Pierre Musitelli

The Correspondence of the Verri Brothers (1766-1797) and Newly Edited Online Resources on the Italian Enlightenment

Posted on March 21, 2016 under Network Events By Pierre Musitelli (École normale supérieure, Paris)

The correspondence between Pietro Verri (1728-1797) and Alessandro (1741-1816) Verri, two brothers from the enlightened Milanese aristocracy, is one of the major epistolary legacies of the Italian Enlightenment. It unfolded over more than 30 years, from 1766 to 1797, and, despite unrecoverable losses, especially in the early years of the 20 th century, it still contains over 3,800 letters. It is an accurate and vivid reflection of the intellectual and political history of the last decades of the Ancien Régime, from the "Age of Reforms" in Italy to the French Revolution and the invasion of Piedmont and Lombardy by Bonaparte. Historian Giuseppe Ricuperati considers these letters "an extraordinary dialogue between two of the sharpest and most enlightened Italian minds, each brother reacting to contemporary events in his own way, making diverging choices out of a common history and culture."1 Throughout these years, the Carteggio had several functions: the first was to spread news and political information along a route between Rome and Southern Italy, on the one hand, and Milan and Vienna, on the other. Some excerpts were read and shown in Milanese aristocratic circles by Pietro and in Roman salons by Alessandro. The Carteggio was well-known for the quality of its information, and although the Verri brothers often used ciphers, their letters often bear traces showing that they were intercepted and opened by Milanese or Roman authorities.

The letters also fulfilled a private role, as the brothers shared news about their sentimental lives, marriages, about births or mourning in the family, moments of doubt or personal crises. They also fulfilled an editorial function, each brother sending his manuscripts for the other to read, amend and revise. It also had an intellectual import, as the brothers sent books along with their letters. Pietro, for example, sent large numbers of pamphlets by Voltaire to Rome, along with Shakespeare's complete works in French, as well as some of the volumes from the Livorno edition of the Encyclopédie. Lastly, and especially for Alessandro, the correspondence had a vital financial function: Pietro sent him money, since he had no resources of his own in Rome.

One of the most fascinating aspects of this correspondence is the debate on the European Enlightenment and its evolution. The debate gathered momentum as the intellectual rift between the two brothers widened. Pietro, a Milanese statesman and ardent reformist, remained a staunch republican throughout his life and hoped the French Revolution would take a constitutionalist turn, while Alessandro, a self-appointed defender of throne and altar, contended that political passions, embodied by the philosophes, were dangerous, and he advocated the values of counter-revolution during the last months of his life.

The first months of the Carteggio, dedicated to Alessandro's trip to Paris and London (with Cesare Beccaria, the author of Crime and Punishments, and at the invitation of the Encyclopédistes), and its last years (1782-1797) were the objects of remarkable critical editions offering reliable and stable versions of the texts. [START_REF]Viaggio a Parigi e Londra[END_REF] For the rest of the correspondence, researchers must work with an incomplete and relatively outdated edition published between 1910 and 1942, in which the text is often erroneous, as it was subjected to alterations and even censored in its crudest or most wanton details. [START_REF] Di | [END_REF] Today, a new prospect for research is the creation of the online platform "illuminismolombardo.it", edited by Gianni Francioni (University of Pavia), an internet archive that will give scholars access to major sources on the Northern Italian Enlightenment. A wider audience will be able to see the results of two recently completed editorial projects: the National Editions of the works of Cesare Beccaria (including his letters) and of Pietro Verri 4 . The platform will include the revised complete edition of the correspondence of the Verri Brothers. The first volume of the Carteggio (1766-1767) is already available online, as well as all the essays from the Caffè, along with the complete collection of the journal Studi settecenteschi (1981-2010, 30 vol.) edited by G. Francioni. Texts and letters by Paolo Frisi, Alfonso Longo, Giuseppe Gorani, Giambattista Biffi will be added later, making this portal the promise of renewed research on this major legacy.

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G. Ricuperati, "L'epistolario dei fratelli Verri", in Nuove idee e nuova arte nel '700 italiano, Rome, Atti dei convegni lincei, 1977, p.


D Bestion

A Ciechocki
email: antoine.ciechocki@cea.fr

S Carnevali
email: sofia.carnevali@cea.fr

The use of System Codes for Scaling Analysis and the use of Scaling tools for the Analysis of Code Predictions

Keywords: Scaling analysis, System thermal-hydraulic codes, ITF design, LOCA transients

Both system codes and experiments are used for simulating nuclear reactor thermal-hydraulic transients in safety analysis. System codes model the whole reactor and must demonstrate their quality by extensive validation against experimental data, which cover the important phenomena. Integral effect test facilities intend to simulate reactor thermal-hydraulic behaviour in reduced scale conditions. Advanced scaling methods exist to define how to respect the dominant phenomena in a scaled experiment and to evaluate distortions. They perform a similarity analysis applied to equations governing the evolution of important parameters at system, component and local levels. They require an evaluation of the order of magnitude of every term of these equations. Today, system codes have reached a good maturity and reliability, and one defines in this work the conditions for a proper use of codes for evaluating terms of the equations used in scaling analyses. It appears that scaling tools at system level can help in the analysis of code predictions, in the establishment of a structured PIRT and for identifying the sources of code predictions differences with experimental data. This paper presents the lessons learnt from a combined use of scaling methods, scaling tools, system codes and integral effect tests data. Some conclusions and recommendations are drawn on the merits of the various scaling methods, on the selection of the equations to be used in the scaling analyses, on the various methods for estimating terms of the selected equations, and on the acceptability limits for the distortions. The applications refer to some PWR LOCA analyses.

INTRODUCTION

Reactor transient thermal-hydraulic simulations uses both experiments and numerical simulation tools such as system codes. Reduced scale Integral Test Facilities (ITF) are used to simulate the whole reactor and scaling methods are necessary to identify and minimize the scale distortions. A state of the art on scaling was published by OECD-NEA-CSNI in 2016 [START_REF] Bestion | Scaling in system thermal-hydraulics applications to nuclear reactor safety: A State of the Art report[END_REF]. Among the conclusions of this report, one may find:

-The use of a well-validated and verified system code can support any scaling analysis, including checking the scaling hierarchy, evaluating the impact of scale distortions and correcting the distortions in reactor applications.

-Counterpart Tests are highly valuable to verify the scaling effects.

-There is a need to identify a qualitative and quantitative framework (precision targets) to judge the quality of a scaling approach. This step is connected with the acceptance criterion for scaling distortions, and with the quantification of uncertainty due to scaling. -Specific scaling-related training is worthwhile. Good training and education of safety analysts should include, in addition to basic single-phase and two-phase thermal-hydraulics, advanced topics of scaling techniques, identification of the dominant phenomena of major transients, code V&V and UQ requirements, and code-scalability requirements. Based on these conclusions, the present work aims to summarize some lessons learnt from applying system codes for scaling analysis with evaluation of scaling distortions, and from using counterpart tests.

Only the Hierarchical Two-Tiered Scaling (H2TS) [START_REF] Zuber | A hierarchical, Two-tiered Scaling Analysis, Appendix D of An Integrated Structure and Scaling Methodology for Severe Accident Technical Issue Resolution[END_REF] and the Fractional Scaling Analysis (FSA) [START_REF]Scaling a Nuclear Power Plant as a Complex System[END_REF][START_REF] Zuber | Application of Fractional Scaling Analysis (FSA) to Loss of Coolant Accidents (LOCA)[END_REF] methods applied with the support of system codes are commented here. The applications are here related to PWR Loss Of Coolant Accident (LOCA) transients. Several authors have reported such analyses (Munoz-Cobo et al., [START_REF] Muñoz-Cobo | Top-down scaling methodology from the LSTF facility to a three loop PWR plant applied to a SBLOCA event -The ROSA 1.2 test[END_REF], Reventos et al. [START_REF] Reventós | Perfecting hybrid models in scaling analysis[END_REF] Martinez-Quiroga & Reventos [START_REF] Martinez-Quiroga | The use of system codes in scaling studies: relevant techniques for qualifying NPP nodalizations for particular scenarios[END_REF]). Examples commented here are based on recent work performed at CEA by Ciechocki et al. [START_REF] Ciechocki | Application of the FSA scaling method to the LSTF ROSA 1.2 test and comparison to an application of the H2TS method[END_REF][START_REF] Ciechocki | Scaling analyses of a SB-LOCA counterpart test between BETHSY and LSTF facilities and a three loops PWR[END_REF]. Some guidelines and recommendations are drawn on the selection of the "scaling equations" to be used in the scaling analyses, on the various methods for estimating terms of the selected equations, and on the acceptability limits for the distortions.

THE SCALING METHODS AND THE BENEFIT OF USING SYSTEM CODES

Scaling methods were initially conceived to define design criteria for ITF without using system codes. The reason was that the experiments resulting from the scaling are used to validate the codes and can only be used after their validation. However, the current system codes are now extensively validated and can be used for scaling analysis as stated in the state of the art report on scaling [START_REF] Bestion | Scaling in system thermal-hydraulics applications to nuclear reactor safety: A State of the Art report[END_REF]. The general process of safety demonstration for a reactor transient is summarized in the Figure 1.

Figure 1. Process of safety demonstration for a reactor transient

A scaling analysis is here performed first for defining design criteria of ITF and IET boundary and initial condition (BIC). An "a posteriori scaling analysis" is added after the code validation to give arguments to the code scalability demonstration. After the validation, one can better identify the origin of differences observed between the IET and reactor calculations, which can be attributed to either code deficiency, IET scaling distortions, code nodalization effects, or bad scalability of some code closure laws. This work intends to bring some guidelines for the steps colored in blue in Figure 1.

The common features of H2TS and FSA

The scaling of an ITF with a top-down approach followed by a bottom-up approach was proposed in the H2TS method [START_REF] Zuber | A hierarchical, Two-tiered Scaling Analysis, Appendix D of An Integrated Structure and Scaling Methodology for Severe Accident Technical Issue Resolution[END_REF] and is also used in the FSA method [START_REF]Scaling a Nuclear Power Plant as a Complex System[END_REF][START_REF] Zuber | Application of Fractional Scaling Analysis (FSA) to Loss of Coolant Accidents (LOCA)[END_REF]. The analysis starts at the system scale and is based on equations written for Parameters of Interest (PoI) characterizing the status of the system, e.g. the primary circuit. It gives some first scaling criteria for the selection of scaling ratios for the circuit volume, the core power, the flow rates and the time scale. Equations are written first for the reactor for such PoI in the following form:

𝑑𝑃𝑜𝐼 𝑗 𝑑𝑡 = ∑ 𝑅 𝑗 𝑘 𝑘 (1)
Each 𝑅 𝑗 𝑘 represents a physical process which is an Agent of Change of 𝑃𝑜𝐼 𝑗 .

The same equation is written for the IET to identify the scaling factor 𝜆 𝑗 applied to 𝑃𝑜𝐼 𝑗 and the scaling factors 𝜆 𝑘,𝑖 applied to 𝑅 𝑗 𝑘 .

𝑑𝑃𝑜𝐼 𝑗 𝑑𝑡 𝜆 𝑗 ≅ ∑ 𝜆 𝑗,𝑘 ⋅ 𝑅 𝑗 𝑘 𝑘 (2)
One may classify the Agents of Change 𝑅 𝑗 𝑘 into the dominant or influent ones 𝑅𝐷𝐼 𝑗 𝑘 and the negligible ones 𝑅𝑁 𝑗 𝑘 , which may be eliminated from the analysis:

𝑑𝑃𝑜𝐼 𝑗 𝑑𝑡 = ∑ 𝑅 𝑗 𝑘 = ∑ 𝑅𝐷𝐼 𝑗 𝑘 𝑘 + ∑ 𝑅𝑁 𝑗 𝑘 𝑘 𝑘 ≅ ∑ 𝑅𝐷𝐼 𝑗 𝑘 𝑘 (3)
If all scaling factors are equal (∀ 𝑖, 𝜆 𝑗 = 𝜆 𝑗,𝑖 ), Eq. ( 3) is almost similar to Eq. ( 1) and the experiment preserves a good prediction of 𝑃𝑜𝐼 𝑗 evolution and preserves the time scale. If 𝜆 𝑗 ≠ 𝜆 𝑗,𝑖 and all 𝜆 𝑗,𝑖 are equal, Eq. ( 3) is almost similar to Eq. ( 1) and the experiment preserves a good prediction of 𝑃𝑜𝐼 𝑗 evolution with a change of time scale. When a scaling factor of a 𝑅𝐷𝐼 𝑗 𝑘 is different from the others, the Agent of Change is distorted and one must give some acceptability limit to this distortion. This analysis may be repeated for several successive phases of a transient since the dominant, influent and negligible terms may change during the transient.

It may happen that the term 𝑑𝑃𝑜𝐼 𝑗 𝑑𝑡 ⁄ is negligible in the equation when there is a quasi-steady state during a given phase of a transient. Some of these 𝑅 𝑗 𝑘 are representing a physical process, which occurs only in a reactor component. A specific analysis may be necessary at another scale, in order to evaluate this term using other scaling equations or existing physical models. This is the top-down approach.

The Hierarchical Two-Tiered Scaling (H2TS) method

In the H2TS method [START_REF] Zuber | A hierarchical, Two-tiered Scaling Analysis, Appendix D of An Integrated Structure and Scaling Methodology for Severe Accident Technical Issue Resolution[END_REF], the scaling equation for a PoI is written for a specific quantity 𝜓 (e.g. 𝜓 = 𝜌, 𝜌𝑣, 𝜌ℎ) within a control volume 𝑉 considering the convective terms 𝑄𝜓, the diffusive terms 𝐽 𝑖 and the volumetric source terms 𝑆 𝑘 .

𝑑𝑉 ⋅ 𝜓 𝑑𝑡 = [𝑄𝜓 ] 𝑖𝑛 -[𝑄𝜓 ] 𝑜𝑢𝑡 + ∑ 𝐽 𝑖 ⋅ 𝐴 𝑖 + ∑ 𝑆𝑘 (4)
A reference time scale is defined: for example a residence time : 𝜏 𝑅 = 𝑉 0 𝑄 0 ⁄ where 𝑄 0 is a reference (e.g. initial) volume flow rate of 𝜓. Then for each process, a non-dimensional process time scale ratio is defined: 𝛱 𝑝𝑟𝑜𝑐 = 𝜏 𝑅 𝜏 𝑝𝑟𝑜𝑐 ⁄ , for example the diffusion time scale:

1 𝜏 𝑑𝑖𝑓𝑓 = 𝜔 𝑑𝑖𝑓𝑓 = 𝐽 𝑖𝑜 ⋅ 𝐴 𝑖𝑜 𝜓 0 ⋅ 𝑉 0 (5)
The behaviour of 𝜓 is controlled by largest 𝛱 𝑝𝑟𝑜𝑐 corresponding to the smallest process time scales. Experiment scaling should respect Π proc exp /Π proc 𝑟𝑒𝑎𝑐 of largest 𝛱 𝑝𝑟𝑜𝑐 . Scaling distortion is quantified by:

𝐷 = 1 - 𝛱 𝑝𝑟𝑜𝑐 𝑒𝑥𝑝 𝛱 𝑝𝑟𝑜𝑐 𝑟𝑒𝑎𝑐 (6)
Some authors (Wulff & Rohatgi, 1999 [10]) have defined limits such as 0.5 < D < 2 for a well-scaled phenomenon. The basic idea is simple: "when several agents of change contribute in parallel to a change of a quantity 𝜓, the dominant agents are those who contribute faster than the others" and one must respect the process time scales of the dominant ones. This is an easy way to identify quickly the impact of some design differences between a reactor and an ITF but also between several reactors and several ITF in a counterpart test. For example for PWR LOCA, one may easily compare the emptying time of the whole circuit by applying a reference critical break flow rate. One may compare the emptying time of the pressurizer to estimate the duration of this phase. One may compare the emptying time of the part of the circuit above the break elevation to estimate the initiation time of the high quality discharge at the break, which makes the primary pressure falling below the secondary pressure in many SBLOCA transients. One may compare characteristic heating or cooling time scales of the core power and of heat losses or heat release from metallic structures during a phase of a transient. One may compare the pressurizing or depressurizing time scales due to core heating or volume flow rate at a break.

The Fractional Scaling Analysis (FSA) method

FSA is an analytical approach for complex problems (economy, ecology, …) based on "fractional analysis" (Zuber et al. [START_REF]Scaling a Nuclear Power Plant as a Complex System[END_REF][START_REF] Zuber | Application of Fractional Scaling Analysis (FSA) to Loss of Coolant Accidents (LOCA)[END_REF]). The FSA method considers state variables 𝜓 that are influenced by processes (convection, diffusion, wave propagation, ...

The FRC 𝜔 𝑗 are shown being the cause 𝜙 𝑗 over the effect:

𝜔 𝑗 = 1 𝜓 ⋅ 𝑑𝜓 𝑑𝑡 = 𝜙 𝑗 𝜓 (8)
The dominant phenomena in the different phases of the transient are identified by ranking them according to the relative magnitude of the FRC. The Effect Metrics Ω 𝑗 are obtained, representing the rate of change of the variable 𝜓 over a reference period 𝜏 𝑟𝑒𝑓 , consequence of the contribution of the Agents of Change:

Ω 𝑗 = 𝜔 𝑗 ⋅ 𝜏 𝑟𝑒𝑓 (9)

IDENTIFICATION OF PARAMETERS OF INTEREST AND BIFURCATING EVENTS

The Figure of Merit(s) (FoM) is (are) the main thermal-hydraulic parameters, which are used for the safety demonstration. For LOCA simulations, the Peak Clad Temperature (PCT) is the first FoM. This PCT depends on the occurrence of Critical Heat Flux (CHF) either by Departure from Nucleate Boiling (DNB) or by Dry-Out. DNB depends on the mass flux 𝐺, the pressure 𝑃 and the thermodynamic quality 𝛼. Then in LBLOCA and IBLOCA where DNB may be reached in the early phase of the transient one should pay a special attention to these three PoI. In SBLOCA the fuel rod temperature excursions may occur by core uncovery in case of lack of water in the core. Therefore, the total primary mass 𝑀 1 and the core mass are important PoI. They depend a lot on the SI injections and the break flow rate, which depend a lot on 𝑃 1 history.

Then the fluid mass repartition in the various components is controlled by some force balance equations, which may be specific to the component and to the physical situations. Table I gives an example for a 6% cold leg break SBLOCA as investigated in [START_REF] Ciechocki | Scaling analyses of a SB-LOCA counterpart test between BETHSY and LSTF facilities and a three loops PWR[END_REF]. For each PoI, the effects are given, as well as the impact on the PCT and the scaling equation that may be used in a scaling analysis. There exist some particular events, which change the phenomenology and the FoM may be very sensitive to the timing of these "bifurcating events". The Table II below illustrates the main bifurcating events and their potential effects for the same SBLOCA transient.

THE SCALING EQUATIONS AT SYSTEM AND COMPONENT SCALES

The mass equation at system scale

The equation for total primary mass is:

𝑀 ̇= 𝑀 ̇𝑙,𝑖𝑛 + 𝑀 ̇𝑣,𝑖𝑛 -𝑀 ̇𝑙,𝑜𝑢𝑡 -𝑀 ̇𝑣,𝑜𝑢𝑡 (10)
In a PWR LOCA there is usually a source of water by SI and a sink of liquid and vapour at the break:

𝑀 1 ̇= 𝑀 ̇𝑆𝐼 -𝑀 ̇𝑙,𝑏𝑟𝑒𝑎𝑘 -𝑀 ̇𝑣,𝑏𝑟𝑒𝑎𝑘 (11)
If the SI flow rate and the break flow rate are reduced by the volume ratio, the equations for PWR and IET are similar with the same timing.

The pressure equation at system or component equation

Based on the phase energy equation, the phase mass equation and the constant fluid volume one may derive the following pressure equation (see Bestion, 2019, [11]).

( 𝜛 being the mass flow rate by vaporization or condensation at the wall (kg/s). All the terms are equivalent to a volume flow rate or Volume Rate of Change (VRC). Since the volume is constant, the sources of fluid volume are compensated by the sinks. The l.h.s term is the change of volume by contraction/expansion in pressure increase/decrease. The two first r.h.s terms are the fluid volume sources entering and the corresponding thermal contraction/expansion due to mixing. The 4 th , 5 th and 6 th r.h.s terms are due to thermal contraction/expansion and boiling/condensation due to wall heat exchanges. The 7 th , 8 th and 9 th r.h.s terms are terms due to thermal contraction/expansion and boiling/condensation due to interfacial heat exchanges. There are situations where both phases are close to saturation in a circuit (e.g. during SBLOCA). One may have a simpler pressure equation assuming perfect thermal equilibrium.

(
This pressure difference is often approximately equal to the difference of liquid head in core and DC:

𝑃 𝑐𝑜𝑟𝑒 -𝑃 𝐷𝐶 ≈ (𝜌 𝑙𝑠 ⋅ 𝐻 𝐶,𝑐𝑜𝑟𝑒 -𝜌 𝑙 ⋅ 𝐻 𝐶,𝐷𝐶 ) ⋅ 𝑔 (19)
However all these equations are coupled to energy equations to predict the evolution of liquid and vapour flow rate and to force balance to predict the void fraction distribution or the slip velocity between phases.

Equations for the void fraction

Elimination of the pressure gradient between liquid and vapour momentum equations results in this force balance:

𝐴 ⋅ 𝜕𝛼 ⋅ (1 -𝛼) ⋅ [𝜌 𝑣 ⋅ 𝑢 𝑣 -𝜌 𝑙 ⋅ 𝑢 𝑙 ] 𝜕𝑡 + 𝜕𝐴 ⋅ 𝛼 ⋅ (1 -𝛼) ⋅ [𝜌 𝑣 ⋅ 𝑢 𝑣 2 -𝜌 𝑙 ⋅ 𝑢 𝑙 2 ] 𝜕𝑧 + 𝐴 ⋅ 𝑝 𝑖 ⋅ 𝜕𝛼 𝜕𝑧 = 𝐴 ⋅ 𝛼 ⋅ (1 -𝛼) ⋅ ∆𝜌 ⋅ 𝑔 𝑧 -𝐴 ⋅ 𝜏 𝑖 -(1 -𝛼) ⋅ 𝐶 𝑣 ⋅ 𝜏 𝑤𝑣 + 𝛼 ⋅ 𝐶 𝑙 ⋅ 𝜏 𝑤𝑙 + 𝐹 𝐴𝑀 + 𝐴 ⋅ 𝛤 𝑖𝑣 ⋅ 𝑊 𝑖 (20)
In quasi-steady-state vertical flow at medium velocity, it simplifies to:

𝜏 𝑖 = 𝐴 𝑖 ⋅ 𝐶𝑓 𝑖 ⋅ 𝜌 𝑣 ⋅ (𝑢 𝑣 -𝑢 𝑙 ) 2 ≈ 𝛼 ⋅ (1 -𝛼) ⋅ (𝜌 𝑙 -𝜌 𝑣 ) ⋅ 𝑔 𝑧 -(1 -𝛼) ⋅ 𝐶 𝑣 ⋅ 𝜏 𝑤𝑣 + 𝛼 ⋅ 𝐶 𝑙 ⋅ 𝜏 𝑤𝑙 (21)
In quasi-steady-state vertical flow at low velocity, it simplifies to:

𝜏 𝑖 = 𝐴 𝑖 ⋅ 𝐶𝑓 𝑖 ⋅ 𝜌 𝑣 ⋅ (𝑢 𝑣 -𝑢 𝑙 ) 2 ≈ 𝛼 ⋅ (1 -𝛼) ⋅ (𝜌 𝑙 -𝜌 𝑣 ) ⋅ 𝑔 𝑧 (22)
In a core with low velocity, one can use this equation to determine the swell level 𝐻 𝑠𝑤 as a function of the collapsed level 𝐻 𝐶 , which corresponds to the core liquid mass (see Figure 2).

The fuel rods energy equations

Before reaching a quasi-steady state with the stabilized 𝑇 𝑣 and 𝑇 𝑤 axial profile, the main inertia is on the rods side controlled by the equation:

𝜌 ⋅ 𝐶 𝑝 ⋅ 𝜕𝑇 𝜕𝑡 =𝜆 ⋅ [ 1 𝑟 ⋅ 𝜕𝑇 𝜕𝑟 + 𝜕 2 𝑇 𝜕𝑟 2 ] +𝑊 𝑣𝑜𝑙 ( 23
)
One may estimate the conduction time scale in the rods and the heating time scale of the rods by:

𝜏 𝑐𝑜𝑛𝑑 = 𝜌 ⋅ 𝐶 𝑝 ⋅ ∆𝑟 2 𝜆 ; 𝜏 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 = 𝜌 ⋅ 𝐶 𝑝 ⋅ ∆𝑇 𝑊 𝑣𝑜𝑙 ( 24
)
τ cond may be respected by respecting bundle geometry and 𝜌 ⋅ 𝐶 𝑝 𝜆 ⁄ . τ inertia may be respected by respecting 𝜆 𝑣 = 𝜆 𝑤 = 𝜆 𝑄 . This imposes 𝜆 𝑧 = 1. One can also estimate the PCT for a stabilized swell level and the time period ∆𝑡 to reach this stabilized clad temperature profile. This is an example of using scaling equations to quantify the impact of distortions on the PCT.

The benefit of using system codes for scaling and for code model improvements

One main benefit of using system codes for scaling analyses is that the terms of the scaling equations can be estimated as function of time and may provide more information than a coarse estimation of terms for a transient phase. Figure 3 shows the CATHARE code simulations (from Ciechocki et al, [START_REF] Ciechocki | Scaling analyses of a SB-LOCA counterpart test between BETHSY and LSTF facilities and a three loops PWR[END_REF]) of a 6% cold leg SBLOCA transient of a PWR and the corresponding 6.2-TC transient in the BETHSY ITF , which was scaled using the power-to-volume scaling method and the height conservation [START_REF] Nahavandi | Scaling Laws for Modeling Nuclear Reactor Systems[END_REF]. A very important bifurcating event is the crossing of primary and secondary pressures since it initiates a second depressurization and has an impact on the timing of the accumulator discharge. The various terms of a primary pressure equation are grouped into four contribution: the volume flow rate at the break 𝑄 𝑏𝑟𝑒𝑎𝑘 , the Volume Rate of Change (VRC) in the core due to heating and vaporization 𝑄 𝑐𝑜𝑟𝑒 , the VRC due to heat transfer in SG 𝑄 𝑆𝐺 and the interfacial and wall heat and mass exchanges in other components 𝑄 𝑜𝑣 . The time of pressure crossing is earlier in BETHSY than in the PWR and this was attributed to different pressure in the SG. However, one may observe (see Figure 3) that the primary pressure evolution is governed by two counteracting dominant processes 𝑄 𝑐𝑜𝑟𝑒 which decreases due to decay power and 𝑄 𝑏𝑟𝑒𝑎𝑘 which decreases due to increasing of steam quality at the break that increases the volume flow rate. Then the heat release by walls 𝑄 𝑜𝑣 is not negligible but has approximately the same relative influence at the time of crossing (called pressure reversal) although one would expect a significant distortion in the IET. This heat release depends on the history of the transient and no coarse evaluation for a transient phase can predict its relative weight better than a code transient simulation. During the first blowdown phase of the 6% cold leg SBLOCA (Ciechocki et al, [START_REF] Ciechocki | Scaling analyses of a SB-LOCA counterpart test between BETHSY and LSTF facilities and a three loops PWR[END_REF]) , the pressurizers empty and their pressures are mainly governed by the exiting liquid volume flow rate 𝑄 𝑙,𝑝𝑟𝑧 and the flashing 𝑄 𝑖,𝑝𝑟𝑧 (Figure 4). During the first one or two seconds 𝑄 𝑙,𝑝𝑟𝑧 is partly compensated by vapour heating at the interface 𝑄 𝑖𝑣,𝑝𝑟𝑧 . Then vaporization starts after a "flashing delay" and 𝑄 𝑙,𝑝𝑟𝑧 is partly compensated by vaporization 𝑄 𝑖,𝑝𝑟𝑧 . One can see that although the full height scaling of the BETHSY pressurizer maximizes a lot the wall thermal capacity and HT area. The wall HT do not induce a significant distortion of the pressure history. Then one may be surprised by the significant impact of the exiting vapour flow rate 𝑄 𝑣,𝑝𝑟𝑧 , which is equivalent to the liquid flow rate in the PWR transient. After analysis, one may attribute this to a too coarse nodalization of the pressurizer with a 2-nodes volume module, which does not predict the void gradient in the liquid with a minimum at the bottom where the surge line is connected and a maximum at the free surface due to rising bubbles created by flashing. Therefore, one may suggest a more detailed 1-D modelling of the pressurizer for transients where the pressure history during the pressurizer emptying may affect the CHF occurrence in the core. This does not apply to SBLOCA but may occur in LBLOCA and in some IBLOCA. This simple use of a "scaling equation" evaluated by a system code illustrates how it can help capturing the physics of the transient phase, identifying quickly the dominant processes and quantifying the impact of distorted phenomena, but also suggesting some weaknesses of the code nodalization. Another distortion was observed in the BETHSY calculation [START_REF] Ciechocki | Scaling analyses of a SB-LOCA counterpart test between BETHSY and LSTF facilities and a three loops PWR[END_REF] due to an accumulator discharge, which is much faster than in the reactor. This is due to the full height design of the BETHSY accumulator. It has the advantage to represent the gravity head in the accumulator even during the discharge but it overestimates the heating of the nitrogen (N2) by walls when pressure decreases. A linear scaling of the accumulator would be more realistic with a distortion of the gravity head but a much lower distortion of the N2 heating. One may use two pressure equations for the primary circuit and for the accumulator to predict the discharge flow rate. One may also expect another distortion of this flow rate due to Direct Contact Condensation (DCC) in the cold leg due to safety injections (SI). When there is a stratified flow in CL, the interfacial heat transfer laws overestimate the condensation for a reduced CL diameter. The two distortions overestimate the discharge flow rate, which may significantly underestimate the PCT. One may then recommend to paying a specific attention on the uncertainty and the scalability of the condensation model to guaranty the code scalability for the reactor calculation.

COMMENTS ON THE ACCEPTABILITY THRESHOLD OF DISTORTIONS

The existing acceptability thresholds for the distortions just give arbitrary limits to ratios 𝑅 𝑗 = Π 𝑗 𝑒𝑥𝑝 Π 𝑗 𝑟𝑒𝑎𝑐 ⁄ of non-dimensional numbers for each distorted process j such as 0.5 < 𝑅 𝑗 < 2 . These criteria are very coarse and do not provide reliable guaranty of sufficient fidelity of the phenomenology. One should check first that the distortions do not miss any important bifurcating event. Then one may estimate the quantitative impact of each distortion on the PoI and on the FoM. This may be achieved by adding calculations with changes in the IET or reactor model to remove one by one the distortions. They were called "hybrid calculations" by Reventos et al. [START_REF] Reventós | Perfecting hybrid models in scaling analysis[END_REF]. An example is given below.

Removing distortions of thermal inertia and heat losses of ITF metallic structures

External wall of circuits have an overestimated thermal capacity in reduced scale ITF. They may also have an overestimated heat transfer area with fluid and external atmosphere. There is a possibility to correct this distortion by modelling fully scaled heat structures in the ITF model as shown in Figure 5. If a power-tovolume scaling is used (𝜆 𝑉 = 𝜆 𝑊 ) for the reduced scale experiment, one should apply a thermal structure to the experiment wall of a pipe that is a fraction of the corresponding reactor pipe wall such as: 𝑅𝑖 = 𝑅𝑖 𝑟𝑒𝑎𝑐 , 𝑅𝑒 = 𝑅𝑒 𝑟𝑒𝑎𝑐 and 𝜑 ⋅ 𝐿 𝑒𝑥𝑝 ⋅ 𝑅𝑖 𝑟𝑒𝑎𝑐 = 𝜆 𝑊 ⋅ 𝐿 𝑟𝑒𝑎𝑐 ⋅ 𝑅𝑖 𝑟𝑒𝑎𝑐 .

(27) This guaranties that the heat transfer areas between fluid and wall and between wall and atmosphere are well scaled and that the conduction time scale is respected.

THE USE OF SYSTEM CODE FOR AN A POSTERIORI SCALING ANALYSIS

An a posteriori scaling analysis is performed after the ITF is built and the transient simulated. A comparison of code predictions with IET results may reveal code weaknesses. Some of them may be corrected by improvements of some models or some changes of the nodalization. Then the scaling analysis may be revisited and one may better identify what difference between IET and PWR results are due to code weaknesses or to IET distortions and what code model requires a specific attention in UQ and scalability demonstration. In this a posteriori scaling analysis, the use of hybrid calculations is very useful to estimate the impact of scaling distortions.

CONCLUSIONS AND RECOMMENDATIONS

The application of scaling methods like H2TS or FSA is of great interest to design ITF with controlled distortions but also to analyze system code predictions of reactor transients and of corresponding IET. The use of process time scales in H2TS is an efficient way to identify the impact of some different designs between several ITF, e.g in case of counterpart tests. For each transient, after a PIRT, it is necessary to identify the Figure of Merit (FoM) and a list of Parameters of Interest (PoI) for which "scaling equations" are written to identify the possible scale distortions of a reduced scale ITF. The circuit fluid mass 𝑀, the system pressure 𝑃 and the circulation loop flow rate 𝑀 ̇ are the main PoI to perform the scaling analysis at the system scale since they may challenge the coolability of the core during transients. It has been shown that the estimation of the order of magnitude of each term of such equations can be better obtained using predictions by mature and extensively validated system codes. It provides a time evolution of each PoI although H2TS and FSA just estimate them for a time period called either "transient phase" or "phenomenological window". This reveals greater accuracy in identifying some distortion or the timing of some bifurcating event.

The cooling loop flow rate is controlled by a mixture momentum equation integrated along the loop. However, the natural circulation flow rate also depends a lot on the void distribution, which requires other specific interfacial force balance equations, which may be specific to the component and to the physical situations. Examples are given. One may recommend to use system codes for an a priori scaling devoted to the design of ITF and then to revisit the scaling analysis with code calculations including comparison to IET data. This helps in identifying not only distortions of IET but also possible weaknesses of code model. As a guidance for safety analysts, for each transient of interest, one should establish a list of most important PoI, a list of most important bifurcating events, and define a "set of scaling equations" to perform the scaling analysis.

Additional calculations (hybrid calculations) removing one by one the differences between IET and reactor models may help quantifying the impact of distortions. There is no general acceptability threshold for distortions and one should give first a targeted accuracy for the FoM and then try to determine the impact of the distortions on the FoM. Finally, the combined use of "scaling equations" and of system codes for IET and reactor calculations has several benefits: -It helps in understanding the phenomenology of a transient.

-It helps identifying weaknesses of the code models.

-It provides more accurate scaling analyses and quantification of distortions.

-Using hybrid calculations can provide an estimation of the effects of distortions on the FoM.

-It helps identifying the code models that require a specific attention in uncertainty quantification and scalability demonstration. -It is a useful tool in the demonstration of the scalability of a system code. This work is in progress and requires further efforts for identifying FoM, PoI, bifurcating events and scaling equations for LOCA and other transients.

NOMENCLATURE

Figure 2 .

 2 Figure 2. Illustration of a core uncovery situation in a PWR SBLOCA with temperature excursion above the swell level By integrating the vapour energy equation along 𝑍 and using the HT law, one may determine 𝑇 𝑣 (𝑍) and 𝑇 𝑤 (𝑍) in a steady state: 𝑇 𝑣 (𝑍) -𝑇 𝑠𝑎𝑡 = + ∫ 4 ⋅ (𝑍 -𝐻 𝑠𝑤 ) 𝐺 v ⋅ 𝐷 ⋅ 𝐶𝑝 𝑣 𝑧

Figure 3 .

 3 Figure 3. Analysis of the pressure crossing in a SBLOCA simulated by CATHARE code. Primary and secondary pressure evolution in a PWR and in the BETHSY IET (left). Contributions to primary pressure decrease for BETHSY (center), and for the PWR (right)

Figure 4 .

 4 Figure 4. Analysis of pressure evolution during the emptying of the pressurizer in a SBLOCA. Contributions to the depressurization for the BETHSY simulation (left) and for the PWR simulation (right).

Figure 5 .

 5 Figure 5. Extraction of a part of a reactor wall to be affected to an experiment to obtain scaled heat transfers -Reactor pipe on the left -ITF pipe on the right-portion of reactor pipe wall affected to ITF pipe

Table I . Example of identification of Parameters of Interest for a PWR Cold leg SBLOCA (see [9])

 I

	Parameter of interest	Effects	Impact on PCT	Scaling equation
	Primary mass 𝑀 1	Impacts 𝑀 𝑐𝑜𝑟𝑒	Big impact on PCT	Eq. (10)
	Primary pressure 𝑃 1	Impacts SI and 𝑄 𝑏𝑟𝑒𝑎𝑘	Big Impact on PCT	Eq. (14) or (15)
	Break flow rate	Impacts 𝑀 1 and 𝑃 1	Big Impact on PCT	Break model
	Mass flow rate in loops	Impacts end of NC and 𝑃 𝑐𝑜𝑟𝑒 -𝑃 𝐷𝐶	Some Impact on PCT	Eq. (17)
	𝑃 𝑐𝑜𝑟𝑒 -𝑃 𝐷𝐶	Impacts core uncovery	Some Impact on PCT	Eq. (18)
	Accumulator flow rate	Impacts 𝑀 1 and 𝑃 1	Big Impact on PCT	Eq. 𝑃 𝑎𝑐𝑐. -𝑃 1
	Mass in core 𝑀 𝑐𝑜𝑟𝑒	Controls core uncovery	Big impact on PCT	Eq. 𝑃 𝑐𝑜𝑟𝑒 -𝑃 𝐷𝐶
	Mass in Intermediate Legs 𝑀 𝐼𝐿	Impacts 𝑀 𝑐𝑜𝑟𝑒 by Loop Clearing (LSC) Seal Plugging (LSP) and	Low Impact on PCT	Eq. (18)

Table II . Example of identification of bifurcating events for a PWR Cold leg SBLOCA (see [9)] Bifurcating events Type of impact Effect on 𝑴 𝟏 and PCT

 II

	SCRAM signal	May impact DNB occurrence in early blowdown	No effect in SBLOCA
	SI signal		Impact on HPSI injection	Some effect on 𝑀 1 No effect on PCT
	Emptying pressurizer	of	Impact on 𝑃 and on 𝑄 𝑏𝑟𝑒𝑎𝑘	Some effect on 𝑀 1 No effect on PCT
	End	of	natural	Change of fluid mass repartition	Effect on a possible core uncover and
	circulation		Loop seal plugging	PCT
	Loop Seal Clearing Increases swell level in core	End of core uncover: effect on PCT
	𝑃 1 falls below 𝑃 2	Necessary to reach accumulator injection and to minimize 𝑄 𝑏𝑟𝑒𝑎𝑘	Significant effect on 𝑀 1 , on timing of core uncovery and on PCT
	Accumulator discharge		Increases 𝑀 1	Significant effect on 𝑀 1 , on timing of core uncovery and on PCT
	End of accumulator discharge	𝑀 1 will decrease again	Significant effect on 𝑀 1 , on duration of core uncovery and on PCT
	LPSI actuation	Increases 𝑀

1 Significant effect on 𝑀 1 , on timing and duration of core uncovery and on PCT

  𝜇 𝑙 ⋅ 𝑀 𝑙 + 𝜇 𝑣 ⋅ 𝑀 𝑣 ) ⋅ 𝑃 ̇= 𝑀 ̇𝑙,𝑖𝑛 ⋅ [𝜈 𝑙 + 𝜈 𝑙,ℎ ′ ⋅ (ℎ 𝑙,𝑖𝑛 -𝐻 𝑙 )] + 𝑀 ̇𝑣,𝑖𝑛 ⋅ [𝑣 𝑣 + 𝑣 𝑣,ℎ ′ ⋅ (ℎ 𝑣,𝑖𝑛 -𝐻 𝑣 )]

	-𝑀 ̇𝑙,𝑜𝑢𝑡 ⋅ 𝜈 𝑙 -𝑀 ̇𝑣,𝑜𝑢𝑡 ⋅ 𝜈 𝑣 + 𝜈 𝑙,ℎ ′ ⋅ 𝑊 𝑤,𝑙 + 𝜈 𝑣,ℎ ′ ⋅ 𝑊 𝑤,𝑣 + 𝑊 𝑖,𝑣 + 𝑊 𝑖,𝑙 -𝜛 With: 𝜈′ 𝑘,𝑝 ≜ 𝜕𝜈 𝑘 𝜕𝑝 | ℎ 𝑘 ; 𝜈 ′ 𝑘,ℎ ≜ 𝜕𝜈 𝑘 𝜕ℎ 𝑘 | 𝑝 ; 𝜇 𝑘 = -𝜈 𝑘,𝑝 ′ -𝜈 𝑘 ⋅ 𝜈 𝑘,ℎ 𝑊 𝑤,𝑖 𝜛 ′	+ 𝜈 𝑙,ℎ ′ ⋅ 𝑊 𝑖,𝑙 + 𝜈 𝑣,ℎ ′ ⋅ 𝑊 𝑖,𝑣 ℎ 𝑣 -ℎ 𝑙 ; 𝜛 ≜ 𝜈 𝑣 -𝜈 𝑙	(12)
	𝑀 𝑖 =	𝑊 𝑖,𝑣 +𝑊 𝑖,𝑙
	𝑀 𝑖,𝑤 =	𝑊 𝑤,𝑖

𝜛

being the mass flow rate at interface by vaporization or condensation (kg/s).

  𝜆 𝑙 ⋅ 𝑀 𝑙 + 𝜆 𝑣 ⋅ 𝑀 𝑣 ) ⋅ 𝑃 ̇= 𝑀 ̇𝑙,𝑖𝑛 ⋅ (𝑣 𝑙𝑠 +This assumes that all wall HT are creating vaporization or condensation. In a PWR SBLOCA, one can use Eq.[START_REF] Nahavandi | Scaling Laws for Modeling Nuclear Reactor Systems[END_REF] or Eq. (13) and write:(𝜇 𝑙 ⋅ 𝑀 𝑙 + 𝜇 𝑣 ⋅ 𝑀 𝑣 ) ⋅ 𝑃 ̇= 𝑄 𝑆𝐼 -𝑄 𝑏𝑟𝑒𝑎𝑘 + 𝑄 𝑐𝑜𝑟𝑒 -𝑄 𝑆𝐺 + 𝑄 𝑜𝑤 + 𝑄 𝑖 (14) (𝜆 𝑙 ⋅ 𝑀 𝑙 + 𝜆 𝑣 ⋅ 𝑀 𝑣 ) ⋅ 𝑑𝑃 𝑑𝑡 = 𝑄 𝑆𝐼 -𝑄 𝑏𝑟𝑒𝑎𝑘 + 𝑄 𝑐𝑜𝑟𝑒 -𝑄 𝑆𝐺 + 𝑄 𝑜𝑤 (15) With: 𝑄 𝑆𝐼 = 𝑀 ̇𝑙,𝑆𝐼 ⋅ [𝜈 𝑙 + 𝜈 𝑙,ℎ ′ ⋅ (ℎ 𝑙,𝑖𝑛 -𝐻 𝑙 )] ; 𝑊 𝑖 = 𝜛 ⋅ 𝑄 𝑖

	4.3. The mixture momentum equation and ΔP equations
	A mixture momentum equation or a ∆𝑃 equation may express the relation between the mass flow rate and
	the pressure differences due to pumps, gravity, and wall friction:
	𝜕𝐺 𝑑𝑡	+	1 𝐴	⋅	𝜕𝐴 ⋅ [𝐺 𝑣 ⋅ 𝑢 𝑣 + 𝐺 𝑙 ⋅ 𝑢 𝑣 ] 𝑑𝑧	+	𝜕𝑃 𝜕𝑧	= 𝜌 𝑚 ⋅ 𝑔 𝑧 -	𝐶 𝐴	⋅ 𝜏 𝑤 -	𝐾 𝑠 𝛿𝑧	⋅	𝐺 2 2 ⋅ 𝐴 ⋅ 𝜌 𝑚	+	∆𝑃 𝑝𝑢𝑚𝑝 𝛿𝑧	(16)
										𝜌 𝑚 = 𝛼 ⋅ 𝜌 𝑣 + (1 -𝛼) ⋅ 𝜌 𝑙
	The two last terms are related to singular pressure losses and to pump head. Integrating this equation along
	a closed cooling loop from 𝑧 0 to 𝑧 0 writes:
		𝜕𝑀 ḋ𝑡	∑	𝑖	𝐿 𝑖 𝐴 𝑖	= ∫ 𝜌 𝑚 ⋅ 𝑔 𝑧 𝑑𝑧 𝑧 0 𝑧 0	-∑	𝑖	𝐶𝑓 𝑖 ⋅ 𝐺 𝑖 2 ⋅ 𝐿 𝑖 2 ⋅ 𝐷 𝑖 ⋅ 𝜌 𝑚,𝑖	-∑	𝑗	2 𝐾 𝑠,𝑗 ⋅ 𝐺 𝑗 2 ⋅ 𝐴 ⋅ 𝜌 𝑚,𝑗	+ ∆𝑃 𝑝𝑢𝑚𝑝	(17)
							-𝑃 𝐷𝐶 = ∫ 𝜌 𝑚 ⋅ 𝑔 𝑧 𝑑𝑧 𝐷𝐶 𝑐𝑜𝑟𝑒	-∑	𝑖	𝐶𝑓 𝑖 ⋅ 𝐺 𝑖 2 ⋅ 𝐿 𝑖 2 ⋅ 𝐷 𝑖 ⋅ 𝜌 𝑚,𝑖	-∑	𝑗	2 𝐾 𝑠,𝑗 ⋅ 𝐺 𝑗 2 ⋅ 𝐴 ⋅ 𝜌 𝑚,𝑗
	With: 𝜆 𝑘 =	ℎ 𝑘𝑠,𝑝 ′ 𝜛 𝑠	-	𝜈 𝑘𝑠 𝜛 𝑠	-𝜈 𝑘𝑠,𝑝 ′	ℎ 𝑙,𝑖𝑛 -ℎ 𝑙𝑠 𝜛 𝑠 -𝑀 ̇𝑙,𝑜𝑢𝑡 ⋅ 𝜈 𝑙𝑠 -𝑀 ̇𝑣,𝑜𝑢𝑡 ⋅ 𝜈 𝑣𝑠 + ) + 𝑀 ̇𝑣,𝑖𝑛 ⋅ (𝑣 𝑣𝑠 + 𝑊 𝑤 𝜛 𝑠 ; 𝜛 𝑠 ≜ ℎ 𝑣𝑠 -ℎ 𝑙𝑠 𝜈 𝑣𝑠 -𝜈 𝑙𝑠 ; ℎ 𝑘𝑠,𝑝 ′ ≜ 𝑑ℎ 𝑘𝑠 𝑑𝑝 ; 𝜈 𝑘𝑠,𝑝 ′ 𝑑𝜈 𝑘𝑠 ≜ 𝑑𝑝	ℎ 𝑣,𝑖𝑛 -ℎ 𝑣𝑠 𝜛 𝑠	)	(13)

The circuit has been split into successive sections of length 𝐿 𝑖 , diameter 𝐷 𝑖 and cross section 𝐴 𝑖 . It is often useful to integrate it from top of core to top of the downcomer (DC) in a quasi-steady approximation for a natural circulation situation:

ACKNOWLEDGMENTS

The authors thank all the contributors to the state of the art report on scaling and particularly F. d'Auria who promoted some ideas used in this work. Thanks go also to the support of the CATHARE development team in CEA.


Never the twain shall meet: the correspondence of Pietro and Alessandro Verri (1766-1797)

Posted on 26 May 2016 by voltairefoundation

When, in October 1766, Alessandro Verri left Milan to go to Paris with Cesare Beccaria at the invitation of André Morellet (the translator of Beccaria's On Crimes and Punishments) and of the encyclopédistes, he concluded an epistolary pact with his brother Pietro. Pietro, who was thirteen years Alessandro's senior, remained in Lombardy and ensured the preservation of the correspondence by having the letters carefully and systematically copied into large in-folio registers.

After his trip to Paris and London, Alessandro settled in Rome in 1767 and he continued his correspondence with his brother until Pietro's death in 1797. The time span it covers and its candid take on private and public matters make it the richest correspondence in eighteenthcentury Italy. It extends over three decades, from the age of Reforms in Lombardy under the rule of Maria Theresa to the eve of the proclamation of the Roman republic in 1798, and encompasses the French Revolution, the invasions of Piedmont and Lombardy by General Bonaparte, and the creation of the Cisalpine republic.

At first, the brothers' dialogue preserved the harmony that characterised the discussions held at the Accademia dei Pugni (the 'Academy of Punches'), the learned society that they had founded in Milan in 1761, and which had been the incubator for two seminal works of the Lombard Enlightenment, namely Beccaria's On Crimes and Punishments and the Caffè periodical (1764-1766). Yet, the brothers' apparent convergence of views in the first months of their correspondence could hardly conceal the fact that they were growing apart intellectually.

Pietro's letters reveal his strong belief in intellectual passions as drivers of historical change, and in the necessity to work towards the happiness of the masses. Throughout his life he remained an ardent reformist and never gave up on his ambition to 'electrify minds' through his writings1 . Alessandro's letters, by contrast, show his individualistic conception of happiness and his wariness of any developments that might upset society's order and stability (free thought, popular passions), as well as his conviction that attempting to create and shape public opinion was a dangerous endeavour. His refusal to envision a society made up of free individualsand above all a secular society -alienated him from the spirit of modernity that was emerging at the end of the century.

In 1782, a dispute cropped up between the brothers over the interpretation of Pope Pius VI's trip to Vienna. The Pontiff had decided to take this unprecedented step to try and dissuade Emperor Joseph II from carrying out a series of religious reforms which would lead to the closure of a number of convents and monasteries, and to the abolition of regular religious orders. Pietro praised the Emperor, whom he saw as an enlightened monarch: 'To make his people more virtuous, humane, industrious and felicitous, he's declaring war on superstition, he inspires and commands tolerance, and leads the way towards a time when vice will be stripped of the treacherous veil which gives it the appearance of virtue.' 2 As for Alessandro, he regarded the Pope's failure as a humiliation inflicted on the Holy See, but expressed pride at the 'veneration' and 'infinite demonstrations of respect' displayed by Italian and Austrian crowds towards the Pope during his journey. Those who 'claim[ed] that their pens commanded public opinion have been proven wrong', he thought. 3 Pietro strove to demonstrate that the political victory of the Emperor was a harbinger of religious regeneration, of the restoration of religion's social import and public efficacy: 'Religion is saved, evil is rooted out.' Above all, he saw this as the outcome of the 'enduring battle' waged by a few superior, isolated but victorious minds, in favour of liberties and science. While he clung to Bayle's belief that 'no religious opinion is essential to civilian government', 4 Alessandro, for his part, regarded religion as part of 'the arcane foundations of the State […] essential to restrain human passions and refrain the multitude, as we have seen men become fierce and mad when the yoke is broken.' 5

After the French Revolution the epistolary dialogue between the two brothers evolved into overt antagonism, as clearly summed up by Pietro in March 1796: 'You hate modern philosophers, who have demonstrated that vigorous passions trigger great actions, while I agree with them.' 6 For Pietro, the 1790s were an exaltated time in an era of political frustration. While he had been disappointed by the reforms carried out by Joseph II in Lombardy, he was filled with enthusiasm for the ideals of liberty and equality proclaimed by the French National Assembly in 1789. However, he understood that the fear awakened by the French Revolution had dealt a lethal blow to the cause of the advancement of reason in Italy: 'Now that we rightly consider that the dissemination of culture among the people sparked off the events in France, we regard the excess of thought, books, thinkers, advocates of all sorts of doctrines as highly suspicious and we cordially hate them all,' he remarked in 1794, adding: 'we live at a time when fierce ignorance is all the rage […]. Reason is losing ground everywhere in Europe.' 7 Alessandro agreed that the revolutionary crisis had been sparked off by the works of enlightened thinkers and he saw this as a vindication of the misgivings he had experienced during his visit to Parisian coteries in the winter of 1766. He was concerned about the way some of the propositions from d'Holbach's Système de la nature were freely debated in Italy, and he accused the 'assassin-philosophers' of undermining the moral and religious principles underpinning civil society: 'As long as this form of thinking is limited to a few silent men, human society won't be affected: but as soon as alleged philosophers take their ideas to the streets and ramble freely, human passions will be unleashed, à la française.' 'The tyranny of the multitude, the emancipated mob is a monster that I dread much more than Tiberius,' he added. 8

By contrast, the strength in Pietro's thinking -which was quite original in the intellectual landscape of late eighteenth-century Italy -was to distance himself from the anti-Jacobin zeal of the Lombard élite and to understand that the transition from 'error' to 'enlightenment,' from tyranny to liberty, could be effected by violent means. For Pietro Verri, the violence in France in 1792 and 1793 could be justified and productive as long as the people managed to put together a constitution. His views echoed those of Beccaria's On Crimes and Punishments, a work to which Pietro had contributed years earlier: 'Whoever reflects on the various histories of nations, which after a certain lapse of time come to resemble each other in their main outlines, will repeatedly find a whole generation sacrificed to the happiness of succeeding generations in the hard-fought but necessary transition from the shadows of ignorance to the light of philosophy and, as a corollary, in the passage from tyranny to freedom.' 9

Pierre Musitelli École normale supérieure -PSL 8 25 August 1794; 15 December 1792. 9 C. Beccaria, On Crimes and Punishments (Cambridge, 1995), p. 106-107.

Carteggio di Pietro e Alessandro Verri, 12 vol. (Milan, 1910-1942), vol.10 (1939), p.

(my translation).


Robert G Hilton
email: r.g.hilton@durham.ac.uk.

Valier Galy

Jérôme Gaillardet

Mathieu Dellinger

Charlotte Bryant

Matt O'regan

Darren R Gröcke

Helen Coxall

Julien Bouchez

Damien Calmels

Erosion of organic carbon in the Arctic as a geological carbon dioxide sink

Soils of the northern high latitudes store carbon over millennial timescales (10 3 yrs) and contain approximately double the carbon stock of the atmosphere [START_REF] Gorham | Northern peatlands: Role in the carbon cycle and probable responses to climatic warming[END_REF][START_REF] Tarnocai | Soil organic carbon pools in the northern circumpolar permafrost region[END_REF][START_REF] Schuur | Vulnerability of permafrost carbon to climate change: Implications for the global carbon cycle[END_REF] . Warming and associated permafrost thaw can expose soil organic carbon and result in mineralisation and carbon dioxide (CO 2 ) release [START_REF] Guo | Mobilization pathways of organic carbon from permafrost to arctic rivers in a changing climate[END_REF][START_REF] Macdougall | Significant contribution to climate warming from the permafrost carbon feedback[END_REF][START_REF] Schädel | Circumpolar assessment of permafrost C quality and its vulnerability over time using long-term incubation data[END_REF] . However, some of this soil organic carbon may be eroded and transferred to rivers [START_REF] Goñi | The supply and preservation of ancient and modern components of organic carbon in the Canadian Beaufort Shelf of the Arctic Ocean[END_REF][START_REF] Vonk | Activation of old carbon by erosion of coastal and subsea permafrost in Arctic Siberia[END_REF][START_REF] Vonk | Permafrost-carbon complexities[END_REF] . If it escapes degradation during river transport and is buried in marine sediments, then it can contribute to a longer-term (>10 4 yrs), geological CO 2 sink [START_REF] Vonk | Activation of old carbon by erosion of coastal and subsea permafrost in Arctic Siberia[END_REF][START_REF] Vonk | Permafrost-carbon complexities[END_REF][START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF] . Despite this recognition, the erosional flux and fate of particulate organic carbon (POC) in large rivers at high latitudes remains poorly constrained. Here, we quantify POC source in the Mackenzie River, the main sediment supplier to the Arctic Ocean [START_REF] Stein | The Organic Carbon Cycle in the Arctic Ocean Springer[END_REF][START_REF] Macdonald | A sediment and organic carbon budget for the Canadian Beaufort Shelf[END_REF] and assess its flux and fate. We combine measurements of radiocarbon, stable carbon isotopes and element ratios to correct for rock-derived POC [START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF][START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] . Our samples reveal that the eroded biospheric POC has resided in the basin for millennia, with a mean radiocarbon age of 5800±800 yr, much older than large tropical rivers [START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF][START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] . Based on the measured biospheric POC content and variability in annual sediment yield [START_REF] Carson | Magnitude and sources of sediment input to the Mackenzie Delta, Northwest Territories[END_REF] , we calculate a biospheric POC flux of 𝟐𝟐. 𝟐𝟐 -𝟎𝟎.𝟗𝟗 +𝟏𝟏.𝟑𝟑 TgC yr -1 from the Mackenzie River, three times the CO 2 drawdown by silicate weathering [START_REF] Gaillardet | Global silicate weathering and CO 2 consumption rates deduced from the chemistry of large rivers[END_REF] . Offshore we find evidence for efficient terrestrial organic carbon burial over the Holocene, suggesting that erosion of organic carbon-rich, high latitude soils may result in a significant geological CO 2 sink.

Photosynthesis and the production of organic carbon by the terrestrial biosphere (OC biosphere ) is a major pathway of atmospheric carbon dioxide (CO 2 ) drawdown. Over millennial timescales (10 3 yrs), some OC biosphere escapes oxidation and contributes to a transient CO 2 sink in soil [START_REF] Tarnocai | Soil organic carbon pools in the northern circumpolar permafrost region[END_REF][START_REF] Schuur | Vulnerability of permafrost carbon to climate change: Implications for the global carbon cycle[END_REF][START_REF] Sundquist | The geologic history of the carbon cycle[END_REF] . Longer-term CO 2 drawdown (>10 4 yrs) can be achieved if OC biosphere is eroded, transferred by rivers and buried in sedimentary basins [START_REF] Vonk | Permafrost-carbon complexities[END_REF][START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Blair | The fate of terrestrial organic carbon in the marine environment[END_REF][START_REF] Hayes | The abundance of 13 C in marine organic matter and isotopic fractionation in the global biogeochemical cycle of carbon during the past 800 Ma[END_REF] . Burial of OC biosphere represents a major geological CO 2 sink (and O 2 source) alongside the chemical weathering of silicate minerals by carbonic acid, coupled to carbonate precipitation [START_REF] Gaillardet | Global silicate weathering and CO 2 consumption rates deduced from the chemistry of large rivers[END_REF][START_REF] Hayes | The abundance of 13 C in marine organic matter and isotopic fractionation in the global biogeochemical cycle of carbon during the past 800 Ma[END_REF] .

These fluxes negate CO 2 emissions from the solid Earth [START_REF] Berner | Atmospheric CO 2 levels over Phanerozoic time[END_REF] and from oxidation of rock-derived OC [START_REF] Hilton | Geological respiration of a mountain belt revealed by the trace element rhenium[END_REF] , contributing to the long-term regulation of global climate [START_REF] Hayes | The abundance of 13 C in marine organic matter and isotopic fractionation in the global biogeochemical cycle of carbon during the past 800 Ma[END_REF][START_REF] Berner | Atmospheric CO 2 levels over Phanerozoic time[END_REF] . Physical erosion is thought to play a significant role in this OC biosphere transfer because it controls the rate of biospheric particulate organic carbon (POC biosphere ) export by rivers [START_REF] Hilton | Climatic and geomorphic controls on the erosion of terrestrial biomass from subtropical mountain forest[END_REF] and influences sediment accumulation and the efficiency of OC burial [START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Blair | The fate of terrestrial organic carbon in the marine environment[END_REF][START_REF] Burdige | Burial of terrestrial organic matter in marine sediments: A re-assessment[END_REF] .

In the northern high latitudes, large amounts of OC biosphere are stored in soil [START_REF] Gorham | Northern peatlands: Role in the carbon cycle and probable responses to climatic warming[END_REF][START_REF] Tarnocai | Soil organic carbon pools in the northern circumpolar permafrost region[END_REF] . The upper three meters of soil in the region of northern circumpolar permafrost are estimated to contain 1035±150 PgC, approximately double the carbon dioxide (CO 2 ) content of the preindustrial atmosphere [START_REF] Sundquist | The geologic history of the carbon cycle[END_REF] . Many of these soils accumulated during the retreat of large continental ice sheets following the Last Glacial Maximum, with a peak expansion between 12,000 and 8,000 cal. yr BP [START_REF] Macdonald | Rapid development of the circumarctic peatland complex and atmospheric CH 4 and CO 2 variations[END_REF] and the OC biosphere can be thousands of years old [START_REF] Vonk | Activation of old carbon by erosion of coastal and subsea permafrost in Arctic Siberia[END_REF] . This vast carbon reservoir is located in a region sensitive to environmental change over glacialinterglacial timescales [START_REF] Macdonald | Rapid development of the circumarctic peatland complex and atmospheric CH 4 and CO 2 variations[END_REF] and to warming over the coming century [START_REF] Schuur | Vulnerability of permafrost carbon to climate change: Implications for the global carbon cycle[END_REF] . Much focus has been placed on its potential to become a CO 2 source [START_REF] Schuur | Vulnerability of permafrost carbon to climate change: Implications for the global carbon cycle[END_REF][START_REF] Guo | Mobilization pathways of organic carbon from permafrost to arctic rivers in a changing climate[END_REF][START_REF] Macdougall | Significant contribution to climate warming from the permafrost carbon feedback[END_REF][START_REF] Schädel | Circumpolar assessment of permafrost C quality and its vulnerability over time using long-term incubation data[END_REF][START_REF] Vonk | Activation of old carbon by erosion of coastal and subsea permafrost in Arctic Siberia[END_REF] . However, geological CO 2 drawdown by POC biosphere erosion at high latitudes has remained poorly constrained [START_REF] Vonk | Permafrost-carbon complexities[END_REF] .

Here we sample POC carried by the major rivers in the Mackenzie Basin and investigate its fate using an offshore sediment core extending over the Holocene (Extended Data Fig. 1). The Mackenzie River is the largest source of sediment to the Arctic Ocean [START_REF] Stein | The Organic Carbon Cycle in the Arctic Ocean Springer[END_REF][START_REF] Macdonald | A sediment and organic carbon budget for the Canadian Beaufort Shelf[END_REF][START_REF] Carson | Magnitude and sources of sediment input to the Mackenzie Delta, Northwest Territories[END_REF] and erosion of mountainous topography in the basin results in a high sediment discharge, similar to the combined total of 16 Eurasian rivers draining to the Arctic [START_REF] Stein | The Organic Carbon Cycle in the Arctic Ocean Springer[END_REF][START_REF] Carson | Magnitude and sources of sediment input to the Mackenzie Delta, Northwest Territories[END_REF] . We collected river depth profiles to characterise POC across the range of grain sizes carried by large rivers [START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF][START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF][START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] at the main conduit for sediment export to the Arctic Ocean in the Mackenzie Delta, at key points on the Mackenzie River and from its major tributaries (Extended Data Fig. 1). To investigate temporal variability of POC composition, river depth profiles were collected shortly after ice-break up at high/rising stage (June 2011) and during falling stage (September 2010), while river surface and bank samples were collected in June 2009. To correct for rock-derived, 'petrogenic' POC (POC petro ), likely to be important in the Mackenzie Basin [START_REF] Goñi | The supply and preservation of ancient and modern components of organic carbon in the Canadian Beaufort Shelf of the Arctic Ocean[END_REF][START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] , we combine measurements of radiocarbon ( 14 C, reported as the 'fraction modern' F mod ), total OC content ([OC total ], %), stable isotopes of OC (δ 13 C org ), nitrogen to OC ratio (N/OC total ) and aluminium to OC ratio (Al/OC total ), which allow us to assess the age and concentration of POC biosphere (Methods) [START_REF] Goñi | The supply and preservation of ancient and modern components of organic carbon in the Canadian Beaufort Shelf of the Arctic Ocean[END_REF][START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF][START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF][START_REF] Hilton | Climatic and geomorphic controls on the erosion of terrestrial biomass from subtropical mountain forest[END_REF] . Published surface samples from the Mackenzie River 4,7 (n=5) have [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C-ages between 6,010 yr and 10,000 yr but the 14 C-depletion caused by POC petro versus aged POC biosphere has not been assessed. We also examine the hydrodynamic behaviour of POC, using the aluminium to silicon ratio (Al/Si) ratio as a proxy of sediment grain size [START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] .

We find that river POC is [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C-depletion, we examine the N/OC total ratio. Degradation of organic matter in soils can increase the relative N abundance [START_REF] Schädel | Circumpolar assessment of permafrost C quality and its vulnerability over time using long-term incubation data[END_REF][START_REF] Kuhry | Fossil carbon/nitrogen ratios as a measure of peat decomposition[END_REF] , differentiating degraded POC biosphere (high N/OC total ) from young, fresh POC biosphere (low N/OC total ). Suspended load samples display a negative relationship between N/OC total and F mod (Fig. 1), similar to measurements from a peat core in the Mackenzie Basin 26 away from permafrost,. There, N/OC total ratios increased with [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C age (1,250 yr to 10,200 yr) and soil depth (0m to 3m). In contrast, river bed materials have lower F mod values and a relatively restricted range of N/OC total values and are distinct from suspended load (Fig. 1). A dominance of POC petro in bed materials [START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] with a N/OC total ratio of ~0.07 can explain their composition.

Together, the F mod and N/OC total values suggest that POC in the Mackenzie River is a mixture of POC petro and POC biosphere , itself varying in [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C age from 'modern' to ~8000 yrs (Fig. 1). The δ 13 C org values and Al/OC total ratios support this inference (Extended Data Fig. 2).

Using an end member mixing analysis [START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF] we quantify POC petro content of sediments (Methods) and find that suspended load at the Mackenzie River Delta is dominated by POC biosphere (~70-90% of the total POC). Having corrected for POC petro , we investigate the source of POC biosphere by estimating its average 14 C age. This varies from 3030±150 yr to 7900±400 yr (Extended Data Fig. 3) with an average 14 C age of POC biosphere = 5800±800 yr (±2SE) in suspended sediments of the Mackenzie River Delta. These values are older than estimates of POC biosphere age from the Amazon River (1120-2750 yr) [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] and Ganges River (1600-2960 yr) [START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF] . The ages reflect mixing of young, fresh POC biosphere present in each of these large river basins, with an older POC biosphere in the Mackenzie Basin (Fig. 1), likely to be peats which expanded between 9,000 yr and 8,000 yr ( 14 C-age) [START_REF] Macdonald | Rapid development of the circumarctic peatland complex and atmospheric CH 4 and CO 2 variations[END_REF] . POC biosphere can be eroded by slumping and landsliding on river banks, across deep soil profiles [START_REF] Guo | Mobilization pathways of organic carbon from permafrost to arctic rivers in a changing climate[END_REF][START_REF] Goñi | The supply and preservation of ancient and modern components of organic carbon in the Canadian Beaufort Shelf of the Arctic Ocean[END_REF] . Sections of the landscape which have discontinuous permafrost and those undergoing permafrost degradation [START_REF] Feng | Differential mobilization of terrestrial carbon pools in Eurasian Arctic river basins[END_REF] may be important sources of aged POC biosphere , in addition to river banks which are undercut during peak water discharge following ice-break up [START_REF] Carson | Magnitude and sources of sediment input to the Mackenzie Delta, Northwest Territories[END_REF] . Our samples suggest that erosion and fluvial transfer of millennial-aged POC biosphere is extensive in the Mackenzie Basin.

Once in the river, POC biosphere is sorted with river depth, revealed by the Al/Si ratio (Fig. 2b) a proxy for grain size [START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] . In bed materials with low Al/Si, POC petro dominates (Fig. 1)

and leads to low F mod values (Fig. 2c). Just above the river bed, during the two sampling campaigns coarse suspended sediments (low Al/Si) hosted the youngest, least degraded POC biosphere (low N/C) leading to a significant contrast in [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C age from the bed materials.

Towards the river surface, older, more degraded POC biosphere appears to dominate, and is transported with fine sediment and clays (high Al/Si) [START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] . The significant contribution of degraded, very old POC biosphere (>5000 yrs) contrasts with large tropical rivers where organic matter turnover in terrestrial ecosystems is more rapid (Fig. 2c) [START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF][START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] .

To assess how erosion in the Mackenzie River may lead to long-term CO 2 drawdown, we estimate POC biosphere discharge. River depth profiles collected at high and falling stage suggest that the [OC total ] of the suspended sediment load did not vary systematically with sediment grain size (Extended Data Fig. 4). Future work should seek to assess temporal variability in POC content and composition. Our samples suggest that changes in grain size with water discharge (Fig. 2b) could be important in setting the variability of POC biosphere age carried by the river (Fig. 2c). The [OC total ] values at the Delta were 1.6±0.5% (n=8, ±1σ), which were similar to the mean measured in the Mackenzie Delta in June-July 1987 of 1.4±0.2 (n=10) [START_REF] Macdonald | A sediment and organic carbon budget for the Canadian Beaufort Shelf[END_REF] . While our sample set is modest in size, it helps us to better constrain the range of POC contents in suspended load of the Mackenzie River. In addition, our end member mixing analysis allows us to provide the first estimates of [OC biosphere ], which varies between 0.7±0.1% and 2.4±0.2%. To estimate POC biosphere discharge, we use the most complete dataset of annual sediment discharge to the Mackenzie Delta (1974-1994) [START_REF] Carson | Magnitude and sources of sediment input to the Mackenzie Delta, Northwest Territories[END_REF] , which ranged from 81 Tg yr -1 to 224 Tg yr -1 . A Monte Carlo approach is used to account for the modest sample size by using the full measured variability in both [OC biosphere ] and sediment discharge (Methods). We estimate POC biosphere discharge = 2.2 -0.9 +1.3 TgC yr -1 which is sustainable over 10 3 -10 4 years, depleting the soil carbon stock by ~0.006% yr -1 (Methods).

We estimate the POC petro discharge = 0.4 -0.1 +0.1 . These estimates do not account for ice covered conditions when <10% of the annual sediment discharge is conveyed [START_REF] Macdonald | A sediment and organic carbon budget for the Canadian Beaufort Shelf[END_REF] . Nevertheless, our estimate of POC biosphere discharge is greater than the combined POC discharge of ~1.9 TgC yr - 1 by the major Eurasian Arctic Rivers (Ob, Yenisei, Lena, Indigirka and Koyma) [START_REF] Stein | The Organic Carbon Cycle in the Arctic Ocean Springer[END_REF][START_REF] Feng | Differential mobilization of terrestrial carbon pools in Eurasian Arctic river basins[END_REF] which cover ~8.6x10 6 km 2 . Based on the available measurements, the Mackenzie River dominates the input of POC biosphere to the Arctic Ocean.

The mobilisation of millennial-aged POC biosphere from soils at high latitudes has been viewed as a short-term source to the atmosphere if decomposition releases greenhouse gases (CH 4 and CO 2 ) 2-6,8 . However, if POC biosphere escapes oxidation during river transport and is buried offshore, erosion acts as a long-term CO 2 sink [START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Blair | The fate of terrestrial organic carbon in the marine environment[END_REF][START_REF] Berner | Atmospheric CO 2 levels over Phanerozoic time[END_REF] . Offshore, aged POC biosphere from the Mackenzie River (Fig. 1) can explain the [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C depletion and δ 13 C of bulk organic matter, and old [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C ages of terrestrial plant wax compounds (up to 20,000 yr) in surface sediments of the Beaufort Sea [START_REF] Goñi | The supply and preservation of ancient and modern components of organic carbon in the Canadian Beaufort Shelf of the Arctic Ocean[END_REF][START_REF] Goñi | Distribution and sources of organic matter in surface marine sediments across the North American Arctic margin[END_REF][START_REF] Drenzek | Constraints on the origin of sedimentary organic carbon in the Beaufort Sea from coupled molecular 13 C and 14 C measurements[END_REF] . We provide new evidence that terrestrial POC is buried efficiently offshore and accumulates in sediments over 10 4 years. Benthic foraminfera [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] ) and >63µm (1.1% to 1.4%) size fractions (Fig. 3). Their N/OC total and δ [START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF] C org values suggest that they are dominated by terrestrial POC with minor marine OC addition (Extended Data Fig. 5). We use the change in OC total /Al ratios offshore to estimate OC burial efficiencies have been 65±27% or more over the Holocene at this site (Methods).

Rapid sediment accumulation and low temperature are likely to promote high POC burial efficiency [START_REF] Blair | The fate of terrestrial organic carbon in the marine environment[END_REF][START_REF] Burdige | Burial of terrestrial organic matter in marine sediments: A re-assessment[END_REF][START_REF] Goñi | Distribution and sources of organic matter in surface marine sediments across the North American Arctic margin[END_REF] . Also, the fluvial transport dynamics of POC biosphere may promote burial (Fig. 2c). The oldest, most-degraded POC biosphere is transported with clays [START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] , whose association with organic matter may enhance burial efficiency [START_REF] Blair | The fate of terrestrial organic carbon in the marine environment[END_REF] , while the youngest, least-degraded POC biosphere is carried near the river bed at the highest sediment concentrations. Our findings suggest that erosion and riverine transfer at high latitudes can lead to the long-term preservation of terrestrial POC in marine sediments (Fig. 3).

Erosion of high latitude soils and riverine export of POC biosphere may represent an important geological CO 2 sink. Our estimate of the modern day POC biosphere discharge of 2.2 -0.9 +1.3 TgC yr -1 in the Mackenzie River may be refined by additional temporal sampling.

However, it is three times the modern rates of CO 2 drawdown by weathering of silicate minerals by carbonic acid in the Mackenzie River [START_REF] Gaillardet | Global silicate weathering and CO 2 consumption rates deduced from the chemistry of large rivers[END_REF] , at ~0.7 TgC yr -1 . Preservation of POC offshore (Fig. 3) suggests that erosion of high latitude soils, riverine POC biosphere transport and export to the ocean acts as the largest geological CO 2 sink operating in the Mackenzie Basin.

It is important to note that these longer-term fluxes are lower than estimates of greenhouse gas emissions from high latitude soils in permafrost zones due to projected warming over the coming century [START_REF] Schuur | Vulnerability of permafrost carbon to climate change: Implications for the global carbon cycle[END_REF][START_REF] Macdougall | Significant contribution to climate warming from the permafrost carbon feedback[END_REF][START_REF] Schädel | Circumpolar assessment of permafrost C quality and its vulnerability over time using long-term incubation data[END_REF][START_REF] Schuur | Climate change and the permafrost carbon feedback[END_REF] . While theses fluxes remain uncertain, recent work [START_REF] Schuur | Climate change and the permafrost carbon feedback[END_REF] has proposed emissions of ~ 1-2 PgC yr -1 which equate to a yield of ~70 tC km -2 yr -1 over 17.8x10 8 km 2 of soils in permafrost zones. This estimate of accelerated release of CO 2 due to anthropogenic warming [START_REF] Schuur | Climate change and the permafrost carbon feedback[END_REF] is more rapid than the natural geological drawdown fluxes, of which we estimate POC biosphere ~ 2-5 tC km -2 yr -1 for the Mackenzie Basin (Methods). Over longer-time periods, we postulate that this geological CO 2 sink may be sensitive to climate conditions in the Arctic. The carbon transfer can operate when high latitudes host significant POC biosphere stocks in soil, and while rivers can erode and transfer sediments to the Arctic Ocean. Over the last 1Ma, the POC biosphere transfer was likely to have been enhanced during interglacials [START_REF] Macdonald | Rapid development of the circumarctic peatland complex and atmospheric CH 4 and CO 2 variations[END_REF] (Fig. 3), whereas during glacial conditions, lower soil POC biosphere stocks and extensive icesheet coverage suggest that POC biosphere erosion may have been supressed. We propose that erosion of terrestrial POC biosphere by large rivers draining the Arctic could play an important role in long-term CO 2 drawdown [START_REF] Hayes | The abundance of 13 C in marine organic matter and isotopic fractionation in the global biogeochemical cycle of carbon during the past 800 Ma[END_REF][START_REF] Berner | Atmospheric CO 2 levels over Phanerozoic time[END_REF] , coupling the carbon cycle to climatic conditions at high latitudes.

collected sediment and carbonate data from the offshore borehole. R.G.H., V.G., M.D., C.B. and D.G. processed the samples and carried out the geochemical analyses. R.G.H. wrote the manuscript with input from all co-authors.

Author Information: Data are found in the Extended Data Tables. Reprints and permissions information is available at www.nature.com/reprints. The authors declare they have no competing financial interests. Correspondence and requests for materials should be addressed to r.g.hilton@durham.ac.uk.

Methods

River sample collection and preparation: River depth-profiles from September 2010 and June 2011 (Extended Data Table 1) were used to collect the full range of erosion products and POC in large river systems, taking advantage of the hydrodynamic sorting of particles [START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF][START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF][START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] . At each sampling site (Fig. 1) channel depth, water velocity and instantaneous water discharge were measured by two or more transects with an Acoustic Doppler Current Profiler (ADCP Rio Grande 600 kHz) before each depth profile was collected at a single point (±10m) in the middle of the channel. On the boat, each sample (~7-8L) was evacuated into a clean bucket and stored in sterilised plastic bags and the procedure repeated depending upon the total water depth. Each bag was weighed to determine the sampled volume, then the entire sample was filtered within 24 hrs through pre-cleaned Teflon filter units through 90 mm diameter 0.2μm PES filters [START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF][START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] . Suspended sediment was immediately rinsed from the filter using filtered river water into clean amber-glass vials and kept cool. River bed materials were collected at the base of the depth transects from the boat, using a metal bucket as a dredge, and decanted to a sterile bag. River bank deposits (June 2009) were collected from fresh deposits close to the channel (Extended Data Table 3) and sieved at 250 µm and 63 µm to investigate the sorting of POC 31 . All sediments were freeze dried upon return to laboratories within 2 weeks, weighed and homogenised in an agate grinder.

Offshore borehole sampling and benthic foraminifera sample preparation: Benthic foraminifera-containing marine sediment samples were obtained from the upper 22 m

Holocene sequence of an 85.1 m MTW01 borehole 32 located at 69 o 20' 53" N, 137 o 59 ' 13" in 45 m water depth in the Mackenzie Trough (Extended Data Fig. 1). Drilled by the Geological Survey of Canada in 1984, the core is currently archived at the GSC-Atlantic core repository. To isolate foraminifera, sediment samples were disaggregated over a <38μm mesh sieve using deionized water. Based on down-hole microfossils counts, 4 samples were selected with sufficient specimens for radiocarbon dating.

Geochemical analyses:

For the river suspended sediments and core samples for organic carbon analyses, inorganic carbon was removed using a HCl fumigation technique to avoid loss of a component of POC which is known to occur during a HCl leach [START_REF] Komada | Carbonate removal from coastal sediments for the determination of organic carbon and its isotopic signatures, 13 C and 14 C: Comparison of fumigation and direct acidification by hydrochloric acid[END_REF] . An adapted method to ensure full removal of detrital dolomite was used [START_REF] Whiteside | Pangean great lake paleoecology on the cusp of the end-Triassic extinction[END_REF] . In summary, samples were placed in an evacuated desiccator containing ~50 ml 12N HCl in an oven at between 60 and Mixed benthic foraminifera samples picked from the MTW01 core were analysed at NOSAMS for [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C analyses. Samples were rinsed and no pre-treatments were used. The samples were directly hydrolyzed with strong acid, H 3 PO 4 , to convert the carbon in the sample to CO 2 . Calibration of the [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C dates was performed using CALIB (version 7.1) [START_REF] Stuiver | Extended 14 C database and revised CALIB radiocarbon calibration program[END_REF] . All Foraminifera dates were calibrated using the MARINE13 dataset [START_REF] Reimer | IntCal09 and Marine09 Radiocarbon Age Calibration Curves, 0-50,000 Years cal BP[END_REF] , with a reservoir age correction (∆R) of 335±85 yrs (Extended Data Table 2). The ∆R value is based on a recent re-analysis of ages from 24 living molluscs collected before 1956 from the northwestern Canadian Arctic Archipelago [START_REF] Coulthard | New marine ∆R values for Arctic Canada[END_REF] . This calibration set does not include specimens from the Beaufort Sea and as such provides only a best available estimate for ∆R in the Mackenzie

Trough.

End member mixing model: The F mod , N/OC total (Fig. 1), δ 13 C org values and Al/OC total values (Extended Data Fig. 2) are consistent with a mixing of POC petro and POC biosphere dominating the bulk geochemical composition of river POC. Autochthonous sources are not significant based on those measured values, which is consistent with the turbid nature of the Mackenzie River (mean suspended sediment concentration of ~300-400 mg/L) meaning that like other turbid river systems (e.g. the Ganges-Brahmaputra) light penetration is minimal. A mixture of POC petro and POC biosphere can be described by governing equations [START_REF] Galy | Efficient organic carbon burial in the Bengal fan sustained by the Himalayan erosional system[END_REF][START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF]31 : 1)

f biosphere + f petro = 1 (Equation
f biosphere × θ biosphere + f petro × θ petro = θ sample ( Equation 2)
where f biosphere and f petro are the fractions of POC derived from biospheric and petrogenic sources, respectively. θ sample is the measured composition (e.g. F mod ) of a river POC sample, and θ biosphere and θ petro are the compositions of biospheric and petrogenic sources. To quantify the f petro in each sample we use the aluminium (Al) to OC total concentration ratio in river sediments. At each locality, a linear trend between F mod and Al/OC total (Extended Data Fig. 2b) can be explained by a mixture of an Al-rich, OC-poor material (rock fragments containing POC petro ) with Al-poor, OC-rich material (soils and vegetation debris as POC biosphere ). Taking advantage of the fact that the POC petro has F mod ~0, the intercept at F mod = 0 gives an estimate of the Al/OC total values and associated uncertainty of the sedimentary rock end member. To estimate the average concentration of OC petro of bedrocks in each basin ([OC petro ], %), we use the Al concentration of river bed materials as a proxy of for the Al concentration in the bedrocks [START_REF] Dellinger | Lithium isotopes in large rivers reveal the cannibalistic nature of modern continental weathering and erosion[END_REF] [START_REF] Johnston | Uncovering the Neoproterozoic carbon cycle[END_REF] . To quantify the average 14 C age of POC biosphere in each sample, Equations 1 and 2 can be solved for θ biosphere , using the f petro and assumed unmeasurable above background [START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C content of POC petro (F mod = 0). The uncertainty mainly derives from that on f petro and [OC total ] and has been propagated through the calculations.

To test if the mixing of POC biosphere and POC petro can describe the composition of the suspended load samples, we predict the δ [START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF] C org measurements which were not used in the mixing analysis. The calculated f petro values and end member values of -26.2±0.5‰ for POC biosphere and -28.6±0.5‰ for POC petro were used, informed by measurements of bedrocks [START_REF] Johnston | Uncovering the Neoproterozoic carbon cycle[END_REF] and vegetation and soil in the basin [START_REF] Bird | The isotopic composition of soil organic carbon on a north-south transect in western Canada[END_REF] . The mixing model (equation 2) can robustly predict the δ 13 C org differences between the Peel and Liard rivers, and between suspended load and bed material δ 13 C org values (Extended Data Fig. 2c), supporting a mixing control on the variables.

Mackenzie River POC discharge:

To quantify the discharge of POC we need to account for the variability in suspended sediment discharge and the variability in the POC biosphere and POC petro content of sediments in the basin. We use the longest, most complete quantification of sediment flux by the Mackenzie River from 1974-1994 [START_REF] Carson | Magnitude and sources of sediment input to the Mackenzie Delta, Northwest Territories[END_REF] , which has an average 127±40 Tg yr -1 (±1σ). Annual sediment yield varied from 81 Tg yr -1 to 224 Tg yr -1 . While the POC samples were not collected at the same time period, our measurements of [OC total ] at the Delta, mean = 1.6±0.5% (n=8, ±1σ) and do not vary systematically between falling and high stage (Extended Data Fig. 4) and are consistent with available data from samples because we: i) account for higher POC biosphere concentrations which may occur in waterlogged POC biosphere near the river bed (Figs. 2c, Extended Data Fig. 4); and ii) account for the potential for very high annual sediment discharge [START_REF] Carson | Magnitude and sources of sediment input to the Mackenzie Delta, Northwest Territories[END_REF] . Based on estimates of soil carbon stock in the Mackenzie Basin 2 of ~50x10 3 tC km -2 and the upstream sediment source area (downstream of the Great Slave Lake, 774,200 km 2 ), the present rate of POC biosphere export represents a depletion of the soil carbon stock by ~0.006% yr -1 , which is sustainable over 10 3 -10 4 years.

OC burial efficiency in MTW01:

To estimate the burial efficiency of terrestrial POC at the MTW01 site, we normalise the measured [OC total ] concentrations (Fig. 3) by Al concentration, an immobile inorganic element hosted by major mineral phases. The OC total /Al normalization allows the effects of dilution to be distinguished from net OC gain (increased ratio) or OC loss (decreased ratio). The mean OC total /Al of the MTW01 samples was 0.17±0.02 (g g -1 , n=4, ±2SE). This is lower than the mean OC total /Al of the suspended load samples from the Mackenzie River delta of 0.26±0.10 (g g -1 , n=8, ±2SE). The decrease in the ratio offshore may suggest a higher relative proportion of POC petro (Extended Data Fig. 2b), however this is not consistent with the less negative δ 13 C org values (Extended Data Fig. 5).

The decrease can therefore be interpreted in terms of OC loss, with the ratio of core to river samples (0.17±0.02 / 0.26±0.10). Assuming that all the change in OC total /Al is driven by OC loss, and taking into account the measurement variability in these values, we estimate that 65±27% of the OC has been preserved. However, we note that the OC total /Al ratios in the core are not statistically different from the river suspended load samples (one-way ANOVA, P>0.1) which suggest the OC burial efficiency could higher (i.e. 100%). In addition, if we use the OC total /Al of finer river sediments carried near the channel surface which may be more easily conveyed offshore of 0.20±0.04 (g g -1 , n=4, ±2SE), we calculate burial efficiency = 85±20%. Future work should seek to better constrain these burial efficiencies with additional terrestrial and marine samples. Nevertheless, despite the remaining uncertainty, these high burial efficiencies [START_REF] Blair | The fate of terrestrial organic carbon in the marine environment[END_REF] are consistent with the high sedimentation rate and low temperature setting. The long-term burial of POC delivered to sites deeper in the Beaufort Sea [START_REF] Goñi | Distribution and sources of organic matter in surface marine sediments across the North American Arctic margin[END_REF] still remains to be assessed, in order to provide a complete picture of source to sink carbon transfers.

Figure 1 :

 1 Figure 1: Source of particulate organic carbon (POC) in the Mackenzie River Basin.

Figure 2 :

 2 Figure 2: Transport of particulate organic carbon (POC) in the Mackenzie River. a.

Figure 3 :

 3 Figure 3: Fate of particulate organic carbon offshore. a. Organic carbon concentration

  65 o C for 60 to 72 hours. Sample were then transferred to another vacuum desiccator charged with indicating silica gel, pumped down again and dried to remove HCl fumes. River sediment samples were analysed for organic carbon concentration [OC total ] on acidified aliquots and nitrogen concentration ([N], %) on non-acidified aliquots by combustion at 1020 o C in O 2 using a Costech elemental analyser (EA) in Durham. For river depth profile samples, acidified aliquots were prepared to graphite at the NERC Radiocarbon Facility of between 1-2 mg C for each sample and standard and 14 C was measured by Accelerator Mass Spectrometry at the Scottish Universities Environmental Research Centre and reported as th May 2015, doi:10.1038/nature14653fraction modern (F mod ) by standard protocol[START_REF] Stuiver | Discussion: Reporting of 14 C data[END_REF] . Process standards (96H humin) and background materials (bituminous coal) were taken through all stages of sample preparation and[START_REF] Bouchez | Source, transport and fluxes of Amazon River particulate organic carbon: Insights from river sediment depth-profiles[END_REF] C analysis and were within 2σ uncertainty of expected values. Stable isotopes of POC (δ 13 C org ) were measured by dual-inlet isotope ratio mass spectrometer (IRMS) on an aliquot of the same CO 2 . These measurements were consistent with δ[START_REF] Galy | Protracted storage of biospheric carbon in the Ganges-Brahmaputra basin[END_REF] C org measurements made by EA-IRMS normalised based on measured values standards (n=7) spanning >30‰ and longterm analytical precision of 0.2‰. River bank samples from 2009 were analysed by similar procedures at the National Ocean Sciences Accelerator Mass Spectrometry Facility (NOSAMS) at Woods Hole Oceanographic Institution.

14 C

 14 dates were normalized to a δ 13 C of -25‰ versus VPDB (http://intcal.qub.ac.uk/calib/).

  , R.G., Galy, A., Hovius, N., Horng, M. J., & Chen, H. The isotopic composition of particulate organic carbon in mountain rivers of Taiwan. Geochimica et Cosmochimica Acta 74, 3164-3181 (2010). 32. Moran, K., Hill. P.R., & Blasco, S.M. Interpretation of piezocone penetrometer profiles in sediment from the Mackenzie Trough, Canadian Beaufort Sea. Journal of Sedimentary Petrology 59, 88-97 (1989).

  since 9183 -156 +125 cal. yr BP, suggesting a high sedimentation rate during the Holocene 2.7±0.1 m ka -1 (Extended Data Table 2, Methods). These marine sediments have [OC total ] values similar to those measured in the Mackenzie River in both the <63µm (1.5% to 1.7%

C ages in a borehole located at the head of the Mackenzie Trough (MTW01) indicates that 21m of sediment have accumulated

  and the Al/OC total value at F mod ~ 0. Following previous work in large rivers, we then assume that the OC petro is well mixed in the water column and has a relatively constant [OC petro ] (refs10,13,14). This method may overestimate f petro if OC petro has been more extensively oxidised in fine grained weathering products carried in the suspended load[START_REF] Hilton | Geological respiration of a mountain belt revealed by the trace element rhenium[END_REF] . f petro is quantified using [OC petro ] and measured [OC total ]. This is consistent with the known presence of POC petro -bearing sedimentary rocks in the Mackenzie River Basin and high OC total contents of bedrocks in the upper Peel River Basin and Mackenzie mountains

	The mixing analysis returns a [OC petro ] = 0.12±0.03% (±2σ) in the Liard River and
	Mackenzie River at Tsiigehtchic, higher values in the Peel River [OC petro ] = 0.63±0.30%,
	with the Mackenzie River at the delta with an intermediate value [OC petro ] = 0.29±0.05%.

  OC petro ] values and the full range of annual sediment yields[START_REF] Carson | Magnitude and sources of sediment input to the Mackenzie Delta, Northwest Territories[END_REF] to quantify POC biosphere and POC petro discharge and associated uncertainty using a Monte Carlo approach. Over 100,000 simulations, we use a 'flat' probability for the range of values for both variables (i.e. equal probability of all measured values). This allows us to fully explore the range of estimates given the available measurements. Future work seeking to expand the number of [OC biosphere ] measurements to assess its flux-weighted mean and variability, while assessing temporal variability in more detail, will allow POC discharge estimates and their uncertainty to be refined. POC biosphere (2.2 -0.9 +1.3 TgC yr -1 ) and POC petro (0.4 -0.1 +0.1 ) discharges are reported as the median (50%) ± 1 s.d. Over the sediment source areas of the Mackenzie (Downstream of the Great Slave Lake 15 ) of 774,200 km 2 , these equate to yields of POC biosphere = 2.9 -1.1 +1.7 tC km -2 yr -1 and POC petro = 0.6 -0.2 +0.2 tC km -2 yr -1 . The total POC discharge is slightly higher than a previous estimate (2.1 TgC yr -1 )[START_REF] Macdonald | A sediment and organic carbon budget for the Canadian Beaufort Shelf[END_REF] based on measurements of POC content made in 1987

	12 collected
	in 1987 (1.4±0.2, n=10). While future work should aim to constrain the variability in POC

composition further, these observations suggest that temporal variability may be less important than the potential variability in [OC total ] with depth at a given time, where we find [OC total ] values can range from 1.0% to 2.7%. We use our measured range of [OC biosphere ] and [

Acknowledgements: Radiocarbon measurements were funded by the Natural Environment Research Council (NERC), UK (Allocation 1611.0312) to R.G.H and C.B. Fieldwork was funded by CNRS (OXYMORE and CANNIBALT) to J.G. and R.G.H., the Woods Hole Oceanographic Institution Arctic Research Initiative to V.G. and an Early Career Research Grant by the British Society for Geomorphology to R.G.H. V.G. was supported by the US National Science Foundation (OCE-0928582) and H.C. by a Royal Society University Fellowship. The research was carried out under Scientific Research Licence No. 14802 issued by the Aurora Research Centre, who we thank for logistical support (in particular D. Ross and J. Gareis). Thanks to I. Peters for preparation of offshore borehole samples. E. Tipper and K. Hilton are thanked for field assistance and discussions and D. Ofukany, G. Lennie, R. Wedel and R. Pilling of Environment Canada for loan of equipment. Three anonymous reviewers are thanked for thoughtful comments which improved the manuscript.