Bao Yu

Anna Perfetto

Luca Allievi

Dr Sabrina Dhambri

Marie-Noelle Rager

Dr Mohamed Selkti

Ilaria Ciofini
email: ilaria.ciofini@chimie-paristech.fr

Marie-Isabelle Lannou
email: marie-isabelle.lannou@parisdescartes.fr

Geoffroy Sorin
email: geoffroy.sorin@parisdescartes.fr

Dr S Yu

Dr M.-I Lannou

Dr M.-N Rager

Silver(I) oxide -/DBU -promoted synthesis of dihydrofuran units through allenyl silver formation

Keywords: Allenyl silver, dihydrofuran, isomerization, intramolecular cyclization

A formal [3+2] cyclization mediated by silver(I) oxide and DBU is described herein. Through a broad variety of carbonyl compounds, this system can promote cyclization reactions with high yield (up to 85%) and diastereoselectivity (up to 95:5) for a straightforward access to complex and congested dihydrofuran derivatives in one step using smooth conditions. Based on DFT studies, the proposed mechanism would involve an allenyl silver intermediate.

B. Y.

Introduction

2,3-Dihydrofuran sub-units can be encountered in a wide range of natural and synthetic products exhibiting, in some cases, biologically interesting profiles. [1] Additionally, such scaffolds can also be regarded as attractive precursors in an array of organic transformations. [2] Based on these considerations, considerable efforts have been devoted to the development of efficient accesses to 2,3-dihydrofuran units by various approaches. [START_REF] Gais | For some reactions using chiral auxiliaries and precursors[END_REF][START_REF] Davies | For some metal-mediated cyclization reactions[END_REF]5] Among them, intra-or intermolecular [3+2] cycloadditions constitute a powerful tool to reach such heterocycles [6] and more generally, heterocyclic 5-membered rings. In this field, the vast majority of transformations encountered implies hetero 1,3-dipolar reagents or their synthetic equivalent for reaching a broad variety of heterocyclic molecules. [7] However, access to cyclic frameworks by means of cycloadditions reveals to be much more challenging in cases where all-carbon 1,3-dipoles are involved due to the difficulty to generate such species. [START_REF]For selected reviews and books[END_REF] To overcome this issue, the activation of double or triple bonds by means of transition metals appears to be an attractive method. However, very specific substrates are required, which constitutes a main drawback and narrows the application field. [START_REF]For selected reviews and books[END_REF] One area of improvement could consist in taking advantage of the strong alkynophilicity of "coinage metals", to carry out such kind of cyclizations. Indeed, in 2007 Zhang's group reported an elegant [3+2] cycloaddition for the diastereoselective synthesis of cyclopentanone enol ethers using enones and allenyl MOM ethers mediated by gold catalysis. [9] One year later, the same group published the synthesis of different 2,5-dihydrofuran motifs through Au-containing all-carbon 1,3-dipoles from propargylic ketals, showing the ability of gold to promote such cyclizations. [10] In the course of their work on dual organocatalysis combining gold and aminocatalysis, Kirsch et al. unexpectedly obtained a tricyclic cis-fused dihydrofuran adduct by using a co-catalytic amount of (Ph3P)AuOTf (10 mol%) and (c-C6H11)(i-Pr)NH (20 mol%) in xylenes at 150°C for 1h (Figure 1). [11] Surprisingly, this transformation has only been underlined in a single case by the authors and, to the best of our knowledge, has not been reported again in literature. Interestingly, in 2010, Zhang et al. demonstrated that the combination of Ag2O and KOH was able to promote formal intramolecular [3+2] cycloaddition reactions, generating all-carbon 1,3-dipoles, under conventional heating [12] or microwaves conditions. [13] However, the scope of this transformation remains restricted to Npropargylamides bearing an electron withdrawing group on the nitrogen atom (Figure 1).

In the work developed herein, we expect to reach bi-or tricyclic complex structures containing 2,3-dihydrofuran cores in a single step from readily accessible compounds bearing an inactivated alkyne moiety. This reaction will proceed via a formal Ag(I) mediated [3+2] cyclization involving all-carbon 1,3-dipoles. This approach appears to be particularly challenging, since we envisioned to attain allenyl silver species through an unprecedented alkyne isomerization mediated by Ag(I) (Figure 1). As far as we know, the formation of an allenyl silver has only been reported by the Vermeer group through a 1,4-conjugated addition of an alkyl silver onto an ene-yne system (path a) or by transmetallation of an allenyl lithium species (path b). [14] Moreover, the reactivity of such species has been little studied and the intramolecular cyclization targeted herein is unprecedented in literature (Figure 1).

Figure 1 Silver/Base mediated alkyne isomerization/[3+2] cyclization reaction

Results and discussion

Silver was chosen as metal catalyst in this transformation since it is both significantly less expensive and more abundant than gold and less prone to undergo side-reactions than copper (i.e. Glaser-Hay coupling). Indeed, due to its [Kr] 4d 10 5s 1 configuration, silver possesses a strong alkynophilicity and its ability to π-coordinate to unsaturated systems having low-lying empty orbitals makes it extremely interesting especially in the field of alkyne-based chemistry. [15] In a preliminary survey, we chose the combination of Ag2O/DBU in acetonitrile and 1,3-keto ester 1a as the substrate (Scheme 1).

Scheme 1 Adduct 2a: Structure and stereochemistry determination

We were pleased to find a complete and clean conversion of 1a into the tricylic compound 2a at 50°C after 2h. The cyclized product was obtained in 85% yield with a total diastereoselectivity in favour of the cis-adduct without detection of the homo-coupling product. [16] The stereochemistry was determined by NOE experiments and further confirmed by a single-cristal X-ray diffraction analysis of p-nitrobenzoyl derivative 3, [START_REF]Crystallographic data are available from the Cambridge Crystallographic Data Centre: a) Compound 3, reference[END_REF] obtained after reduction of ester 2a and subsequent esterification of the resulting alcohol with p-nitrobenzoyl chloride (90% yield over two steps, Scheme 1).

Encouraged by this preliminary result, we focused on the optimization the reaction conditions (Table 1). A short screening of silver salts highlighted the ability of inexpensive silver(I) oxide to cleanly promote the cyclization after 2 h with a satisfying yield (85%, Entry 1). Silver nitrate and silver antimonate afforded 2a albeit in moderate yields, respectively 64% and 63% (Entry 2 and 4) whereas silver sulfate furnished only 32% of the desired product (Entry 3). The reaction proceeded smoothly with K2CO3 as a base (80%, Entry 5) and pyrrolidine (80%, Entry 6) but only traces of the cyclized product were detected after 20h by using NEt3 (Entry 7). Interestingly, the reaction is able to proceed in a wide range of solvents, providing 2a in satisfying yields (Entry 8-10, see SI). Finally, without DBU (Entry 11) or silver(I) oxide (Entry 12) no reaction was observed, establishing that the combination of silver reagent and base is mandatory to promote such cyclization (Entry 11 and 12). Using 0.25 eq. of Ag2O under air atmosphere afforded 2a (78%, Entry 13) after a prolonged reaction time (48 h) which can be dramatically reduced to 24 h by conducting the reaction under O2 atmosphere (83%, Entry 14). Interestingly, reactions can also be run with a catalytic amount of DBU (0.1 eq) without significant lengthening of the reaction time (3 h instead of 2 h). However, for accuracy and reproducibility considerations, we chose to perform the reactions with 1 equivalent of DBU instead of 0.1. Optimized conditions do not provide the product when the reaction is conducted at r.t (Entry 15). Also, we performed the reaction under conditions close to those reported by Kirsch but no product was detected (Entry 16)

Next, the scope of the reaction was examined: Our optimized conditions (Ag2O: 0.5 eq., DBU: 1 eq. in MeCN, [C] = 0.24 M) were first applied to various cyclic ketones (Scheme 2).

Scheme 2 Cyclization reactions of cyclic ketones *2g was obtained under Condition B (see Scheme 3 )

The cyclization of five (1a) or six-membered ring substrates (1b) occured under mild conditions and good yields were obtained (up to 85%) with a total diastereoselectivity in favour of the cis-adduct. We assumed that the selectivity is imposed by the configuration at the spiro carbon. Hence, for geometry and distance considerations, the ketone can be only attacked in a syn-periplanar fashion leading to the (+/-)-cis-isomer. Surprisingly, sevenmembered ring analog 1c was reluctant to cyclize and two equivalents of silver were necessary to attain a good conversion. Additionally, the cyclized products 2c/4c were obtained as a mixture of two diastereoisomers (55:45 by 1 H NMR) probably due to a higher conformational flexibility. Starting from the naphtalenone derivative 1d, no cyclization occured even under heating in a sealed tube at 100°C. [START_REF]Only traces of 2d along with degradation products were observed by 1 H NMR[END_REF] Tricyclic ketone 2e was obtained in 78% yield from the corresponding symmetrical diketone 1e. Noteworthy, these cyclization conditions were also suitable to heterocyclic compounds and piperidine derivative 2f could be isolated in satisfying yield (80%). To our delight, using conditions B (see Scheme 3) allowed the extension to substrate with longer chain (1g) in order to reach six-membered-ring analog with good yield and high selectivity for the cis-isomer 2g. The reactivity of linear ketones was next surveyed by varying the substitution of the malonate, the nature of the ketone and/or the influence of the functional group in beta position. We first studied the behaviour of different linear compounds towards the optimized cyclization conditions by tuning the nature of the substituent in alpha position to the ketone (R 2 ). Substitution by an allylic chain (compound 5a) as well as neutral-(5c) or enriched-(5d) benzylic chains was not deleterious to the reaction and good reactivities were observed along with modest selectivities in favour of the antiadduct (63:37, 67:33 and 63:37, respectively). Astonishingly, the diastereo-Scheme 3 Cyclization reactions of linear ketones -meric ratio raised to 83:17 when the aromatic moiety was substituted by a p-nitro group (5g). [START_REF]Crystallographic data are available from the Cambridge Crystallographic Data Centre: a) Compound 3, reference[END_REF] A bulkier isopropyl chain 5b delivered the corresponding bicyclic compound in 67% yield with inversion of the diastereomeric ratio in favour of the syn-adduct (7b), evidencing that the steric hindrance at this position is not detrimental to the reaction but strongly influences the selectivity. Modification of the size of the ester provided the expected 2,3dihydrofurans in good yields, however without notable improvement of the diastereoselectivity. On the contrary, the anti isomer 6h was exclusively obtained with 58% yield when the lactone 5h was used. [START_REF]Crystallographic data are available from the Cambridge Crystallographic Data Centre: a) Compound 3, reference[END_REF] Unlike benzylic ketone 5j, ketone 5i was reluctant to cyclize under conditions A while applying conditions B afforded good conversion, providing bicyclic compounds 6i/7i and 6j/7j (in 42 and 57% yield, respectively) with a similar selectivity towards the anti-isomer, along with degradation products. Finally, the substitution pattern of the triple bond was examined. To this aim, the reaction was conducted onto different substituted alkynes bearing either methyl (1h), phenyl (1i) or TMS (1j) groups (Scheme 4)

Scheme 4 Influence of the substitution of the triple bond

Reactions onto 1h or 1i were ineffective under both conditions (A or B). On the contrary, when trimethylsilylalkyne 1j was submitted to conditions A, it was smoothly converted into the cyclized product 2a (R = H). This result suggests a preliminary insertion of silver into the C-Si bond, which is consistent with the well-known propensity of terminal alkynes and TMS alkynes to undergo a metalation step under similar conditions. [START_REF] Vitérisi | [END_REF]20] In order to provide insights into the mechanism and to provide a rationale for the diastereoselectivity, we studied the first five steps of the Ag(I) mediated cycloaddition reaction mechanism by means of Density Functional Theory (DFT) approaches. [21] The computed reaction path together is presented in Figure 2, while computational details together with the labelling scheme are reported in Supporting Information. Two most probable isomers of BH + and the two most probable approaches of B to Ag have been computed (see supporting information for the related structures). We would like to underline that other trials starting from other B to Ag geometry always lead to one of the two species here reported. The data discussed here correspond to the arrangements of the B/BH + and reactants providing the lowest energies. A multistep pathway is proposed to explain the reaction mechanism up to the first cyclization process involving the carbonyl group and carbons of alkynyl residue (Figure 1). The analyzed reaction mechanism proceeds through five steps. Starting from the very stable silver alkynyl intermediate (I), the initial step corresponds to the proton transfer from conjugated acid of DBU (BH + in Figure 2) to I leading to the formation of intermediate II. This step proceeds with a barrier of 14.40 kcal/mol (Table 1 in Supporting Information) and leads to the formation of stable intermediate II. A subsequent proton transfer from II to DBU (B in Figure 2) generates the silver allenyl intermediate (III). The energetic barrier to overcome for its formation is found to be 12.07 kcal/mol (Figure 2). Of note the back reaction from II to I is characterized by a very small barrier so that we can assume that I is easily reformed. Following the energetic span model, [22] we can therefore assume that the energy span is related to the difference between the energy of I and TS2, leading to an effective barrier of 22.45 kcal/mol determining the reaction turnover. The III allenyl intermediate is characterized by practically equivalent C1-C2 and C2-C3 bonds (1.30 Å). From this intermediate, a first cyclization occurs via the TS3 transition state: the C3-C4 bond decrease achieving a value In the corresponding transition state (TS4), the Ag atom is found to bridge the oxygen and the alkynyl lateral group (C1-Ag, C2-Ag, Ag-O bonds in the 2.30-2.40 Å range), while the C4-O bond is elongated up to 1.41 Å and the O-H bond is forming (1.21 Å). Indeed, the V intermediate shows a proton bonded to the oxygen (O-H bond 1.02 Å) and the silver atom coordinated to the carbons of alkynyl residue (C1-Ag, C2-Ag around 2.30-2.35 Å). At this stage, the reaction proceeds via a second cyclization process involving the alkynyl lateral group and the oxygen to form the dihydrofuran cycle [START_REF] Alcaide | For such cycloetherification mediated by silver see for example[END_REF] which actually we computed to be extremely thermodynamically favored (∆E= -119.16 Kcal/mol, see SI). In order to disclose the origin of the diastereoselectivity, we computed the first three steps of the reaction mechanism following two approaches leading to anti or syn diasteroisomers (Figure 3; all computational details are reported in the Supporting Information). All computed energy barriers are similar for both diasteroisomers paths nonetheless we observe that IIa is effectively more stable than IIb of 4.19 kcal/mol. Indeed, the destabilizing interaction between the oxygen lone pairs of both carbonyls (O-O bond 2.80 Å) disfavors the syn adduct. With the aim of further validating the hypothesis that regioselectivity may be related to the presence (or absence) steric interactions for one of the adducts, we extended the study of reaction step II to compound 5e. (see Figure 3 in the Supporting Information). The reaction then proceeds with similar barriers both along the a and b pathway, suggesting that the final product depends on the most favorable initial intermediate. Indeed, with the aim of further validating our hypothesis, we extended the study of the reaction steps of compound 5d. In this case, comparing the two optimized diasteroisomers, it is worth to notice that while the energy difference between the anti/syn adducts for the first step is similar to the value observed for compound 5e (3.43 kcal/mol), it decreases down to 1.39 kcal/mol at step II. Since such energy difference is very low, we can assume that this is the origin of the absence of diastereoselectivity. Indeed, for this compound, the destabilizing interaction is substantially decreased upon substitution with less sterically-hindered methyl group, since the oxygen lone pairs of both carbonyls are located further apart in this case (O-O bond 3.00 Å see Figure 3 in the SI). Experimentally, we ran the reaction on bulkier amide 8 moiety leading to a drastic enhancement of the diastereomeric ratio (anti/syn: > 95:5) ( ).

Scheme 5 Selectivity on β-keto amide versus β-keto ester

However, we were unable to confirm the stereochemistry at this stage. In order to determine the relative configuration of the cyclized product, the amide 9 was subsequently reduced into the corresponding amine 10 in 73% yield over two steps. A NOE experiment unambiguously evidenced anti adduct 10 as the major compound. In parallel, performing the reaction on ethyl ester 5e under conditions B did not allow improvement of the diastereomeric ratio (50:50), underlining the primary importance of the nature of the electron withdrawing group on the diastereoselectivity. Noteworthy, conditions B allowed a better conversion without influencing the diastereoselectivity.

Conclusion

This unprecedented formal silver mediated [3+2] cyclization appears as a valuable tool for the synthesis of complex and/or congested polycyclic dihydrofurans. As an example, the approach developed herein would permit a straightforward access to precursor natural compounds, which possess an angularly fused tricyclic lactone backbone like alliacol or teucrolivin, under smooth conditions (Figure 4) In summary, we have developed a new formal [3+2] cyclization promoted by silver(I) oxide and DBU from readily available starting ß-dicarbonyl compounds. The reaction is broad in scope, proceeds in mild conditions and allows an access to complex and congested molecules in a single step. This transformation is highly tolerant towards a wide range of solvents and bases (in stoichiometric or in catalytic amounts), thus facilitating its adaptation to structurally diverse, and potentially sensitive substrates. DFT calculations have demonstrated the formation of an allenyl silver intermediate by isomerization of the corresponding alkyne. Although the description of this species remains particularly scarce in the literature, its formation is a prerequisite in the promotion of the reaction. Moreover, DFT calculations have also allowed to determine the thermodynamic origin of the syn/anti diastereoselectivity for β-dicarbonyl compounds bearing an ester group. More generally, through this methodology, we highlighted a new reactivity of silver salts towards terminal alkyne. Thanks to this unprecedented isomerization reaction, we opened the way to a new field of transformations. The peculiar reactivity of allenyl silver species is currently under investigation in our laboratory.

Experimental Section

See supporting informations.

Figure 2

 2 Figure 2 Computed reaction mechanism (∆E in kcal/mol of all stationary points are reported with respect to the sum of all isolated species participating to the reaction).

of 2 .

 2 20 Å and suggesting the closure of the cycle. TS3 lies at 10.61 kcal/mol higher in energy with respect to the III and leads to the formation of intermediate IV, characterized by a C3-C4 bond of 1.57 Å and thus by a fully formed 5-carbon cycle. Of note in IV the silver atom approaches the oxygen (Ag-O bond of 2.15 Å), while C1-Ag and C2-Ag are elongating up to 2.70 Å. Comparing the atomic (Mulliken) charges computed for the TS3 and the IV, it is found that C4 becomes more electrophilic (from -1.91 |e-| to -0.22 |e-|), while the oxygen becomes more nucleophilic (from -0.34 |e-| and -0.49 |e-|). This makes possible to envisage a proton transfer from the conjugated acid BH + of DBU to the oxygen atom of IV. The very low energy barrier (2.23 kcal/mol) computed for this reaction confirmed this hypothesis.

Figure 3

 3 Figure 3 Computed ∆E of the first three steps of the reaction mechanism for the anti/syn adducts.

Figure 4

 4 Figure 4 Dihydrofuran core as useful synthetic intermediate in natural products synthesis

Table 1 :

 1 Reaction optimization. Full data optimization is available in the ESI. Each attempt was conducted with 1 eq. of base. b Isolated Yield.

	Entry	Metal (eq.)	Base a	Solvent	Time, Yield b %
	1	Ag2O (0.5)	DBU	MeCN	2h, 85%
	2	AgNO3 (1.0)	DBU	MeCN	2h, 64%
	3	Ag2SO4 (0.5)	DBU	MeCN	3h, 32%
	4	AgSbF6 (1.0)	DBU	MeCN	30h, 63%
	5	Ag2O (0.5)	K2CO3	MeCN	12h, 80%
	6	Ag2O (0.5)	Pyrrolidine	MeCN	20h, 80%
	7	Ag2O (0.5)	NEt3	MeCN	20h, -c
	8	Ag2O (0.5)	DBU	DMF	2h, 76%
	9	Ag2O (0.5)	DBU	Toluene	2h, 85%
	10	Ag2O (0.5)	DBU	THF	2h, 84%
	11	Ag2O (0.5)	-	MeCN	24h, n.r
	12	-	DBU	MeCN	24h, n.r
	13	Ag2O (0.25)	DBU	MeCN	48h, 78% d
	14	Ag2O (0.25)	DBU	MeCN	24h, 83% e
	15	Ag2O (0.5)	DBU	MeCN	24h, n.r f
	16	P(Ph)3AuOTf (0.1)	pyrrolidine	toluene	2h, n.r g
	a

c

Only traces of expected product were detected by 1 H NMR after 20h. d Reaction carried out under ambient atmosphere. e Reaction carried out with O2 balloon. f Reaction performed at r.t instead of 50°C. g Reaction performed at 110°C

Acknowledgements

thanks the China Scholarship Council for Ph.D grant. I.C. and A.P thank the European Research Council (ERC) for funding under the European Union's Horizon 2020 research and innovation programme (grant agreement No 648558, STRIGES CoG grant). We would like to thank Pr. P. Belmont for fruitful discussions and single-cristal X-ray diffraction service from ICMMO-UMR8182 Orsay.

Entry for the Table of Contents

An unprecedented isomerization/allenyl silver formation/[3+2] cyclization promoted by a combination of silver salts and base are presented. The reaction allows the straightforward and diastereoselective (up to 95:5 d.r.) synthesis of complex dihydrofuran frameworks, under mild conditions from easily accessible substrates and is broad in scope.


Yufang Tao

Marcello Brigante

Hui Zhang

Gilles Mailhot
email: gilles.mailhot@uca.fr

Pr Gilles Mailhot

Phenanthrene degradation using Fe(III)-EDDS photoactivation under simulated solar light: A model for soil washing effluent treatment

Keywords: Advanced oxidation processes, sulfate radical, iron complexes, soil depollution, photochemistry

In this work, for the first time, the nonionic surfactant polyoxyethylene-(20)-sorbitan monooleate (Tween 80, C 64 H 124 O 26 ) aided soil washing effluent was treated by enhanced activation of persulfate (PS) using Fe(III)-EDDS (EDDS: ethylenediamine-N, N-disuccinic acid) complexes under simulated solar light irradiation. The performance of this system was followed via the production and reactivity of radical species (SO 4

) and degradation of phenanthrene (PHE) used as a model pollutant in soils. Different physico-chemical parameters such as the concentration of reactive species and pH were investigated through the PHE degradation efficiency. The second-order rate constants of the reactions for generated radicals with PHE and Tween 80 in solution were identified through competitive reaction experiments under steady-state conditions and application of nanosecond laser flash photolysis (LFP) as well. A kinetic approach was applied to assess the selectivity 3 and reactivity of photo-generated radicals in aqueous medium in order to explain the observed degradation trends. This work proposes an innovative technology of management of soil washing solutions using Fe(III)-EDDS complexes and solar light for the activation of persulfate.

Introduction

Soil contamination is known to be a severe and ubiquitous problem worldwide. Specially, soil pollution caused by hydrophobic organic compounds (HOCs) have attracted more attention and been taken seriously by governments and enterprises due to the characters of high-toxicity and long-refractory of this kind of pollutant. Polycyclic aromatic hydrocarbons (PAHs) are typical representatives of HOCs, which are notable and distinguished by their high health risks of mutability, carcinogenicity and teratogenicity through long-term bioaccumulation in the environment [START_REF] Yap | Fenton based remediation of polycyclic aromatic hydrocarbons-contaminated soils[END_REF][START_REF] Li | Electrokinetics-enhanced biodegradation of heavy polycyclic aromatic hydrocarbons in soil around iron and steel industries[END_REF][START_REF] Trellu | Removal of hydrophobic organic pollutants from soil washing/flushing solutions: A critical review[END_REF]. The characteristics of low solubility in water and poor mobility in soil of PAHs impede the elimination of this type of compound from soils. Comprehensive researches of in-situ technologies such as bioremediation [START_REF] Tiehm | Surfactant-enhanced mobilization and biodegradation of polycyclic aromatic hydrocarbons in manufactured gas plant soil[END_REF],phytoremediation [START_REF] Huang | A multi-process phytoremediation system for removal of polycyclic aromatic hydrocarbons from contaminated soils[END_REF]) and electrokinetic remediation [START_REF] Reddy | Enhanced electrokinetic remediation of contaminated manufactured gas plant soil[END_REF] technologies are available in the literature (Gan et al., 2009a). In Ye's [START_REF] Ye | Degradation of polynuclear aromatic hydrocarbons by Sphingomonas paucimobilis[END_REF] study of biodegradation of PAHs by Sphingomonas paucimobilis, after 16 h of incubation of pyrene, benz [a]anthracene (B[a]A), benzo[a]pyrene (B[a]P), benzo [b]fluoranthene (B[b]F) for each with 10mg/L , the degradation percentage was 1.3± 0.4, 1.8±0.1, 9.6±1, 5.0±0.6, respectively. Lee's group [START_REF] Lee | Degradation of phenanthrene and pyrene in rhizosphere of grasses and legumes[END_REF] compared four native Korean plant species for phytoremediation of phenanthrene (PHE) and pyrene, the maximum removal efficiency of phenanthrene was >99%, for pyrene was 94% in planted soil after 80 days of experiments. Maturi's group [START_REF] Maturi | Simultaneous removal of organic compounds and heavy metals from soils by electrokinetic remediation with a modified cyclodextrin[END_REF]) investigated modified cyclodextrin enhanced electrokinetic remediation of PHE and Nickel contaminated soil, the optimal 10% removal efficiency of PHE was achieved with 1% cyclodextrin addition after around 166 days elapsed duration. However, all these in-situ techniques present at least one non-negligible disadvantages such as time-consuming treatment requirements, low remediation efficiency and high economic input. In this case, ex-situ techniques represented by surfactant aided soil washing processes appeal to more and more researchers. During the past several decades, ex-situ soil washing (SW) and in-situ soil flushing (SF) techniques, improved by application of surfactants as the extracting agents, have exhibited high performances for PAH elimination from soils (Gan et al., 2009b;Mousset et al., 2014a).

Nonionic surfactant Polyoxyethylene (20) sorbitan monooleate (Tween 80), Dodecyl polyglycol ether(Brij 35), Polyoxyethylene (10) isooctylphenyl ether (Triton X-100), anionic surfactant sodium dodecyl sulfate (SDS) and sodium dodecyl benzene sulphonate (SDBS) are commonly used in the literature of surfactant enhanced soil remediation [START_REF] Mao | Use of surfactants for the remediation of contaminated soils: A review[END_REF]. Among these surfactants, the application of Tween 80 (TW80) was more extensive and prominent due to its lower toxicity and higher solubility capacity [START_REF] Cheng | Effects of pig manure compost and nonionic-surfactant Tween 80 on phenanthrene and pyrene removal from soil vegetated with Agropyron elongatum[END_REF].

Except for the common application for PAHs, TW80 was supposed to be an excellent alternative for numerous hydrophobic contaminants such as PAHs anthracene, fluoranthene, pyrene and phenanthrene [START_REF] Alcántara | Combined treatment of PAHs contaminated soils using the sequence extraction with surfactant-electrochemical degradation[END_REF][START_REF] Peng | Removal of PAHs with surfactant-enhanced soil washing: influencing factors and removal effectiveness[END_REF], Petroleum Hydrocarbons [START_REF] Huguenot | Combination of surfactant enhanced soil washing and electro-Fenton process for the treatment of soils contaminated by petroleum hydrocarbons[END_REF], p-cresol [START_REF] Rosas | Enhancing p-cresol extraction from soil[END_REF], Decabromodiphenyl ether (Zhou et al., 2007), 4,4'-dichlorobiphenyl (Chu and[START_REF] Chu | Remediation of contaminated soil by a solvent/surfactant system[END_REF], p-Nitrochlorobenzene [START_REF] Guo | The feasibility of enhanced soil washing of p-nitrochlorobenzene (pNCB) with SDBS/Tween80 mixed surfactants[END_REF] and pesticides like Diazinon [START_REF] Hernández-Soriano | Desorption of two organophosphorous pesticides from soil with wastewater and surfactant solutions[END_REF], DDT [START_REF] Zheng | Enhanced Solubilization and Desorption of Organochlorine Pesticides (OCPs) from Soil by Oil-Swollen Micelles Formed with a Nonionic Surfactant[END_REF]. The application of TW80 in the remediation of PAHs contaminated soil is especially ubiquitous owing to its better solubilization characteristic than other surfactants. It was reported that TW80 was efficient for solubilizing a plenty of PAHs such as Pyrene,PHE,Fluorene, Acenaphthene,Naphthalene and TW80 exhibited superior capacity than other nonionc surfactants such as Tween 20, Triton X-100, Triton X-305, Triton X-405 [START_REF] Zhu | Partitioning of polycyclic aromatic hydrocarbons to solid-sorbed nonionic surfactants[END_REF]. In another work, TW80 was suggested as the highest solubilization capacity than Brij35, TX100 and SDS for PHE on the basis of molar solubilization ratios (MSR) [START_REF] Zhao | Solubilization and biodegradation of phenanthrene in mixed anionic-nonionic surfactant solutions[END_REF]. Nevertheless, as a consequence of these soil washing/flushing processes, a complicated mixture of waste liquids containing great loads of extractants, PAHs and other accompanying organics is generated and should be further treated to reduce the potential environment concern of discharge the effluent into environment. Owing to the complexity of the mixed components of SW solutions, the management and disposal are a tremendous challenge for environmental engineering [START_REF] Rosas | Soil remediation using soil washing followed by Fenton oxidation[END_REF][START_REF] Trellu | Removal of hydrophobic organic pollutants from soil washing/flushing solutions: A critical review[END_REF]. Soil washing effluent could be treated with biological method, adsorption techniques as well as Advanced oxidation process (AOPs). However, biological treatments bear the drawbacks of long duration and low efficiency [START_REF] Gharibzadeh | Reuse of polycyclic aromatic hydrocarbons (PAHs) contaminated soil washing effluent by bioaugmentation/biostimulation process[END_REF], while adsorption technologies possess potential risks since contaminants could not be completely discomposed [START_REF] Ahn | Soil washing using various nonionic surfactants and their recovery by selective adsorption with activated carbon[END_REF][START_REF] Zhou | Removal of polycyclic aromatic hydrocarbons from surfactant solutions by selective sorption with organo-bentonite[END_REF][START_REF] Li | Selective removal of polycyclic aromatic hydrocarbons (PAHs) from soil washing effluents using biochars produced at different pyrolytic temperatures[END_REF]. In this case, AOPs including electrochemical oxidation [START_REF] Gómez | Remediation of polluted soil by a two-stage treatment system: desorption of phenanthrene in soil and electrochemical treatment to recover the extraction agent[END_REF]Mousset et al., 2014c, b;Trellu et al., 2017a), ozone oxidation [START_REF] Liu | Soil remediation using soil washing followed by ozone oxidation[END_REF], photocatalytic oxidation [START_REF] Zhang | Heterogeneous photocatalytic degradation of phenanthrene in surfactant solution containing TiO2 particles[END_REF][START_REF] Bai | Remediation of phenanthrene contaminated soil by coupling soil washing with Tween 80, oxidation using the UV/S2O82-process and recycling of the surfactant[END_REF] have been developed to the most popular and common applied technology for PAHs and various surfactant involved soil washing effluent treatments. As reviewed in Table S1, AOPs based on radical oxidation mechanism could effectively decompose PAHs into various products during the processes of soil washing effluent treatment with PAHs dosage from 1mg L -1 to around 200 mg L -1 and optimal surfactants ranging from 1g L -1 to 10g L -1 . Among multiple AOPs, Sulfate radicals based advanced oxidation processes (SR-AOPs) have been widely applied as effective and practicable techniques for remediation of contaminated waters and soils [START_REF] Yen | Application of persulfate to remediate petroleum hydrocarbon-contaminated soil: Feasibility and comparison with common oxidants[END_REF][START_REF] Wacławek | Chemistry of persulfates in water and wastewater treatment: A review[END_REF]. Persulfate (S 2 O 8 2-, PS) is an ideal alternative oxidant (E 0 = 2.01 V), which can be activated to produce stronger sulfate radicals (SO 4 •-

) (E 0 = 2.6 V) by various chemical or physical methods [START_REF] Usman | Application of magnetite catalyzed chemical oxidation (Fenton-like and persulfate) for the remediation of oil hydrocarbon contamination[END_REF]. In addition to its strong oxidation capacity, sulfate radical has revealed several advantages over other oxidants. SO 4

•-is more stable than hydroxyl radical (HO • ) and exhibits a longer half-life time as well as a wider pH range for the application [START_REF] Anipsitakis | Radical generation by the interaction of transition metals with common oxidants[END_REF].

Iron, one of the chemical activators of PS, is the most abundant transition metal present in the earth's crust and exists in a variety of forms in water such as soluble, colloidal and particulate forms. In natural water, insoluble iron-containing oxides account for a large proportion [START_REF] Faust | Photochemistry of aqueous iron(III)-polycarboxylate complexes: roles in the chemistry of atmospheric and surface waters[END_REF], while dissolved iron is only found in a small percentage and most of them exists in the form of complex with organic ligands [START_REF] Zhou | Fe(III)-oxalate complexes induced photooxidation of diethylstilbestrol in water[END_REF]. Polycarboxylates and (amino-)polycarboxylates like citrate, malonate, oxalate, EDTA, etc. can react with ferric ions (Fe 3+ ) to form stable and strong complexes and improve its solubility and stability in natural water [START_REF] Chen | Photodegradation of glyphosate in the ferrioxalate system[END_REF]. Moreover, such (amino-)polycarboxylate complexes are reported to have strong and fast photochemical reactivity under sunlight irradiation giving rise to oxidative species [START_REF] Zhang | Degradation of atrazine photoinduced by Fe(III)-pyruvate complexes in the aqueous solution[END_REF]. Those iron complexes such as Fe(III)-EDTA, Fe(III)-EDDS, Fe(III)-oxalate or Fe(III)-citrate have exhibited high efficiency as photo-catalysts and as persulfate and hydrogen peroxide activators for the elimination of a variety of contaminants [START_REF] Manenti | Insights into solar photo-Fenton process using iron(III)-organic ligand complexes applied to real textile wastewater treatment[END_REF][START_REF] Miralles-Cuevas | EDDS as complexing agent for enhancing solar advanced oxidation processes in natural water: Effect of iron species and different oxidants[END_REF][START_REF] Soriano-Molina | Assessment of solar raceway pond reactors for removal of contaminants of emerging concern by photo-Fenton at circumneutral pH from very different municipal wastewater effluents[END_REF]. Ethylenediamine-N,N-disuccinic acid (EDDS) is a structural isomer of EDTA, and

[S,S]-EDDS, [R,R]-EDDS and [R,S/S,
R]-EDDS are three stereo isomers of EDDS. Among those isomers, [S,S]-EDDS is recognized and employed as a environmentally safe and friendly substitution for EDTA for environmental restoration since it is easier biodegradable than EDTA [START_REF] Nagaraju | Determination of copper and iron using [S-S']-ethylenediaminedisuccinic acid as a chelating agent in wood pulp by capillary electrophoresis[END_REF].

In this work, Fe(III)-EDDS complex was used to improve the PS photo-activation for the phenanthrene (PHE) degradation in soil washing solution. Main physico-chemical parameters such as concentration of chemical species and pH were investigated using simulated solar radiation to generate the sulfate radical from PS. This work provides a novel proposal of application of Fe(III)-EDDS for soil washing effluent treatments.

Experimental Details

Chemicals

Tween 80 (TW80), phenanthrene (PHE) and ferric perchlorate (Fe(ClO 4 ) 3 •9H 2 O) were obtained from Sigma-Aldrich. Ethylenediamine-N,N-disuccinic acid trisodium salt solution (35% in water) was purchased from Fluka. All the other used reagents and chemicals were of analytical grade and all the solutions employed in this work were prepared with purified water from Millipore Ultra-Pure System (18.2 M cm). The stock solutions of Fe(III)-EDDS were a mixture of appropriate amount of newly prepared aqueous solutions of ferric perchlorate and EDDS at a stoichiometry ratio of 1:1.

Preparation of the synthetic SW solution

TW80 is one of the suggested and most applied enhancing-solubility nonionic surfactants that increases the solubility of HOCs due to its notable features of lower soil sorption capacity and critical micellar concentration (CMC), low toxic property as well as higher solubilization ability and higher cost-effectiveness [START_REF] Mulligan | Surfactant-enhanced remediation of contaminated soil: a review[END_REF]. It is known that above CMC, the surface tension of HOCs varies with a much smaller slope and the solubility of HOCs is strongly improved [START_REF] Trellu | Removal of hydrophobic organic pollutants from soil washing/flushing solutions: A critical review[END_REF]. The CMC of TW80 is equal to 0.016 g L -1

(López-Vizcaíno et al., 2012) which is much lower compared to other surfactants. Due to the above advantages, TW80 was chosen as the aiding compound and its solubilization capacity of PHE was investigated.

Irradiation experiments and degradation kinetic

cooling system and placed in a rectangular wood box to prevent the interference from the external light sources. Four tubular lamps (Sylvania, F15W/350BL), used as irradiation sources, were fixed on the top of the wood box. The irradiation energy was determined employing an optical fiber with a charge coupled device (CCD) spectrophotometer (Ocean Optics USD 2000 + UV-vis) previously calibrated with a DH-2000-CAL reference lamp. The energy reaching the reactor determined between 300 and 400 nm (1390 µW cm -2 ) was very closed to the typical solar emission in the same wavelength range [START_REF] Bianco | Tryptophan and tryptophan-like substances in cloud water: Occurrence and photochemical fate[END_REF].

In Figure 1 emission spectrum of light sources and UV-vis absorption spectra of chemical species (carried out with a Cary 300 UV-visible spectrophotometer) were presented. An overlap between the emission spectrum of the lamps and the absorption spectrum Fe(III)-EDDS complex was highlighted. PHE absorbed in the ultraviolet region with two maxima at 252 nm and 295 nm. The main absorption region was located at λ < 250 nm for PS and λ < 300 nm for PHE in TW80 solution. Fe(III)-EDDS clearly exhibited a broad but gradual decrease absorption at the region of wavelengths up to approximately 400 nm and the overlapping absorption with the lamp emission spectrum from 300 nm and 400 nm as well.

All experiments were carried out at 293 ± 2 K and solutions were magnetically stirred with a magnetic bar to insure the homogenous mixture of the solution. At regular intervals, 0.5 mL of sample was withdrawn and analyzed by an UPLC-UV system (Alliance) equipped with a photodiode array detector. The eluent was a mixture of water and acetonitrile (30/70, v/v) with isocratic mode in a flow rate of 1 mL min -1 . The column was a Nucleodur 100-5 C18 of 150 mm × 4.6 mm with particle size 5 μm. The pseudo-first-order decay of PHE was determined using the following exponential equation:

) ' exp( ] [ ] [ t k PHE PHE O  
Where [PHE] 0 and [PHE] are the initial and residual concentrations of PHE at time t, k' is the pseudo-first-order apparent rate constant (s -1

). The initial conversion rate of PHE (R PHE ) was

calculated as R PHE (M s -1 ) = k' × [PHE] 0 .
The retention time and the analysis of PHE were not modified by the presence of TW80. The pHs of the solutions were adjusted with 0.1 M of HClO 4 or NaOH solutions and no significant pH variation was observed during irradiation.

Laser flash photolysis

The determination of the second order rate constants were carried out with the nanosecond laser flash photolysis (LFP) apparatus from Applied Photophysics (LKS60). A 266 nm excitation was used to generate radicals from solution and apparatus description as well as competition kinetic approaches have been reported elsewhere and a brief description is given in the supplementary materials section [START_REF] Huang | Homogeneous and heterogeneous Fenton and photo-Fenton processes : impact of iron complexing agent ethylenediamine-N,N'-disuccinic acid (EDDS). Université Blaise Pascal[END_REF].

Results and discussion

PHE degradation under various systems

The solubility of PHE was tested using different concentration of TW80 in water. The results reported in Figure SM1 indicate that PHE solubility increased with the concentration of TW80 and a maximum of PHE solubility was reached around 16 mg L -1 (corresponding to a concentration of 90 µM). This maximum of solubility of PHE was obtained from 0.5 g L -1 of TW80. This concentration of TW80 was then used to prepare PHE solution for all experiments.

The PHE degradation under simulated solar light at pH 3.5, using different mixture of Fe(III)-EDDS, Fe(III) and PS is reported in Figure 2. Control experiments showed that very low degradation of PHE (less than 5%) occurred with the involvement of Fe(III)-EDDS complex and PS in the dark. Under irradiation, the direct photolysis (without PS and iron complexes) of PHE was negligible, less than 8% and addition of Fe(III)-EDDS (0.5 mM) not significantly changed the efficiency of PHE degradation after 150 min of irradiation. As a contrary, about 30% removal of PHE was being achieved in the presence of PS (5 mM) with UVA light due to the photogeneration of sulfate radicals that contributed to PHE degradation.

Under UVA irradiation with addition of Fe(III) to PS solution, almost 35% of PHE removal was obtained under irradiation. So, although Fe(III) can be photo-reduced to Fe(II) [START_REF] Brand | Degradation photoinduced by Fe(III): Method of alkylphenol ethoxylates removal in water[END_REF] which is able to activate PS leading to the generation of sulfate radicals [START_REF] Anipsitakis | Radical generation by the interaction of transition metals with common oxidants[END_REF][START_REF] Liang | Persulfate oxidation for in situ remediation of TCE. II. Activated by chelated ferrous ion[END_REF], only 5% of PHE degradation was promoted compared to PS/UV system. In the system with replacement of Fe(III) by Fe(III)-EDDS (0.5 mM) to PS solution, the performance was much more pronounced and PHE removal was remarkably strengthened under irradiation, especially in the first 30 minutes. This difference can be explained by the lower photoreactivity of Fe(III) aquacomplexes in comparison with Fe(III)-EDDS at pH around 3.5 [START_REF] Catastini | Degradation of amitrole by excitation of iron(III) aquacomplexes in aqueous solutions[END_REF][START_REF] Huang | Homogeneous and heterogeneous Fenton and photo-Fenton processes : impact of iron complexing agent ethylenediamine-N,N'-disuccinic acid (EDDS). Université Blaise Pascal[END_REF] since Fe(II) formation from Fe(III) aquacomplexes is lower than from Fe(III)-EDDS at  > 300 nm. Therefore, in the presence of PS, sulfate radical formation is more important with Fe(III)-EDDS (reactions 1 and 2):

Fe(III)-EDDS + hν →[Fe(III)-L] * → Fe(II) + EDDS • (1) Fe(II) + S 2 O 8 2-→Fe(III) + SO 4 2-+ SO 4 •- (2)
Fe(OH) 2+ + hν → Fe(II) + HO • (3)

Under sun-simulated irradiation in the presence of Fe(III)-EDDS, PHE kinetic exhibited two different decay steps (Figure 2). In the first 30 mins of irradiation, Fe(III)-EDDS was almost completely degraded at pH 3.5 (Figure 3) to produce Fe(II) which represented the rate-limiting parameter for SO 4 •-formation. In the second step, after complete decomposition of Fe(III)-EDDS, iron remained in the form of Fe(III) aquacomplexes, and under the influence of UVA light, hydroxyl radical and Fe(II) could be generated as well (reaction 3) [START_REF] Mailhot | Fe(III)-solar light induced degradation of diethyl phthalate (DEP) in aqueous solutions[END_REF][START_REF] Krýsová | Comparative kinetic study of atrazine photodegradation in aqueous Fe(ClO4)3 solutions and TiO2 suspensions[END_REF]. In the presence of PS, Fe(II) led to Fe(III) again (reaction 2) and we were in the presence of the (photo)chemical cycle involving Fe(III) and Fe(II). In this system, the degradation of PHE was mainly attributed to the formation of SO 4 •-and HO • through analogous reactions obtained with Fe(III) aquacomplexes in solution and the subsequent cycle of Fe(III)/Fe(II). In fact, the degradation kinetic after 30 minutes followed the same profile compared with those obtained where PS was mixed with Fe(III) aquacomplexes.

Same experiments conducted at pH 7.0 (Figure SM2) indicated that an enhancement of PHE degradation was still acquired with Fe(III)-EDDS, suggesting that the iron complex is photoactive also at higher pH [START_REF] Wu | Toward a better understanding of Fe(III)-EDDS photochemistry: Theoretical stability calculation and experimental investigation of 4-tert-butylphenol degradation[END_REF]. As a contrary, a strong inhibition of PHE degradation occurred with the dosing of Fe(III) aquacomplexes and PS. This effect could be ascribed to the precipitation of Fe(III) at pH > 4.0, resulting in absorption of light by iron particles and a potential screening effect for the photolysis of PS and also a very low photoreactivity of precipitated iron species.

What should be noted is on the contrary of many studies involving iron species, in our system UV/PS/Fe(III)-EDDS no significant effect of pH in the range 3.0 to 8.5 was obtained (Figure SM3). This observation was quite surprising, if we compare with former studies [START_REF] Wu | Toward a better understanding of Fe(III)-EDDS photochemistry: Theoretical stability calculation and experimental investigation of 4-tert-butylphenol degradation[END_REF][START_REF] Wu | Sulfate radical photogeneration using Fe-EDDS: influence of critical parameters and naturally occurring scavengers[END_REF] but could be due to the presence of TW80 which modified the reactivity of radical species (see following results).

Effect of Fe(III)-EDDS and PS concentrations

The effects of persulfate (PS) and Fe(III)-EDDS concentrations were investigated on the PHE degradation efficiency. PHE degradation involving different concentrations of Fe(III)-EDDS with 5 mM PS are reported in Figure 3A. An enhancement of PHE degradation, after 30 min of irradiation, more than 3 times was achieved with the Fe(III)-EDDS concentration increased from 0 to 0.5 mM (Figure 3B). However, the degradation efficiency of PHE decreased with the further increase of iron complex concentration. In fact, a scavenging effect of Fe(III)-EDDS and Fe(II) toward SO 4 •-was expected at higher concentrations [START_REF] Peng | Degradation performance and mechanism of decabromodiphenyl ether (BDE209) by ferrous-activated persulfate in spiked soil[END_REF][START_REF] Yu | Degradation of phenanthrene in aqueous solution by a persulfate/percarbonate system activated with CA chelated-Fe(II)[END_REF]. Considering that a second order rate constant of 9.9 × 10 9 M -1 s -1 has been determined for the reaction between sulfate radical and Fe(II) (E. et al., 1966), it was possible to argue that scavenging effect of ferrous ions, that were more stable in solution at pH 3.5, became significant at higher dosage of Fe(III)-EDDS. The maximum degradation of PHE (33%) after 30 mins was reached using 0.5 mM Fe(III)-EDDS, while after 150 minutes the highest degradation efficiency (55%) was obtained with 0.25 mM of Fe(III)-EDDS (Fig. 3B). This shift of the maximum efficiency after 30 min was attributed to the higher load of Fe(II) and EDDS at higher concentration of Fe(III)-EDDS and so stronger scavenging effect of sulfate radicals.

To further explore the degradation performance of PHE in the system PS/Fe(III)-EDDS, the effect of persulfate concentration was investigated. As shown in Figure 4, PHE degradation efficiency was strongly promoted by increasing PS loading from 0 to 30 mM both for the photolysis of PS alone in solution and in the presence of Fe(III)-EDDS. The influence of Fe(III)-EDDS was much pronounced at higher concentration of persulfate. In the presence of Fe(III)-EDDS and after 150 min of irradiation, the enhancement of PHE degradation with 2 mM of PS was less than 2% while with 5 mM of PS the enhancement was around 18% and more than 50% with 15 and 30 mM of PS.

These results suggested that Fe(III)-EDDS had stronger impact at higher concentration of PS

showing that the generation of sulfate radical through reaction between PS and Fe(II) was the main contribution of the efficiency of the system.

Effect of Fe(III)/EDDS ratio

According to previously reported works; the ratio of Fe(III) and EDDS concentrations represented a key parameter for the efficiency of the pollutants removal in water [START_REF] Li | Photochemical efficiency of Fe(III)-EDDS complex: OH radical production and 17β-estradiol degradation[END_REF]. The concentration of PS was kept constant at 5 mM while various initial loads of Fe(III)

were applied. As shown in Figure SM4, the initial degradation rate of PHE increased with a higher concentration of Fe(III). At 1 mM of Fe(III) the final elimination of PHE was higher than at 0.5 or 0.25 mM. This observation was resulted from the fact that at 1 mM of Fe(III), there were in solution 0.5 mM of Fe(III)-EDDS and 0.5 mM of Fe(III) aquacomplexes. The two iron species were photo-transformed into Fe(II) which could activate PS to generate sulfate radicals and moreover Fe(III) aquacomplexes generated photochemically also hydroxyl radicals. So in these conditions, a higher amount of sulfate and hydroxyl radicals were generated increasing the degradation of PHE. At concentrations of Fe(III) of 0.25 or 0.5 mM, the final removals of PHE were very close with a little higher efficiency at 0.25 mM.

During the first step of reactions, Fe(III) was reduced into Fe(II) and EDDS was oxidized.

However if there were some residual EDDS remaining in solution, Fe(II) species were easily re-oxidized to Fe(III) and combined with EDDS to generate Fe(III)-EDDS complex [START_REF] Li | Photochemical efficiency of Fe(III)-EDDS complex: OH radical production and 17β-estradiol degradation[END_REF]. As a consequence, the formation of SO 4 •-and the degradation of PHE photoinduced by Fe(III)-EDDS complex could continue.

Effect of TW80 concentration.

To better explore the role of TW80 in this system, experiments with 0.5, 0.75 and 1 g L -1 of TW80 were carried out. The increase of the surfactant concentration from 0.5 to 1.0 g L -1

inhibited significantly PHE degradation (Figure SM5A). In fact, when TW80 concentration increased to a factor of 1.5 and 2, the apparent rate constant of PHE degradation was reduced by a factor 1.8 and 3.3 (Figure SM5B). The negative effect acquired at higher TW80 concentrations could be ascribed to the competition between TW80 and PHE for the reactivity with photogenerated sulfate radicals. In fact, as reported in Table 1, a second order rate constant between SO 4

•-and TW80 ( ) of 4.6±0.2×10 9 M -1 s -1 has been determined

(see SM for more details). At 0.5 g L -1 of TW80 and considering the reactivity between SO 4

•and Fe(III)-EDDS and PS [START_REF] Neta | Rate constants for reactions of inorganic radicals in aqueous solution[END_REF][START_REF] Bianco | Disinfection of water inoculated with Enterococcus faecalis using solar/Fe(III)EDDS-H 2 O 2 or S 2 O 8 2process[END_REF] reported in literature, we can argue that about 33 % of photogenerated SO 4 •-reacted with TW80 while only 7 % with PHE in solution. However, increasing the concentration of TW80 to 1 g L -1 , the values were respectively 49 and 5 %. Moreover, in previously works it has been suggested that PHE was trapped into the micelle core of the surfactant, leading to a lower availability toward oxidizing radicals [START_REF] Trellu | Removal of hydrophobic organic pollutants from soil washing/flushing solutions: A critical review[END_REF]Trellu et al., 2017b). The significant sulfate radical selectivity between TW80 and PHE clearly demonstrated that the concentration of TW80 was the major parameter determining the efficiency of oxidative treatment adopted for PHE removal.

Effect of chloride ions addition

Inorganic species such as chloride ions are widely present in soil, resulting in the presence of such ions into the effluent of the procedure of soil washing. However, the existence of chloride ions (Cl -) in wastewater has been reported to affect the degradation efficiency of contaminants during the advanced oxidation processes and more particularly in PS oxidation treatments [START_REF] Huang | Homogeneous and heterogeneous Fenton and photo-Fenton processes : impact of iron complexing agent ethylenediamine-N,N'-disuccinic acid (EDDS). Université Blaise Pascal[END_REF][START_REF] Zhang | Impact of chloride ions on UV/H 2 O 2 and UV/persulfate advanced oxidation processes[END_REF]. Different roles of chloride ions (Cl -) are reported in the literature on the efficiency of SO 4 •-and HO • based oxidations. Generally, it is reported that Cl -acts as a radical scavenger that substantially inhibits the efficiency of UV based activation of H 2 O 2 and S 2 O 8 2- [START_REF] Tsuneda | Inhibition effect of chlorine ion on hydroxyl radical generation in UV-H 2 O 2 process[END_REF]. However, some researchers find that reactive species produced from chloride ions, chlorine and dichlorine radicals (

Cl • /Cl 2 •- ) ,
show high selectivity when react with organic pollutants and increase the efficiency of removal [START_REF] Yang | Comparison of halide impacts on the efficiency of contaminant degradation by sulfate and hydroxyl radical-based advanced oxidation processes (AOPs)[END_REF]. Moreover, in other investigations it is suggested that Cl -could react with SO 4

•-to generate HO • in circumneutral and alkaline pH [START_REF] Kiwi | Mechanism and kinetics of the OH-radical intervention during Fenton oxidation in the presence of a significant amount of radical scavenger (Cl -)[END_REF]. In addition, the involvement of Cl -is reported to modify the degradation mechanism using SO 4

•-based oxidation processes [START_REF] Fang | Sulfate radical-based degradation of polychlorinated biphenyls: Effects of chloride ion and reaction kinetics[END_REF]. As reported in Figure 5, complete degradation of PHE was achieved in 150 min of irradiation with 10 mM of Cl -, while with 50 mM or 100 mM of Cl -, the same efficiency could be obtained within 60 min, demonstrating that the presence of higher dosages of Cl -remarkably enhanced PHE degradation using Fe(III)-EDDS under UVA irradiation. This result was totally distinguished with other studies where Cl -acted as an inhibitor or where its effect was negligible [START_REF] Yang | Effect of matrix components on UV/H 2 O 2 and UV/S 2 O 8 2advanced oxidation processes for trace organic degradation in reverse osmosis brines from municipal wastewater reuse facilities[END_REF]. To better understand the mechanism, for the first time, we determined the second order rate constants of Cl 2 •-with PHE ( ) and TW80 ( ) respectively at 4.6±0.3×10 8 M -1 s -1 and 7.1±0.1×10 6 M -1 s -1 (table 1). This results suggested that second order rate constant between PHE and dichlorine radical anion ( ) was almost 65 times higher than the one between TW80 and the same radical . Therefore, Cl 2 •-generated by Cl - exhibited much more selectivity to PHE in our soil washing solution. Moreover, the form of kinetics demonstrated that reactivity between SO 4 •-and Cl -played an important role to maintain PHE degradation, especially after 30 min of irradiation.

If we simplified our chemical system considering that generated radical species (HO ) could react only with TW80 or PHE in solution, it was possible to explain, from a chemical kinetic point of view the impact of chloride on the system efficiency. In fact, in the absence of chloride ions mainly SO 4

•-was generated in solution trough reaction (2) and photolysis of persulfate ions. Considering the concentration of chemical species (i.e. TW80

and PHE) and reactivity constant, about 82% of SO 4 •-were trapped by TW80. However, if we considered that all SO 4 •-was significantly converted into Cl 2

•-(about 90% using 100 mM of Cl -), in this case, the stronger reactivity between Cl 2 •-and PHE compared to those determined for TW80 suggested that only 6% of Cl 2 •-react with the surfactant and 94% with PHE.

Radical species involvement

In order to clearly clarify the significance of the different radicals formed during the process and their roles in this system and so further elucidate the oxidation mechanism of PHE degradation, free radicals scavenging tests were carried out. Methanol (MeOH) was employed as scavenger for both HO • ) and SO 4

•-(

, and tert-Butanol (t-But) has higher selectivity for reactivity with HO • than with SO 4

•- [START_REF] Buxton | Critical review of rate constants for reactions of hydrated electrons, hydrogen atoms and hydroxyl radicals ( • OH/ • O -) in aqueous solution[END_REF][START_REF] Neta | Rate constants for reactions of inorganic radicals in aqueous solution[END_REF]. With the addition of 50 mM of tert-Butanol, PHE

Conclusion

A novel application of Fe(III) complexes was studied to improve the photochemical activation of persulfate for treatments of soil washing effluent. Experimental parameters such as Fe(III)-EDDS concentration, PS concentration,TW80 concentration exhibited great effects on the PHE photodegradation in this system. The PHE degradation included two stages and the first period of Fe(III)-EDDS decomposition was the rate-limiting step. Quenching experiments proved that SO 4

•-played a more important role than HO • on PHE degradation more particularly in the presence of surfactant which completely modified the reactivity of radical species. Moreover, we provided the evidence that the presence of chloride ions could significantly improve the PHE removal in this system. The higher selectivity of Cl 2 •-between PHE and TW80 was responsible for the efficiency increase.

Table 1: Second-order rate constants (M -1 s -1 ) of TW80 and PHE with different radicals determined using LFP and competition kinetic method.

Compounds

  Figures Caption

Figure 6

 6 Figure 6

Acknowledgments

We gratefully acknowledge the Ministry of Education of the PR of China for providing financial support for Yufang TAO to stay at the Institute of Chemistry of Clermont-Ferrand and Clermont Auvergne University in France. Authors acknowledge financial support from the Region Council of Auvergne, from the "Féderation des Recherches en Environnement" through the CPER "Environment" founded by the Region Auvergne, the French government, FEDER from the European Community from PRC program CNRS/NSFC n°270437 and from CAP 20-25 I-site project.


Felix Gerick
email: felix.gerick@univ-grenoble-alpes.fr

D Jault

J Noir

Fast Quasi-Geostrophic Magneto-Coriolis Modes in the Earth's core

Magneto-Coriolis modes of periods close to torsional Alfvén modes could be present in Earth's core model without stratification.

• The magnetic field changes of such modes show properties similar to geomagnetic observations, with fast changes localized near the equator.

• Our model could allow core-flow inversions from geomagnetic field data to the flow and simultaneously the magnetic field within the core.

1 Introduction [START_REF] Hide | Free hydromagnetic oscillations of the earth's core and the theory of the geomagnetic secular variation[END_REF] proposed that temporal changes of Earth's magnetic field, called secular variations (SV), could originate from linear modes present in the Earth's liquid outer core. These modes are separated into modes dominated by a balance of magnetic, Coriolis and pressure forces, known as Magneto-Coriolis modes (MCM), and modes dominated by inertial, Coriolis and pressure forces. The latter are often referred to as quasi-geostrophic (QG) inertial modes, or Rossby modes (RM). Torsional Alfvén modes (TM), consisting of geostrophic motions [START_REF] Braginsky | Torsional magnetohydrodynamics vibrations in the Earth's core and variations in day length[END_REF], complete the set of incompressible and diffusion-less magnetohydrodynamic (MHD) modes. They obey a balance between inertia and the magnetic force. In this study, we use a reduced model, based on the QG assumption for the velocity and a three-dimensional (3-D) magnetic field that is compatible with an insulating mantle, to investigate the SV associated with such QG modes, with a focus on MCM, in the Earth's core.

Monitoring the radial magnetic field component at the core-mantle boundary (CMB) is the main way of probing flows in the liquid outer core of Earth. A large number of studies are concerned with the inversion of the downward projected geomagnetic field to flows in the outer core, a process referred to as core-flow inversion (see [START_REF] Holme | 8.04 -Large-Scale Flow in the Core[END_REF], for a review). The most commonly applied core-flow inversions are based on geostrophic flows tangential to the CMB [START_REF] Le Mouël | Outer-core geostrophic flow and secular variation of Earth's geomagnetic field[END_REF][START_REF] Chulliat | Local computation of the geostrophic pressure at the top of the core[END_REF]. These inversions give the flow field local to the CMB. Several works have then inferred core dynamics from the inverted flow field at the CMB. [START_REF] Zatman | Torsional oscillations and the magnetic field within the Earth's core[END_REF] and more recently [START_REF] Gillet | Fast torsional waves and strong magnetic field within the Earth's core[END_REF] have inverted these surface flows to the mean radial magnetic field component within Earth's core through TM. [START_REF] Buffett | Geomagnetic fluctuations reveal stable stratification at the top of the Earth's core[END_REF] correlated Magneto-Archimedes-Coriolis (MAC) waves in a stably-stratified layer at the top of the outer core with the inferred surface flows.

Our model potentially serves as a new forward model to invert geomagnetic field observations. In this approach a reduced set of MHD equations is solved in the bulk of the fluid. It is based on the QG assumption for the velocity, where a balance between Coriolis and pressure gradient forces is dominant, while allowing linear axial dependence of the flow field. This assumption is appropriate to investigate fluids under rapid rotation at time scales much larger than the rotation period. Different studies have shown that a large part of the inferred surface core flow is equatorially symmetric and may account for a large part of the observed secular variations [START_REF] Gillet | Ensemble inversion of time-dependent core flow models[END_REF][START_REF] Gillet | Rationale and geophysical evidence for quasi-geostrophic rapid dynamics within the Earth's outer core[END_REF]. Additionally, recent 3-D high resolution numerical simulations revealed a largely columnar flow structure, in agreement with the QG assumption (e.g. [START_REF] Schaeffer | Turbulent geodynamo simulations: a leap towards Earth's core[END_REF]. To establish and maintain such a columnar flow within the Earth's core different mechanisms have been proposed, e.g. 3-D inertial-Alfvén waves that transport energy along the rotation axis [START_REF] Bardsley | Inertial-Alfvén waves as columnar helices in planetary cores[END_REF]. Such built up of columns occurs on diurnal periods and are thus not captured in the QG model, where a columnar structure is assumed to be already established. Previously, consistently derived QG models that include a magnetic field were limited to magnetic fields that treated the CMB as a perfectly conducting boundary [START_REF] Busse | of planetary magnetism by convection[END_REF][START_REF] Canet | Hydromagnetic quasi-geostrophic modes in rapidly rotating planetary cores[END_REF][START_REF] Labbé | On magnetostrophic inertia-less waves in quasi-geostrophic models of planetary cores[END_REF][START_REF] Gerick | Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle[END_REF]. For such magnetic fields the radial component at the boundary must vanish, rendering them unsuitable to associate core flows with magnetic field imprints at the CMB. There have been approaches to combine a QG model with a magnetic field that has a non-zero radial magnetic field component at the surface, but they rely on the neglect of surface terms in the induction equation that is difficult to justify [START_REF] Canet | Forward and adjoint quasi-geostrophic models of the geomagnetic secular variation[END_REF][START_REF] Maffei | Kinematic validation of a quasi-geostrophic model for the fast dynamics in the Earth's outer core[END_REF]. Here we present a hybrid model that combines QG velocities with a 3-D insulating magnetic field. We follow the approach presented in [START_REF] Gerick | Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle[END_REF] with a new basis for the magnetic field that satisfies the insulating boundary condition at the CMB. Both the QG velocity and the magnetic field basis vectors are expressed in Cartesian polynomials. This methodology has been fruitful to model modes and instabilities in rapidly rotating ellipsoids [START_REF] Vantieghem | Inertial modes in a rotating triaxial ellipsoid[END_REF][START_REF] Vidal | Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits[END_REF][START_REF] Vidal | Fossil field decay due to nonlinear tides in massive binaries[END_REF][START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF]. This Cartesian presentation of the basis vectors is particularly useful for the Galerkin approach used here, due to the easy integration of Cartesian monomials over the volume [START_REF] Lebovitz | The stability equations for rotating, inviscid fluids: Galerkin methods and orthogonal bases[END_REF]. We derive a basis for the magnetic field in Cartesian polynomials, that exploits the properties of spherical harmonics.

2 A Hybrid Quasi-Geostrophic Model and Columnar Modes

Magnetohydrodynamic Equations

The equations governing the incompressible flow u and the magnetic field B in a rapidly rotating planetary core of volume V, here assumed to be a full sphere without an inner core, are given in non-dimensionalized form by

∂u ∂t + (u • ∇) u = - 2 Le 1 z × u -∇p + Pm Lu ∇ 2 u + (∇ × B) × B, (1a)
∂B ∂t =∇ × (u × B) + 1 Lu ∇ 2 B. (1b)
The non-dimensional Lehnert, Lundquist and magnetic Prandtl number are given by

Le = B 0 ΩR 0 √ µ 0 ρ , Lu = R 0 B 0 η √ µ 0 ρ , Pm = ν η , (2)
with Ω = Ω1 z the rotation vector, ρ the fluid density, p the reduced pressure, ν the kinematic viscosity, µ 0 the permeability of vacuum, η the magnetic diffusivity, R 0 the core radius and B 0 the characteristic strength of the magnetic field. The characteristic time scale is the Alfvén time scale

T A = R 0 /u A , where u A = B 0 / √ ρµ 0
is the characteristic Alfvén velocity. Equations (1) are subject to the non-slip boundary condition u = 0 and the continuity of the magnetic field across the boundary [B] = 0, where [•] denotes a jump.

For parameters relevant for Earth's core, Le ∼ 10 -4 , Lu ∼ 10 5 and Pm ∼ 10 -6 [START_REF] Wijs | The viscosity of liquid iron at the physical conditions of the Earth's core[END_REF][START_REF] Gillet | Fast torsional waves and strong magnetic field within the Earth's core[END_REF][START_REF] Pozzo | Thermal and electrical conductivity of solid iron and iron-silicon mixtures at Earth's core conditions[END_REF]. Thus, if we additionally consider time scales on the order of T A , it is appropriate to neglect viscous and diffusive effects in the bulk. In the next step, since we are interested in the linear response of the system, the velocity and magnetic field are perturbed around a background state with no motion and steady magnetic field B 0 . In the Earth's core, the characteristic mean velocity field is thought to be negligible compared to the Alfvén wave velocity [START_REF] Gillet | Planetary gyre, time-dependent eddies, torsional waves, and equatorial jets at the Earth's core surface[END_REF][START_REF] Bärenzung | Modeling and Predicting the Short-Term Evolution of the Geomagnetic Field[END_REF]. Hence, the equations describing the evolution of the velocity and magnetic perturbations [ũ, B] are given by

∂ ũ ∂t + 2 Le 1 z × ũ = -∇p + (∇ × B 0 ) × B + (∇ × B) × B 0 , (3a)
∂ B ∂t =∇ × (ũ × B 0 ) . (3b)
In the limit Pm → 0 the boundary conditions on the velocity reduces to the nonpenetration condition u • n = 0, with n the vector normal to the boundary, and the magnetic boundary condition is not modified [START_REF] Stewartson | The dispersion of a current on the surface of a highly conducting fluid[END_REF][START_REF] Hide | Hydromagnetic oscillations of the Earth's core[END_REF]. Previous studies allowed for a jump in the tangential component of the magnetic field across a diffusive boundary layer [START_REF] Braginsky | Torsional magnetohydrodynamics vibrations in the Earth's core and variations in day length[END_REF]Jault & Finlay, 2015). Here, we assume that the motions that we are investigating are able to eliminate any current layer on the fluid surface.

Quasi-Geostrophic Velocity Basis

Assuming that the equatorial components of the velocity are independent of the coordinate z along the rotation axis; the non-penetration boundary condition, u•n = 0 on the core-mantle boundary ∂V holds and the flow is incompressible, ∇ • u = 0, the quasi-geostrophic (QG) velocity takes the form [START_REF] Amit | Helical core flow from geomagnetic secular variation[END_REF][START_REF] Schaeffer | Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers[END_REF][START_REF] Bardsley | Could hydrodynamic Rossby waves explain the westward drift[END_REF]

u = ∇ψ × ∇ z h , (4)
with h the half height of the fluid column and ψ a scalar stream function depending only on the horizontal coordinates.

In Cartesian coordinates the stream function can be expressed as [START_REF] Maffei | Propagation and reflection of diffusionless torsional waves in a sphere[END_REF][START_REF] Gerick | Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle[END_REF])

ψ i = h 3 Π i , (5)
with Π i being a monomial in the equatorial Cartesian coordinates x and y of degree N , so that i ∈ [0, N 2 ] with N 2 = N (N + 1)/2. The QG basis vectors u i are given by

u i = h 2 ∇Π i × 1 z + 3Π i ∇G × 1 z -z∇Π i × ∇G, (6)
with ∇G = h∇h = -x1 x -y1 y .

Magnetic Field Basis

In this section we present a set of basis vectors for the 3-D magnetic field, satisfying insulating boundary conditions at the CMB. Unlike in classical geodynamo simulations, where the boundary condition is enforced at each time step of the forward iteration, the boundary condition is included in the basis elements [START_REF] Zhang | Hydromagnetic waves in rapidly rotating spherical shells generated by poloidal decay modes[END_REF][START_REF] Li | An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere[END_REF][START_REF] Chen | The optimal kinematic dynamo driven by steady flows in a sphere[END_REF]. The detailed derivation of such a basis is given in Appendix A.

We write the magnetic field B in the toroidal-poloidal expansion, so that

B = B t + B p = ∇ × T r + ∇ × ∇ × P r. (7)
The toroidal and poloidal scalars are written for each spherical harmonic degree l, order m and radial degree n, so that

T lmn = (1 -r 2 )r 2n R m l , (8)
P lmn = -r 2(n+1) R m l 2(n + 1)(2(l + n) + 3) . ( 9
)
with R m l = r l Y m l (θ, φ) the solid spherical harmonics. We have |m| ≤ l and l ∈ [1, N ], n ∈ [0, (Nl)/2 for the toroidal basis and l ∈ [0, N -1], n ∈ [0, (N + 1l)/2 -1 for the poloidal basis, resulting in a total of N 3 = 1 6 N (N + 1)(2N + 7) basis vectors. The toroidal part of B is given by

B t,lmn = ∇ × T lmn r. ( 10
)
The poloidal magnetic field has to satisfy the continuity at the core-mantle boundary ∂V, so that ∇Φ i + B p = ∇Φ e , with Φ i and Φ e the interior and exterior potential field, respectively. The exterior potential field must vanish at infinity, if the source of the magnetic field lies within the interior. The poloidal basis vectors are thus given by

B p,lmn = ∇ × ∇ × P lmn r + ∇Φ i lmn , (11)
with

Φ i lmn = - (l+1) (2l+1)(2n+2) r l Y m l .
Together with the toroidal component (10), this basis can be transformed to Cartesian coordinates by representing the spherical harmonics in terms of unit Cartesian coordinates, as presented in Appendix B. We choose the Schmidt-semi normalization for the spherical harmonics, but any other may be chosen, as we normalize the basis vectors afterwards, so that V B i • B i dV = 1.

Projection Method

We introduce a hybrid quasi-geostrophic (QG) model with QG velocities and 3-D magnetic field , following [START_REF] Gerick | Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle[END_REF]. The linearized momentum equation (3a) and induction equation (3b) are projected onto a QG basis u of the form of (4) and a 3-D magnetic field basis B of the form ( 10) and ( 11), respectively. This method is essentially a variational approach, which consists in finding solutions [ũ, B] satisfying

V u • ∂ ũ ∂t dV = - V u • 2 Le 1 z × ũ + ∇p dV + V u • (∇ × B 0 ) × B + (∇ × B) × B 0 dV ∀ u (12a) V B • ∂B ∂t dV = V B • ∇ × (ũ × B 0 ) dV ∀ B (12b)
This set of equations may be reduced to a scalar evolution equation for the stream function ψ accompanied by the 3-D induction equation, as shown in equation ( 48) of [START_REF] Gerick | Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle[END_REF]. This hybrid model has been verified against a fully 3-D model at moderate polynomial truncation (see also [START_REF] Gerick | Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle[END_REF].

When replacing the test functions u in (12a) by the subset of purely geostrophic velocities u G = u G (s)1 φ we obtain the equation for the diffusion-less torsional Alfvén modes (TM) initially discovered by [START_REF] Braginsky | Torsional magnetohydrodynamics vibrations in the Earth's core and variations in day length[END_REF]. The one-dimensional (1-D) TM equation is written

s 3 H ∂ 2 ξ ∂t 2 = ∂ ∂s s 3 Hv 2 A ∂ξ ∂s (13)
with ξ = u G (s)/s and the mean squared cylindrical Alfvén velocity

v 2 A (s) = 1 4πsH (B 0 • 1 s ) 2 s dzdφ. ( 14
)
For more details on the derivation we refer the reader to [START_REF] Jault | Electromagnetic and topographic coupling, and LOD variations[END_REF]. Equation ( 13) diverges near the equator as s → 1, but solutions exist, if v 2 A (1) = 0 [START_REF] Maffei | Propagation and reflection of diffusionless torsional waves in a sphere[END_REF]. Since we can compute solutions to the 1-D equation for a given background magnetic field satisfying these conditions, TM suit well as a benchmark of our hybrid QG model capable of capturing TM. [START_REF] Lehnert | Magnetohydrodynamic Waves Under the Action of the Coriolis Force[END_REF] introduced two distinct families of MHD modes as solutions to the linearized MHD equations (3), namely slow MCM and fast, slightly modified RM.

Quasi-geostrophic Inertial and Magneto-Coriolis Modes

The phase velocity is prograde for slightly modified RM and retrograde for MCM. [START_REF] Malkus | Hydromagnetic planetary waves[END_REF] has shown that for an idealized magnetic field B 0,M = s1 φ of uniform current density analytical solutions exists for these two mode families. The dispersion relations are given by [START_REF] Labbé | On magnetostrophic inertia-less waves in quasi-geostrophic models of planetary cores[END_REF])

ω ± n,m = 1 2Le λ n,m 1 ± 1 + 4Le 2 m(m-λ n,m ) λ 2 n,m 1/2 , (15)
with ω + n,m , ω - n,m and λ n,m the frequencies of the slightly modified RM, MCM and hydrodynamic (HD) inertial modes of azimuthal wave number m and radial scale n, respectively. The dispersion relation shows that the difference between ω + n,m and λ n,m /Le is small, if Le 1. The frequencies λ n,m are scaled by the rotation frequency and can be obtained as solutions to a univariate polynomial in the sphere [START_REF] Zhang | On inertial waves in a rotating fluid sphere[END_REF]). An approximate value for the equatorially symmetric inertial modes is given by [START_REF] Zhang | On inertial waves in a rotating fluid sphere[END_REF])

λ n,m ≈ - 2 m + 2 1 + m(m + 2) n(2n + 2m + 1) 1/2 -1 . ( 16
)
As the magnetic field perturbations in the Malkus field satisfy the perfectly conducting boundary condition, with B • n = 0 at ∂V, the solutions cannot be associated with the SV at the CMB and a more suitable background magnetic field needs to be introduced.

Numerical Calculation of the Modes

The velocity and magnetic field perturbations are assumed to be periodic in time, i.e. ũ(r, t) = ũ(r) exp(iωt), (17a) B(r, t) = B(r) exp(iωt).

(17b)

Enumerating the QG velocity basis ũi , with i = 1, ..., N 2 and magnetic field basis Bi , with i = 1, ..., N 3 , the projections (12) discretize to iωMx = Dx,

with x = (α j , ζ j ) ∈ C N2+N3 , and M, D ∈ R N2+N3×N2+N3 of the form

M = U ij 0 0 B ij , D = C ij L ij V ij 0 . (19)
Here, the coefficient matrices

U ij , C ij ∈ R N2×N2 , L ij ∈ R N2×N3 , B ij ∈ R N3×N3 and V ij ∈ R N3×N2
correspond to the inertial acceleration, Coriolis force, Lorentz force, time change of magnetic field and magnetic induction respectively. This form is referred to as a generalized eigen problem solvable for eigen pairs (ω k , x k ). A large part of M and D is zero, which allows us to use large polynomial degrees whilst keeping computational efforts small. To avoid numerical inaccuracies we use quadruple precision numbers (see supplementary material for more details). Our model extends the code available at https://github.com/fgerick/Mire.jl by the magnetic field basis introduced in 2.3.

Results

We choose a poloidal magnetic field of low polynomial degree, given by with the constant c = 5 368 4725 , so that V B 0 • B 0 dV = 1. This field has a non-zero mean radial magnetic field at the equator. Its mean squared radial Alfvén velocity profile is given by

B 0 =∇ × ∇ × (P 100 + P 110 )r + ∇ α 100 Φ i 100 + α 110 Φ i 110 = 1 c   x 2 + xz + 2y 2 + 2z 2 -5/3 -xy + yz -2x 2 -xz -2y 2 -z 2 + 5/3   , (20)
v 2 A (s) = - 189 1840 s 4 - 42 115 s 2 + 1281 1840 , (21)
which decays slowly enough to ensure that TM are captured with low polynomial degrees. The Lehnert number is chosen to be Le = 10 -4 , corresponding to a mean magnetic field strength of about 3 mT within Earth's core [START_REF] Gillet | Fast torsional waves and strong magnetic field within the Earth's core[END_REF].

Properties of the Mode Spectrum

The density of the eigen solution spectrum is shown in Figure 1 (left) for different degrees of truncation up to N = 29. The band limitation of the spectrum is easily seen on the fast end of the spectrum with a similar upper end of frequencies for all truncation degrees. The fastest mode in the spectrum approximately corresponds to |ω + 1,5 | ≈ 0.26Le -1 (or 0.26Ω) instead of |ω| < 2Ω for inertial modes [START_REF] Greenspan | The theory of rotating fluids[END_REF]). For MCM the slowest frequency is affected by the truncation. This is due to the fact that the convergence of magnetic modes depends on the truncation, as the Lorentz force alters the polynomial degree of the modes, unlike the Coriolis operator [START_REF] Ivers | Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere[END_REF]. For a truncation N ≤ 13 TM are separated from the MCM and the fast modes. At larger N some MCM are present also in the frequency range of TM. At low truncation the classification of MCM, RM and TM is straightforward by the difference in frequencies. At higher truncation we classify the modes by their kinetic and magnetic energies.

The kinetic and magnetic energies are respectively given by

E kin = 1 2 V u • u dV, ( 22
)
E mag = 1 2 V B • B dV. ( 23
)
At the degree of N = 29 both RM and MCM reach frequencies around the TM frequency range, but their energy ratio E kin /E mag is still different from unity (see Figure 1, right). At the considered polynomial degrees more MCM have periods comparable to those of TM than RM. This bias can be explained by the periods of Malkus modes as a function of n and m (compare middle Figure 2

2015)

. For RM the periods only decrease towards unity as a function of n, not of m (unless m 1). For MCM both an increase in m and n leads to an increase of the mode period towards unity. This explains why at a certain truncation level the fastest MCM is closer to the periods of TM than the slowest RM. Compared to the Malkus field, MCM (RM) spread out to higher (lower) frequencies and higher (lower) energy ratio.

Torsional Modes

Given the profile of v 2

A (s) we compute the TM by integrating (13) using finite differencing. The 1-D equation implies that ∂ξ/∂s = 0 at s = 1 and it is automatically satisfied in our solver, if v 2 A (1) = 0. The selection of TM in the spectrum of modes of the hybrid solution is done by considering the frequency range indicated by the 1-D solutions and by a unit ratio of kinetic to magnetic energy of the eigen solutions. The comparison between the 1-D solutions (dashed black) and the hybrid model (solid colors) is shown in Figure 2. The six largest scale TM calculated by the hybrid model are in excellent agreement with the 1-D solutions. The frequencies obtained from the two models have a relative difference of O(10 -3 ). We see that for both models ∂ξ/∂s = 0 at s = 1, as expected. For the hybrid model the resolution of this boundary condition depends on the radial wave number of the TM in s, the spatial heterogeneity of B 0 and the polynomial truncation. At a polynomial truncation of N = 29 at least the six largest scale TM are well resolved by the basis. The spatial structure of the magnetic field component of the TM depends only on the structure of B 0 and the complexity of the TM in s (not presented here, see e.g. [START_REF] Cox | The observational signature of modelled torsional waves and comparison to geomagnetic jerks[END_REF].

Slightly Modified Rossby Modes

For the range of polynomial degrees studied here, RM are only slightly influenced by the presence of the magnetic field. Their spatial structure and frequency remain comparable to that of the RM in the purely HD case. We compared the frequencies of some of the largest scale RM to the frequencies of the RM when including magnetic forces with the Malkus field B 0,M = s1 φ and (20). The relative differences between Table 1. Non-dimensional frequency ω, dimensionalized period T in years (for |B0| = 3 mT), and ratio of kinematic to magnetic energy E kin /Emag of the six slowest TM (displayed in Figure 2), the three fastest RM and MCM (displayed in Figure 3). the frequencies of the three models are below 10 -5 for all modes up to a truncation of N = 29. This difference can increase at N > 29, as may be anticipated by the dispersion relation ( 15). The spatial structure also remains mostly unchanged and their phase velocity is prograde, as observed in the HD case. We show the velocity and the radial magnetic field at the surface of three RM in Figure 3a. The slowest modes are associated with m = 1 and increasingly large N (see the top mode). This can be seen also by a careful analysis of the approximated dispersion relation ( 16). We found even slower RM in our model, but we display three modes that are well converged from degree N = 29 to N = 35 (see details in supplementary material). In both the HD and MHD case an increase of the velocity amplitudes near the equator is observed (compare with RM in [START_REF] Kloss | Time-dependent low-latitude core flow and geomagnetic field acceleration pulses[END_REF]. Their ratio of kinetic to magnetic energy is O(10), suggesting that their surface magnetic field perturbation might be observable in the future.

Magneto-Coriolis Modes

MCM are strongly influenced by the background magnetic field. They are not easily compared between different magnetic fields, e.g. between the idealized Malkus field and the magnetic field B 0 . Instead we focus on the MCM of relatively high frequency, that are of particular interest here. From Figure 1 we see that some MCM evolve on time scales similar to those of TM. We select three MCM (MCM1-3) with dimensionalized periods of a few years (see exact figures in Table 1), that show a polynomial complexity below that of the truncation degree and a converged structure (see supplementary material). The spatial structure of these selected MCM is presented in Figure 3b, showing that they have a large complexity along the cylindrical radius and a relatively small azimuthal wave number. The short length scale in cylindrical radius is even more evident in the radial profile of the azimuthal velocity in the equatorial plane, shown in Figure 3c. Similarly to the slow RM, the fast MCM concentrate their kinetic and magnetic energy near the equator. All MCM observed here travel retrograde, compared to the prograde direction of RM, as predicted by [START_REF] Hide | Free hydromagnetic oscillations of the earth's core and the theory of the geomagnetic secular variation[END_REF]. For an azimuthal wave number m = 2 in the equatorial band, the magnetic field perturbations of the three displayed MCM have a phase velocity of ωm -1 u A ≈ 680 -2400 km/yr. However, we highlight that other wave numbers contribute to each mode with different phase velocities. 1.

Discussion

We have shown, for Le = 10 -4 and a mean magnetic field strength in the core interior of about 3 mT, that changes of the magnetic field on periods as short as a few years could be explained by MCM. Since the periods of these fast MCM are only a few years they may be associated with the periodic secular acceleration impulses inferred from recent satellite observations [START_REF] Chulliat | Fast equatorial waves propagating at the top of the Earth's core[END_REF][START_REF] Kloss | Time-dependent low-latitude core flow and geomagnetic field acceleration pulses[END_REF][START_REF] Chi-Durán | Decomposition of geomagnetic secular acceleration into traveling waves using complex empirical orthogonal functions[END_REF]. These observations have been interpreted as the signature of MCM in the presence of a strong azimuthal magnetic field [START_REF] Hori | Slow magnetic Rossby waves in the Earth's core[END_REF] or of MAC waves in a stratified layer at the top of the core [START_REF] Knezek | Influence of magnetic field configuration on magnetohydrodynamic waves in Earth's core[END_REF][START_REF] Buffett | Equatorially trapped waves in Earth's core[END_REF]. We find there is no need to introduce a magnetic field stronger than inferred from TM or a stratified layer to account for fast wave propagation in the equatorial region of the core. A key result of our study is the presence of large horizontal scales of B r at the core surface next to the equator associated with MCM, while the mode structure itself remains small scale in the cylindrical radial direction. Such large magnetic features near the equator should be captured by satellite observations. Meanwhile, [START_REF] Aubert | Geomagnetic jerks and rapid hydromagnetic waves focusing at Earth's core surface[END_REF] have linked the secular variation impulses to so-called QG Alfvén waves arising near strongly heterogeneous magnetic fields of buoyant plumes in their numerical simulations. Whether or not our fast MCM are in agreement with their explanation remains to be investigated. Our model could be used to invert geomagnetic observations for such a possible excitation mechanism, as described by [START_REF] Buffett | Inversion of torsional oscillations for the structure and dynamics of Earth's core[END_REF] for TM.

Fast MCM show a concentration of energy near the equator, similar to the slowest RM. Equatorially trapped waves have been much discussed either from observations [START_REF] Chulliat | Fast equatorial waves propagating at the top of the Earth's core[END_REF] or from physical models [START_REF] Bergman | Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core[END_REF]. The surface core flow calculations of [START_REF] Gillet | A reduced stochastic model of core surface dynamics based on geodynamo simulations[END_REF] also show the largest core flow acceleration pattern in an equatorial belt below 10 • of latitude. Concentration of energy of the modes could be favored by the weaker intensity of B 0 in the equatorial region, which [START_REF] Knezek | Influence of magnetic field configuration on magnetohydrodynamic waves in Earth's core[END_REF]; [START_REF] Buffett | Equatorially trapped waves in Earth's core[END_REF] also found to be important for the focusing of MAC waves in the equatorial region of a stratified layer. A systematic study over a wider range of magnetic field geometries would be needed to make this statement quantitative. [START_REF] Bergman | Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core[END_REF] and [START_REF] Buffett | Equatorially trapped waves in Earth's core[END_REF] have shown that equatorially trapped MAC modes are strongly affected by damping. We haven't included diffusion in our study and an investigation into how a diffusive layer at the top of the core may influence the observed fast MCM is necessary, even though they are of large spatial scale at the equator. The new basis presented here potentially allows us to include magnetic diffusion, at a substantial computational cost.

Previously, dynamics in the bulk of the core have been linked to inverted surface flows, but not directly to the observed changes in the magnetic field. Being able to associate at once MCM, as well as TM, to magnetic field changes occurring with periods of 10 years or less and yet with large horizontal scale in an equatorial band at the CMB opens new perspectives for data assimilation and analyses of the dynamics occurring in the Earth's outer core.

Figure 1 .

 1 Figure 1. An overlap of the density of eigen solution spectrums at different truncation 9 ≤ N ≤ 29 and Le = 10 -4 (left). Ratio of kinetic to magnetic energy at N = 29 (right).

Figure 2 .

 2 Figure 2. The six largest scale TM of the hybrid model at N = 29 and Le = 10 -4 (solid colors) and of the 1-D equation (dashed black).

Figure 3

 3 Figure 3. a) Core surface flows (top row) and associated radial magnetic field perturbation at the surface (bottom row) of the three fastest, converged, RM (RM1-3). b) Core surface flows (top row) and associated radial magnetic field perturbation at the surface (bottom row) of three selected MCM (MCM1-3) in the TM frequency range. Colors indicate the azimuthal velocity magnitude with blue being prograde and red being retrograde and inward (blue) and outward (orange) magnetic flux. The arrows indicate tracers of the surface velocity. c) Cylindrical-radial profile of the azimuthal velocity in the equatorial plane at φ = 0 of MCM1-3. The modes are computed for a maximum polynomial degree N = 35. The frequencies and periods of the modes are given inTable 1.

  in Labbé et al.,

		1.0					ω 1 =2.67
		0.8					ω 2 =4.39 ω 3 =6.05
		0.6					ω 4 =7.68
							ω 5 =9.30
	ξ(s)	0.4					ω 6 =10.92
		0.2
		0.0
		-0.2
		-0.4	0.0	0.2	0.4	0.6	0.8	1.0
					s

Acknowledgments

The authors like to thank Henri-Claude Nataf and two anonymous reviewers for their help in improving this work. FG was partly funded by Labex OSUG@2020 (ANR10 LABX56). Support is acknowledged from the European Space Agency through contract 4000127193/19/NL/IA. This work has been carried out with financial support from CNES (Centre National d' Études Spatiales, France). JN was partly funded by SNF Grant #200021 185088. Computations were performed on ETH Zurich's Euler cluster. The data and the code to calculate it can be found at https://dx.doi.org/ 10.5281/zenodo.4008396.

Appendix A Derivation of Magnetic Field Basis

Let us write the current density j = ∇ × B in the toroidal-poloidal expansion, so that j = ∇ × Qr + ∇ × ∇ × Sr.

(A1)

We can also write the magnetic field in the toroidal-poloidal expansion, with

It follows [START_REF] Backus | Foundations of Geomagnetism[END_REF] that

We construct the toroidal and poloidal scalars for the basis of j following the velocity basis introduced by [START_REF] Ivers | Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere[END_REF], so that

with R m l = r l Y m l (θ, φ) the solid spherical harmonics. We have |m| ≤ l and l ∈ [1, N ], n ∈ [0, (Nl)/2 for the toroidal basis and l ∈ [0, N -1], n ∈ [0, (N + 1l)/2 -1 for the poloidal basis. These N 3 = 1 6 N (N + 1)(2N + 7) elements form a complete basis for the current density in the set of polynomial vector fields of degree N in the volume V [START_REF] Ivers | Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere[END_REF]. Then, the toroidal part of B is directly given by B t,lmn = ∇ × S lmn r.

(A7)

For the poloidal part

we need to solve the Poisson equation

We can use a slightly modified version of equation (3.1.9) in [START_REF] Backus | Foundations of Geomagnetism[END_REF])

to find that

It remains to ensure that the internal magnetic field can be matched to a potential field at the boundary. The poloidal component has to satisfy

with Φ i and Φ e the interior and exterior potential field, respectively. The exterior potential field must vanish at infinity, if the source of the magnetic field lies within the interior. We can solve for Φ i and Φ e by considering the radial component of (A12) and the horizontal component of (A12)

Using the properties of the spherical harmonics, we find that

with L 2 P = ∂ r (r 2 ∂P/∂r)r 2 ∇ 2 P and the system simplifies to

Since this is linearly independent of l, m, and n we can consider this system for each B p,lmn individually, so that

we need to search for potentials of the form

The system to be solved for each l, n is then

with α = 2(n + 1)(2(l + n) + 3), for each l, n. The solutions to (A19) are

so that the poloidal basis vectors are given by B p,lmn = ∇ × ∇ × P lmn r + ∇Φ i lmn .

(A21) [START_REF] Li | An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere[END_REF] presented a similar basis to express the magnetic field for an insulating mantle, which involves slightly more complicated expressions for the poloidal and toroidal scalars with Jacobi polynomials. We have not proven any weighted orthogonal inner products of the poloidal and toroidal scalars, but our basis vectors show the same orthogonality for the unweighted inner product between vectors of different harmonic order and degree as does the basis of [START_REF] Li | An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere[END_REF]. Orthogonal bases, based on Jacobi polynomials and spherical harmonics, have been presented in [START_REF] Chen | The optimal kinematic dynamo driven by steady flows in a sphere[END_REF]; [START_REF] Li | Taylor state dynamos found by optimal control: axisymmetric examples[END_REF]. These bases are desirable for reducing computational efforts, but without an orthogonal QG basis, no such computational advantage is given for the hybrid model presented here.

Appendix B Spherical Harmonics in Cartesian Coordinates

The unnormalized spherical harmonics are defined as Ỹ m l (θ, φ) = e imφ P m l (cos θ), (B1)

with the so-called associated Legendre polynomials

The unit Cartesian coordinates are given by x = x/r = cos φ sin θ, (B4) ŷ = y/r = sin φ sin θ, (B5) ẑ = z/r = cos θ.

(B6)

We can rewrite (B2), so that

Using the trigonometric identities (B10)

The solid spherical harmonics are Rm l (x, y, z) = r l Ỹ m l (x, ŷ, ẑ), (B11) which is polynomial in x, y and z.