Marie Poulain

Matthieu J Mercier

Laurent Brach

Marion Martignac

Corinne Routaboul

Emile Perez

Marie Christine Desjean

Alexandra Ter Halle

Small microplastics as a main contributor to plastic mass balance in the North Atlantic subtropical gyre

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

We demonstrate that SMP buoyancy is significantly decreased compared to the l arg e microplastics (LMP, 1-5 mm) and consequently more susceptible to vertical transport. The uncorrected LMP concentration in count was between 13 000 and 174000 pieces 1an-2 , and was between 5 and 170 times more abundant for SMP. With a wind driven vertical mixing correction, we estimated that SMP were 300 to 70 000 times more abundant than LMP. When discussing this in terms of weight after correction, LMP concentrations were between 50 and 1000 g 1an-2 , and SMP concentrations were between 5 and 14000 g 1an-2 _

• INTRODUCTION

Plastic production has had an amazing exponential growth since its industrialization in the 1950s. However, only 40 years later, we realized that we do not control plastic's end oflife and have not put forth suflicient effort to promote its recycling. In the 20 10s, the annual plastic input into the ocean was estimated to be 10 million tons. The scientific community has agreed that the character ization of SMP by visual detection and manual sorting is prone to significant errors. 6 • 10 -12 For the detection and quantification of SMP, micro Fourier transform infrared (µ FTIR) and micro Raman spectroscopy are promising tools that should be considered. The development ofµ FTIR and its suitability were demonstrated with spiked natural samples. [START_REF] Tagg | Identification and Quantification of Microplastics in Wastewater Using Focal Plane Array Based Reflectance Micro FT IR Imaging[END_REF] This techni qu e was also applied successfully to address SMP concentration in count in environmental samples. However, the data are still scarce, and studies have been conducted with marine sediments, [START_REF] Loeder | Focal plane array detector based micro Fourier transform infrared imaging for the analysis of microplastics in environmental samples[END_REF] lagoon sediment, [START_REF] Vianello | Microplastic particles in sediments of Lagoon of Venice, Italy: First observations on occurrence, spatial patterns and identification[END_REF] coastal waters [START_REF] Song | A comparison of microscopic and spectroscopic identification methods for analysis of microplastics in environmental samples[END_REF] and wastewater treatment plant [START_REF] Mintenig | Identification of microplastic in effluents of waste water treatment plants using focal plane array based micro Fourier trans form infrared imaging[END_REF] samples. From these studies, it is apparent that SMP outnumber LMP, but the SMP weight concentration basis was not discussed. SMP weight estimates have only been discussed on the basis of emission scenarios. [START_REF] Van Wezel | Release of primary microplastics from consumer products to wastewater in the Netherlands[END_REF] Another specific issue discussed here is how to properly estimate the abundance of plastic particles at sea in the water column at a given location by the extrapolation of sea surface samples. The sea surface state, with turbulence induced by wind and waves or some specific flows, [START_REF] Kukulka | Evidence for the Influence of Surface Heat Fluxes on Turbulent Mixing of Microplastic Marine Debris[END_REF] can mix the upper water column and transport plastic particles vertically. This effect has been rationalized by turbulent models based on the assumption of a balance between the upward flux of particles due to buoyancy and the downward flux due to turbu lence. [START_REF] Kukulka | The effect of wind mixing on the vertical distribution of buoyant plastic debris[END_REF][START_REF] Kukulka | Passive buoyant tracers in the ocean surface boundary layer: 1. Influence of equilibrium wind waves on vertical distributions[END_REF][START_REF] Enders | Abundance, size and polymer composition of marine microplastics ≥ 10 μm in the Atlantic Ocean and their modelled vertical distribution[END_REF] These models rely on the rise velocity of plastic, W b , and turbulence modeling. The simplest approach consists of using a mean rise velocity of 0.01 m s -1 for all plastic particles. [START_REF] Kukulka | The effect of wind mixing on the vertical distribution of buoyant plastic debris[END_REF] However, laboratory measurements of W b for plastic debris between 0.5 and 207 mm are evidence of important variations with ratios up to a factor 40; smaller particles rising slower. [START_REF] Reisser | The vertical distribution of buoyant plastics at sea: An observational study in the North Atlantic Gyre[END_REF] Enders et al used a sphere to model microplastic (25 μm to 5 mm) buoyancy and demonstrated that LMP can be mixed over a few meters, whereas SMP could be transported down to hundreds of meters at the same sea state. [START_REF] Enders | Abundance, size and polymer composition of marine microplastics ≥ 10 μm in the Atlantic Ocean and their modelled vertical distribution[END_REF] Improvements to current modeling approaches are still highly necessary to better quantify concentration and mass estimates in open oceans. 4,[START_REF] Law | Plastics in the Marine Environment[END_REF] The present study investigates SMP and LMP composition, count and weight concentrations in the North Atlantic subtropical gyre at the sea surface. SMP were characterized and numbered by μ FTIR. A wind driven vertical mixing correction was developed to account for individual particle properties (density and geometry). Owing to the very dispersed geometry and density of microplastics, our modeling approach provided lower and upper bounds for the concentrations. The results with and without wind driven mixing were discussed.

■ MATERIALS AND METHODS

Chemicals. High density polyethylene (PE) pellets (CAS# 9002-88-4), poly(ethylene terephthalate) (PET) pellets (CAS# 25038-59-9, density of 1.68 g mL -1 ), poly(vinyl chloride) (PVC) powder (CAS# 9002-86-2, density of 1.4 g mL -1 at 25 °C), poly(methyl methacrylate) (PMMA) (CAS# 90011-14-7, an average molecular weight of 120 000), Nylon 6/6 pellets (CAS# 25038-54-4, density of 1.14 g mL -1 at 25 °C) and Nylon 12 (CAS# 25038-74-8, density of 1.14 g mL -1 at 25 °C) were purchased from Sigma Aldrich (Saint Louis, MO). Polystyrene (PS) pellets (3.5 mm) (CAS# 9003-53-6 PS), polypropylene (PP) pellets (3 mm) (CAS# 9003-07-0, melting flow index of 0.4 g per 10 min), low density PE pellets (1 mm), nonexpanded polyurethane (PU), and foam were purchased from Goodfellow (Huntingdon, UK). Sodium dodecyl sulfate (ACS reagent, >99%) and sodium hydroxide pellets were purchased from Sigma Aldrich (Saint Louis, MO).

Microplastic Sampling and Extraction. Plastic particles were collected from the sailing vessel Guyavoile in the North Atlantic subtropical gyre in June 2015 during the French sea campaign Expedition 7 th Continent. [START_REF] Ter Halle | Nanoplastic in the North Atlantic Subtropical Gyre[END_REF] LMP were collected using a Neuston net with a rectangular frame (0.5 × 0.4 m) fitted with a 2 m long net with a standard mesh size of 300 μm.

LMP were collected from the surface layer at a depth of [0-20] cm. Tow durations were set to 30 min, and tows were all undertaken while the vessel was traveling at a speed of 1-2.5 knots. On the boat, the content of the tows was filtered on 300 μm sieves. Most of the plastic particles were removed with tweezers and stored at -5 °C in glass vials. In total, 40 nets were towed: 12 outside the accumulation area and 28 within the accumulation area. The GPS locations and sea states associated with these 28 stations are supplied in the Supporting Information (SI), Table S1. SMP were collected using a Neuston net with a standard mesh size of 25 μm. The net was fixed to a 0.3 × 0.1 m rectangular frame and was 3 m long. SMP were collected from the surface layer at a depth of [0-6] cm. Tow durations were set to 10 min and were all undertaken while the vessel was traveling at a speed of less than 1 knot. Indeed, the speed of the boat was strictly restricted to below 1 knot to avoid clogging the 25 μm net. The net was towed from a zodiac boat, and for security reasons, it was only possible when the sea was relatively calm. Thus, only eight measure ments were performed within the accumulation area. The content of the tows was filtered through a cellulose acetate membrane (5 μm) in a closed filtration unit and immediately stored in closed glass vials at -18 °C. Both manta nets were equipped with flow meters from which sea concentrations in count were calculated and expressed as numbers of particles per square kilometer.

Microplastic Isolation. The sorting and numbering of LMP were carried out classically and described elsewhere. [START_REF] Ter Halle | Nanoplastic in the North Atlantic Subtropical Gyre[END_REF] To calculate LMP sea surface concentrations (in count and weight), only those having at least one dimension above 1 mm were considered. For SMP isolation, the acetate filters obtained on the boat were immersed under laboratory conditions in 80 mL ultrapure water in a closed glass bottle at ambient temperature under gentle agitation for 1 h. After removal of the filter, the biogenic matter was eliminated by the addition of 20 mL of a 10 mol L -1 NaOH solution and 300 μL of a 50 g L -1 sodium dodecyl sulfate solution. The solution was gently agitated for 4 h and stored for 1 week at ambient temperature. After 1 week, the solution was filtered through glass filters in a closed unit (Whatman GF/F; 0.7 μm; 47 mm) and rinsed abundantly with ultrapure water. The filters were then stored in closed glass Petri dishes prior to analysis. A control experiment under these conditions was performed to ensure that the plastic was not altered. (PET, PVC, PS, PE, PP, PU expanded or foam, Nylon6, Nylon12, and PMMA) were grinded at 200 μm using a ZM 200 grinder (Retsch, Haen, Germany). The plastic powder (300 mg) was treated with sodium hydroxide under the same conditions.

SMP Characterization by μ-FTIR. Micro FTIR spectros copy was performed using a Thermo Fisher Scientific Nicolet iN10 apparatus in reflection mode equipped with a liquid nitrogen cooled MCT detector. The particle bigger than 30 × 30 μm visually detected by the operator under the microscope were analyzed. We cannot estimate how many particles could be missed by the operator, this is the most problem of working with an operator. All the work done those past two years by Primpke et al. are a huge improvement of the method. In the future we will work like this team. The spectra were recorded as the average of 16 scans in the spectral range of 650-4000 cm -1 at a resolution of 8 cm -1 . The library was from the Thermo Scientific software (database: Hummel Polymer library, HR Polymer and additives, HR Polymer additives and plasticizers). Each particle was identified and analyzed individually. The microscope aperture was adapted to each particle. If the particle was suspected to be plastic, several measurements at different spots were undertaken to prevent false signals due to either local impurities or the rough and irregular shapes of the particles, which could alter their spectra. The plastic particles were identified using a polymer spectral library if the match was greater than 80% (by the software OMNIC PICTA with no normalization or derivation).

Microplastic Geometrical Description. We have identi fied two families of objects: lines and pieces. Lines were less numerous than pieces (14,6% for the LMP and 6.6% for the SMP). Here, we only consider pieces, and lines were excluded from the calculation.

We measured the length (L), width ( ) and mass (M) to the nearest 0.01 mg using a precise scale (Genius shatorius, Gottingen, Germany). In addition, for 415 microplastics (the sum of LMP and SMP) collected during a previous sea campaign in the North Atlantic subtropical gyre, [START_REF] Ter Halle | Understanding the Fragmentation Pattern of Marine Plastic Debris[END_REF] we also measured the thickness (h) and the apparent surface area (S a ) using image analysis (ImageJ © ). A complete description is given in SI Section S2.

Rise Velocity Measurements. Rise velocity measure ments of LMP samples collected were done in the laboratory. A cylindrical tank (diameter, 13 cm and height, 100 cm) was filled with 18 L of fresh (1.001 g cm -3 ) or salt water (1.025 g cm -3 ). A single particle was inserted at the bottom of the column using a double valve system. The particle raised in the quiescent fluid (no residual flow observed). Passing times of the particle at three marks (50, 70, and 80 cm away from the bottom) were recorded in order to calculate W b . We verified that the particle had reached a steady velocity, and each measurement was repeated four times, leading to less than 5% of uncertainty on W b .

■ RESULTS AND DISCUSSION Microplastic Characterization. LMP and SMP have been sampled at the sea surface of the North Atlantic subtropical gyre during the cruise Expedition 7 th Continent. LMP extraction and numbering are classical procedures and have been described previously. [START_REF] Brach | Anticyclonic eddies increase accumulation of microplastic in the North Atlantic subtropical gyre[END_REF] For SMP characterization, an essential first step of purification is needed to eliminate organic matter (algae, plankton) and inorganic particles (sand and silt) before detection. Very diverse purification approaches have been developed; all methods consist of several steps using density separation together with degradation of the organic matter. [START_REF] Imhof | highly efficient method for the separation and quantification of plastic particles in sediments of aquatic environ ments[END_REF][START_REF] Claessens | Occurrence and distribution of microplastics in marine sediments along the Belgian coast[END_REF][START_REF] Nuelle | A new analytical approach for monitoring microplastics in marine sediments[END_REF] Organic matter is eliminated by strong acidic [START_REF] Enders | Abundance, size and polymer composition of marine microplastics ≥ 10 μm in the Atlantic Ocean and their modelled vertical distribution[END_REF][START_REF] Imhof | Pigments and plastic in limnetic ecosystems: A qualitative and quantitative study on microparticles of different size classes[END_REF] or alkaline [START_REF] Foekema | Plastic in North Sea Fish[END_REF] solutions, by oxidation agents [START_REF] Tagg | Identification and Quantification of Microplastics in Wastewater Using Focal Plane Array Based Reflectance Micro FT IR Imaging[END_REF] or by the use of enzymes. [START_REF] Cole | Isolation of microplastics in biota rich seawater samples and marine organisms[END_REF] We underline that the purification step needs to be as efficient as possible to facilitate spectroscopic detection without damaging the plastic particles by breaking them down or by altering the structure of the polymer, that is, conditions should remain mild. For SMP isolation, a sodium hydroxide treatment was chosen. As a control, 10 different polymers were ground (500 μm) and tested with the sodium hydroxide treatment. Recoveries were higher than 95% (±5%). Under the microscope, the treated polymer did not appear to be altered (no yellowing), and the μ FTIR spectra of the treated plastics were not modified. Procedural blank experiments have been conducted: no plastic particles or cloth fibers have been detected in these control experiments. After purification and filtration, the particles deposited on the filter are analyzed by microspectroscopy. Both FTIR and Raman provide vibrational fingerprint spectra with information about the shape and dimension of the particles. The techniques present a spatial resolution in the micrometric range and both methods have advantages and disadvantages for the detection of SMP. Their use in combination has been proposed by Kaeppler et al. [START_REF] Kaeppler | Analysis of environmental microplastics by vibrational microspectroscopy: FTIR, Raman or both? Anal[END_REF] The detection and characterization of SMP by μ FTIR can be envisioned three ways. A first approach consists of a visual presorting; in this case, the attenuate total reflection (ATR) crystal is focused on every single particle. [START_REF] Song | A comparison of microscopic and spectroscopic identification methods for analysis of microplastics in environmental samples[END_REF][START_REF] Imhof | highly efficient method for the separation and quantification of plastic particles in sediments of aquatic environ ments[END_REF][START_REF] Enders | Abundance, size and polymer composition of marine microplastics ≥ 10 μm in the Atlantic Ocean and their modelled vertical distribution[END_REF][START_REF] Cauwenberghe | Microplastic pollution in deep sea sediments[END_REF] The visual presorting and focusing is very time consuming. In a second approach, the automated image analysis of the filter is proposed. This consists of the detection and localization of the particles prior to characterization. [START_REF] Primpke | An automated approach for microplastics analysis using focal plane array (FPA) FTIR microscopy and image analysis[END_REF] The third option is a spectroscopic mapping of the whole surface of the filter. [START_REF] Loeder | Focal plane array detector based micro Fourier transform infrared imaging for the analysis of microplastics in environmental samples[END_REF][START_REF] Vianello | Microplastic particles in sediments of Lagoon of Venice, Italy: First observations on occurrence, spatial patterns and identification[END_REF] In this later case, the bands selected to detect the plastic have to be chosen among polymer specific regions that are insensitive to variations in the particle shape or state of oxidation. [START_REF] Harrison | The applicability of reflectance micro Fourier transform infrared spectros copy for the detection of synthetic microplastics in marine sediments[END_REF] To control the amount of data generated, the region for the IR scanning is also restricted to some specific polymer bands, and the number of scans per measurement, thus the resolution of the measurements, is lower. All these options affect the signal/ noise ratio, and these adjustments are detrimental to the reliability of the measure. Primpke et al. developed software to analyze and interpret the very voluminous data generated by the mapping. [START_REF] Primpke | An automated approach for microplastics analysis using focal plane array (FPA) FTIR microscopy and image analysis[END_REF] Here, we opted for the μ FTIR characterization with a visual presorting and a systematic crystal focus on every single particle. The spectra were much better compared to the automated method. We decided to analyze the whole surface of the filter because it presented very important heterogeneity. In contrast, this option utilized an operator for a long time and thus was accompanied by an operator bias that was difficult to estimate. Lines with diameters a few hundreds of micron and a few millimeter big were present in the LMP samples. Line probably comes from the degradation of fishing lines or gears. Textile fibers present much smaller diameters, tens of microns, they belong to the SMP category. But because fibers do not deposit flat on the surface of the filter (a part of the fiber is out of focus of the IRbeam) they are hardly detectable and their infrared spectra does not allow a correct correspondence with the library. [START_REF] Peeken | Arctic sea ice is an important temporal sink and means of transport for microplastic[END_REF] We presume that this method underestimated SMP abundance, and we acknowledge that this method needs further methodological improvement to make the analysis more reliable.

LMP (960 particles isolated in total) were made of 90% PE and 10% PP; this is in agreement with reported data from the open ocean. [START_REF] Ter Halle | Understanding the Fragmentation Pattern of Marine Plastic Debris[END_REF] SMP (1 100 particles isolated in total) were made of a greater diversity of polymers, as has already reported. [START_REF] Enders | Abundance, size and polymer composition of marine microplastics ≥ 10 μm in the Atlantic Ocean and their modelled vertical distribution[END_REF] SMP were made of PE (70%), PP (17%), PVC (7%), PS (1%), PET (1%) and to a lesser extent polyurethane, poly aryl ether sulfone and phenoxy resins (Figure 1). However, polyolefins were still predominant. SMP repartition among polymer types presented variation along the sampling locations (SI Table S1). For example, at station 12, 98% of the SMP were made of PE, while at station 20, only 41% were made of PE and 51% of PP. A contribution of 13% of PVC particles was identified at station 24. Finally, for all sampling points, PS and PET proportions never exceeded 5%.

Modeling Approach and Validation. LMP concen trations at the sea surface are usually corrected using the modeling approach introduced by Kukulka et al. [START_REF] Kukulka | The effect of wind mixing on the vertical distribution of buoyant plastic debris[END_REF] This model is based on the number of microplastics collected at the surface (N tow ), the rise velocity (W b ) and the surface forcing (wind and waves), which is based on the friction velocity of water slower than the sphere with R eq = L eq . Thus, to encompass the dynamics of microplastics, we attributed to the ellipsoid density the upper limit (1.005 g cm -3 ), and to the sphere density, the lower limit (0.900 g cm -3 ). We note that M e and M s are the corresponding masses for the ellipsoid and sphere geometries. The drag coefficient models and corresponding resolution of eq 2 for these objects are presented in SI Section S4.

As a first validation of the model, we compared the mass of the LMP to the calculated mass of the sphere and the ellipsoid. The majority of LMP mass (85%) fell within the delimited boundaries (Figure 2A). Since we could not weigh the SMP, we assumed that the two geometries described the SMP well, and we used the average mass, M m = (M e + M s )/2, to provide a mass estimate for the direct observations. As a second validation, we compared the calculated rise velocity of the ellipsoid, W b e , and the sphere, W b s , to the rise velocity of 35 arbitrarily selected LMP. Their rise velocities were measured in laboratory conditions with a system that avoided the generation of any turbulence during the introduction of the particle into the water column (Figure 2B). The majority (97%) had a rise velocity between W b e and W b s . Only one object was out of bounds, it turned out to have an aspect ratio larger than 10, which is consistent with a smaller rising velocity. We calculated W b for a wide range of microplastic sizes. However, our calculations of W b were in the same range as those measured previously (>0.5 mm). [START_REF] Kukulka | The effect of wind mixing on the vertical distribution of buoyant plastic debris[END_REF][START_REF] Reisser | The vertical distribution of buoyant plastics at sea: An observational study in the North Atlantic Gyre[END_REF] It was not possible to measure the rise velocity of SMP because they were too small to be handled. Nonetheless, eq 2 is also valid in the micrometric range. For SMP, the calculated values for W b are between 3 × 10 -3 and 10 -4 m s -1 . These are 1-2 orders of magnitude below the mean value for LMP. This finding well illustrates the importance of integrating the variations of W b into the turbulence model.

Using eq 1, we define f e and f s as the correction factors associated with the ellipsoid and the sphere, respectively (Figure 3A andB). As smaller particles present lower buoyancy at a given sea state, their correction factors are higher. For example, at Beaufort 1, 100% of the plastic particles with L eq > 1 mm are within the first 20 cm (f e = f s = 1), while when L eq = 100 μm, only between 20 and 100% are within the first 20 cm (f e = 5 and f s = 1). The corrections proposed by previous studies for LMP fell within the interval proposed here (data not shown). [START_REF] Kukulka | The effect of wind mixing on the vertical distribution of buoyant plastic debris[END_REF][START_REF] Reisser | The vertical distribution of buoyant plastics at sea: An observational study in the North Atlantic Gyre[END_REF] It must be noted that this difference between small and large microplastic distributions in the water column increases with the Beaufort. For instance, at Beaufort 5, between 5.5 and 65% of plastic particles with L eq = 1 mm are located within the surface layer, but only between 1 and 5% of plastic particles with L eq = 100 μm are (inset plots Figures 3A andB). In these conditions, the corrections could lead to major error estimates, which is why we excluded stations at Beaufort 5 from the following discussion (Tables 1 and2).

Our modeling approach allows us to relate count and weight concentrations evaluations. For each sample, an estimated mass can be provided, and for the sake of simplicity, we chose the average mass of the two geometries, M m . For a given L eq , the corrected concentrations in weight are obtained by multiplying M m with the correction factors f e and f s (Table 2).

Measured Count and Weight Concentrations. Aver aged LMP and SMP concentrations in count without correction were 50 000 particles km -2 and 1 630 000 particles km -2 , respectively. LMP concentrations were between 13 000 and 174 000 particles km -2 , while SMP concentrations were between 630 000 and 6 000 000 particles km -2 (Table 1). The highest concentrations for both LMP and SMP were measured for station 24 at low wind speed (Beaufort 2). LMP concentrations in count were typical of the ones encountered in subtropical gyres. [START_REF] Law | Plastic Accumulation in the North Atlantic Subtropical Gyre[END_REF][START_REF] Law | Distribution of surface plastic debris in the eastern pacific ocean from an 11 year data set[END_REF][START_REF] Cozar | Plastic debris in the open ocean[END_REF][START_REF] Eriksen | Plastic Pollution in the World's Oceans: More than 5 Trillion Plastic Pieces Weighing over 250,000 Tons Afloat at Sea[END_REF] Concerning SMP concentrations, there are no data for comparison. The global trend over these 8 sampling stations was that there were between 5 and 171 times is the corrected concentration in count, where superscript i refers to the model used for correction (s for the sphere model and e for the ellipsoid model).

  ABSTRACT: Estimates of cumulative plastic inputs into the oceans are e:xpressed in hundred million tons, whereas the total R = ,,fITF mass of microplastics afl oat at sea is 3 orders of magnitude twt In couni. l.ottrge pla.<_; tki, (1-f> mm) below this. This l arg e gap is evidence of our ignorance about the fate of plastics, as well as transformations and sinks in the oceans. One of the current challenges consists of identifying and quantifying plastic particles at the microscale, the small µm ) . The aim of the present study is to investigate SMP concentration in count and in mass at the sea surface in the North Atlantic subtropical gyre during the sea campaign Expedition i h Continent After isolation, From plastic dcbris SMP were characterized by micro Fourier transform infrared to models Rise velocities 1\, ficroplo.stics corroetcd concentrations spectroscopy. Microplastic distribution was modeled by a wind driven vertical mixing correction mode! taking into account individual particle properties ( dimension, shape and density).

Figure 3 .

 3 Figure 3. Correction factors estimated for (A) the sphere model ( f s ) and (B) the ellipsoid model (f e ) calculated for a net immersion depth of 20 cm. Each color corresponds to a Beaufort (blue: Beaufort 1-2, green: Beaufort 3, orange: Beaufort 4, and magenta: Beaufort 5). Inset plots correspond to the theoretical fraction of surface captured plastic particles ( f -1 ) in percent at [0-20] cm.

1 None of the commonly used plastics are biodegradable, and thus, they tend to accumulate once discarded in the environment. 2 With solar illumination and mechanical forcing, plastic waste is oxidized and fragmented into smaller and smaller particles, reaching the nanoscale. 3 Currently, one important scientifi c question is how to balance the annual input into the oceans, which is one hundred of times more than the amount we are actually able to locate. This large gap is evidence of our critical Jack of knowledge about the fate of plastic in the ocean. Efforts to detect plastic particles at the micro and nanoscales are needed. 4 ,s Published works state that small microplastics (SMP, 25-1000 µm ) are present in the natural environment 6 • 7 and that they are genuinely more abundant than l arg e microplastics

(LMP, 1-5 mm).

until now, there has been no estimation of the weight SMP could represent at sea.

8 • 9 However,

Table 1 .

 1 Concentration in Count (×10[START_REF] Gigault | What is a nanoplastic ?[END_REF] Particles km -2 ) for LMP (1 5 mm) and SMP (25-1000 μm) at Eight Sampling Stations (detailed information available in SI TableS1), Lines and Fibers Excluded a

				1 5 mm			25 1000 μm
	#	Beaufort	N tow	N corr s	N corr e	N tow	N corr s	N corr e
	6	5	0.14			24
	8	5	0.18			12
	11	3	0.13	0.13 0.27	7.9	870 41 000
	12	5	0.14			11
	17	3	0.38	0.40 1.13	6.3	730 34 000
	20	3	0.98	0.98 3.20	5.0	600 24 000
	24	2	1.74	1.74 1.88	60	750 36 000
	27	3	0.34	0.34 0.66	3.8	110 45 000
	mean		0.50	0.72 1.43	16.3	610 36 000

a N tow is the uncorrected concentration in count, and N corr i


Kjetil Thøgersen
email: kjetil.thogersen@fys.uio.no

Henrik Andersen Sveinsson

Julien Scheibert

François Renard

Anders Malthe-Sørenssen

The moment duration scaling relation for slow rupture arises from transient rupture speeds

A Burridge-Knopoff model with only two dimensionless parameters; the homogeneous stress on a fault and a velocity strengthening friction term.

• The simplicity of the model allows for both numerical and analytical calculations of moment versus duration scaling relationships during fault slip.

• Moment versus duration scaling relation for slow events arises from transient rupture speeds.

Plain language summary

Observations have shown that the duration of earthquakes is related to the seismic moment through a power law. The power law exponent is different for regular earthquakes and slow aseismic rupture, and the origin of this difference is currently debated in the literature. In this letter, we introduce a minimal mechanical friction model that contains both slow and regular earthquakes, and demonstrate that the different power laws emerge naturally within the model because the propagation speed of slow earthquakes decays as a power law in time whereas the propagation speed of regular earthquakes remains fairly constant.

Introduction

Over the last decades, an increasing number of slow slip events on faults have been reported [START_REF] Bürgmann | The geophysics, geology and mechanics of slow fault slip[END_REF]. A measure that is viewed as a key to unravelling the mechanism of slow and fast rupture is the relation between seismic moment M 0 and slip event duration T . Regular fast earthquakes have long been known to follow a moment duration scaling relation of M 0 ∝ T 3 . Ide et al. suggested that slow events follow a unified scaling relation M 0 ∝ T [START_REF] Ide | A scaling law for slow earthquakes[END_REF]. They suggested that the linear relation between moment and duration for slow events can be explained in two ways: (1) the average slip is proportional to the fault length as for fast propagation, and the stress drop is constant for all events, which gives the relation M 0 ∝ T . (2) the slip amount is constant for all events, and the fault area increases linearly with time L 2 ∝ T , which results in M 0 ∝ T . Peng et. al [START_REF] Peng | An integrated perspective of the continuum between earthquakes and slow-slip phenomena[END_REF]) elaborated on the ideas of Ide et. al [START_REF] Ide | A scaling law for slow earthquakes[END_REF] and reached a different conclusion; that rupture should span a continuum between fast and slow velocity end-members. However, almost 10 years after the suggestion of Peng et. al, observations of events between the fast and slow end-members are still sparse. Later studies have reported on a variety of scalings between moment and duration ranging from M 0 ∝ T to M 0 ∝ T 2 [START_REF] Ide | Bridging the gap between seismically and geodetically detected slow earthquakes[END_REF][START_REF] Aguiar | Moment release rate of cascadia tremor constrained by gps[END_REF][START_REF] Frank | Revealing the cluster of slow transients behind a large slow slip event[END_REF][START_REF] Gao | Scaling relationships of source pa-rameters for slow slip events[END_REF], although the definition of events can vary somewhat between different studies.

The shape of slip patches can also influence the observed scaling. [START_REF] Ben-Zion | Episodic tremor and slip on a frictional interface with critical zero weakening in elastic solid[END_REF] argued that fractal slip patches can result in a scaling relation M 0 ∝ T 2 / log(T ) because the average displacement is approximately constant rather than proportional to the rupture dimension. Bounded propagation can also play an important role [START_REF] Ben-Zion | Episodic tremor and slip on a frictional interface with critical zero weakening in elastic solid[END_REF][START_REF] Gomberg | Reconsidering earthquake scaling[END_REF]. Gomberg et. al [START_REF] Gomberg | Reconsidering earthquake scaling[END_REF] suggested that the scaling relation between moment and duration is the same for slow and fast events, but that a transition occurs between a two-dimensional scaling and a one-dimensional scaling when the rupture propagation switches from unbounded to bounded in one direction. Assuming the fault displacement can be approximated using dislocation theory, this results in a transition from T 3 to T . They suggest that there should be a bimodal but continuous distribution of slip modes, and that a difference in scaling relations alone does not imply a fundamental difference between fast and slow slip. The above mentioned theoretical considerations implicitly assume constant rupture velocity. However, this contradicts observations by Gao et. al [START_REF] Gao | Scaling relationships of source pa-rameters for slow slip events[END_REF] that show that the average rupture speed for slow events decreases with increasing seismic moment, which is a strong indication of transient rupture speeds.

Slow slip events emerge in numerical models with rate-and-state friction. Colella et. al [START_REF] Colella | Multi-event simulations of slow slip events for a cascadia-like subduction zone[END_REF]) simulated a Cascadia-like subduction zone using rate-andstate friction. They analyzed a large number of slip events, and found that the seismic moment M w scales as M w ∝ T 1.5 for M w ≤ 5.6 with a transition to M w ∝ T 2 for M w > 5.6. Shibazaki et. al [START_REF] Shibazaki | Modeling of slow slip events along the deep subduction zone in the kii peninsula and tokai regions, southwest japan[END_REF] modeled the subsuction zone of southwest Japan with rate-and-state friction. For slow events, they found a scaling M 0 ∝ T 1.3 . Liu et. al [START_REF] Liu | Source scaling relations and along-strike segmentation of slow slip events in a 3-d subduction fault model[END_REF] used rate-and-state friction on a 3D subduction fault model and found a scaling M 0 ∝ T 1.85 . Romanet et. al [START_REF] Romanet | Fast and slow slip events emerge due to fault geometrical complexity[END_REF] highlighted the role of interactions between faults. They argue that the scaling relationships of slow slip events and earthquakes emerge from geometrical complexities due to interactions between fault segments. The moment duration scalings have not only been addressed using rate-and-state friction. Ide et. al [START_REF] Ide | A brownian walk model for slow earthquakes[END_REF] introduced a Brownian walk model for slow rupture, where the assumption is that there is a random expansion and contraction of the fault area, so that its radius can be described as a Brownian walk with a damping term. This model predicts M 0 ∝ T for large T .

Here, our goal is to answer the following two questions: (1) Is there a separation of two distinct classes [START_REF] Ide | A scaling law for slow earthquakes[END_REF], or is there a continuum of possible scaling relations between the fast and slow end-members [START_REF] Peng | An integrated perspective of the continuum between earthquakes and slow-slip phenomena[END_REF])? (2) Can a difference in M 0 -T scaling relations alone be attributed to different physical mechanisms behind slow and fast rupture? We address both (1) and ( 2) through a Burridge-Knopoff type model with Amontons-Coulomb friction with a velocity strengthening friction term that has previously been shown to contain a large variety of rupture phenomena, including sub-shear, super-shear and slow propagation [START_REF] Thøgersen | A minimal model for slow, sub-rayleigh, supershear and unsteady rupture propagation along homogeneously loaded frictional interfaces[END_REF]. Velocity strengthening friction has been shown to be a generic feature of dry friction [START_REF] Bar-Sinai | On the velocity-strengthening behavior of dry friction[END_REF], and has been reported in Halite shear zones at low slip speeds or large confining pressures [START_REF] Shimamoto | Transition between frictional slip and ductile flow for halite shear zones at room temperature[END_REF]. The friction law can also be interpreted as a transition from a dry contact to a lubricated sliding regime with increasing velocity (a Stribeck curve) under the additional assumption that the transition from dry to contact to lubricated sliding occurs at a small sliding speed [START_REF] Gelinck | Calculation of stribeck curves for line contacts[END_REF][START_REF] Olsson | Friction models and friction compensation[END_REF].

For homogeneously stressed faults, the model can be reduced to only two dimensionless parameters τ and ᾱ representing the prestress and a velocity strengthening friction term, respectively. The advantage of such approach is that the simplicity of the model allows us to calculate moment duration scaling relations both through numerical simulations and through analytical calculations. Through numerical simulations, we demonstrate that there exists a continuum of rupture modes between the slow and fast end-members, but that the most likely selection of τ in nature produces two distinct classes separating sub-shear and slow rupture velocities. Through analytical calculations, we show that the scaling relation for slow fronts arises due to long-lived transients in the rupture velocity. Such transient rupture velocity has been observed in nature through a dependence on the average rupture speed on the seismic moment for slow fronts [START_REF] Gao | Scaling relationships of source pa-rameters for slow slip events[END_REF]. In addition, the model predicts a separate scaling for super-shear rupture not previously reported in the literature.

A one-dimensional Burridge-Knopoff containing slow and fast rupture

We solve the one-dimensional Burridge-Knopoff model [START_REF] Burridge | Model and theoretical seismicity[END_REF] with Amontons-Coulomb friction with a linear velocity strengthening term. The dimensionless equation of motion for a chain of N blocks can be written as (a detailed derivation can be found in the supplementary information)

üi -ūi-1 -ūi+1 + 2ū i + ᾱ ui -τ ± = 0, ∀i ∈ [0, N ],
(1) which is integrated using the Euler-Cromer method with d t = 10 -3 . ū is the dimensionless displacement, and

τ ± = τ /σ N ∓ µ k µ s -µ k (2)
is the dimensionless prestress where σ N is the normal stress, τ is the initial shear stress, and µ s and µ k are the static and dynamic coefficients of friction, respectively.

± denotes the sign of the block velocity. For positive velocities, we only need to consider τ + , but negative velocities can occur in a small subset of our simulations. In such situations, we need to prescribe the relation between µ s and µ k , which we set to µ s = 2µ k , so that τ -= τ + + 2. In the rest of the paper we will use τ as a reference to τ + . The second dimensionless parameter

ᾱ = α √ ρE (3)
is a viscous term, where ρ is the density, E is the elastic modulus, and α is a velocity strengthening term with units [Pa s/m]. ᾱ can range from 0 to infinity, where ᾱ = 0 recovers the ordinary Amontons-Coulomb friction without viscosity. τ has an upper limit of 1, where the prestress equals the static friction threshold. For τ < 0, steady state The model can be written in dimensionless units with only two parameters: τ representing the prestress on the fault, and ᾱ representing the velocity strengthening friction term. This simple model produces a large variety of slip, including, slip pulses, cracks, sub-Rayleigh rupture, super-shear rupture, slow rupture, and arresting fronts (c). The colorbars show the fault length L of arresting fronts, and the steady state rupture speed vc,∞ for given τ and ᾱ (adapted from [START_REF] Thøgersen | A minimal model for slow, sub-rayleigh, supershear and unsteady rupture propagation along homogeneously loaded frictional interfaces[END_REF]). Each event consists of a single simulation, which gives the block sliding velocity u as a function of position l and time t (d), from which we extract the front position L (e). We also measure the block displacement ū( l, t) (f), from which we extract the average displacement ū (g). Using equation 6 we obtain the seismic moment and the duration of the event marked with a star in (h).

propagation does not occur [START_REF] Amundsen | Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces[END_REF]. This corresponds to a prestress below the dynamic friction level. We thus simulate τ ∈ [10 -7 , 1] and ᾱ ∈ [10 -3 , 10].

The boundary conditions assuming triggering through soft tangential loading (small driving velocity V and driving spring stiffness K) are given by ū-1 = ū0 + 1 -τ . The rightmost block is fixed so that ūN = 0. Blocks start to move once the static friction threshold is reached, which in dimensionless units can be written as

ūi-1 + ūi+1 -2ū i ≥ 1 -τ (4)
Moving blocks restick if the velocity changes sign. The system is sketched in Figure 1a.

This model has previously been used to determine the steady state rupture velocity which includes sub-shear, supershear, and slow rupture, as well as an arresting region at low τ and intermediate ᾱ [START_REF] Thøgersen | A minimal model for slow, sub-rayleigh, supershear and unsteady rupture propagation along homogeneously loaded frictional interfaces[END_REF]. The steady state front speed vc,∞ can be found exactly when ᾱ = 0 [START_REF] Amundsen | Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces[END_REF]. For ᾱ > 0 we can use the empirical expression [START_REF] Thøgersen | A minimal model for slow, sub-rayleigh, supershear and unsteady rupture propagation along homogeneously loaded frictional interfaces[END_REF]

vc,∞ ≈ 1 -e -τ ᾱ(1-τ ) √ 1 -τ 2 . ( 5
)
3 Moment duration scaling relations

Model results

For each simulation we predefine τ and ᾱ, and each simulation consists of a single event. From each event we extract the displacement ū and the fault length L, which is found from the position of the rightmost block that has ruptured. We run the simulations until all blocks are immobile, or until the average velocity reaches 0.1% of the steady state slip speed τ /ᾱ [START_REF] Thøgersen | A minimal model for slow, sub-rayleigh, supershear and unsteady rupture propagation along homogeneously loaded frictional interfaces[END_REF]. In dimensionless units the zeroth order moment for rupture propagation along a line is

M0,1D = ū L, ( 6
)
where ū is the average displacement on a fault of length L. The seismic moment and the duration are measured when 99% of the total displacement has been reached.

Figure 1 shows how M0,1D and the event duration T is measured in the simulations.

For each simulation, we measure the duration, as well as the fault length L and the average block displacement ū . This results in a single point in the ( M0,1D , T ) diagram for each event.

Figure 2 shows the measured M0,1D and event duration T for 1120 simulations with N ∈ 5×2 {0,7} . If the stress drop is small compared to the absolute shear stress, as is often found for faults [START_REF] Shearer | Comprehensive analysis of earthquake source spectra in southern california[END_REF], τ should often lie close to the dynamic level, which corresponds to τ 0. For low values of τ , the arresting region in (τ ,ᾱ)

gives rise to a separation of these scaling relations, so that fast and slow rupture fall into two distinct lines in the moment duration diagram (Figure 2a). This is in line with the ideas of [START_REF] Ide | A scaling law for slow earthquakes[END_REF]. This separation occurs because steady state propagation at small τ and intermediate ᾱ is forbidden (Figure 1a). If we include also larger prestress values we obtain a continuum of slip modes in the moment duration diagram (Figure 2d), in line with the suggestions of [START_REF] Peng | An integrated perspective of the continuum between earthquakes and slow-slip phenomena[END_REF]. The model also predicts a second scaling relations for super-shear rupture, which is found at large τ , that has not previously been reported (Figure 2b).

Origin of scaling relations -analytical calculations

The simplicity of the model allows an analytical treatment of several aspects which helps explain the various scaling relations between seismic moment and event duration. We summarize the analytical predictions for slip, front speed and event duration, and explain why the different scaling relations appear. A detailed derivation is given in the supplementary information.

First, we can determine the average slip on a fault. If the stress is at the dynamic level after rupture (the stress drop equals τ ), we can calculate ū exactly

ū = τ L2 3 + (1 -τ ) L 2 . ( 7
)
Equation 7 is derived for soft tangential loading, and we stress that a different boundary conditions could lead to different dependencies between L, ū and τ . Combining equation 7 with equation 6 we find that the seismic moment can be written as

M0,1D = τ L3 3 + (1 -τ ) L2 2 , (8)
which only depends on the prestress τ and the length of the fault L. To obtain the moment duration scaling relation we need to determine L( T ), and thus have to combine equation 8 with information about the rupture propagation and the afterslip (i.e. the amount of slip after the propagation has stopped).

A key observation on the rupture propagation is shown in Figure 3. While fast fronts exhibit short transients and quickly reach the steady state propagation speed given by equation 5, slow rupture contains long transients where the propagation speed decays. In the figure, we have illustrated this effect as a decay in the average rupture speed vc ≡ L T with increasing seismic moment M0 . This result is in line with observations by [START_REF] Gao | Scaling relationships of source pa-rameters for slow slip events[END_REF].

Fast regime

The short transients in the fast regime indicate that we can approximate the fault length by

L = trupture 0 vc (t )dt ≈ vc,∞ trupture , (9)
where t rupture is the time it takes for a rupture front to reach the end of the fault.

The time it takes to arrest completely depends upon the existence of a backward propagating arresting front. If we assume that this backward propagation occurs at roughly the same speed as the forward propagation we obtain

T ≈ 2 L vc,∞ (10)

Slow regime

For slow fronts, vc ( t) is transient, and we observe that vc ( t) is well described by a power law with exponent β for large ᾱ and small τ . To obtain an approximation for vc ( t, ᾱ, τ ), we plot vc ( t) for a selection of τ and ᾱ in Figure 3. All curves collapse when we subtract the steady state front velocity v c,∞ and scale with the initial rupture velocity v c,0 given in equation S32. We can then write down vc,slow

≈ (v c,0 -vc,∞ )(v c,0 t) -β + vc,∞ (14)
Figure 3 shows that this relation fits well with simulations for small τ and large ᾱ, and we measure empirically the exponent β = 0.685 ± 0.002 (using τ = 10 -3 and ᾱ = 10).

To obtain a parametric equation for M0 and T , we need to find T ( L). T has two main components; the time it takes to rupture a fault of length L, trupture , and the time it takes for all motion to stop. Unlike for fast fronts, the arresting phase in the slow regime is not governed by a backward propagating arresting front, but rather a slow exponential decay in velocity. We denote this time tafterslip , and define

T = trupture + tafterslip . (15)
trupture can be found from equation 14

L = trupture 0 vc ( t )d t (16) = (v c,0 -vc,∞ ) (1 -β)v β c,0 t1-β rupture + vc,∞ trupture , (17)
The afterslip time can also be found analytically, and the detailed calculation is given in the supplementary information. The result is

tafterslip = log(100) 2 L2 ᾱ π 2 (18)
where log(100) indicates that we take the time when 99% of the slip has been accumulated (which is necessary because the afterslip is exponentially decaying). The prediction of seismic moment versus duration can then be found using equation 8 for the seismic moment, equation 17 for trupture (this has to be solved numerically for nonzero τ ), and equation 18 for tafterslip , with T = trupture + tafterslip . The excellent agreement between the analytical approach and the numerical simulations is demonstrated in Figure 4.

We can determine the bound on the slow front scaling relation by noting that for infinitesimal τ , vc,∞ ≈ 0 and M0,1D ≈ L2 2 . This yields Tτ=0 = vc,0 (1 -β)

1 1-β L 1 1-β + log(100) 2 L2 ᾱ π 2 , (19)
where the first term will dominate over the second term (negligible afterslip) for large L because 1 1-β > 2 for the measured β = 0.685 ± 0.002. We can then solve for

L ≈ T 1-β τ =0 (1 -β)v 1-β c,0 (20)
which gives us

M0,slow,1D,τ=0 ≈ L2 2 ∝ T 2(1-β) (21)
with 2(1 -β) ≈ 0.63. We also observe a transition to a different scaling at large M0,1D

when τ is nonzero. To obtain the exponent in this regime, we note that in this limit the steady state rupture velocity is reached, so that

T ≈ L vc,∞ + log(100) 2 L2 ᾱ π 2 . ( 22
)
For large L and nonzero τ , the afterslip will dominate, so that L ∝ T 1 2 . The cubic term in equation 8 will eventually dominate, which results in

M0,slow,1D, L 1,τ >0 ∝ T 3 2 (23)
This means that the moment duration scaling relation in the slow regime is expected to follow a power law with exponent 2(1 -β) with a possible transition to 3 2 at large M0,1D

Discussion

We have demonstrated that a simple Burridge-Knopoff model with Amontons-Coulomb friction is capable of reproducing the range of power law scaling relations between seismic moment and duration observed in nature. The simplicity of the model means that we can calculate the scaling relations analytically, and we find the onedimensional exponents 2(1 -β) with a transition to 3 2 for large seismic moments for slow rupture, 2 for sub-shear rupture, and 3 for super-shear rupture, where β is the power law exponent of the transient slow rupture velocity.

In this letter, we aimed to address two questions. First, whether there is a separation of two distinct classes, or is there a continuum of possible scaling relations between the fast and slow end-members. We argue that the most likely value for τ is close to 0, which corresponds to shear stress at the dynamic level, or to ruptures where the stress drop is small compared to the background stress like in faults [START_REF] Shearer | Comprehensive analysis of earthquake source spectra in southern california[END_REF]. If this is indeed the case, the moment duration scaling should contain a continuum of slip modes between the slow and fast end-members. However, because large τ would in this case be unlikely, it would result in a distinction of fast and slow scalings simply because this is more likely. This would indicate that both the interpretations by [START_REF] Ide | A scaling law for slow earthquakes[END_REF] and by [START_REF] Peng | An integrated perspective of the continuum between earthquakes and slow-slip phenomena[END_REF] hold in the sense that there is a continuum of slip modes, but the natural variation of τ could result in more frequent events along the end-member scalings. In our simulations, the separation into the slow and sub-shear scaling relations occurs spontaneously under the assumption that τ ≈ 0.

The second question we aimed to address was whether a difference in M 0 -T scaling relations alone could be attributed to different physical mechanisms behind

(25) M0,slow,2D ∝ T {3(1-β),2} (26)
where 3(1 -β) ≈ 0.945 is the exponent that is dominant for τ = 0 at large L. This is remarkably close to the hypothesized exponent of 1 from observations [START_REF] Ide | A scaling law for slow earthquakes[END_REF]. However, it should be noted that the prestress it not expected to be radially symmetric, which puts limitations on this extension. We stress that future studies should incorporate two-dimensional simulations to adress these scaling relations without such limitation.

The transition from 3(1-β) to 2 also indicates that a simple linear scaling relation between seismic moment and duration for slow events is not appropriate, because it is only valid at τ = 0. We find it likely that a scaling in the approximate range M 0 ∝ T to T 2 should be observed for slow events, depending also on the decaying exponent β. For a constant ᾱ, this variation in the power law exponent occurs due to changes in the stress state of the interface. This is in line with observations, where different studies have reported on scaling exponents ranging from approximately 1 to 2 [START_REF] Ide | A scaling law for slow earthquakes[END_REF][START_REF] Ide | A brownian walk model for slow earthquakes[END_REF][START_REF] Aguiar | Moment release rate of cascadia tremor constrained by gps[END_REF][START_REF] Frank | Revealing the cluster of slow transients behind a large slow slip event[END_REF][START_REF] Gao | Scaling relationships of source pa-rameters for slow slip events[END_REF].

From our results in Figure 3 we are in a position to explain the observed relation between average rupture speed and seismic moment [START_REF] Gao | Scaling relationships of source pa-rameters for slow slip events[END_REF]. Here, we have assumed that propagation is not bounded. [START_REF] Gomberg | Reconsidering earthquake scaling[END_REF] demonstrated that there will be a change from a two-dimensional scaling to a onedimensional scaling when the rupture propagation goes from unbounded to bounded in one of the directions. While we have demonstrated that different scalings can originate without such effect, a bounded system would add a number of possible transitions in moment duration, and would in principle allow for scaling relations following both the two-dimensional and the one-dimensional exponents.

Linear elasticity and Amontons-Coulomb friction with a viscous term is sufficient to produce a large variety in scaling exponents between seismic moment and duration.

This suggests that different scaling relations for fast and slow slip events do not require complex underlying physical mechanisms. However, our findings do suggest that whether rupture is dominated by inertia or not plays an important role because fast inertial rupture fronts propagate at fairly constant speeds while slow non-inertial rupture fronts contain long-lived transients. Our findings also suggest that there exists a continuum of slip modes between the slow and fast slip end-members, but that the natural selection of stress on faults can cause less frequent events in the intermediate range. We find that the sub-shear scaling follows M 0 ∝ T 2 (which corresponds to T 3 in 2D), while the slow scaling follows T 2(1-β) (which corresponds to T 3(1-β) in 2D) with a transition to T 3 2 (T 2 in 2D) for larger seismic moments depending on the prestress.

β ≈ 0.685 corresponds to the power law decay in the slow rupture velocity with time.

The model also predicts a separate scaling for super-shear rupture with M 0 ∝ T 3 (T 4 in 2D).

Figure 1 .

 1 Figure 1. (a) We solve the one-dimensional Burridge-Knopoff model with Amontons-Coulomb friction with velocity-strengthening dynamic friciction for homogeneously loaded faults. V is the driving velocity, K is the driving spring constant, m is the block mass, k is the spring constant, and fi is the friction force on block i. The friction law is given by a static friction coefficient µs, and a dynamic friction coefficient µ d plus a velocity strengthening term αv (b). To obtain the seismic moment and duration for a given maximum fault length we fix the block at position N .

Figure 2 .

 2 Figure 2. One-dimensional seismic moment M0,1D and event duration T obtained from simulations. The color of the markers show the average front speed vc . The origin of the four different scaling exponents {2(1 -β), 3 2 , 2, 3} is discussed in detail in the text. (a) In the limit of small τ , there is a separation in two distinct scalings for fast and slow events. (b) For large τ the model predicts super-shear rupture, which has a different scaling exponent than regular sub-shear earthquakes. (c) For intermediate τ , the central part of the diagram is populated. (d) Results from the entire range of τ and ᾱ show that moment duration can exhibit a continuum of slip modes in between the slow and fast end-members.

Figure 3 .Figure 4 .

 34 Figure 3. (a) Average propagation speed vc as a function of seismic moment for τ ∈ [10 -7 , 10 -3 ] and ᾱ ∈ [10 -3 , 10]. Yellow markers show fast fronts while blue show slow fronts. Grey lines show predictions for ᾱ = 10 and τ ∈ [10 -5 , 10 -3 ] from equation 14. The prediction for τ = 0 follows vc ∝ M -β 2(1-β) 0,1D

.

  A transient rupture speed with a decaying exponent β would result in a two-Gao et. al[START_REF] Gao | Scaling relationships of source pa-rameters for slow slip events[END_REF] observed that slow events follow the approximate relation vc ∝ M -0.5±0.05 0 , which indicates that β ≈ 0.6 ± 0.025.Using equation 26 yields a moment duration scaling relation for slow rupture following M0 ∝ T {1.1,1.3} , which is fully consistent with their observed linear relationship between seismic moment and duration.

  slow and fast rupture. Our model contains only two dimensionless parameters, which highlights that the observed scaling relations do not necessitate complex underlying mechanisms. The same friction law with different values for the coefficients and a varying prestress can explain the entire range of scaling relations, and the slow scaling regime arises simply because slow rupture speeds are transient. We have previously shown that fast rupture is governed by inertia, while slow rupture is non-inertial

	(Thøgersen et al., 2019), which has consequences for whether the slow and fast scaling
	relations can be attributed to different underlying physical mechanisms. While the
	derivation of the scaling relations presented in this letter does not require specifica-
	tion of the underlying physical mechanism causing transient rupture speeds, transient
	rupture speeds are only observed in the non-inertial regime. This suggests that the
	different scalings observed in the model originate because fast rupture is inertial while
	slow rupture is not.
	To compare our results to observations on faults, it is instructive to discuss
	relations that would be obtained for rupture on a 2D plane. If we can assume radial
	symmetry, we can use the same expression for ū as in 1D, but M0,2D = ū L2 , which
	changes the scaling by a term L. This changes the scaling relations to
	M0,sub-shear,2D ∝ T 3	(24)
	M0,super-shear,2D ∝ T 4

Acknowledgements

We thank David Morgan, Agnès Helmstetter, Michel Bouchon and Yehuda Ben-Zion for providing useful suggestions. K.T acknowledges support from EarthFlows -A strategic research initiative by The Faculty of Mathematics and Natural Sciences at the University of Oslo. H.A.S acknowledges support from the Research Council of Norway through the FRINATEK program, grant number 231621. Code to generate

the data used for the figures in the manuscript is available on Github (Thøgersen, 2019).