Naouel Boulenoir

Guillaume Turc

Hilde Henon

Nadia Laksiri

François Mounier-Véhier

Isabelle Girard-Buttaz

Duc-Long Duong

Jérémie Papassin

Marion Yger

Aude Triquenot

Aïcha Lyoubi

MD-MSc Adrien Ter Schiphorst

Christian Denier

Jean-Claude Baron

Pierre Seners
email: pierre.seners@gmail.com

Wagih Ben Hassen

Bertrand Lapergue

Chantal Lamy

Ludovic Lucas

Frédéric Philippeau

Michael Obadia

Michel Piotin

Omar Bennani

Laura Mechtouff

Frédéric Klapczynski

Olivier Detante

Caroline Arquizan

Gioia Mione

Denis Sablot

Sébastien Gazzola

Séverine Debiais

Serkan Cakmak

Valer Grigoras

Ruben Tamazyan

Cécile Preterre

Charlotte Rosso

Philippe Niclot

Pierre Garnier

Sébastien Soize

François Lun

Amélie Tuffal

Marion Boulanger

Mathieu Zuber

Fernando Pico

Frédéric Clarençon

Igor Sibon

Sonia Alamowitch

Laurie Fraticelli

Anne-Evelyne Vallet

Vincent Costalat

Catherine Oppenheim

Jean-Philippe Cottier

Yves Berthezene

Hubert Desal

Pierre Agius

Claire Boutet

Nicolas Bricout

Didier Leys

Mohamed Chbicheb

Arnaud Malbranque

Laurent Spelle

Serge Bracard

Early neurological deterioration following

Keywords: Cerebral ischemia, Thrombolysis, Thrombectomy

INTRODUCTION

Isolated cervical or intracranial internal carotid artery occlusion (iICAo), i.e., not involving the circle of Willis, is found in ~5% of acute stroke patients admitted in an early time window [START_REF] Turc | Clinical Scales Do Not Reliably Identify Acute Ischemic Stroke Patients With Large-Artery Occlusion[END_REF]. Clinical severity is highly variable, in part as a function of collateral circulation [START_REF] Shuaib | Collateral blood vessels in acute ischaemic stroke: a potential therapeutic target[END_REF]. The early clinical course of iICAo patients treated or not with reperfusion therapies has been reported in a few articles only [START_REF] Damania | Factors associated with recurrent stroke and recanalization in patients presenting with isolated symptomatic carotid occlusion[END_REF][START_REF] Mazya | Minor stroke due to large artery occlusion. When is intravenous thrombolysis not enough? Results from the SITS International Stroke Thrombolysis Register[END_REF][START_REF] Jadhav | Angioplasty and stenting for symptomatic extracranial nontandem internal carotid artery occlusion[END_REF][START_REF] De Castro-Afonso | Endovascular Reperfusion for Acute Isolated Cervical Carotid Occlusions: The Concept of "Hemodynamic Thrombectomy[END_REF][START_REF] Gliem | Outcome and Treatment Effects in Stroke Associated with Acute Cervical ICA Occlusion[END_REF], and remains poorly known. Following intravenous thrombolysis (IVT) -the recommended therapy for acute ischemic stroke in the early time window [START_REF] Powers | Guidelines for the Early Management of Patients With Acute Ischemic Stroke: A Guideline for Healthcare Professionals From the American Heart Association/American Stroke Association[END_REF]-, a worrying 17% rate of early neurological deterioration (END) within the first 24 hours was recently reported in a multicentric cohort of iICAo patients with mild baseline symptoms [START_REF] Mazya | Minor stroke due to large artery occlusion. When is intravenous thrombolysis not enough? Results from the SITS International Stroke Thrombolysis Register[END_REF].

Considering that END is strongly associated with poor 3-month functional outcome [START_REF] Mazya | Minor stroke due to large artery occlusion. When is intravenous thrombolysis not enough? Results from the SITS International Stroke Thrombolysis Register[END_REF][START_REF] Seners | Incidence, causes and predictors of neurological deterioration occurring within 24 h following acute ischaemic stroke: a systematic review with pathophysiological implications[END_REF], better understanding the mechanisms underlying END after acutely symptomatic iICAo may inform preventative decisions. Apart from symptomatic intracranial haemorrhage, two main underlying mechanisms, each with widely distinct therapeutic implications, may explain END in this context [START_REF] Seners | Revisiting 'progressive stroke': incidence, predictors, pathophysiology, and management of unexplained early neurological deterioration following acute ischemic stroke[END_REF]: i) thrombo-embolic events, including distal artery-to-artery embolism and in situ extension of the original thrombus up the post-Willisian arteries, and ii) 'hemodynamic' failure, particularly in case of incomplete circle of Willis. For instance, while the former could be prevented using antithrombotics [START_REF] Damania | Factors associated with recurrent stroke and recanalization in patients presenting with isolated symptomatic carotid occlusion[END_REF], the latter might be preventable via endovascular therapy (EVT) [START_REF] Jadhav | Angioplasty and stenting for symptomatic extracranial nontandem internal carotid artery occlusion[END_REF]. However, the predictors and mechanisms of END in this setting are largely unknown, and notably were not addressed in the above-mentioned study [START_REF] Mazya | Minor stroke due to large artery occlusion. When is intravenous thrombolysis not enough? Results from the SITS International Stroke Thrombolysis Register[END_REF].

Here, we aimed to i) identify the incidence and predictors of END in a multicentric sample of iICAo patients with mild baseline symptoms treated with IVT, and ii) decipher the mechanisms of END in these patients.

METHODS

Study design and data sources

The MINOR-STROKE collaboration retrospectively collected the data from all consecutive acute stroke patients admitted to 45 French stroke centers between 2006 and 2018 who fulfilled the following criteria [START_REF] Seners | Bridging therapy or IV-thrombolysis in minor stroke with large vessel occlusion[END_REF]: [START_REF] Turc | Clinical Scales Do Not Reliably Identify Acute Ischemic Stroke Patients With Large-Artery Occlusion[END_REF] In accordance with French legislation, each patient was informed of his/her participation in this study and was offered the possibility to withdraw. However, as this study only implied retrospective analysis of anonymized data collected as part of routine care, approval by an Ethics Committee was not required.

Clinical data

The following variables were collected: age, gender, vascular risk factors, pre-stroke anti-thrombotic medication, blood pressure on admission, time between symptom onset and start of IVT, time between IVT and END, NIHSS score on admission and at 24h, 3-month modified Rankin Scale (mRS) score and stroke etiology according to the Trial of Org 10172 in Acute Stroke Treatment classification. Excellent functional outcome was defined as 3-month mRS<2 and good functional outcome as mRS<3.

Radiological data

All included patients underwent either MRI with MR-angiography or CT with CT-angiography before IVT start, and follow-up MRI/MRA or CT/CTA ~24hrs following admission. Additional MRI/MRA, CT/CTA or digital subtraction angiography was obtained in case of END. To ensure homogeneity in radiological evaluation, two stroke neurologists (NB and PS) reviewed independently all pre-IVT and follow-up imaging blinded to clinical outcomes, including occurrence of END. Discrepancies were resolved by consensus.

On baseline imaging, the location of iICAo -dichotomized as bulbar and supra-bulbar, i.e., within or upstream the ICA bulb, respectively-and the presence of anterior and posterior communicating arteries downstream the iICAo were collected. Perfusion imaging (CT-or MR-perfusion) was part of routine admission protocol in some centers. When available, the time-to-maximum (Tmax)>6s and >10s volumes were automatically segmented from perfusion imaging using Olea Sphere software, with manual correction whenever necessary [12]. Severity of hypoperfusion was assessed using the hypoperfusion intensity ratio (HIR), defined as the proportion of Tmax>6s volume with Tmax>10s (i.e., HIR=[Tmax>10s volume / Tmax>6s volume] X100) [START_REF] Olivot | Hypoperfusion intensity ratio predicts infarct progression and functional outcome in the DEFUSE 2 Cohort[END_REF], low HIR indicating milder hypoperfusion.

On follow-up imaging (CT/CTA, MRI/MRA or digital subtraction angiography), the occurrence of (1) intracranial haemorrhage based on European Cooperative Acute Stroke Study imaging classification [START_REF] Hacke | Intravenous thrombolysis with recombinant tissue plasminogen activator for acute hemispheric stroke. The European Cooperative Acute Stroke Study (ECASS)[END_REF], (2) intracranial arterial occlusion ipsilateral to the initially occluded ICA, and (3) carotid recanalization defined as 2 or 3 on the Arterial Occlusive Lesion scale [START_REF] Zaidat | Recommendations on angiographic revascularization grading standards for acute ischemic stroke: a consensus statement[END_REF], were recorded. If several follow-up imaging sessions were performed, e.g. in case of END, we only considered the session closest in time to the occurrence of END. In this retrospective study plain CT was used as admission and/or follow-up parenchymal imaging in an important proportion of patients, and accordingly the topography of new infarcts was not systematically assessed.

Definition and classification of END

In this study, END refers to neurological deterioration with a NIHSS score increase ≥4 points within 24 hours after IVT [START_REF] Seners | Incidence, causes and predictors of neurological deterioration occurring within 24 h following acute ischaemic stroke: a systematic review with pathophysiological implications[END_REF]. For each END case, the presumed END mechanism was determined based on a careful review of baseline and follow-up clinical and radiological data. Apart from symptomatic intracranial haemorrhage, which was defined as END with parenchymal haemorrhage on follow-up imaging, END was categorized as thrombo-embolic or not. END of presumed thrombo-embolic origin was defined on follow-up vascular imaging as presence of new intracranial occlusion (middle, anterior or posterior cerebral arteries or anterior choroïdal artery) ipsilateral to the initially occluded iICAo, and was sub-categorized as i) embolic (artery-to-artery embolism) if the new intracranial occlusion was not contiguous with the persistent carotid occlusion or if it was associated with ICA recanalization, ii) due to in situ extension of the carotid thrombus if in continuity, or iii) undetermined if it was not possible to determine whether the new intracranial occlusion was contiguous with the persistent ICA occlusion. If no new intracranial occlusion was identified, the END was considered of presumed non thrombo-embolic mechanism, which includes a hemodynamic origin.

Statistical analysis

Continuous variables were described as mean ± standard deviation or median (interquartile range

[IQR]), as appropriate, and categorical variables as numbers and percentages. Univariable relationships between END and pre-or post-IVT variables were assessed using Student t test or Mann-Whitney U test for continuous variables and χ2 or Fisher's 'exact' test for categorical variables, as appropriate. Regarding pre-IVT variables, to adjust for potential confounders, multivariable binary logistic regression analyses were subsequently conducted, with END as dependent variable. Pre-IVT variable selection was performed stepwise, whereby candidate variables entered the model at P<0.20 and were retained only if they remained associated at P<0.05 with the dependent variable. Statistical analyses were performed using SPSS 16.0 (SPSS, Inc). Two-tailed P<0.05 was considered statistically significant.

RESULTS

Study population

Out of 1072 patients included in the MINOR-STROKE cohort, a total of 74 IVT-treated patients with minor stroke and iICAo were included in the present study. Reasons for exclusion were no iICAo (n=992), lack of follow-up NIHSS score (n=1) or direct transfer for additional EVT (n=5). Mean age was 64 ±10 yrs, median NIHSS score was 3 (IQR 2-4) and median onset-to-IVT time was 170 mins (IQR 140-210). In 10 (14%) patients, iICAo was diagnosed on intracranial vascular imaging but the exact iICAo location (i.e., bulbar or supra-bulbar) could not be determined because extracranial vascular imaging was not performed. In the remaining, iICAo was supra-bulbar in 27/64 (42%). After completion of etiological work-up, cause of stroke was large artery atherosclerosis, cardio-embolic, dissection and other/undetermined in 55%, 14%, 26% and 5% of patients, respectively. Perfusion imaging was available in 27 (36%) patients.

Incidence and characteristics of END

END occurred in 22 (29.7%, 95%CI: 19.3-40.1) patients. Median NIHSS score increase in END patients was [START_REF] Hacke | Intravenous thrombolysis with recombinant tissue plasminogen activator for acute hemispheric stroke. The European Cooperative Acute Stroke Study (ECASS)[END_REF][START_REF] Zaidat | Recommendations on angiographic revascularization grading standards for acute ischemic stroke: a consensus statement[END_REF][START_REF] Seners | Unexplained early neurological deterioration after intravenous thrombolysis: incidence, predictors, and associated factors[END_REF][START_REF] Kim | Early recanalization rates and clinical outcomes in patients with tandem internal carotid artery/middle cerebral artery occlusion and isolated middle cerebral artery occlusion[END_REF], and the timing of END was ≤2, 2-6, 6-12, and 12-24 hrs after IVT start in 55%, 14%, 14% and 18% of END patients, respectively. Following END, 14 (64%) patients were referred for rescue EVT, which was performed in 12 (2 patients had first-run angiography but no EVT performed because of clinical recovery following END).

Pre-IVT variables associated with END

The pre-IVT characteristics of patients with and without END and the results of the univariable analyses are presented in Table 1. END patients less frequently had diabetes and bulbar iICAo than no-END patients. Perfusion parameters did not significantly differ between the two groups, but the sub-sample was small. After stepwise variable selection, supra-bulbar iICAo was the sole variable retained in the model (OR=4.0; 95%CI: 1.3-12.2; P=0.015).

Post-IVT variables associated with END

The results of the univariable analyses for post-IVT variables for which an association with END were assessed are presented in Table 2. Occurrence of END was associated with non-atheromatous iICAo etiology (i.e., cardio-embolic, dissection or other/undetermined), with END rates of 5/41 (12%) and 17/33 (52%) for atheromatous vs. non-atheromatous iICAo, respectively. No instance of parenchymal haemorrhage was recorded in either group. Follow-up vascular imaging was performed more frequently and earlier in the END as compared to no-END group, and revealed the occurrence of an intracranial occlusion in 16 (76%) of END vs. 0 (0%) of no-END patients (P<0.001). Location of new-occurrence intracranial occlusion was intracranial carotid + first MCA segment, first MCA segment, second or more distal MCA segments, and anterior choroïdal artery in 5, 7, 3 and 1 patients, respectively. There was a strong trend towards higher ICA recanalization rate in END than no-END patients (29% vs. 9%, respectively, P=0.07). The rates of excellent and good functional outcomes were significantly lower in the END vs. no-END groups. However, ICA recanalization was not associated with 3-month outcome (mRS 0-1: 44% vs. 48% in recanalized and non-recanalized ICA patients, respectively, P=0.86; and mRS 0-2: 56% vs. 64%, respectively, P=0.71).

Presumed END mechanisms

The presumed END mechanisms are summarized in Figure 1, and the relevant clinical, imaging and rescue treatment data of the 22 END patients are detailed in Table 3. The END mechanism could not be determined in 1 patient without follow-up vascular imaging. Presumed END mechanism was thrombo-embolic in 16/21 (76%) patients with new-occurrence intracranial occlusion. Among these patients, END was sub-classified as embolic (artery-to-artery embolism) in 13 (81%) patients: 6 with new MCA occlusion associated with ICA recanalization and 7 with new MCA occlusion discontinuous from persistent ICA occlusion; and as in situ thrombus extension in one patient. In the remaining 2 patients, both with new-occurrence intracranial ICA + first MCA artery occlusion, the exact thrombo-embolic mechanism was unknown since it was not possible to determine whether the new intracranial occlusion was contiguous with the persistent cervical ICA occlusion.

Given the lack of intracranial occlusion on follow-up imaging, END was classified as non thromboembolic in 5/21 patients (24%). A hemodynamic origin appears likely in three of these patients considering, in two patients, the lack of both anterior and posterior communicating arteries downstream the iICAo, and, in a third patient, the presence of an extensive severe hypoperfusion on post-END CT-perfusion, but remains uncertain for the two remaining patients. Blood pressure drops before END were not documented in any of these patients and newly appeared borderzone infarcts could not be documented as MRI both before and after END was performed in none. Three typical patients with END are shown in Figure 2.

DISCUSSION

Our aim in this multicenter study was to determine the incidence, predictors and mechanisms of post-IVT END in a sample of patients with acute mild stroke and iICAo. The study disclosed 3 key findings: (1) END affected 30% of patients and was strongly associated with poor 3-month outcome;

(2) supra-bulbar iICAo site was the only independent predictor of END; and (3) a thrombo-embolic origin was the underlying mechanism in 3 out of 4 END patients, of which at least 81% was referable to artery-to-artery embolism.

Incidence of END and impact on outcome

We found that END affected 30% of patients with mild stroke and iICAo intended for IVT alone.

Thus far, END in IVT-treated minor stroke patients with iICAo has been assessed in one previous study only [START_REF] Mazya | Minor stroke due to large artery occlusion. When is intravenous thrombolysis not enough? Results from the SITS International Stroke Thrombolysis Register[END_REF], which reported an incidence of 17%, which is high yet not as high as the present study.

This difference is likely explained by the exclusion of patients treated with rescue EVT (i.e., performed because of END) in this previous study [START_REF] Mazya | Minor stroke due to large artery occlusion. When is intravenous thrombolysis not enough? Results from the SITS International Stroke Thrombolysis Register[END_REF], inevitably implying an underestimated incidence of END.

Similar to all earlier studies on IVT-treated populations [START_REF] Mazya | Minor stroke due to large artery occlusion. When is intravenous thrombolysis not enough? Results from the SITS International Stroke Thrombolysis Register[END_REF][START_REF] Seners | Incidence, causes and predictors of neurological deterioration occurring within 24 h following acute ischaemic stroke: a systematic review with pathophysiological implications[END_REF][START_REF] Seners | Unexplained early neurological deterioration after intravenous thrombolysis: incidence, predictors, and associated factors[END_REF], END was strongly predictive of poor functional outcome, with only ~25% of END patients reaching 3-month functional independence. This further underlines the critical need to understand the predictors and underlying mechanisms of END in order to develop preventative strategies.

Predictors of END

Supra-bulbar ICA occlusion was the only independent predictor of END. This novel association likely mainly reflects the etiology underlying iICAo. Accordingly, stroke etiology, as established post-hoc following full etiological work-up, was strongly associated with END -non-atheromatous being at much higher END risk than atheromatous iICAo (52% vs. 12%, respectively). How could stroke etiology influence END risk in this population? We propose this observation may reflect a different response of the originally occluding carotid thrombus to IVT. Thus, on one hand the double association of END with both ICA recanalization and new intracranial occlusion (Table 2) might suggest that carotid thrombi that respond well to IVT might proceed to fragmentation causing new intracranial occlusion, while, on the other hand, strokes due to large-artery atherosclerosis are known to be less prone to post-IVT recanalization as compared to other stroke subtypes [START_REF] Kim | Early recanalization rates and clinical outcomes in patients with tandem internal carotid artery/middle cerebral artery occlusion and isolated middle cerebral artery occlusion[END_REF][START_REF] Molina | Differential pattern of tissue plasminogen activator-induced proximal middle cerebral artery recanalization among stroke subtypes[END_REF]. Also, it is possible that a fraction of the isolated ICA occlusions included in our study might in fact be chronicparticularly those of atheromatous origin-, and thereby less prone to post-IVT recanalization and fragmentation. The lack of association found between ICA recanalization and 3-month outcome (see Results) might suggest that post-IVT ICA recanalization is a dual edge sword in this setting: on one hand it might favor thrombus fragmentation, new intracranial occlusion and therefore END in some patients, while on the other hand it may improve cerebral perfusion and clinical recovery in other patients.

Mechanisms of END

To our knowledge, this is the first study to address END mechanisms in patients with iICAo. To determine presumed mechanism, we carefully analyzed for each END patient the available admission and follow-up imaging dataset. Interestingly, the predominant END mechanism turned out to be thrombo-embolic. This was evidenced by the presence of new-occurrence intracranial occlusion in 76% of END patients, as compared to 0% of no-END patients. A thrombo-embolic origin of END was reported in one previous study, which showed significantly more frequent thrombus extension in END than in no-END patients treated with IVT [START_REF] Seners | Is Unexplained Early Neurological Deterioration After Intravenous Thrombolysis Associated With Thrombus Extension?[END_REF]. However, the vast majority of patients in this study had MCA occlusion, not iICAo. Two main underlying thrombo-embolic processes may lead to the occurrence of an intracranial occlusion in the setting of IVT-treated iICAo. The first is artery-toartery embolism, which may result from fragmentation of the carotid thrombus involving either its entirety or its distal tail. This was found to be the primary mechanism in thrombo-embolic ENDs, occurring in 81% of thrombo-embolic END cases. The second thrombo-embolic mechanism, identified in one patient only, was in situ extension of the original carotid thrombus up the post-Willisian arteries. This secondary event may result from disrupted thrombosis pathways, such as increased coagulation activity and resistance to fibrinolysis, or activation of the physiological coagulation cascade because of blood stasis adjacent to the original thrombus, for instance as a result of poor collateral flow [START_REF] Qazi | Thrombus Characteristics Are Related to Collaterals and Angioarchitecture in Acute Stroke[END_REF].

To our surprise, a presumed non thrombo-embolic, and therefore potentially hemodynamic, mechanism underlying END in our population was relatively rare, affecting at most 1 in 5 END patients only. Causes for such putative hemodynamic event could not be clearly determined in any of our patients, for instance blood pressure drops were not documented in any patient. 'Collateral failure', defined as insufficient endurance of collateral circulation to maintain cerebral perfusion pressure, has been proposed as a potential mechanism of END [START_REF] Liebeskind | Collateral failure? Late mechanical thrombectomy after failed intravenous thrombolysis[END_REF], yet its operational definition and exact underlying processes remain unclear [START_REF] Shuaib | Collateral blood vessels in acute ischaemic stroke: a potential therapeutic target[END_REF][START_REF] Seners | Revisiting 'progressive stroke': incidence, predictors, pathophysiology, and management of unexplained early neurological deterioration following acute ischemic stroke[END_REF].

Importantly, none of the END patients in our cohort had symptomatic intracranial haemorrhage, consistent with previous findings in mild stroke [START_REF] Mazya | Minor stroke due to large artery occlusion. When is intravenous thrombolysis not enough? Results from the SITS International Stroke Thrombolysis Register[END_REF].

Therapeutic implications: can END be prevented?

Given the apparent major role of thrombo-embolic factors in END, preventing distal embolization and in situ thrombus extension would appear as logical measures in the setting of IVT-treated iICAo.

Three main approaches might be considered, namely i) avoiding IVT, ii) early administration of antithrombotics, and iii) direct referral for EVT. First, the effect of IVT on the thrombus in the setting of iICAo may be a double-edged sword: although IVT may prevent in situ thrombus extension and related ENDs, it could also cause thrombus fragmentation and subsequent distal embolism -the most frequent END mechanism based on the present study-, particularly in the event of ICA recanalization as observed here. Considering the very short half-life of alteplase, that it may favour embolic ENDs is further supported by the observation that most ENDs occurred within 2hrs from IVT start in our study. If true, this could mean that, at least in case of supra-bulbar iICAo, withholding IVT might prevent END. As a caveat, this scenario cannot be inferred from our dataset as a control group (i.e., non IVT-treated iICAo) was not available. Although IVT likely is beneficial in acute minor strokes with intracranial occlusions [START_REF] Tsivgoulis | Intravenous thrombolysis for large vessel or distal occlusions presenting with mild stroke severity[END_REF], further studies are warranted specifically regarding minor strokes with iICAo. Note that in the observational ICARO study [START_REF] Paciaroni | Systemic thrombolysis in patients with acute ischemic stroke and Internal Carotid ARtery Occlusion: the ICARO study[END_REF], which reported better outcomes in IVTas compared to non IVT-treated patients with ICA occlusion, the proportion of patients with iICAo is unknown as occurrence of tandem ICA + MCA occlusions is not presented. Second, early post-IVT antithrombotic administration may be an attractive option to prevent thrombo-embolic ENDs.

Regarding antiplatelet therapy, a post hoc analysis of the ARTIS trial, a randomized trial comparing ultra-early (within 90min of IVT start) addition of aspirin after IVT vs. IVT alone in an unselected acute stroke population, found that aspirin increased the risk of END due to intracranial hemorrhage, and had no effect on incidence of non-hemorrhagic END [START_REF] Zinkstok | Early deterioration after thrombolysis plus aspirin in acute stroke: a post hoc analysis of the Antiplatelet Therapy in Combination with Recombinant t-PA Thrombolysis in Ischemic Stroke trial[END_REF]. However, this approach may still be of interest in populations at high risk of post-IVT thrombo-embolic END and low risk of intracranial hemorrhage, such as supra-bulbar iICAo with mild symptoms, and should be further tested. As another antithrombotic approach, a recent observational study in patients not candidates for IVT suggested that, as compared to antiplatelet therapy, anticoagulation was associated with lower 7-day END rate in iICAo patients admitted within 24hrs from onset (7% vs. 39%, respectively, P=0.03) [START_REF] Damania | Factors associated with recurrent stroke and recanalization in patients presenting with isolated symptomatic carotid occlusion[END_REF].

However, in the setting of IVT, the considerably higher risk of intracranial hemorrhage strongly limits such therapy. Last, regarding direct referral for EVT, two recently published case series showed that immediate EVT, which includes acute angioplasty and stenting, is feasible in the setting of iICAo [START_REF] Jadhav | Angioplasty and stenting for symptomatic extracranial nontandem internal carotid artery occlusion[END_REF][START_REF] De Castro-Afonso | Endovascular Reperfusion for Acute Isolated Cervical Carotid Occlusions: The Concept of "Hemodynamic Thrombectomy[END_REF] and afforded high rates of successful recanalization, but was also associated with a high rate (22%) of distal embolism [START_REF] Jadhav | Angioplasty and stenting for symptomatic extracranial nontandem internal carotid artery occlusion[END_REF]. However, the vast majority of iICAo patients in these studies had high baseline NIHSS scores [START_REF] Jadhav | Angioplasty and stenting for symptomatic extracranial nontandem internal carotid artery occlusion[END_REF][START_REF] De Castro-Afonso | Endovascular Reperfusion for Acute Isolated Cervical Carotid Occlusions: The Concept of "Hemodynamic Thrombectomy[END_REF], and it may not be warranted to extrapolate their findings to our population. Randomized trials are needed to test these different strategies.

Regarding hemodynamic ENDs, apart from routine physiological measures to maintain collateral flow [START_REF] Bang | Collateral Circulation in Ischemic Stroke: Assessment Tools and Therapeutic Strategies[END_REF], the most logical measures to efficiently prevent hemodynamic failure in iICAo would appear to be direct EVT, including angioplasty-stenting, and collateral enhancing strategies such as induced hypertension, lying flat head position or volume expansion [START_REF] Bang | Collateral Circulation in Ischemic Stroke: Assessment Tools and Therapeutic Strategies[END_REF].

Limitations

Our study has several limitations. First, despite the multicenter design, the overall sample was relatively small, thereby limiting the statistical analysis. This can be explained by the relative rarity of iICAo in the acute stroke setting [START_REF] Turc | Clinical Scales Do Not Reliably Identify Acute Ischemic Stroke Patients With Large-Artery Occlusion[END_REF]. Second, considering the retrospective nature of the study, baseline and follow-up imaging was not standardized. Consequently, only few patients had baseline perfusion imaging, and the lack of prognostic value of perfusion parameters found here should be interpreted cautiously. Third, due to the intrinsic limitations of non-invasive vascular imaging, some patients with baseline severe carotid stenosis may have been misclassified as iICAo, in particular for those 10 patients without extracranial vascular imaging. Fourth, transcranial Doppler continuous-monitoring, which may have helped to determine the exact END mechanism, was not available in the participating centres. Fifth, as our study focused on iICAo with mild baseline symptoms treated with IVT, the results cannot be generalized to iICAo patients with more severe symptoms and/or not treated with IVT. Last, as a control group -i.e., acute iICAo with mild symptoms not treated with IVTwas not available, it is unknown whether the clinical course described here was modified by IVT.

Further research is warranted.

CONCLUSIONS

END is a frequent and highly deleterious event after IVT for minor stroke with iICAo. This study identified a thrombo-embolic origin as mechanism underlying 3/4th of the ENDs, with artery-toartery embolism involved in the vast majority of these patients. The strong association with iICAo site, which reflects underlying stroke etiology, might reflect a different thrombus response to thrombolysis. Our findings may have potential implications regarding prevention of END in this setting. However, further studies are needed at this stage to confirm and expand our findings. END of presumed thrombo-embolic origin was defined on follow-up vascular imaging as presence of new intracranial occlusion ipsilateral to the initially occluded iICAo, and was sub-categorized as i) embolic (arteryto-artery embolism) if the new intracranial occlusion was not contiguous with the persistent carotid occlusion or if it was associated with ICA recanalization, ii) due to in situ extension of the carotid thrombus if in continuity, or iii) undetermined if it was not possible to determine whether the new intracranial occlusion was contiguous with the persistent ICA occlusion. If no new intracranial occlusion was identified, the END was considered of presumed non thrombo-embolic mechanism, which includes a hemodynamic origin. See Table 3 for details and

  baseline admission National Institutes of Health Stroke Scale (NIHSS) score ≤5; (2) large vessel occlusion on pre-treatment vascular imaging (internal carotid artery [ICA], first or second segment of middle cerebral artery [MCA], basilar artery), and (3) treated with IVT (alteplase only), with or without additional EVT. The present study focuses on patients with iICAo intended for IVT alone, including those who eventually received rescue EVT because of post-IVT END. The patients immediately intended for additional EVT following IVT (i.e., bridging therapy) were excluded. iICAo was defined as cervical or intracranial internal carotid occlusion not involving the circle of Willis or the arteries distal to it.

Figure 1 :

 1 Figure 1: Presumed END mechanisms

Figure 2

 2 Figure 2 for illustrative cases.

Figure 2 :Patient 1 :Patient 2 :Patient 3 :

 2123 Figure 2: Illustrative END cases.

Table 2 .

 2 Univariable relationships between post-IVT variables and END Cardio-embolic, dissection, other or undetermined. ‡: If several follow-up imaging were performed, e.g. in case of END, we only considered the closest in time to the occurrence of END. §: A follow-up vascular imaging was performed in 55 patients(21 END and 34 no-END). ¶: Available for 69 patients (20 END and 49 no-END).

	END n=22	No-END n=52	P value

Abbreviations: END indicates early neurological deterioration; ICA, internal carotid artery; IVT, intravenous thrombolysis; mRS, modified Rankin scale.

Acknowledgements: none.

Source of funding: none.

Data availability statement:

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest: the authors report no competing interest concerning this manuscript. ¶: On baseline imaging this patient had a supra-bulbar iICAo with clot visible up to the distal end of ICA on susceptibility weighted imaging upstream the anterior choroïdal artery origin. MRI performed following END showed distal extension of the ICA thrombus on susceptibility weighted imaging now occluding the anterior choroïdal artery, with corresponding territory infarct. † †: The exact thrombo-embolic mechanism (i.e., artery-to-artery embolism vs. in situ thrombus extension) is unknown in these patients since it is not possible to determine whether the new intracranial ICA + first MCA segment occlusion was contiguous with the persistent cervical ICA occlusion. ‡ ‡: The END mechanism (thrombo-embolic vs. non thrombo-embolic) is undetermined because of the lack of follow-up vascular imaging. § §: A CT-perfusion was performed following END, showing a large severe hypoperfusion (Tmax>6s=140ml, Tmax>10s=3ml, HIR=2%).

Abbreviations: ACoA indicates anterior communicating artery; CE, cardio-embolic; D, dissection; END, early neurological deterioration; HIR, hypoperfusion intensity ratio; iICAo, isolated internal carotid occlusion; IVT, intravenous thrombolysis; LAA, large artery atherosclerosis; MCA, middle cerebral artery; mRS, modified Rankin scale; MT, mechanical thrombectomy; mTICI, modified thrombolysis in cerebral infarction scale; mRS, modified Rankin scale; NA, not available; O: other or undetermined; PCoA, posterior communicating artery.


T Le Reun

B Gallet

B Favier

M Le Bars

Near-resonant instability of geostrophic modes: beyond Greenspan's theorem

We explore the near-resonant interaction of inertial waves with geostrophic modes in rotating fluids via numerical and theoretical analysis. When a single inertial wave is imposed, we find that some geostrophic modes are unstable above a threshold value of the Rossby number kRo based on the wavenumber and wave amplitude. We show this instability to be caused by triadic interaction involving two inertial waves and a geostrophic mode such that the sum of their eigen frequencies is non-zero. We derive theoretical scalings for the growth rate of this near-resonant instability. The growth rate scaled by the global rotation rate is proportional to (kRo) 2 at low kRo and transitions to a kRo scaling for larger kRo. These scalings are in excellent agreement with direct numerical simulations. This instability could explain recent experimental observations of geostrophic instability driven by waves.

Introduction

Rotating turbulent flows are ubiquitous in geo-and astrophysical systems such as stellar interiors, planetary cores, oceans and atmospheres. In a large number of numerical simulations and experiments (see the review by [START_REF] Godeferd | Structure and Dynamics of Rotating Turbulence: A Review of Recent Experimental and Numerical Results[END_REF]), rotating turbulence is observed to develop a strong anisotropy and to spontaneously form vortices that are invariant along the rotation axis. The latter correspond to a first-order balance between the Coriolis force and pressure gradients and are called "geostrophic modes". Yet, the systematic observation of strong geostrophic modes is at odds with various evidence suggesting that rotating turbulence could as well be dominated by inertial waves that are sustained by the restoring action of the Coriolis force. Recent numerical [START_REF] Le Reun | Inertial Wave Turbulence Driven by Elliptical Instability[END_REF] and experimental [START_REF] Le Reun | Experimental study of the nonlinear saturation of the elliptical instability: Inertial wave turbulence versus geostrophic turbulence[END_REF][START_REF] Brunet | Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability[END_REF] studies have shown that injecting energy in waves solely creates a turbulent state comprising of inertial waves only when the forcing amplitude is sufficiently small, i.e. a discrete version of inertial wave turbulence [START_REF] Galtier | Weak inertial-wave turbulence theory[END_REF]. It is only at larger forcing amplitudes that a secondary instability leads to the classical geostrophic-dominated turbulence. Asymptotic theories describing rotating turbulence in the limit of vanishing forcing amplitude and dissipation also suggest that waves could dominate over geostrophic modes in such a regime [START_REF] Bellet | Wave turbulence in rapidly rotating flows[END_REF][START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF]. Hence, although bi-dimensionalisation in the form of geostrophic eddies has been commonly observed, it may not be the only equilibrium state of rotating turbulence, be it at moderate [START_REF] Yokoyama | Hysteretic transitions between quasi-two-dimensional flow and three-dimensional flow in forced rotating turbulence[END_REF][START_REF] Favier | Subcritical turbulent condensate in rapidly rotating Rayleigh-Bénard convection[END_REF] as well as small (van Kan & Alexakis 2019) forcing amplitudes. In addition, the nature of the forcing seems fundamental in determining the equilibrium state of rotating turbulence.

These results altogether call for a better understanding of the fundamental processes by which waves give rise to balanced geostrophic modes. The studies of Le [START_REF] Le Reun | Experimental study of the nonlinear saturation of the elliptical instability: Inertial wave turbulence versus geostrophic turbulence[END_REF] and [START_REF] Brunet | Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability[END_REF] suggest that such a transfer occurs through an instability. Although wave-to-wave interactions are primarily governed by triadic resonance [START_REF] Bordes | Experimental evidence of a triadic resonance of plane inertial waves in a rotating fluid[END_REF][START_REF] Vanneste | Wave Interactions[END_REF], they cannot account for wave-to-geostrophic transfers [START_REF] Greenspan | On the non-linear interaction of inertial modes[END_REF], at least in the asymptotic limit of vanishing velocity amplitude and dissipation. Several alternative mechanisms, outside the framework of Greenspan's theorem, have been proposed. Four-modes interactions can transfer energy from waves to geostrophic flows, either directly [START_REF] Newell | Rossby wave packet interactions[END_REF][START_REF] Smith | Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence[END_REF] or through an instability mechanism [START_REF] Kerswell | Secondary instabilities in rapidly rotating fluids: Inertial wave breakdown[END_REF][START_REF] Brunet | Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability[END_REF]. The growth rate of such an instability scales like Ro 2 , with Ro the dimensionless wave amplitude or Rossby number. It develops over longer timescales that triad-type interactions between waves. The other inviscid mechanism that has been proposed to account for wave-geostrophic transfer is quasi-resonant triadic interaction [START_REF] Newell | Rossby wave packet interactions[END_REF][START_REF] Smith | Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence[END_REF], that is, a triad between waves whose frequencies do not exactly satisfy the resonance condition [START_REF] Bretherton | Resonant interactions between waves. The case of discrete oscillations[END_REF]. Their presence and their role in the bi-dimensionalisation of rotating turbulence has been assessed by several numerical studies [START_REF] Smith | On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number[END_REF][START_REF] Alexakis | Rotating Taylor-Green flow[END_REF][START_REF] Di Leoni | Quantifying resonant and near-resonant interactions in rotating turbulence[END_REF]. While it has been shown that such triads can transfer directly energy from two pre-existing waves to geostrophic modes, we show that this transfer can arise spontaneously through an instability mechanism. More precisely, we show with direct numerical simulations (DNS) and theoretical analysis that there exists a linear mechanism by which a single inertial wave drives exponential growth of geostrophic modes through near-resonant triadic interaction.

The stability of a single inertial wave

Governing equations and numerical methods

Let us consider an incompressible fluid rotating at rate Ωe z . We investigate the stability of a single plane inertial wave with wave vector k and eigen frequency ω k . Its amplitude is proportional to U w with

U w = h s k k exp i(k • x -ω k t) + c.c. (2.1)
h s k k is a helical mode, that is, when k is not parallel to the axis of rotation e z [START_REF] Cambon | Spectral approach to non-isotropic turbulence subjected to rotation[END_REF][START_REF] Waleffe | The nature of triad interactions in homogeneous turbulence[END_REF])

h s k k = 1 √ 2 (k × e z ) × k | (k × e z ) × k| + is k k × e z |k × e z | , (2.2)
where s k = ±1 is the sign of the helicity of the plane wave. If k is parallel to e z , h k = e x + is k e y . U w automatically satisfies the incompressibility condition since

∇ • U w ∝ k • h s k k = 0. U w satisfies the linearised rotating Euler equation, ∂ t U w + 2Ω × U w = -∇π w , (2.3)
provided that ω s k k and k are related by the dispersion relation of inertial waves maintained at a constant amplitude via the following set of equations

ω k = 2s k Ω k z k = 2s k Ω cos θ, ( 2
∂ t u + Ro (U w • ∇u + u • ∇U w ) + u • ∇u + 2e z × u = -∇π + E∇ 2 u (2.5a) ∇ • u = 0 (2.5b)
where time is scaled by Ω -1 and length by the domain size L. We have introduced the Ekman number E = ν/(L 2 Ω), ν being the kinematic viscosity, and an input Rossby number Ro quantifying the dimensionless amplitude of the plane wave. Equations (2.5) are solved numerically in a triply periodic cubic box using the code Snoopy [START_REF] Lesur | On the relevance of subcritical hydrodynamic turbulence to accretion disk transport[END_REF]. The dynamics of the perturbation flow {u, π} is determined with pseudo-spectral methods, that is, {u, π} is decomposed into a truncated sum of Fourier modes {û q , πq } e iq•x . A wave-vector q writes 2π/L (n x , n y , n z ) where n x,y are integers varying from -N to N and n z is an integer ranging from 0 to N because of Hermitian symmetry. In the following, N = 96; higher resolutions have been tested and yield the exact same results. The temporal dynamics of the modes ûq is solved using a third order Runge-Kutta method. Note that the size of the box L is artificial and we thus expect our results to depend on the intrinsic Rossby number based on the imposed wavelength kRo rather than Ro alone.

Numerical results

Keeping the Ekman number to E = 10 -6 , two simulations of the stability of the inertial wave k = 2π [4, 0, 8] (with s k = 1) are performed at low (Ro = 2.83 × 10 -3 ) and moderate (Ro = 2.83 × 10 -2 ) wave amplitudes. They are both initiated with a random noise comprising wavenumbers ranging between 0 and 15π, with very small initial amplitude. The use of spectral methods allows separating the kinetic energy of the perturbation flow u into a two-dimensional component E 2d , accounting for all modes q with q z = 0, and a complementary three-dimensional component E 3d . In addition, performing Fourier transform in space and time allows projecting the kinetic energy of u in the sub-space of the dispersion relation of inertial waves to draw spatio-temporal spectrum E(θ, ω) [START_REF] Yarom | Experimental observation of steady inertial wave turbulence in deep rotating flows[END_REF][START_REF] Le Reun | Inertial Wave Turbulence Driven by Elliptical Instability[END_REF]. Note that, in these maps, θ is restricted to [0, π/2] since the flow is real and only the wavevectors q with q z 0 are simulated due to the Hermitian symmetry. Moreover, the spectra are symmetrised with respect to ω = 0 and the maps are shown as a function of |ω| to be more compact.

The kinetic energy time series and maps are shown for both simulations in figure 1. At low imposed wave amplitude (figure 1 a-b), three-dimensional perturbations dominate the growth of the instability. The energy map displays two spots aligned on the dispersion relation with negative frequencies ω 1,2 such that ω k + ω 1 + ω 2 = 0 which is indicative of several waves undergoing triadic resonance with the imposed mode. The growth of two-dimensional modes is delayed and their growth rate is approximately twice larger than the rate of three-dimensional modes. Removing the non-linear term u • ∇u (see equation (2.5)) suppresses the growth of two-dimensional modes which are thus not unstable themselves, at least on the timescale of the growth of waves. In fact, we find that E 2d ≃ 3 × 10 -2 (kE 3d ) 2 (see figure 1a) in the growth phase which suggests that two-dimensional modes' growth is due to direct forcing by non-linear interaction of two growing waves involved in triadic resonances, with close frequencies and opposed vertical wavenumbers. This mechanism corresponds to the direct excitation of geostrophic modes by two waves identified by [START_REF] Newell | Rossby wave packet interactions[END_REF] and [START_REF] Smith | Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence[END_REF].

At larger wave amplitude (figure 1c-d), this picture changes: two-and threedimensional modes grow at the same rate from the start of the simulation. The vertical vorticity field of the growing perturbation and its vertical average are displayed in figure 2a-b. The frequency of two-dimensional modes (θ = π/2) is close to 0 according to figure 1d, they are thus geostrophic, that is, slow as well as invariant along the z axis. The geostrophic unstable flow contains the wavevector p g = 2π [0, 5, 0] (and -p g for hermiticity), which grows along with other three-dimensional structures whose energy location in the (θ, ω) space is reminiscent of triadic resonant instability, but with significantly more spreading. Note that the properties of the transition and the instability we report are robust to changes in the aspect ratio of the box, which discards any spurious effect of the discretisation [START_REF] Smith | On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number[END_REF].

To further characterise the growth of geostrophic modes, we carry out simulations with the same imposed wave, but we use as initial condition only one wavevector, p = 2π [0, p y , 0] with p y ∈ {1, 3, 5} and s p = ±1, instead of random noise. The Ekman number is set to 10 -8 to discard effects of viscosity in the mode growth. The growth rate of geostrophic modes is reported in figure 2c where Ro is systematically varied. At large wave amplitude, the growth rate increases linearly, but it goes to zero at a finite value of Ro ∼ 10 -2 . This value is very large compared to the viscous damping k 2 E ∼ 10 -5 which plays no role here, hence suggesting an inviscid mechanism.

On the kinetic energy map E(θ, ω) (figure 2d) two particular spots appear, one at the location of geostrophic modes and the other at the angle of the vectors closing the triads between k and ±p g , which is indicative of triad-type interactions. This may seem a priori in contradiction with the result of [START_REF] Greenspan | On the non-linear interaction of inertial modes[END_REF]. Yet, our results are outside Greenspan's theorem framework in two aspects. First, our study is necessarily limited to finite Rossby number. Second, the frequency of the closing mode -k ± p (figure 2d) is outside its eigen frequency. In fact, the discrepancy between the observed and eigen frequencies of the closing mode corresponds to the sum of the eigen frequencies of the three modes, ∆ω kpq . Moreover, as Ro is decreased, the fastest growing mode p g in figure 2c goes from p y = 5 with ∆ω kpq = 0.22 to p y = 1 with ∆ω kpq = 0.01: the triad is nearly resonant and draws closer to exact resonance Two mechanisms may be considered to explain the growth of geostrophic modes. One hypothesis is that the growth of the geostrophic modes is a by-product of a classical resonance between, say, k and -q that forces q (by Hermitian symmetry) and hence p. However, in this case, the mode closing the wave triad would have an eigen frequency -ω k + ω q = ∆ω kqp -2ω k which is outside the range of inertial waves for small ∆ω kpq since ω k ≃ 1.78. Instead, our results point towards a near-resonant triadic instability that transfers energy from an inertial wave to a z-invariant geostrophic mode.

3. Near-resonance of geostrophic modes: theoretical approach

The low Rossby number limit

To provide theoretical insight into the inertial-wave destabilisation observed in the numerical simulations, we turn to linear stability analysis. The flow is U = u + Ro U w where Ro U w is the base flow (Ro being the finite Rossby number) and u ≪ Ro U w is now an infinitesimal perturbation. We consider a Cartesian domain with periodic boundary conditions and we decompose U into a superposition of plane waves h

db sp p dt = k+p+q=0 s k ,sq =±1 C sps k sq pkq b s k * k b sq * q exp (i∆ω kpq t) (3.1) with C sps k sq pkq ≡ 1 2 (s q q-s k k) h sp * p • h s k * k × h sq * q
and ∆ω kpq ≡ ω s k k +ω sq q +ω sp p , (3.2) and where k, p = |k|, |p|. ∆ω kpq is the sum of the eigen frequencies of the three modes involved in the triad. In general, maximum energy transfer between the three modes is ensured when the oscillations due to the detuning in the right hand side of (3.1) are cancelled, that is, when ∆ω kpq → 0. If all three modes k, p and q are inertial waves, this leads to the well known mechanism of triadic resonance [START_REF] Bretherton | Resonant interactions between waves. The case of discrete oscillations[END_REF][START_REF] Vanneste | Wave Interactions[END_REF][START_REF] Bordes | Experimental evidence of a triadic resonance of plane inertial waves in a rotating fluid[END_REF]. When the mode k is imposed with an amplitude Ro and helicity s k , p and q grow exponentially with a rate proportional to kRo. This picture is changed when one of the modes, say p, is geostrophic (i.e. ω sp p = 0 and p z = 0, regardless of s p ). The spatial interaction condition k + p + q = 0 forces k z = -q z and the resonance condition becomes

∆ω kpq = k z s k k - s q q = ω k 1 q q - s q s k k → 0 . (3.3)
It may be fulfilled only when s q = s k and q -k → 0. At exact resonance, these conditions impose p to be located on a circle centred on

-k ⊥ = -[k x , k y ] with radius |k ⊥ |.
Moreover, in the governing equation for ḃsp p , the slow oscillation terms involve a coupling coefficient C sps k s k pkq ∝ k -q → 0. At exact resonance, the coupling coefficient vanishes: there is no energy transfer from waves to geostrophic modes, as proved by [START_REF] Greenspan | On the non-linear interaction of inertial modes[END_REF].

Nevertheless, wave-to-geostrophic transfer is still possible when the detuning ∆ω kpq is small but non-zero [START_REF] Newell | Rossby wave packet interactions[END_REF][START_REF] Smith | Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence[END_REF][START_REF] Alexakis | Rotating Taylor-Green flow[END_REF], and we investigate instabilities of geostrophic modes driven by this mechanism. Let us assume that the wave k with helicity s k is imposed with a small constant amplitude Ro, and that p is geostrophic. To infer from (3.1) the time evolution of the geostrophic mode amplitude, we proceed to an asymptotic expansion using a two-time method involving a fast time τ = t and a slow time T . The hierarchy between them must be a power of kRo, the intrinsic Rossby number based on the imposed wavelength. Because the waveto-geostrophic transfer coefficient C sps k s k pkq vanishes as ∆ω kpq → 0, we find via a heuristic analysis detailed in appendix A that T = (kRo) 2 t, instead of (kRo) t for classical wave triads. In addition, it imposes the amplitude of p to be smaller by a factor kRo compared to the mode closing the triad q = -kp. The amplitudes of the modes interacting with the imposed waves (p and q with both helicity signs s p,q ) are thus expanded as

b sq q = (kRo)B sq q1 (T, τ ) + (kRo) 3 B sq q2 (T, τ ) b sp p = (kRo) 2 B sp p1 (T, τ ) + (kRo) 4 B sp p2 (T, τ ) (3.4)
where the B j i are all O(1). The hierarchy between orders is imposed by the need to match the slow time derivative of the leading order with the next order in the multiple scale expansion. Since the slow derivation introduces a factor (kRo) 2 , there must be a (kRo) 2 hierarchy between the first and second orders.

To find the equations governing the leading order coeffcients, we follow the method of [START_REF] Bretherton | Resonant interactions between waves. The case of discrete oscillations[END_REF] and inject the ansatz (3.4) into (3.1). First, the leading order coefficients are found to be independent of τ and their long time evolution with T is determined at next order. As noted by [START_REF] Bretherton | Resonant interactions between waves. The case of discrete oscillations[END_REF], for fast triads with detuning larger than O((kRo) 2 ), the imposed wave only drives fast and bounded oscillations of the second order terms, and the leading order terms must be zero to avoid secular growth. It is only for slow triads, i.e. when ∆ω kpq = O((kRo) 2 ), that the exponential term in equation (3.1) drives slow oscillations that contribute to secular growth. Such a condition on the detuning is consistent with the numerics: the fastest growing modes at Ro = 1 × 10 -2 (kRo ∼ 0.6) in figure 2c is p = 2π [0, 3, 0] for which ∆ω kpq ≃ 0.1 ∼ 0.3(kRo) 2 . The condition ∆ω kpq = O((kRo) 2 ) may be fulfilled only for the modes p with s p = ±1 and q with s q = s k and, as noted earlier, the wave-to-geostrophic transfer coefficient C sps k s k pkq is also O((kRo) 2 ). Cancelling the secular growth terms at second order gives the following amplitude equations, is real. The expression of the real part of growth rate, σ k (p; Ro), is consistent with the numerical findings: for a given near-resonant triad, it drops to zero at a finite value of Ro and it is proportional to Ro at larger Ro. However, in the small Rossby number limit, because the detuning ∆ω kpq and the coupling coefficient C sps k s k pkq are both O((kRo) 2 ), the maximum geostrophic growth rate remains O((kRo) 2 ) at most. The (kRo) 2 scaling governing the maximum growth rate of geostrophic near-resonance is found quantitatively by expanding the frequency detuning, the transfer coefficients and then σ k (p; Ro) in the neighbourhood of exact resonance. The fact that the growth rate is non-zero only close to the exact resonance is illustrated qualitatively in figure 3a. We thus introduce p = p 0 + (kRo) 2 δp where δp is a O(1) vector in the geostrophic plane. Consider the closing modes q 0 = -(k + p 0 ) and q = -(k + p), then q = q 0 -(kRo) 2 δp. The imposed mode k is left unperturbed. Since ∆ω kpq = C sps k s k p0kq0 = 0 at exact resonance, the leading orders of the frequency detuning and the transfer coefficients are found from (3.2) and (3.3) to be O((kRo) 2 ). The perturbation of the wavevectors is thus consistent with the asymptotic expansion. At leading order,

∂ T B sq q1 = sp C s k s k sp qkp B sp * p1 e i ∆ω kpq (
∆ω kpq (kRo) 2 ≃ s k ω k q 0 • δp k 2 and C sps k s k pkq (kRo) 2 ≃ 1 2 s k h sp * p0 • h s k * k × h s k * q0 q 0 • δp k , (3.7)
where we have used that q 0 = k and s q = s k . In the growth rate σ k (p; Ro), the product of the coupling coefficient is

sp C sps k s k pkq C s k s k sp * qkp ≃ 1 4 (k Ro) 2 k 2 q 0 • δp k 2 sp 1 -s p p 0 k h sp p0 • h s k k × h s k q0 2 . (3.8)
Therefore, at leading order in powers of Ro, the growth rate is

4σ k (p; Ro) 2 = (C k (p 0 )X -ω 2 k X 2 )(kRo) 4 , (3.9)
where X ≡ (q 0 • δp)/k 2 and C k (p 0 ) is the sum in the right hand side of (3.8).

When C k (p 0 ) > 0, the growth rate reaches an optimum at X = C k /(2ω 2 k ) with value (kRo) 2 C k (p 0 )/|4ω k |, which remains to be maximised over all exactly resonant wavevectors p 0 . The coefficient C k (p 0 ) is shown in figure 3b for several wavevectors k with different frequencies ω k . When plotted against p 0 /k ⊥ , all the curves C k collapse on a master curve that reaches a maximum value of 1 at p 0 → 0, regardless of the helicity sign s k of the imposed wave. As (kRo) → 0, the unstable geostrophic mode becomes large-scale and stems from the interaction between k and q ≃ -k. In the low Rossby number limit, the maximal growth rate is then

σ max k (Ro) = 1 4 (kRo) 2 |ω k | . (3.10)
We confirm that kRo, the Rossby number based on the wavelength, is the relevant parameter to describe the growth rate of the instability. While each geostrophic mode follows the law (3.6), all the growth rate curves lie below a (kRo) 2 upper envelope. More details are given below in section 3.3 where we compare the law (3.10) to exact computation of the growth rate curves from (3.6).

The moderate to large Rossby number regime

The Ro 2 law governing the growth rate in the small Rossby number limit cannot hold as the wave amplitude is increased since the growth rate has an upper bound following a Ro law. This is proven directly by multiplying (2.5) by u and integrating over the fluid domain V , thus giving:

σ = 1 2 d ln u 2 2 dt = - Ro u 2 2 V u • ∇U w • u (3.11)
where • n denotes the L n -norm. In virtue of Hölder's inequality [START_REF] Gallet | Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows[END_REF],

|σ| Ro ∇U w ∞ kRo (3.12)
This upper bound applies in particular to the low Ro scaling (3.10), which thus holds up to kRo ∼ 1 at most. Note that, beyond that point, non-triad type instabilities (shear, centrifugal) may add to near-resonance in driving the dynamics of the flow excited by the maintained wave.

Comparison with exact computation

In figure 3, we sample the growth rate curves given by (3.6) of many geostrophic modes in near resonant interaction with k = 2π [4, 0, 8], as functions of the Rossby number. We notice that each growth rate curve follows a Ro scaling at sufficiently large Ro, which is consistent with the asymptotic expansion carried out in section 3.1. However, because for all modes the growth rate vanishes at a finite value of Ro (which decreases to 0 close to exact resonance), the upper envelope is a Ro 2 law that perfectly matches the theoretical prediction (3.10). This instability is thus fundamentally different from fourmodes interactions for which each growth rate curve follows a Ro 2 law [START_REF] Kerswell | Secondary instabilities in rapidly rotating fluids: Inertial wave breakdown[END_REF][START_REF] Brunet | Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability[END_REF]. Besides, in agreement with the upper bound (3.12), we observe the Ro 2 maximum growth rate law to break down above kRo ∼ 1. Beyond that, the nearresonance growth rate derived from (3.6) remains below the upper bound and follows a Ro law. However, additional instabilities (shear, centrifugal, etc.) may also drive the growth of geostrophic modes in this regime.

Finite size and viscous effects

Finite size domain translates into discretisation of the modes. Since the exactly resonant geostrophic modes lie on a finite radius circle, they cannot be approached with arbitrarily low detuning ∆ω kpq as Ro → 0. Hence, discretisation implies the existence of a finite value of Ro below which the near-resonant instability vanishes. Regarding viscous effects, at finite but low Ekman number, i.e. when (kRo) 2 ≫ k 2 E, the low Rossby number scaling is unaltered since the near-resonant modes p and q have at most similar wavenumbers to k. The O(Ro) upper bound on the growth rate is also unaltered by the inclusion of viscosity.

A refined model: the double near-resonant triad

Although promising, the model of the previous section needs to be refined to fully account for numerical results. It misses by a factor two the growth rate curves of figure 2c and (3.6) predicts a frequency ∆ω kpq /2 = 0 for the geostrophic modes, while it is 0 according to figure 2d. We must include in our model that not only k, but also -k (with the same helicity sign s) is imposed due to Hermitian symmetry. Consider two triads k + p + q = 0 and -k + p + q ′ = 0 (see figure 4) with helicity signs s q = s q ′ = s and s p = ±1. As shown schematically in figure 4a, the exact resonance circles associated to ±k coincide at p = 0. In a neighbourhood of this point (the dark grey intersection in figure 4a), a geostrophic mode p is in near-resonance with both ±k with both detunings ∆ω kpq and ∆ω -kpq ′ small. This is all the more important that p = 0 corresponds to the largest near-resonant growth rate as Ro → 0.

In a very similar way to the two-time asymptotic expansion of section 3.1, we can retrieve the amplitude equations governing the slowly varying envelopes B sp p1 and B s Q1 , with K = ±k, Q = -Kp, and s p = ±1. Similarly to (3.5), the four amplitude equations are

           ∂ T B s k Q1 = sp=±1 C s k s k sp QKp B sp * p1 e i ∆ω KpQ (kRo) 2 T , ∂ T B sp p1 = K=±k C sps k s k pKQ (kRo) 2 B s k * Q1 e i ∆ω KpQ
(kRo) 2 T .

(4.1)

The characteristic polynomial P(σ) of this set of four linear differential equations in terms of the Rossby number Ro and the detunings ∆ω K is:

P(σ) = σ 2 -i∆ω kpq σ -Ro 2 S k σ 2 -i∆ω -kpq ′ σ -Ro 2 S -k -Ro 4 P 0 , (4.2)
with S K and P 0 two real coefficients defined as

S K = sp C sps k s k pKQ C s k s k sp * QKp and P 0 = sp C s k s k sp qkp C sps k s k p-kq ′ sp C s k s k sp q ′ -kp C
sps +k s k pkq .

(4.3) In general, the growth rate σ k (p; Ro) is found by numerical computation of the roots of P. Nevertheless, an estimate of the maximum growth rate can be obtained in the limit |p x | ≪ |p y |, which is relevant to our DNS. In this case, symmetries impose ∆ω kpq = -∆ω -kpq ′ , S k = S -k and P 0 = S 2 k . The polynomial P then has a purely real root,

σ k (p; Ro) = 2Ro 2 S k (p) -∆ω 2 kpq (4.4)
and the frequency of the growing geostrophic mode is then 0. We show in figure 4b the excellent agreement between this theoretical law and the DNS data of figure 2a.

An expansion similar to section 3.1 reveals that the maximum growth rate follows the exact same (kRo) 2 law as in the case of the single triad (3.10). This is confirmed by the systematic computation of the growth rate in the double triad case, as shown in figure 4c. Note that although this instability is driven by two imposed modes, it remains different from the four-modes interaction mechanism detailed in [START_REF] Brunet | Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability[END_REF]. The latter features intermediate, non-resonant modes that are absent in the double triad mechanism. Moreover, the near-resonant growth rate (4.4) is proportional to Ro when Ro is sufficiently large, whereas it always follows a Ro 2 law in the case of four-modes interaction.

We compare in figure 4c our theoretical predictions with the geostrophic growth rate extracted from DNS initiated with a large-scale noise, as in section 2, down to Ro = 5 × 10 -3 . We use the same imposed wave vector as previously (k = 2π [4, 0, 8]) but also 1.5 and 2 times longer wavevectors to confirm that the growth rate is a function of the intrinsic Rossby number kRo. Such a dependence is expected since in the infinite-domain limit, L → ∞, L becomes irrelevant and 1/k is the only remaining length scale, and the associated Rossby number is kRo. As shown in figure 4c, the numerical growth rate coincides with the maximum near-resonant growth rate in the low Ro regime but also in the moderate Ro regime where it is proportional to kRo. Note that for the three lowest kRo points, the noise has been implemented on two-dimensional modes (p z = 0) to facilitate the isolation of the geostrophic instability. It delays the growth of twodimensional modes with non-zero frequency by direct forcing at the lowest values of Ro. Despite their rapid growth, the latter are subdominant in the saturation of wave-driven flows, and do not prevent the long-term growth of unstable geostrophic modes under the mechanism examined here.

Lastly, our analysis allows us to understand the transition in the stability of the geostrophic flow observed in the numerical study (see figure 1). Ro = 2.83 × 10 -2 (kRo ≃ 1.6) corresponds to the transition zone where the geostrophic growth rate is O(kRo), as the wave-only triadic resonances (see figure 4c). When the Rossby number is decreased by one order of magnitude, as shown in figure 4c, the growth rate of geostrophic modes scales like (kRo) 2 , i.e. a factor kRo smaller than the growth rate of wave-only triadic resonances. This is why the geostrophic instability is not observed in the simulation at Ro = 2.83 × 10 -3 .

Conclusion

By means of numerical simulations and theoretical analysis, we have described a new instability mechanism by which inertial waves excite z-invariant geostrophic modes. We have proved that this instability is driven by near-resonant triadic interaction and derived its theoretical growth rate. When normalised by the global rotation rate Ω, the growth rate follows a (kRo) 2 law and a kRo law at small and moderate wave amplitude, respectively, k being the imposed wavenumber. It translates into Ω -1 and Ω 0 laws, respectively, for the dimensional growth rate. The near-resonant instability completes the picture proposed in [START_REF] Brunet | Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability[END_REF] where another inviscid geostrophic instability based on two imposed modes and four-modes interaction is detailed. Although of different nature, both instabilities have a (kRo) 2 growth rate in the limit of small Rossby numbers which makes them possibly difficult to distinguish. On the one hand, in the linear growth phase, the eigenmode of the four-mode instability consists of waves and geostrophic flow of comparable amplitude, whereas the geostrophic flow is kRo times smaller than the waves in the present mechanism. On the other hand, the near-resonant instability achieves a growth rate of order kRo for kRo 1. Our analysis may thus explain the experimental results of Le [START_REF] Le Reun | Experimental study of the nonlinear saturation of the elliptical instability: Inertial wave turbulence versus geostrophic turbulence[END_REF]: they found at moderate Rossby number a geostrophic instability with a growth rate proportional to Ro, which thus matches the moderate Rossby number law derived here.

Our work paves the way for new studies dealing with the energy transfers from waves to geostrophic modes in rotating fluids, in particular in rotating turbulence. First, it remains to be seen under which conditions the near-resonant instability may excite slow nearly geostrophic modes with small but non-zero frequencies. These modes have been proved to be important for the development of anisotropy in rotating turbulence [START_REF] Smith | On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number[END_REF] even in the asymptotic limit of small Rossby numbers (van Kan & Alexakis 2019).

With a heuristic analysis similar to the one developed in section 3.1 and appendix A, we can predict again a (kRo) 2 growth rate, but a detailed analysis is difficult and remains to be done. Lastly, the instability we describe in the present article could play an important role in the rotating turbulence dynamics that remains to be fully deciphered. Since wavewave triadic interactions grow at rate O(kRo) while wave-geostrophic interactions have a O((kRo) 2 ) growth rate, we may speculate that, at sufficiently low forcing amplitudes, rotating turbulence may remain purely three-dimensional as the wave amplitudes would remain below the threshold of geostrophic instabilities. Such states have been observed by Le [START_REF] Le Reun | Inertial Wave Turbulence Driven by Elliptical Instability[END_REF][START_REF] Le Reun | Experimental study of the nonlinear saturation of the elliptical instability: Inertial wave turbulence versus geostrophic turbulence[END_REF] and [START_REF] Brunet | Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability[END_REF], but a systematic investigation of their existence in various realisations of rotating turbulence remains to be carried out. In the future, our study could help bridging the gap between finite Rossby number experiments and simulations of turbulence [START_REF] Godeferd | Structure and Dynamics of Rotating Turbulence: A Review of Recent Experimental and Numerical Results[END_REF] and asymptotic models of rotating turbulence [START_REF] Galtier | Weak inertial-wave turbulence theory[END_REF][START_REF] Bellet | Wave turbulence in rapidly rotating flows[END_REF][START_REF] Van Kan | Critical transition in fast-rotating turbulence within highly elongated domains[END_REF]. spectral version of the Euler equation using a two-timing methods involving a fast time τ = t and a slow time T . The hierarchy between τ and T may be derived heuristically from an analysis of the amplitude equation in a triad involving a geostrophic mode. Let us assume the amplitudes b p,q to scale like ε p,q and that the slow time is given by the small detuning, ∆ω kpq = O(β). We further assume that β, ε p,q = O(kRo) at least. The two amplitude equations stemming from (3.1) governing b p,q give the following scaling relationships: βε p = kRoβε q and βε q = kRoε p (A 1)

where we have used that the wave-to-geostrophic coupling coefficient is proportional to the detuning, hence β, as explained in section 3.1 below equation (3.3). We have also used that the coupling coefficients scale like the wavenumber k. For the hierarchy between ε p,q and β to hold, the determinant of the system (A 1) with unknowns ε p,q must vanish. This condition is satisfied when β = (kRo) 2 . When this condition is satisfied, the equation (A 1) yields ε p = kRoε q . Therefore, we chose the amplitudes b sp p and b sq q to scale like (kRo) 2 and kRo respectively at leading order.

Figure 1 .

 1 Figure 1. Kinetic energy time series (a and c) and heat map of log(E (θ, ω)) (b and d) resulting from two numerical simulations of the stability of the inertial wave k = 2π [4, 0, 8] at Ro = 2.83 × 10 -3 and Ro = 2.83 × 10 -2 . In panels (a) and (c), the labels indicate the slope of the best fit for the exponential growth. In panel (b) and (d), the plain line materialises the dispersion relation of inertial waves and the horizontal dashed line the frequency of the imposed wave (ω k ≃ 1.78). For the spectral energy maps, the temporal Fourier transforms have been performed until t = 800 for panel (b) and t = 100 for panel (d). In panel (b), we have indicated the extremal frequencies of the two energy locations.

Figure 2 .

 2 Figure 2. (a) Three-dimensional vertical vorticity field of the growing perturbation at Ro = 2.83 × 10 -2 and (b) its vertical average. The fields are normalised by their maximum values. (c) Growth rate of the geostrophic modes p g = 2π [0, py, 0] with py ∈ {1, 3, 5} as a function of the imposed wave amplitude Ro. The vertical line materialises kRo = 1. The lines joining the markers are used to facilitate the identification of each curve. (d) Heat map of log(E (θ, ω)) in the case py = 3 and Ro = 1.3 × 10 -2 . The same lines as in figures 1 (b and d) are reported, and a vertical line materialises the angle of the modes closing the triad, that is -k ± p.

  sp p exp i(p• x -ω sp p t) with time-dependent amplitudes b sp p (t). The Euler equation governing U is then equivalent to a set of ordinary differential equations governing the amplitudes b sp p (Smith & Waleffe 1999):

Figure 3 .

 3 Figure 3. (a) Map of the growth rate of geostrophic modes at Ro = 7.5 × 10 -3 computed from (3.6). The imposed wave is 2π [4, 0, 8]. The white dashed circle locates the exactly resonant geostrophic modes. The color scale gives the amplitude of the growth rate. Where it is maximum (px = 0, py ≃ ±2), the detuning is about 0.04 ∼ 0.24(kRo) 2 . (b) Plot of C k (p 0 ) on the exact resonant circle against p0 normalised by the horizontal wavenumber k ⊥ for several wavevectors k with different frequencies ω k . The curve is the same regardless of the imposed helicity sign s. (c) Growth rate curves of the geostrophic modes as a function of the Rossby number. The geostrophic modes are sampled over 15 circles whose centers are the same as the exact resonance circle with 5 points on each circle. The line colour codes the frequency detuning |∆ω k |. The upper envelope is compared to the law (3.10) and the upper bound (3.12).

Figure 4 .

 4 Figure 4. (a) Schematic cartoon of a geostrophic mode p in near-resonance with both imposed modes ±k at the same time, based on the map of figure 3a. The ⊥ indices denote the horizontal component of wavevectors. (b) Comparison between theory (4.4) and the direct numerical simulations of figure 2c. The imposed wave is k = 2π [4, 0, 8] and the geostrophic modes are p = 2π [0, py, 0], py being given in the legend. (c) Samples of the growth rate curves σ k (p; Ro) of modes p interacting with ±k = 2π [4, 0, 8] including simple (dark grey) and double (light grey) triad mechanisms. The double triad growth rate curves are determined by finding the roots of P (see equation (4.2)) for wavevectors p restricted to |px| < π and |py| < 12π. We recall the Ro 2 law (3.10) and the growth rate upper bound (3.12). The dots represent the geostrophic growth rate found in the DNS with imposed wave vector k = 2π [4, 0, 8] (DNS 1), k = 1.5 × 2π [4, 0, 8] (DNS 1.5) and k = 2 × 2π [4, 0, 8] (DNS 2).

  kRo) 2 T and ∂ T B

					sp p1 =	C pkq sps k s k (kRo) 2 B	sq * q1 e	i	∆ω kpq (kRo) 2 T , (3.5)
	the rescaled quantities C pkq sps k s k have exponentially growing solutions and the complex growth rate of the instability /(kRo) 2 and ∆ω kpq /(kRo) 2 being O(1). These equations
	normalised by the rotation rate is
	σ = i	∆ω kpq 2	+	1 2	4 sp C

sps k s k pkq C s k s k sp * qkp Ro 2 -∆ω 2 kpq ≡ i ∆ω kpq 2 + σ k (p; Ro) . (3.6)

Note that the product of coupling coefficients C

sps k s k pkq C s k s k sp * qkp

† Email address for correspondence: tl402@cam.ac.uk

Acknowledgement. The authors acknowledge funding by the European Research Council under the European Union's Horizon 2020 research and innovation program through Grant No. 681835-FLUDYCO-ERC-2015-CoG and FLAVE 757239. TLR is supported by the Royal Society through a Newton International Fellowship (Grant reference NIF\R1\192181). Declaration of Interests. The authors report no conflict of interest.

Appendix A. Heuristic justification of the (kRo) 2 slow timescale

To investigate near-resonant instability, we proceed to an asymptotic expansion of the