: by

text: δx

text: in

text: x:

text: δx(x,

text: y)

text: (x

text: δx,

text: y).

text: x

text: R

text: T

text: (b)

text: Reﬂection

text: in

text: x

text: combined

text: with

text: a

text: discrete

text: shift

text: of

text: half

text: a

text: period

text: in

text: y:

text: w(x,

text: y)

text: x

text: y

text: x,

text: y

text: w).

text: R

text: R

text: (c)

text: Reﬂections

text: in

text: both

text: x

text: and

text: y:

text: (x,

text: y)

text: x,

text: y).

text: R

text: Rx

text: y

text: Note

text: that

text: the

text: double

text: reﬂection

text: is

text: equivalent

text: to

text: a

text: rotation

text: by

text: angle

text: π

text: about

text: the

text: x

text: y

text: R

text: T

text: z-axis,

text: while

text: the

text: square

text: of

text: the

text: symmetry

text: operation

text: w

text: corresponds

text: to

text: a

text: discrete

text: T

text: R

text: T

text: T

text: x

text: y

text: shift

text: in

text: the

text: y-direction,

text: i.e.,

text: For

text: the

text: DPS,

text: δx

text: L

text: so

text: the

text: y

text: x

text: y

text: y

text: x

text: ##LTLine##

text: ##LTLine##

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text: 10 J. Tithof, B. Suri, R. K. Pallantla, R. O. Grigoriev, M. F. Schatz ical simulations. The ﬂow over the doubly-periodic domain is simulated using a pseudo- spectral method in the vorticity-stream function formulation, as described in Mitchell (2013). This simulation is henceforth referred to as the “doubly-periodic simulation,” abbreviated DPS. For the singly-periodic and the non-periodic domains, numerical sim- ulations have been performed using a ﬁnite-diﬀerence scheme, described in Armﬁeld & Street (1999). These simulations are hereafter referred to as the “singly-periodic sim- ulation” (SPS) and the “non-periodic simulation” (NPS), respectively. Details of the spatiotemporal discretization and the integration schemes employed in all the numerical simulations can be found in Appendix . 4. Comparison of Experiment and Simulations In this section we present the results of our comparison between the experiment and the numerical simulations on the three domains described above. First, we discuss the straight uniform ﬂow found at lower Reynolds numbers, with a special emphasis on the eﬀect of boundaries. We then perform linear stability analysis to demonstrate that equation (1.4) describes the primary instability more accurately than equation (1.2) on an unbounded domain, but still substantially underpredicts the experimentally observed critical Reynolds number. Next, we describe and compare the steady ﬂow states found in the experiments and simulations above the primary instability. Finally, we discuss the secondary instability which gives rise to a time-periodic ﬂow. 4.1. Straight Flow For weak driving, the ﬂow mimics the forcing closely, with spatially alternating bands ± of ﬂuid ﬂow along the x-directions, as can be seen in ﬁgure 3 for Re = 8.1. In this ﬁgure, black vectors represent the velocity ﬁeld u and the colour indicates the vorticity ∇ × · ω = ( u) ˆz. For the experiment (ﬁgure 3 (d)), the y-component of the velocity measured in the central region of the domain is close to zero. However, there are regions of strong recirculation near the end walls, characterized by a nonzero y-component of velocity. A closer inspection of the ﬂow shows a slight tilt in the alignment of the ﬂow bands. This tilt is due to the global circulation, resulting from conﬁnement and the ﬂuid ± ﬂowing in opposite directions over the end magnets at y = 6.5. Figures 3 (a) and (b) show the straight ﬂows found in the DPS and SPS. It can be seen that ﬂow ﬁelds in the DPS and SPS reproduce the experimental ﬂow qualitatively away from the lateral walls. Futhermore, the SPS captures the turnaround ﬂow near the end walls. However, neither the SPS nor the DPS displays the tilt of the ﬂow bands observed in the experiment since the periodic ﬂows are devoid of global circulation. In contrast, the NPS generates a ﬂow ﬁeld that looks indistinguishable from the experimental one (cf. ﬁgure 3 (c)). For a quantitative description of the straight ﬂow proﬁle, we have plotted in ﬁgure 4 (a) the longitudinal component uexp of the velocity along the line x = 0 in the experiment. x The location of this cross section is indicated by the vertical dashed line in ﬁgure 3 (d). The diﬀerence in u between the experiment and the numerical simulations along x this line is shown in ﬁgure 4 (b). As can be seen, the DPS and SPS, which are only | | ≤ deﬁned for y 4, show systematic deviation from the experiment as high as 18% since they do not capture global circulation. In comparison, the NPS agrees to within about 5% over the same region, with no clear systematic deviation. The disagreement between the experiment and NPS in this region, we believe, is a result of the dipole summation not accounting for the variation in the strength of each individual magnet. Closer to ≈ ≈ − the boundaries, at y 7 and y 6, the largest diﬀerence between the NPS and the

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text: using

text: a

text: pseudo-

text: spectral

text: method

text: in

text: the

text: vorticity-stream

text: function

text: formulation,

text: as

text: described

text: in

text: Mitchell

text: This

text: simulation

text: is

text: henceforth

text: referred

text: to

text: as

text: the

text: “doubly-periodic

text: simulation,”

text: abbreviated

text: DPS.

text: For

text: the

text: singly-periodic

text: and

text: the

text: non-periodic

text: domains,

text: numerical

text: sim-

text: ulations

text: have

text: been

text: performed

text: using

text: a

text: ﬁnite-diﬀerence

text: scheme,

text: described

text: in

text: Armﬁeld

text: Street

text: These

text: simulations

text: are

text: hereafter

text: referred

text: to

text: as

text: the

text: “singly-periodic

text: sim-

text: ulation”

text: (SPS)

text: and

text: the

text: “non-periodic

text: simulation”

text: (NPS),

text: respectively.

text: Details

text: of

text: the

text: spatiotemporal

text: discretization

text: and

text: the

text: integration

text: schemes

text: employed

text: in

text: all

text: the

text: numerical

text: simulations

text: can

text: be

text: found

text: in

text: Appendix

text: Comparison

text: of

text: Experiment

text: and

text: Simulations

text: In

text: this

text: section

text: we

text: present

text: the

text: results

text: of

text: our

text: comparison

text: between

text: the

text: experiment

text: and

text: the

text: numerical

text: simulations

text: on

text: the

text: three

text: domains

text: described

text: above.

text: First,

text: we

text: discuss

text: the

text: straight

text: uniform

text: ﬂow

text: found

text: at

text: lower

text: Reynolds

text: numbers,

text: with

text: a

text: special

text: emphasis

text: on

text: the

text: eﬀect

text: of

text: boundaries.

text: We

text: then

text: perform

text: linear

text: stability

text: analysis

text: to

text: demonstrate

text: that

text: equation

text: describes

text: the

text: primary

text: instability

text: more

text: accurately

text: than

text: equation

text: on

text: an

text: unbounded

text: domain,

text: but

text: still

text: substantially

text: underpredicts

text: the

text: experimentally

text: observed

text: critical

text: Reynolds

text: number.

text: Next,

text: we

text: describe

text: and

text: compare

text: the

text: steady

text: ﬂow

text: states

text: found

text: in

text: the

text: experiments

text: and

text: simulations

text: above

text: the

text: primary

text: instability.

text: Finally,

text: we

text: discuss

text: the

text: secondary

text: instability

text: which

text: gives

text: rise

text: to

text: a

text: time-periodic

text: ﬂow.

text: Straight

text: Flow

text: For

text: weak

text: driving,

text: the

text: ﬂow

text: mimics

text: the

text: forcing

text: closely,

text: with

text: spatially

text: alternating

text: bands

text: of

text: ﬂuid

text: ﬂow

text: along

text: the

text: x-directions,

text: as

text: can

text: be

text: seen

text: in

text: ﬁgure

text: for

text: Re

text: In

text: this

text: ﬁgure,

text: black

text: vectors

text: represent

text: the

text: velocity

text: ﬁeld

text: u

text: and

text: the

text: colour

text: indicates

text: the

text: vorticity

text: ω

text: u)

text: ˆz.

text: For

text: the

text: experiment

text: (ﬁgure

text: (d)),

text: the

text: y-component

text: of

text: the

text: velocity

text: measured

text: in

text: the

text: central

text: region

text: of

text: the

text: domain

text: is

text: close

text: to

text: zero.

text: However,

text: there

text: are

text: regions

text: of

text: strong

text: recirculation

text: near

text: the

text: end

text: walls,

text: characterized

text: by

text: a

text: nonzero

text: y-component

text: of

text: velocity.

text: A

text: closer

text: inspection

text: of

text: the

text: ﬂow

text: shows

text: a

text: slight

text: tilt

text: in

text: the

text: alignment

text: of

text: the

text: ﬂow

text: bands.

text: This

text: tilt

text: is

text: due

text: to

text: the

text: global

text: circulation,

text: resulting

text: from

text: conﬁnement

text: and

text: the

text: ﬂuid

text: ﬂowing

text: in

text: opposite

text: directions

text: over

text: the

text: end

text: magnets

text: at

text: y

text: Figures

text: (a)

text: and

text: (b)

text: show

text: the

text: straight

text: ﬂows

text: found

text: in

text: the

text: DPS

text: and

text: SPS.

text: It

text: can

text: be

text: seen

text: that

text: ﬂow

text: ﬁelds

text: in

text: the

text: DPS

text: and

text: SPS

text: reproduce

text: the

text: experimental

text: ﬂow

text: qualitatively

text: away

text: from

text: the

text: lateral

text: walls.

text: Futhermore,

text: the

text: SPS

text: captures

text: the

text: turnaround

text: ﬂow

text: near

text: the

text: end

text: walls.

text: However,

text: neither

text: the

text: SPS

text: nor

text: the

text: DPS

text: displays

text: the

text: tilt

text: of

text: the

text: ﬂow

text: bands

text: observed

text: in

text: the

text: experiment

text: since

text: the

text: periodic

text: ﬂows

text: are

text: devoid

text: of

text: global

text: circulation.

text: In

text: contrast,

text: the

text: NPS

text: generates

text: a

text: ﬂow

text: ﬁeld

text: that

text: looks

text: indistinguishable

text: from

text: the

text: experimental

text: one

text: (cf.

text: ﬁgure

text: (c)).

text: For

text: a

text: quantitative

text: description

text: of

text: the

text: straight

text: ﬂow

text: proﬁle,

text: we

text: have

text: plotted

text: in

text: ﬁgure

text: (a)

text: the

text: longitudinal

text: component

text: uexp

text: of

text: the

text: velocity

text: along

text: the

text: line

text: x

text: in

text: the

text: experiment.

text: x

text: The

text: location

text: of

text: this

text: cross

text: section

text: is

text: indicated

text: by

text: the

text: vertical

text: dashed

text: line

text: in

text: ﬁgure

text: (d).

text: The

text: diﬀerence

text: in

text: u

text: between

text: the

text: experiment

text: and

text: the

text: numerical

text: simulations

text: along

text: x

text: this

text: line

text: is

text: shown

text: in

text: ﬁgure

text: (b).

text: As

text: can

text: be

text: seen,

text: the

text: DPS

text: and

text: SPS,

text: which

text: are

text: only

text: deﬁned

text: for

text: y

text: show

text: systematic

text: deviation

text: from

text: the

text: experiment

text: as

text: high

text: as

text: since

text: they

text: do

text: not

text: capture

text: global

text: circulation.

text: In

text: comparison,

text: the

text: NPS

text: agrees

text: to

text: within

text: about

text: over

text: the

text: same

text: region,

text: with

text: no

text: clear

text: systematic

text: deviation.

text: The

text: disagreement

text: between

text: the

text: experiment

text: and

text: NPS

text: in

text: this

text: region,

text: we

text: believe,

text: is

text: a

text: result

text: of

text: the

text: dipole

text: summation

text: not

text: accounting

text: for

text: the

text: variation

text: in

text: the

text: strength

text: of

text: each

text: individual

text: magnet.

text: Closer

text: to

text: the

text: boundaries,

text: at

text: y

text: and

text: y

text: the

text: largest

text: diﬀerence

text: between

text: the

text: NPS

text: and

text: the