Walid Larbi

Jean-François Deü

Roger Ohayon

Absorbing interfaces in structural-acoustic coupled problems

published or not. The documents may come

Introduction

In recent years, considerable research has been devoted to the development and testing of noise reduction techniques by passive damping treatments. Many approaches have been proposed in the literature, in particular to model the absorbing material. These approaches are often based on poroelastic material modeling [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF][START_REF] Atalla | A mixed displacement-pressure formulation for poroelastic materials[END_REF][START_REF] Göransson | A 3-D, symmetric, finite element formulation of the Biot equations with application to acoustic wave propagation through an elastic porous medium[END_REF][START_REF] Davidsson | Structure-acoustic analysis; Finite element modelling and reduction methods[END_REF]. In this work, we focus on the formulation of structural-acoustic problems with interface damping using a wall impedance approach. In this context, let us mention the paper of [START_REF] Kehr-Candille | Elastoacoustic damped vibrations -finite element and modal reduction methods[END_REF], where a frequency-dependent impedance is introduced to describe the absorbing material at the fluid-structure interface, and where a substructuring method is used to solve the dissipative structuralacoustic system. In that work, the fluid is described by a scalar unknown field (pressure or fluid displacement potential) and the problem is numerically solved in frequency domain by the finite element method. Using the same kind of approach, Bermúdez and Rodríguez present in [START_REF] Bermúdez | Modelling and numerical solution of elastoacoustic vibrations with interface damping[END_REF] a finite element method to compute the dynamic response of an elastoacoustic system with dissipative interface subject to external harmonic excitations. In their paper, a displacement formulation is used for both media, requiring a particular attention to the discretization of the admissible class of irrotational motions of the fluid. More recently, an original formulation for interior structural-acoustic dissipative problems, based on the introduction of the normal fluid displacement field at the fluid-structure interface, has been proposed by the authors [START_REF] Deü | Dissipative interface modeling for vibroacoustic problems -A new symmetric formulation[END_REF][START_REF] Larbi | A new finite element formulation for internal acoustic problems with dissipative walls[END_REF]. In the present paper, the main purpose is to establish the link between dissipative models by wall acoustic impedance and by poroelastic approach based on the Biot-Allard theory [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF] (cf. Figure 1).

Structure

(u) The dissipative structural-acoustic coupled problem consists of an elastic structure covered by a thin layer of absorbing material and filled with an inviscid, compress-ible and barotropic fluid, gravity effects being neglected. In order to take the effect of a thin layer of absorbing material at the fluid-structure interface into account, a wall impedance approach is used. The question is here how to model and identify this impedance. A first strategy consists in using experimental measurements in a particular frequency range [START_REF] Pierce | Acoustics: An Introduction to Its Physical Principles and Applications[END_REF]. Another choice, which is developed in this paper, is to use refined poroelactic approaches. In this context, we present in the first part of this paper the finite element modeling of poroelastic absorbing material based on Biot-Allard theory. The associated acoustic wall impedance can then be predicted from this numerical approach or by an analytical expression in the case of laterally infinite poroelastic materials excited by unit amplitude normal incidence plane wave. Next, we present the finite element formulations of structural-acoustic problem with absorbing interface considering (i) general acoustic impedance model and (ii) approximate Kelvin-Voigt model. In this last case, the associated matrix system can be solved in frequency and time domains using direct time integration methods and/or modal reduction approaches.

Fluid (p) Porous medium (u s , p f ) Kelvin-Voigt wall impedance Z(ω) Fluid (p) Structure (u) (a)

Poroelastic modeling of absorbing materials based on Biot-Allard theory

In the literature, several 3D finite element models based on Biot-Allard theory are developed for the forced response of a poroelastic material [START_REF] Göransson | A 3-D, symmetric, finite element formulation of the Biot equations with application to acoustic wave propagation through an elastic porous medium[END_REF][START_REF] Atalla | A mixed displacement-pressure formulation for poroelastic materials[END_REF][START_REF] Davidsson | A reduction method for structure-acoustic and poroelastic-acoustic problems using interface-dependent Lanczos vectors[END_REF]. We recall in this section the pressure/displacement (u s ,p f ) finite element formulation developed by Atalla [START_REF] Atalla | A mixed displacement-pressure formulation for poroelastic materials[END_REF]. This numerical approach will be used to model and identify the normal incidence surface impedance Z(ω).

Finite element formulation

We consider a single isotropic porous material described by the Biot-Allard approach [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF]. This model is defined by (i) five geometrical parameters (the porosity φ, the flow resistivity σ, the tortuosity α ∞ , the viscous characteristic length Λ, and the thermal characteristic length Λ ′ ), (ii) the mechanical characteristics of the skeleton, and (iii) the saturating fluid properties. A detailed description of this model can be found for example in [START_REF] Atalla | A mixed displacement-pressure formulation for poroelastic materials[END_REF].

The Biot-Allard poroelasticity equations are written here in terms of the solid phase displacement u s and the interstitial fluid pressure p f . It is important to note that the small movement of the fluid is not irrotational and the dissipation effects are expressed by complex quantities which depend on the angular frequency ω (cf. signs "~" in the following equations).

For harmonic motion, the local equations of the poroelastic medium are given in terms of (u s ,p f ) by divσ s (u s )+ω 2 ρu s +γ∇p f = 0

[1]

∆p f + ω 2 ρ22 R p f -ω 2 ρ22 φ 2 γdivu s =0 [2]
where σ s is the in vacuo stress tensor of the poroelastic aggregate. Moreover, we have

ρ =ρ 11 -ρ2 12 /ρ 22 , γ = φ(ρ 12 /ρ 22 -Q/ R), Q =( 1-φ) Ke , R = φ Ke
where Ke is the bulk modulus of the fluid phase; ρ11 is the corrected mass density for the solid phase; ρ12 is the inertial coupling factor; and ρ22 is the corrected mass density of the fluid phase. All these parameters are defined in terms of the previously mentioned geometrical and mechanical properties of the porous medium.

From Equations [1] and [2], the variational formulation can be written in the following form

Ω p σ s : ε s (δu s ) dv -ω 2 Ω p ρu s • δu s dv - Ω p γ∇p f • δu s dv - ∂Ω p (σ s n) • δu s ds =0 ∀ δu s [3] Ω p φ 2 ω 2 ρ22 ∇p f •∇δp f dv - Ω p φ 2 R p f δp f dv - Ω p γu s •∇δp f dv + ∂Ω p γu s • n - φ 2 ω 2 ρ22 ∂p f ∂n δp f ds =0 ∀ δp f [4]
where Ω p and ∂Ω p refer to the poroelastic domain and its boundary surface, n is the external normal vector of the boundary ∂Ω p , ε s is the strain tensor of the solid phase.

The discretization of the previous variational formulation leads to the following matrix equation

K - C 0 H U s P f -ω 2 M0 CT G U s P f = F s F f [5]
where U s and P f are the vectors of nodal values of u s and p f respectively, and where the submatrices of Equation [5] are given by

Ω p σ s (u s ):ε s (δu s ) dv ⇒ δU sT KU s ; Ω p ρu s • δu s dv ⇒ δU sT MU s Ω p γ∇p f • δu s dv ⇒ δU sT CP f ; Ω p γu s •∇δp f dv ⇒ δP f T CT U s Ω p φ 2 ρ22 ∇p f •∇δp f dv ⇒ δP f T HP f ; Ω p φ 2 R p f δp f dv ⇒ δP f T GP f
In structural-acoustic problem with absorbing material, this approach leads to a coupling between porous material degrees of freedom and the surrounding fluid and structure degrees of freedom. Therefore, the whole problem is very large and thus computationally time consuming. One of the best ways to avoid this problem is to use a reduced or impedance models as described in the next sections.

Computation of normal incidence surface impedance

The surface impedance of laterally infinite poroelastic material predicted by the presented (u s ,p f ) model is investigated in this section. The configuration under study is depicted in Figure 2. The porous layer, of thickness L =0.1m and material properties given in Table 1, is bonded onto a rigid wall at x =0 . A normal incidence plane wave of unit amplitude excites the absorbing material at x = -L. The normal incidence surface impedance (Figure 3a) is calculated using the displacement of fluid and solid phases, u s and u f respectively, at the input surface (x = -L) for the unit acoustic pressure excitation by the following equation

Z(ω)= 1 iω [φu f (-L)+(1-φ)u s (-L)] [6]
where the displacement of fluid and solid phases are calculated using the previous (u s ,p f ) formulation. In frequency range [0 -300 Hz], the normal incidence surface impedance can be approached by a Kelvin-Voigt model, i.e. sum of a constant real part and an imaginary part inversely proportional to the frequency:

Z(ω)=d I + ik I /ω [7]
The parameters k I and d I (k I ≈ 1.1 × 10 6 Pa/m and d I ≈ 1290 Pa.s/m), characterizing respectively the elastic and the viscous aspect of the absorbing layer, are deduced from the acoustic surface impedance by a least squares method (Figure 3b).

Finite element formulations of structural-acoustic problems with absorbing interface

This section concerns the finite element modeling of structural-acoustic problems with absorbing interfaces. The absorbing material is modeled by a frequency-dependent wall acoustic impedance. We present here a pressure/displacement formulation with (i) general acoustic impedance and (ii) Kelvin-Voigt impedance models.

General acoustic impedance model

We consider an elastic structure occupying the domain Ω S at the equilibrium. The structure is clamped on a part Γ u and subjected to surface force density F d on the complementary part of its external boundary Γ t . The damping interface Σ between the fluid and the structure corresponds to a third domain without thickness. This domain is modeled by a wall acoustic impedance Z(ω). The local equations of this structuralacoustic coupled problem with damped interface can be written in terms of structure displacement u and fluid pressure field p by

divσ(u)+ρ S ω 2 u = 0 in Ω S [8] σ(u)n S = F d on Γ t [9] σ(u)n S = pn on Σ [10] u =0 on Γ u [11] ∆p + ω 2 c 2 F p = 0 in Ω F [12] ∇p • n = ρ F ω 2 u • n + i ρ F ω Z(ω) p on Σ [13]
where n S and n are the external unit normal to Ω S and Ω F ; ρ S and ρ F are the structure and fluid mass densities; c F is the speed of sound in the fluid; and σ is the structure stress tensor.

The variational formulation of the problem is obtained using the test-function method. For this purpose, we introduce the spaces C u and C p of sufficiently smooth functions associated with the field variables u and p respectively.

Let δu be the test function associated to u, belonging to the admissible space

C ⋆ u = {δu ∈ C u | δu = 0 on Γ u }. Multiplying Equation [8] by δu ∈ C ⋆ u
, applying Green's formula, and finally taking Equations [9] and [10] into account, we have:

Ω S σ(u):ε(δu) dv - Σ pn • δu ds -ω 2 Ω S ρ S u • δu dv = Γ t F d • δu ds ∀δu ∈ C * u [14]
Similarly, let δp be the test function, associated to p, belonging to the admissible space C p . Multiplying Equation [12] by δp ∈ C p , applying Green's formula, and finally taking Equation [13] into account, we obtain:

1 ρ F Ω F ∇p •∇δp dv -i ω Z(ω) Σ pδp ds -ω 2 Σ u • nδp ds - ω 2 ρ F c 2 F Ω F pδpdv =0 ∀δp ∈ C p [15]
Thus, the variational unsymmetric formulation of the elastoacoustic problem with interface damping consists, for given appropriate initial conditions, in finding (u,p) ∈ (C ⋆ u ,C p ) such that, ∀(δu,δp) ∈ (C ⋆ u ,C p ) Equations [14] and [15] are satisfied. After discretizing by the finite element method the bilinear forms in Equations [14] and [15], we obtain the following matrix system of the coupled problem:

K u -C up 0K p -i ω Z(ω) 00 0D p -ω 2 M u 0 C T up M p U P = F 0 [16]
where U and P are the vectors of nodal values of u and p respectively; F is the vector of external forces defined by Γ t F d • δu ds ⇒ δU T F; and the real and frequencyindependent submatrices of Equation [16] are given by:

Ω S σ(u):ε(δu) dv ⇒ δU T K u U ; Ω S ρ S u • δu dv ⇒ δU T M u U 1 ρ F Ω F ∇p •∇δp dv ⇒ δP T K p P ; 1 ρ F c 2 F Ω F pδp dv ⇒ δP T M p P Σ pn.δu ds ⇒ δU T C up P ; Σ pδp ds ⇒ δP T D p P

Kelvin-Voigt impedance model

Due to the frequency complex dependence of the acoustic wall impedance Z(ω), the previous unsymmetric formulation can be written only in frequency domain and allows to nonlinear system in terms of angular frequency ω. To avoid this problem, a new finite element formulation based on the introduction of an additional scalar unknown, namely the normal fluid displacement field η at the dissipative interface, has been recently presented by the authors [START_REF] Deü | Dissipative interface modeling for vibroacoustic problems -A new symmetric formulation[END_REF][START_REF] Larbi | A new finite element formulation for internal acoustic problems with dissipative walls[END_REF]. In this case, the acoustic wall impedance Z(ω) is approximated by a Kelvin-Voigt rheological model, i.e. sum of a constant real part and an imaginary part inversely proportional to the frequency: Z(ω)=d I + ik I /ω (see Section 2). The local equations of the structural-acoustic problem with absorbing material can then be written in terms of structure displacement u, fluid pressure p and normal fluid displacement at the interface η:

-Structure divσ(u) -ρ S ∂ 2 u ∂t 2 =0 in Ω S [17] σ(u)n S =F d on Γ t [18] u =0 on Γ u [19] σ(u)n S = k I (η -u • n)+d I ( ∂η ∂t - ∂u ∂t • n) n on Σ [20] -Fluid ∆p - 1 c 2 F ∂ 2 p ∂t 2 =0 in Ω F [21] ∇p • n = -ρ F ∂ 2 η ∂t 2 on Σ [22] -Interface p + k I (u • n -η)+d I ( ∂u ∂t • n - ∂η ∂t )=0 on Σ [23]
The variational formulation of the problem is obtained using the test-function method.

Multiplying Equation [17] by δu ∈ C ⋆ u , applying Green's formula, and finally taking Equations [18] and [20] into account, we have:

Ω S σ(u):ε(δu) dv + k I Σ (u • n)n • δu ds + d I Σ ( ∂u ∂t • n)n • δu ds -k I Σ ηn•δu ds-d I Σ ∂η ∂t n•δu ds+ Ω S ρ S ∂ 2 u ∂t 2 •δu dv = Γ t F d •δu ds [24]
Similarly, multiplying Equation [21] by δp ∈ C p , applying Green's formula, and finally taking Equation [22] into account, we obtain:

1 ρ F Ω F ∇p •∇δp dv + 1 ρ F c 2 F Ω F ∂ 2 p ∂t 2 δp dv + Σ ∂ 2 η ∂t 2 δp ds =0 [25]
Finally, we consider the space C η of sufficiently regular functions η defined in Σ. Multiplying Equation [23] by δη ∈ C η ,wehave:

k I Σ ηδηds -k I Σ u • n δη ds + d I Σ ∂η ∂t δη ds -d I Σ ∂u ∂t • n δη ds - Σ pδηds =0 [26]
Thus, the variational unsymmetric formulation of the elastoacoustic problem with interface damping consists, for given appropriate initial conditions, in finding [25] and [26] are satisfied.

(u,η,p) ∈ (C ⋆ u ,C η ,C p ) such that, ∀(δu,δη,δp) ∈ (C ⋆ u ,C η ,C p ), Equations [24],
After discretizing by the finite element method the bilinear forms in Equations [24], [25] and [26], we obtain the following matrix equation of the coupled system:

⎛ ⎝ M u 00 000 0C T ηp M p ⎞ ⎠ ⎛ ⎝ Ü Ḧ P⎞ ⎠ + ⎛ ⎝ d I G u -d I G uη 0 -d I G T uη d I G η 0 00 0 ⎞ ⎠ ⎛ ⎝ U Ḣ Ṗ⎞ ⎠ + ⎛ ⎝ K u + k I G u -k I G uη 0 -k I G T uη k I G η -C ηp 00 K p ⎞ ⎠ ⎛ ⎝ U H P ⎞ ⎠ = ⎛ ⎝ F 0 0 ⎞ ⎠ [27]
where U, H and P are the vectors of nodal values of u, η and p respectively and the new submatrices are given by

Σ pδηds ⇒ δH T C ηp P ; Σ ∂ 2 η ∂t 2 δp ds ⇒ δP T C T ηp Ḧ Σ (u • n)n • δu ds ⇒ δU T G u U ; Σ ηδηds ⇒ δH T G η H Σ ηn • δu ds ⇒ δU T G uη H ; Σ u • n δη ds ⇒ δH T G T uη U
This formulation has the advantage that it can be solved in frequency and in time domain by introducing only one unknown per node on the damping interface (normal fluid displacement η). On the other hand, it has the disadvantage of yielding unsymmetric matrices. The symmetric formulation can be obtained through the introduction of an intermediate unknown field, namely fluid displacement potential ϕ defined up to an additive constant [START_REF] Morand | Fluid-Structure Interaction[END_REF][START_REF] Deü | Dissipative interface modeling for vibroacoustic problems -A new symmetric formulation[END_REF].

Example: dissipative acoustic pipe submitted to a pressure load

In this example, a pressure load is applied to the left edge of a straight pipe (of length A =5m and width B =0 .5 m) containing air (density ρ F =1kg/m 3 , speed of sound c F = 340 m/s) and with an absorption wall at the right edge (Figure 4). This absorbing boundary is modeled by the previously described wall acoustic impedance Z(ω)=d I + ik I /ω. The damping parameters, given in Section 2.2, are chosen in order to attenuate the vibration amplitudes without shifting the frequencies in a too significant way. The considered harmonic excitation has the following form p d (t)= p 0 sin(2πt/T 1 ) with 1/T 1 = 200 Hz and p 0 = 1000 Pa. Moreover, the transient response of the acoustic pipe is computed using a direct time integration method. Figure 4 shows the frequency responses of the damped and undamped acoustic pipe at the point M (2.5 m,0.4 m). This figure indicates that the damping significantly reduced the first resonant peaks, with the largest reductions achieved after the excitation (at 200 Hz). Moreover, it can be observed that the absorbing boundary caused a small shift of the resonant frequencies. This is due to the spring effect of the Kelvin-Voigt model used for the impedance.

Conclusions

Finite element formulations of interior vibroacoustic coupled problems with absorbing interfaces have been presented in this work. The dissipative interface is modeled by wall acoustic impedance in a limited frequency range. The used technique is based on the prediction of the surface impedance parameters from a Biot-Allard poroelastic model. This approach, combined with an original structural-acoustic finite element formulation developed by the authors, is tested on an acoustic pipe example with damping interface.

Figure 1 .

 1 Figure 1. Structural-acoustic problem with absorbing interface modeled by: a) porous medium, b) acoustic wall impedance

Figure 2 .

 2 Figure 2. Geometry of the normal acoustic impedance problem

Figure 3 .

 3 Figure 3. Real and imaginary parts of normal incidence surface impedance: (a) normal incidence surface impedance computed from Biot-Allard theory, (b) identification of normal impedance parameters k I and d I in frequency range [0 -300 Hz]

Figure 4 .

 4 Figure 4. Straight pipe submitted to a pressure load at the left end

Figure 5 .

 5 Figure 5. Pipe frequency responses to harmonic pressure in the damped and undamped cases

Table 1 .

 1 Physical properties of the porous material φσ (Ns/m 4 ) α ∞ Λ (μm) Λ ′ (μm) E s (kPa) ν s η s ρ s (kg/m 3 )

	.94	40000	1.06	56	110	4400	0 .1	130


Amann Benjamin
email: benjamin.amann@giub.unibe.ch

Szidat Sönke
email: szidat@dcb.unibe.ch

Grosjean Martin
email: martin.grosjean@oeschger.unibe.ch

Benjamin Amann

A millennial-long record of warm season precipitation and flood frequency for the North-western Alps inferred from varved lake sediments: implications for the future

Keywords: Proglacial Lake, varve chronology, sedimentology, climate change, paleoclimatology, paleofloods

The recent warming of the global climate is well recognized. However, does a warmer climate also mean a moister climate? Does dry get drier and wet get wetter? This is an important question as it relates to changes in the water cycle and impacts the water resources as well as the frequency and intensity of storms and floods in the near future. In Europe, regional climate models do not show consistent and robust results for future hydroclimatic changes and how extreme events will evolve in response to future climate change.

Paleo-hydroclimatic data from natural archives are one of the few means to assess such changes in the longer context. Here, we present an annually-resolved record of warm season (MJJA) precipitation and summer flood frequency from the varved (annually laminated) sediments of proglacial Lake Oeschinen (46°30'N -7°44'E, 1580 m, NW Swiss Alps) back to AD 884. These records are inferred from the thickness of annual sediment deposits and the occurrence of flood event layers in the sediments. The chronology of the sediment record is based on multiple varve counts and validated with historical floods chronicled in written documents (back to the 14 th century) and 14 C AMS dates.

. Driest conditions are reconstructed for AD 960 -1080, AD 1250 -1300 and for AD 1880 -1900. Our precipitation record is consistent with the few multi-centennial warm-season precipitation records available for Europe.

We did not find a persistent relationship between warm-season precipitation and temperature. In contrast, results show that the relation between precipitation and temperature has oscillated between positive correlations (warmer gets wetter, cooler gets drier) and negative correlations (warmer gets drier, cooler gets wetter) with a highly significant (χ 2 = 99%) multidecadal (60 -70 yrs) periodicity over the last millennium. Possible explanations for this phenomenon are changes in the weather type statistics or the within-weather-type variability, which influence the combinations between precipitation and temperature over Europe and operate at multidecadal scales.

Such multidecadal effects might also be important for precipitation scenarios in the Alpine area under future warming.

Our record of flood frequency suggests more frequent floods under cool and humid climates.

This picture is consistent with other studies from small and medium size catchments at mid-and high elevations in the Alpine area. However, the 13 th century reveals a period with high flood frequency during warm and moderately dry (average precipitation) conditions. This anomalous situation is currently not understood; nonetheless, this is also one out of several possible scenarios for the future.

From the different combinations found in our record, we conclude that the relation between floods, precipitation and temperature and, in consequence, future projections remain poorly constrained.

Introduction

Our climate is warming. The recent warming is unequivocal and many of the observed changes are unprecedented over decades to millennia. This is one of the key findings of the latest comprehensive scientific assessments of climate change (IPCC 2013) and based on observations and multiple lines of other independent evidence. However, does a warmer climate also mean a moister climate? A theoretical response to the warming is an increase in evaporation and precipitation following the Clausius-Clapeyron relation.

Although at slower rates, [START_REF] Held | Robust Responses of the Hydrological Cycle to Global Warming[END_REF] showed that 'moistening' is the case at global and hemispheric scales. Yet, this thermodynamic relationship between temperature, moisture and precipitation can be complicated at regional scales and a pattern "dry gets dryer and wet gets wetter" is rather expected [START_REF] Held | Robust Responses of the Hydrological Cycle to Global Warming[END_REF][START_REF] O'gorman | How closely do changes in surface and column water vapor follow Clausius-Clapeyron scaling in climate-change simulations[END_REF]. However, in a most recent study [START_REF] Greve | Global assessment of trends in wetting and drying over land[END_REF] found that only 10.8% of the global land area show a robust 'dry gets drier, wet gets wetter' pattern (DDWW paradigm); 9.5% of global land area show robust opposite patterns ('dry gets wetter and wet gets drier'), and for most of the land areas the relationship is not conclusive. Thus, large uncertainties still remain due to differences in hydroclimatic conditions among regions and among seasons, and due to spatial and temporal data scarcity [START_REF] Trenberth | Relationships between precipitation and surface temperature[END_REF][START_REF] Huntington | Evidence for intensification of the global water cycle: Review and synthesis[END_REF]).

Changes in the hydroclimate are most relevant since they relate to changes in regional water resources directly. Moreover, changes in the hydroclimate may also lead to changes in the frequency and intensity of storms and floods in the near future [START_REF] Köplin | Seasonality and magnitude of floods in Switzerland under future climate change[END_REF]. Changes in the properties of such extreme events are of particular concern to society, especially in vulnerable Alpine areas [START_REF] Ipcc | Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation. A Special Report of Working Groups I and II of the Intergovernmental Panel on Climate Change[END_REF][START_REF] Christensen | Climate Phenomena and their Relevance for Future Regional Climate Change[END_REF][START_REF] Smith | Environmental hazards: Assessing risk and reducing disaster[END_REF]). However, it remains poorly understood how changes in temperature, precipitation and regional synoptic-scale weather patterns interact and how properties and statistics of extreme events (floods but also droughts) might eventually change. In Europe for instance, future projections of strong 72 h rainfall events from high-resolution regional climate models do not show a consistent and robust climate change signal (CH2011, 2011. This leads to the following questions we want to address in the present study:

 Has the climate in the region of the NW Swiss Alps responded according to the Clausius-Clapeyron relation (positive correlation between temperature and precipitation) and to the 'wet gets wetter' scheme over the past millennium? Or is the climatological relation between temperature and precipitation not stationary over time and modulated by other effects?

 Was there a relationship between flood frequency, air temperature and precipitation in the past? For instance, were floods more frequent in warmer, colder, wetter or drier climates?

Over the past few decades, the study of past precipitation variability has received relatively little attention, while considerable efforts have been devoted to investigate past temperature changes at regional scales (PAGES 2k, 2013). This can be attributed to the paucity of precipitation-sensitive proxies and a lack of data. This makes comprehensive assessments of long-term regional changes in precipitation conditions difficult. This data situation has been described as "low confidence", "geographical inconsistency", "difficult spatial interpretations", "syntheses too limited to support regional assessment", or "substantial uncertainty due to large regional differences" [START_REF] Büntgen | Assessing the spatial signature of European climate reconstructions[END_REF]Wilson2013;[START_REF] Masson-Delmotte | Information from Paleoclimate Archives[END_REF]PAGES 2k 2013).

Scarcity of data is also the case in Europe where only a few multi-centennial-to millennial-long precipitation reconstructions exist. These were produced from documentary data and moisturesensitive tree rings mainly [START_REF] Brázdil | Fir tree-ring reconstruction of March -July precipitation in southern Moravia (Czech Republic), 1376 -1996[END_REF][START_REF] Linderholm | Central Scandinavian winter precipitation variability during the past five centuries reconstructed from Pinussylvestris tree rings[END_REF][START_REF] Wilson | A 500 year dendroclimatic reconstruction of springsummer precipitation from the lower Bavarian Forest region, Germany[END_REF][START_REF] Wilson | A millennial long March-July precipitation reconstruction for southern-central England[END_REF][START_REF] Jönsson | Scots Pine (pinussylvestris L.) on Shingle Fields: A Dendrochronologic Reconstruction of Early Summer Precipitation in Mideast Sweden[END_REF][START_REF] Büntgen | 2500 years of European climate variability and human susceptibility[END_REF][START_REF] Dobrovolný | Precipitation reconstruction for the Czech Lands, AD 1501-2010[END_REF], and from a few lake sediment records [START_REF] Trachsel | Mineralogy-based quantitative precipitation and temperature reconstructions from annually laminated lake sediments (Swiss Alps) since AD 1580[END_REF][START_REF] Romero-Viana | Reconstruction of annual winter rainfall since A.D.1579 in central-eastern Spain based on calcite laminated sediment from Lake La Cruz[END_REF].

Besides, growing attention has been paid to the reconstruction of floods and flood frequency in Europe, notably using lake sediments. Lakes are excellent natural archives that offer the great opportunity to study past flood events [START_REF] Glur | Frequent floods in the European Alps coincide with cooler periods of the past 2500 years[END_REF]. Detrital flood layers found in the sediment cores can be linked directly to flood events recorded in instrumental or documentary sources [START_REF] Debret | North western Alps Holocene paleohydrology recorded by flooding activity in Lake Le Bourget, France[END_REF][START_REF] Giguet-Covex | Frequency and intensity of high-altitude floods over the last 3.5 ka in northwestern French Alps (Lake Anterne)[END_REF]Wilhelm et al. 2012a[START_REF] Wilhelm | Palaeoflood activity and climate change over the last 1400 years recorded by lake sediments in the north-west European Alps[END_REF][START_REF] Gilli | Lake sediments as archives of recurrence rates and intensities of past flood events[END_REF]Wirth et al. 2013b).

Further, particularly valuable records have been uncovered from varved (annually laminated) lake sediments. These varves provide up-to-seasonal resolutions and may reduce chronological uncertainties significantly [START_REF] Zolitschka | Varved lake sediments[END_REF][START_REF] Czymzik | A 450-year record of spring-summer flood layers in annually laminated sediments from Lake Ammersee (Southern Germany)[END_REF][START_REF] Czymzik | Orbital and solar forcing of shifts in Mid-to Late Holocene flood intensity from varved sediments of pre-alpine Lake Ammersee (southern Germany)[END_REF][START_REF] Stewart | Reconstructions of late Holocene paleofloods and glacier length changes in the Upper Engadine, Switzerland (ca. 1450 BC-AD 420)[END_REF][START_REF] Swierczynski | A 1600 yr seasonally resolved record of decadal-scale flood variability from the Austrian Pre-Alps[END_REF][START_REF] Swierczynski | Mid-to late Holocene flood frequency changes in the northeastern Alps as recorded in varved sediments of Lake Mondsee (Upper Austria)[END_REF]Wirth et al. 2013a;[START_REF] Corella | Annually-resolved lake record of extreme hydro-meteorological events since AD 1347 in NE Iberian Peninsula[END_REF].

Recently, Lake Oeschinen, a high elevation (1580 m a.s.l.) proglacial varved lake in the Northwestern Swiss Alps has shown to be an excellent archive to record past warm-season precipitation and flood frequency variability with a reliable and precise varve chronology [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF]. For the calibration period 1901 -2008, varve thickness was found to be a quantitative predictor for warm season (MJJA and MJ) precipitation (rMJJA = 0.6; rMJ = 0.7, p <0.01, n = 108). This calibration model 'varve thickness -precipitation' is the basis for the precipitation reconstruction back to AD 884 presented in this study. The aims are to investigate the relationship between warm season precipitation, temperature and flood frequency over the past millennium. For this purpose, we will first extend the varve thickness record from Lake Oeschinen to reconstruct past spring/summer precipitation conditions in the North-western Swiss Alps. Then, we will compare this record with temperature variability reconstructed from tree ring data for the same region. Finally, we will use the occurrence of flood layers in the sediment of Lake Oeschinen to investigate, for the first time, the long-term relationship between flood frequency and the amount of precipitation from the same location and the same natural archive.

Study site and sediment formation

Lake Oeschinen (46°30'N, 7°44'E) is a proglacial lake situated at 1580 m a.s.l. in the Northwestern Alps, 55 km south of Bern, Switzerland (Fig. 1a). A Holocene rockslide formed the lake basin [START_REF] Niklaus | Geomorphologische und limnologische Untersuchungen am Öschinensee[END_REF]. The 56-m deep lake has a sub-surface outflow through the rockslide and drains into the Kander River (Figs. 1b andd). In winter, ice cover develops from December to late April generally.

Mesozoic limestone predominates in the catchment; however, it is interrupted by two sizable belts of Tertiary Flysch deposits (Fig. 1d) that consist of easily erodible siliciclastic-rich sandstones [START_REF] Suchy | Vitrinite reflectance and shear-induced graphitization in orogenic belts: a case study from the Kandersteg area, Helvetic Alps, Switzerland[END_REF]. Glacier cover accounts for 30% of the entire lake catchment.

The mineralogical composition and distribution of the bedrock (limestone in the high-elevation glaciated areas, siliciclastics in the lower elevation Flysch areas) is important for the varve formation [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF]. During spring snowmelt and summer rainfall, a coarse-grained dark basal layer is formed. This layer is enriched in siliciclastics eroded from the northern Flysch catchment. During winter, clay-size bright calcite particles settle through the water column under very calm conditions when the lake is frozen. These fine particles are produced by glacier abrasion in the limestone part of the catchment. The winter layer has very little influence on the varve thickness [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF].

Climate variability in the North-western Alps is influenced by both Atlantic and continental regimes with the North Atlantic Oscillation (NAO) as the dominant climate mode [START_REF] Wanner | Global climate change and variability and its influence on Alpine climate -concepts and observations[END_REF][START_REF] Sodemann | Seasonal and inter-annual variability of the moisture sources for Alpine precipitation during 1995-2002[END_REF]. Highest amounts of precipitation are recorded between May and August (nearby station at Kandersteg; Fig. 1d) with 1173 mm of total annual precipitation. Monthly mean temperatures range from -2.9°C in January to 15°C in July (AD 1900(AD -2009;;MeteoSwiss). Local thunderstorms associated with heavy rainfall are relatively frequent in the region (Schmocker-Fackel and Naef 2010). These heavy rainfall events occur within a few hours and predominate from May to August (MJJA), which is the season of interest for our study. Summer rainstorms induce high daily river discharge which is chronicled by the gauging station Kander-Hondrich (Fig. 1d).

Material and methods

Sediment core collection and sampling

Sediment cores with overlapping segments were retrieved with an UWITEC piston corer from the deepest point of Lake Oeschinen in August 2011. The coring site was chosen based on bathymetric and seismic surveys (Fig. 1b,c). The core segments were split lengthwise. A master composite core of 580 cm length was generated by stratigraphic correlation of key sediment strata from visual core inspection and high resolution pictures. One core-half was used to develop the chronology (varve counting and 14 C AMS dates). The other core-half was used to establish the multi-proxy data set.

The core was sliced at 1-cm resolution for geochemical data. Water content was calculated after freeze-drying for 24 h. Particle-size distributions of homogenized samples were derived from laser diffraction (Malvern Mastersizer 2000) after removal of organic matter by H2O2. Organic carbon and calcite contents were obtained from loss on ignition (LOI) at 550°C and 950°C respectively, following [START_REF] Heiri | Loss on ignition as a method for estimating organic and carbonate content in sediments: reproducibility and comparability of results[END_REF]. The residual after LOI950 is summarized as the siliciclastic fraction (mainly quartz, feldspar and clay minerals).

The relative proportions of the two main sediment fractions (calcite and siliciclastics) was also measured by Visible Reflectance Spectroscopy scans (VIS-RS, 380 -730 nm) using a portable Gretag-Spectrolino (GretagMcBeth, Switzerland). We used the mean 1 st -derivative of each VIS-RS spectrum (overall spectral slope) and calibrated these to the LOI950-based calcite/siliciclastics ratios [START_REF] Debret | Spectrocolorimetric interpretation of sedimentary dynamics: The new "Q7/4 diagram[END_REF]. This allows us to rapidly infer proportions of calcite(bright)/siliciclastics(dark) from the entire sediment sequence at a very high (2 mm) resolution (see supplementary material S1 for more details).

Sediment dating

Thin sections were taken from overlapping sediment slabs (7 x 1.5 x 0.5 cm) using freeze-drying and resin-epoxy embedding techniques [START_REF] Lamoureux | Varve chronology techniques[END_REF]. Laminae identification and turbidite distinction followed [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF]. The master chronology was generated by collating and merging optical microscopy pictures (magnification x10) of the thin sections. The thickness of the varves (laminae couplets) was semi-automatically measured using the software package WinGeol Lamination Tool (0.01-mm resolution; [START_REF] Meyer | The wingeol lamination tool: new software for rapid, semiautomated analysis of laminated climate archives[END_REF]. Varve counting uncertainties were assessed from three independent counts. The sediment chronology for the upper 50 cm of the core had already been established using a 210 Pb-CRS model constrained with the AD 1963 137 Cs peak, which was consistent with varve counting [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF]. To validate the varve counting in the rest of the sequence, we used six terrestrial plant macrofossils for 14 C dating using Accelerator Mass Spectrometry at the LARA Laboratory of the University of Bern (Szidat et al. 2014). Macrofossils consisted of 4-to 10-mm leaf fragments, fragments of deciduous periderm and other wooden remains (Table 1). Two samples were replicated. 14 C ages were calibrated using the Intcal13 calibration curve [START_REF] Reimer | Intcal13 and Marine13 radiocarbon age calibration curves 0-50,000 yearscal BP[END_REF]). In addition, the varve chronology was validated by prominent flood layers in the sediments which were compared with spring-summer flood disasters documented in written archives of the Kander region back to AD 1480 [START_REF] Pfister | Wetternachhersage: 500 Jahre Klimavariationen und Naturkatastrophen[END_REF][START_REF] Bütschi | Gefürchtet, gebändigt und neu gedacht -die Kander[END_REF].

Statistics and mapping

With a reliable and precise chronology, we applied the calibration model 'varve thicknessprecipitation' developed by [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF] to the 580-cm sediment profile. We used inverse linear regression (OLS) from the varve thickness record to reconstruct warm season (MJ and MJJA) precipitation conditions back in time.

In addition to the annual resolution, a triangular filter of 31 years was also applied to emphasise decadal-multidecadal variability (climatology). Significance tests (p-values) were corrected for serial autocorrelation (pc; [START_REF] Bayley | The ''effective'' number of independent observations in an autocorrelated time series[END_REF].

We compared our reconstruction with other multi-centennial European precipitation records to assess the regional coherency. However, the various precipitation records differed in the seasons reconstructed and represent different areas across Europe. Because it is unclear how the different records are expected to relate to each other, we produced spatial correlation maps for the proxy sites and the respective seasons targeted. For instance, how May-to-August precipitation from Kandersteg (our study) correlates to April-to-June precipitation reconstructed for Central Europe [START_REF] Büntgen | 2500 years of European climate variability and human susceptibility[END_REF]. For this purpose, we used the dataset CRU TS 3.21 [START_REF] Jones | CRU TS3.21[END_REF] and mapped the correlations at 0.5° x 0.5° spatial resolution.

To investigate the synoptic weather patterns that may trigger floods in Lake Oeschinen and the North-western Swiss Alps, we produced European-scale composites of wind vector fields, precipitation rates and sea level pressure conditions from nine well-documented flood days between 1981 and 2007. To explore to which extent the composites of the daily heavy rainfall events can be scaled-up to very wet seasons, we also produced these composites for the mean May-to-August conditions of the corresponding years for comparison. Here, we used the 20 th Century Reanalysis dataset.

Results

Lithological description and multi-proxy data set

Mineroclastic sediments recovered from Lake Oeschinen generally consist of rhythmically laminated silty loam, intercalated by several dark and thick (up to 46 mm) sandy layers (Fig. 2a). The sediment composition (proportion of wt calcite/ wt siliciclastics) stays fairly stable throughout the entire section with a mean ratio of 3.1 (Fig. 2b). Mean water contents amount to 29.2 % with minor fluctuations (3.4 % standard deviation). The wt clay/ wt sand proportion fluctuates slightly around the mean value of 2.5 from the top of the core to approx. 300 cm sediment depth, and drops to 0.5 in the bottom section that is dominated by sands. This leads to a mean grain size that ranges between 32.9 µm for the core section 0 -300 cm and 80.6 µm for the bottom section 300 -580 cm.

The thin sections and layer counting (validated by 14 C dates; see section 4.2) revealed that the record is continuously varved. These varves consist of layer couplets, whereby the layers of graded sand-to-silt deposits are systematically capped by a layer with clay-size particles. An average thickness of 2.7 mm with a standard deviation of 1.1 mm can be reported. Three sub-types of varves were found that differ in their internal structure (Fig. 2a). Type I is characterized by very thin (1-1.5 mm) fining upward sequences; Type II is thicker (>3 mm) and comprises coarser particles; while multiple internal micro-laminations compose varve Type III.

Flood layers are distinguished from the varves in the thin sections (Fig. 2a). These special layers show a high sedimentation with systematic grading of the particle sizes from sand to clay and they often show an erosive base. Fig. 2b shows that flood layers are also marked by a darker colour (core stratigraphy), low wt calcite/ wt siliciclastics proportions (<2.5), low water content and high proportions of sand-size material. Comprehensively, these criteria allows us to distinguish a flood layer (Facies B; Fig. 2a) from a thick varve of type II (Facies A; Fig. 2a). This is of prime importance for the construction of the chronology.

Sediment chronology

Fig. 3 shows the age-depth model from varve counting placing the base of the core to AD 884 ± 22. The varve counting uncertainty amounts to 9 years from the top of the core to the year AD 1450 (300-cm depth); it reaches a maximum of 30 varve years (c. 2.5%) in the lowest core section where the varves are less well preserved. The robustness of the varve chronology of the top section can be validated by documentary data on floods starting from AD 1480 onwards (Fig. 3). The comparison shows consistency between the varve-ages of the flood layers found in Lake Oeschinen sediments and the ages of the flood disasters documented in written sources from the Kander River.

The overall chronology inferred from varve counts is consistent with the independent radiometric chronology; i.e., thirty points for 210 Pb, five out of six 14 C dates and two 14 C replicates. The replication of the first two radiocarbon samples ( 14 C1 and 14 C2) shows slightly different probability structures (uncertainty) of the calibrated ages. One 14 C date ( 14 C5) was excluded from the record because it conferred a too old and inconsistent date, suggesting reworked material. The varve chronology, validated by 14 C dates, indicates stable sedimentation rates throughout the entire sediment record. 2. Significant positive correlations (pc < 0.01) were found between lake Oeschinen (this study) and the Greater Alpine Area precipitation (data from [START_REF] Casty | Temperature and precipitation variability in the European Alps since 1500[END_REF], Central Europe (data from [START_REF] Büntgen | 2500 years of European climate variability and human susceptibility[END_REF]) and S England/N France (data from [START_REF] Wilson | A millennial long March-July precipitation reconstruction for southern-central England[END_REF]).

However, correlation tests did not show significant correlations between Lake Oeschinen precipitation and precipitation in SE Germany (data from [START_REF] Wilson | A 500 year dendroclimatic reconstruction of springsummer precipitation from the lower Bavarian Forest region, Germany[END_REF], and with the Czech Lands particularly (data from [START_REF] Dobrovolný | Precipitation reconstruction for the Czech Lands, AD 1501-2010[END_REF]. The correlation patterns from the reconstructions are consistent with the spatial correlation maps derived from 20 th Century Reanalysis data (maps in Fig. 4). Fig. 5a shows the comparison between the warm season precipitation reconstruction from Lake Oeschinen and Greater Alpine summer temperature (31-year filtered data). Over the entire last millennium, there is no relationship between the two time-series (r = -0.09). However, on shorter (multidecadal) time scales, phases with strong positive correlations ('warmer gets wetter -colder gets drier') alternate with phases of strong negative correlations ('warmer gets drier and colder gets wetter'). 31-yr running correlations between warm-season precipitation and temperature oscillate between negative values (up to r = -0.6) and positive values (up to r = 0.5) with a highly significant (Chi 2 99%) periodicity between 58 -73 years throughout the last millennium (Fig. S2, supplementary material). Furthermore, periods with cool and wet conditions coincide with advances of glaciers in the region (around AD 1100, AD 1220, AD 1600, the 17 th century, and the 1 st half of the 19 th century), while warm and dry conditions lead to glacier retreats (Fig. 5a). Seven of these eight flood periods coincide with wetter and cooler conditions in the Alps, while only the period P2 (AD 1190-1240) exhibits high flood frequencies during generally very warm conditions (among the warmest during the last millennium). No significant correlations (p> 0.05) were found between the thickness of flood layers (often interpreted as flood intensity) and precipitation (r = -0.053) or temperature (r = -0.083).

Paleoflood record

Discussion

Sediment properties -Is the calibration model applicable?

An important issue in paleoclimate studies is the assumption of stationarity; i.e. the question whether or not the proxy-climate calibration model established and validated in the calibration period can be extended back in time. Moreover, the question arises whether or not a calibration of a single proxy (here varve thickness) to a single climate variable (here precipitation) for a particular season (here warm season) is appropriate or might mask the 'true geo-ecological complexity' of varve formation. The dilemma is that a common denominator (climate variable for a given season) must be determined for validation purposes, for comprehensive multi-site and multi-proxy regional climate reconstructions (e.g. PAGES 2k, 2013) or for data-model comparisons.

Several studies have demonstrated the complexity of lacustrine systems and how sediment formation is influenced by a combination of several variables (temperature, precipitation, periods of snow accumulation, glacial melt, among others) rather than a single one [START_REF] Cockburn | Century-scale variability in late-summer rainfall events recorded over seven centuries in sub annually laminated lacustrine sediments, White Pass, British Columbia[END_REF][START_REF] Hodder | Glaciolacustrine varved sediment as an alpine hydroclimatic proxy[END_REF][START_REF] Kaufman | Long-term river discharge and multidecadal climate variability inferred from varved sediments, southwest Alaska[END_REF]. This may also apply to the varves of Lake Oeschinen, notably for specific climate periods such as the Medieval Climate Anomaly (MCA) or the Little Ice Age (LIA). [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF] have shown that the calibration model 'varve thickness -precipitation' is valid if precipitation in the ice-free season falls mainly as rain (not as snow). In consequence, the model is valid if the sediments of the summer layer are mainly composed of silty sand with limited proportions of clay. Therefore, threshold levels for the calibration model were established as wt% clay/ wt% sand <10 and mean grain size >10 µm [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF]). These conditions are fulfilled for most of the sediment sections considered here (thresholds shown as red vertical dashed lines in Fig. 2b). The horizontal blue line in Fig. 2b highlights the 7-year period (AD 1907(AD -1913) ) where the sediment data exceed these thresholds and the calibration model does not apply [START_REF] Amann | Quantitative high-resolution warm season rainfall recorded in varved sediments of Lake Oeschinen, northern Swiss Alps: calibration and validation AD 1901-2008[END_REF]. Moreover, the calibration period spans a very broad hydroclimatic range from very dry to very humid conditions. This range covers most of the conditions reconstructed for the past millennium except the very dry phases around AD 1000, AD 1300 and AD 1890. These droughts are outside the calibration range and, thus, their amplitudes remain uncertain.

Sediment chronology

The good preservation of varves in Lake Oeschinen allows for a precise and reliable chronology which is crucial when aiming at high-resolution reconstructions (Fig. 3). The maximum varve counting uncertainty is rather small (c. 2.5 %) and on the order of other high-quality varve chronologies [START_REF] Ojala | Characteristics of sedimentary varve chronologies -a review[END_REF].

Of particular interest is the comparison between the varve counts and 14 C dates. As it is expected for young 14 C dates, 14 C1 and 14 C2 show a very large range of possible calibrated ages (Fig. 3). More precisely, they show complex structures of probability with two to three relatively narrow discrete possible time-windows, while the age mid-point (50% summed probability) is highly unlikely (Fig. 3). In this case, neither the mid-point nor the time window with the highest probability allows for an unambiguous determination of the correct calendar age from the complex probability density functions. To test different possibilities, we replicated the two first 14 C dates. Although the replicate 14 C1 confirms the three narrow discrete possible ages, the highest probability (arrows in Fig. 3) moves from the middle cluster to the oldest cluster of possible ages and matches with the varve chronology.

A similar effect is observed in replicate 14 C2: one of the two possible time-windows disappeared in the replicate and was transformed into a unique well-confined age cluster with 95.4 % probability. This is a very good tie point for the 14 C chronology (Table 1), which suggests that the varve chronology and varve counting uncertainty are very robust.

In addition, the reliability of the varve chronology is confirmed by the consistency between flood layers and documentary data. More than 90 % of the thick flood layers found in the sediments of Lake Oeschinen can be related to flood disasters documented in written archives from the Kander River (Fig. 1d). The difference between the 'best varve age' and the 'real age' (documentary) of these marker floods is typically smaller than ± 5 to 6 years, which remains well within the varve counting uncertainty (± 9 years until AD 1480).

Precipitation reconstruction -Reliability and regionalisation

Here we provide the first millennial annually resolved quantitative warm-season precipitation reconstruction for the North-western Alps. The spatial correlation map of May-to-August precipitation shows that the reconstruction from Lake Oeschinen is representative for most of continental Western Europe except for the Mediterranean (Fig. 4a).

Our reconstruction is consistent with independent precipitation data produced from long instrumental data and documentary proxies for the same area back to AD 1659 (Casty et al. 2005, grid cell: 46.25N/6.75E; Fig. 4a and Table 2).

Figs. 4b to 4e show the comparisons of our record with (i) Central European April to June (AMJ) precipitation reconstructed using fir trees [START_REF] Büntgen | 2500 years of European climate variability and human susceptibility[END_REF]), (ii) Southern England/Northern France March to July (MAMJJ) precipitation reconstructed from oak trees [START_REF] Wilson | A millennial long March-July precipitation reconstruction for southern-central England[END_REF]), (iii) Southern Bavaria March to August (MAMJJA) precipitation reconstructed from oak trees [START_REF] Wilson | A 500 year dendroclimatic reconstruction of springsummer precipitation from the lower Bavarian Forest region, Germany[END_REF], and (iv) Czech Lands June to August (JJA) precipitation reconstructed from documentary indices and long homogenized instrumental series [START_REF] Dobrovolný | Precipitation reconstruction for the Czech Lands, AD 1501-2010[END_REF].

The maps in Fig. 4 show the 20 th century spatial correlations between warm season May to August precipitation at Lake Oeschinen (black arrows; our study) and precipitation for the respective seasons reconstructed at the sites used for comparison (black boxes). These correlation maps help assess to which degree one would expect consistency between the record of Lake Oeschinen (this study) and the records used for comparison (Fig. 4 and Table 2).

The best correlation and regional consistency with our reconstruction is found for AMJ precipitation reconstructed for Central Europe [START_REF] Büntgen | 2500 years of European climate variability and human susceptibility[END_REF]. Similarities between the two records are found for the wetter periods I, III, V, VI (Fig. 4a,4b). The wet periods in the 12 th century (period II) and around 1600 AD (period IV) are not present in the [START_REF] Büntgen | 2500 years of European climate variability and human susceptibility[END_REF] record but documented in the record of Northern France [START_REF] Wilson | A millennial long March-July precipitation reconstruction for southern-central England[END_REF]. Our reconstruction compares also well with hydroclimatic conditions in Southern England/Northern France, especially for the wet periods II, IV, V, VI (Table 2; Fig. 2c). Similarly, this reconstruction shows a wetting trend after AD 1950. Finally, precipitation reconstructed in the North-western Alps (this study) does not correlate with the two more continental records from Bavaria and the Czech Lands (Table 2; Fig. 2d and2e). This is coherent with results from the 20 th century correlation maps and particularly true for the Czech Lands. This supports the idea that the West-East moisture gradients over Europe and also the different sources of moisture [START_REF] Wanner | Global climate change and variability and its influence on Alpine climate -concepts and observations[END_REF][START_REF] Sodemann | Seasonal and inter-annual variability of the moisture sources for Alpine precipitation during 1995-2002[END_REF] operated in a similar way during the past 500 years.

The warm season precipitation reconstruction from Lake Oeschinen does not show any relationship with summer temperature reconstructed from the European Alps [START_REF] Büntgen | Summer Temperature Variations in the European Alps, A.D. 755-2004[END_REF] over the past millennium (Fig. 5a). This suggests that neither the Clausius-Clapeyron relation nor a 'dry gets dryer -wet gets wetter' scheme does apply in its pure form [START_REF] Held | Robust Responses of the Hydrological Cycle to Global Warming[END_REF], and that the relation between precipitation and temperature (PP-TT) might be more complicated at regional scales [START_REF] Greve | Global assessment of trends in wetting and drying over land[END_REF]. Our data suggest that the PP-TT relation in the NW Alps was persistently cyclic (multi-decadal ca. 60 -70 years) throughout the past millennium (Fig. S2, supplementary material). This observation is in line with the finding by Knudsen et al. (2014), who have reported increasing evidence for multidecadal oscillations in paleoclimate archives in the Atlantic realm, and assigned this feature to the Atlantic Multidecadal Oscillation (AMO). At the current state of research, it is premature to speculate about possible mechanisms how AMO would modulate the PP-TT relation in the NW Alps. One possible explanation is a change in the frequency of weather types, another possible explanation might be found in a large within-weather-type variability [START_REF] Jacobeit | Atmospheric circulation variability in the North-Atlantic-European area since the mid-seventeenth century[END_REF][START_REF] Küttel | Multidecadal changes in winter circulation-climate relationship in Europe: frequency variations, within-type modifications, and long-term trends[END_REF][START_REF] Küttel | Multidecadal changes in winter circulation-climate relationship in Europe: frequency variations, within-type modifications, and long-term trends[END_REF] reported that "[…] large parts of the changes in European temperatures and precipitation are likely due to within-type variations […]". These authors also found that the within-type variability operates at multidecadal scales and they emphasized the role of North Atlantic sea surface temperatures, which are known to influence European climate strongly. Our data also show that multidecadal cyclicity persisted in pre-industrial time, which would argue against an anthropogenic source for this kind of variability and agree with the conclusions from [START_REF] Küttel | Multidecadal changes in winter circulation-climate relationship in Europe: frequency variations, within-type modifications, and long-term trends[END_REF] and Knudsen et al (2014).

Furthermore, the 'precipitation -temperature' relationship for the past millennium provides novel insight into the climatic conditions that governed glacier length variations of two well documented glaciers in that region: the Great Aletsch Glacier [START_REF] Holzhauser | Glacier and lake-level variations in west-central Europe over the last 3500 years[END_REF]) and the Lower Grindelwald Glacier [START_REF] Nussbaumer | Alpine climate during the Holocene: a comparison between records of glaciers, lake sediments and solar activity[END_REF]Fig. 5a). Our data suggest that substantial glacier advances in the 12 th century, in the 14 th century, in the 17 th and 19 th centuries coincided with cool and wet periods. This is in great detail consistent with the findings by [START_REF] Steiner | Sensitivity of European glaciers to precipitation and temperature -two case studies[END_REF], who analysed the sensitivity of the Lower Grindelwald Glacier to seasonal precipitation and temperature since the 16 th century. In the context of the Maunder Minimum (AD 1645(AD -1715;;[START_REF] Luterbacher | The Late Maunder Minimum (1675-1715) -A key period for studying decadal scale climatic change in Europe[END_REF], it is interesting to note that the extremely cold conditions around AD 1690 did not lead to a massive advance due to very dry conditions during that period.

Climatic control on flood frequency

The millennial-long flood frequency record from Lake Oeschinen reveals that, in the Northwestern Alps, intense rainfall events were generally more frequent during wet and cool summers. This is true for seven of the eight periods with enhanced flood frequencies since AD 884. A positive relation between flood frequency and wet periods is plausible since saturated soils during long rainy seasons lead to enhanced surface runoff and erosion. Moreover, frequent cyclonic weather with moisturebearing westerly and north-westerly flow to the Alps lead to cooler summers.

Our results support recent findings from an increasing number of studies in the Swiss, German, French and Austrian Alps showing that floods were consistently more frequent during cool periods (Smocker-Fackel and Naef 2010;[START_REF] Stewart | Reconstructions of late Holocene paleofloods and glacier length changes in the Upper Engadine, Switzerland (ca. 1450 BC-AD 420)[END_REF][START_REF] Giguet-Covex | Frequency and intensity of high-altitude floods over the last 3.5 ka in northwestern French Alps (Lake Anterne)[END_REF][START_REF] Swierczynski | A 1600 yr seasonally resolved record of decadal-scale flood variability from the Austrian Pre-Alps[END_REF]Wilhelm et al. 2012a;[START_REF] Glur | Frequent floods in the European Alps coincide with cooler periods of the past 2500 years[END_REF]Wirth et al. 2013b). However, due to the lack of robust regional precipitation reconstructions, the relation between flood frequency and moist climate has remained speculative or information was inferred from low-resolution lake level changes. For the first time, our precipitation reconstruction allows for a direct comparison. Our results show that flood frequency increased during wetter conditions for most of the time during the last millennium. The period around AD 1200 (P2 in Fig. 5a) is enigmatic in the sense that phases with high flood frequency coincided with very warm summer temperatures and relatively dry conditions. Although this flood peak is missing in the Central and eastern Swiss compilation [START_REF] Glur | Frequent floods in the European Alps coincide with cooler periods of the past 2500 years[END_REF], it is present in the French Alps (Wilhelm et al. 2012b[START_REF] Wilhelm | Palaeoflood activity and climate change over the last 1400 years recorded by lake sediments in the north-west European Alps[END_REF] and seems to be a regional phenomenon. It is also suggested that different synopticscale weather patterns with rather convective rain might have triggered flood layers during this warm and relatively dry period in the Medieval Climate Anomaly (MCA). In contrast to all the other lakes studied in the Alps, Lac Blanc-Chamonix in the NW French Alps is the only lake that recorded more frequent floods during generally warm periods (MCA and 20 th century; [START_REF] Wilhelm | Palaeoflood activity and climate change over the last 1400 years recorded by lake sediments in the north-west European Alps[END_REF].

In contrast to Wilhelm et al. (2012a), we could not find a statistical relationship between flood layer thickness (in the sediments) and flood/discharge intensity during the instrumental period (Kander river data through the 20 th century, Fig. 1). In addition, no significant correlation could be found between flood layer thickness and summer precipitation (r = -0.05) or temperature (r = -0.08) during the past millennium. However, according to [START_REF] Jenny | A 4D sedimentological approach to reconstructing the flood frequency and intensity of the Rhône River (Lake Bourget, NW European Alps)[END_REF], one might argue that, due to inhomogeneities of the flood deposit geometry in Alpine lakes, our results based on one sediment core might not be conclusive for this specific purpose.

To better understand what triggers flood events at Lake Oeschinen, we explored synoptic-scale weather-types using composites of wind vectors, sea level pressure fields and precipitation rates for nine well documented flood days between 1981 and 2007 (Fig. 6a). The composites for the respective seasons with the flood events are shown in Fig. 6b. The daily and seasonal composites show enhanced zonal flow and the maximum precipitation is found in the E/NE of the Alps, which is typical for cyclonic weather types, and weather type Vb in particular [START_REF] Nissen | Vb cyclones and associated rainfall extremes over Central Europe under present day and climate change conditions[END_REF]. These situations seem to be typical for heavy precipitation and floods in the northern Alps [START_REF] Sodemann | Seasonal and inter-annual variability of the moisture sources for Alpine precipitation during 1995-2002[END_REF] and have been identified as important triggers for flood deposits [START_REF] Stewart | Reconstructions of late Holocene paleofloods and glacier length changes in the Upper Engadine, Switzerland (ca. 1450 BC-AD 420)[END_REF][START_REF] Glur | Frequent floods in the European Alps coincide with cooler periods of the past 2500 years[END_REF]).

Conclusions and implications for future climate change

We have investigated the varve thickness and flood layer properties in a 580-cm long sediment sequence from Lake Oeschinen, NW Swiss Alps, (i) to reconstruct annually-resolved warm season precipitation, and (ii) to assess the flood history back to AD 884. The underlying research questions are:

1.

What is the relationship between climatological precipitation and temperature?

2.

Is there a relationship between flood frequency, precipitation and temperature variability?

From our analysis we conclude:

-The varve record from Lake Oeschinen provides a robust well-calibrated quantitative record Such multidecadal effects in the relation between precipitation and temperature might also operate and be important for precipitation scenarios in the Alpine area under future warming.

-There is an increasingly consistent picture from smaller and medium size catchments in midand high elevations in the Alpine area that warm season floods were generally more frequent during cool and moist periods. However, the 13 th century has also documented a period with high flood frequency during warm and moderately dry (average precipitation) conditions.

Currently, it is not understood why the 13 th century showed a different behaviour with regard to floods than the rest of the past millennium and how the relation floods-precipitationtemperature can be explained in a consistent framework. This would be the cornerstone to assess whether floods in the Alpine area would likely be more or less frequent in a future warmer climate.

Table 1 Details for radiocarbon dating. 14 C ages were calibrated online from OxCal. Calibrated ages are given in year AD at 2σ range with cumulative probability in % Table 2 Correlation coefficients between Lake Oeschinen MJJA precipitation reconstruction and the different precipitation records of Europe, at annual resolution and for 31 years filtered data

4. 3 .

 3 Fig. 4a shows the reconstruction of warm season (MJ -MJJA; sum) precipitation at annual and 31-year filtered resolutions from Lake Oeschinen sediments back to AD 884. This millennial record indicates substantial centennial-scale variability but also decadal-scale fluctuations. Periods with apparent increased wetness (labelled from I to VII) can be noticed in the following time intervals: AD 920 -950 (I), AD 1100 -1180 (II), AD 1300 -1400 (III), AD 1590 -1650 (IV), AD 1700 -1790 (V), AD 1820 -1880 (VI), and AD 1960 -2008 (VI). Driest conditions are reconstructed for AD 960 -1080, AD 1250 -1300 and for AD 1880 -1900. The amplitude of reconstructed precipitation remains in the range observed for the calibration period (AD 1901 -2008, calibration range) except for the two very dry periods AD 960 -1080 and AD 1250 -1300. The comparisons between the precipitation record from Lake Oeschinen and a set of precipitation reconstructions for slightly different seasons and different areas across Europe is shown in Figs. 4b-e and Table2. Significant positive correlations (pc < 0.01) were found between lake

Fig. 5b shows

  Fig. 5b shows the frequency of flood layers (>5 mm) in the sediments of Lake Oeschinen and how this compares with the flood frequency in the Northern Alpine area (composite of ten sites; data from Glur et al. 2013) and with local paleoclimate variability (Alpine warm-season precipitation from this study, and temperature; Fig. 5a). Eight periods with higher flood frequency characterize the sediment record, each about 60 years long (labelled as P1 to P8). These periods are dated to AD 1090 -1140 (P1), AD 1190 -1240 (P2), AD 1300 -1380 (P3), AD 1430 -1510 (P4), AD 1590 -1650 (P5), AD 1690 -1750 (P6), AD 1800 -1870 (P7) and, more recently, starting at AD 1950 (P8).

  of warm season precipitation back to AD 884. The varve chronology is validated by independent 14 C AMS dates and documentary sources of historical floods. The chronological uncertainty is between 5 and 10 years back to ca. AD 1400, and between max. 20 to 30 years prior to AD 1250.-The warm season precipitation record shows interannual to centennial scale variability with particularly humid periods in the 12 th and 14 th century, repeated humid peaks during the Little Ice Age and in the 20 th century. The precipitation record is validated with independent precipitation reconstructions for the same area. Correlation maps from the different seasons of European precipitation records show the spatial and seasonal pattern expected from 20 th century Reanalysis data.-Over the entire past millennium, there is no positive correlation between precipitation and temperature (climatology), contrary to what would be expected from the Clausius-Clapeyron relation or from the 'dry gets dryer -wet gets wetter' scheme. In contrast, the relation between precipitation and temperature has persistently oscillated with a highly significant multidecadal periodicity (60 -70 yrs) over the last millennium. Although the relation with the Atlantic Multidecadal Oscillation (AMO) remains not clear at the current state of research, changes in the statistics of weather patterns or within-weather-type variability might offer possible explanations. The within-type variability has been documented to operate at multidecadal scales and to influence precipitation and temperature significantly over Europe.

Figure captions Fig. 1 Fig. 3 Fig. 4 Fig. 5 Fig. 6

 captions13456 Figure captions

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6

 123456 Figure 1

Acknowledgements

This research was carried out within the Swiss National Science Foundation grant 200020-134945/1. We thank the Hirschi Family for granting access to Lake Oeschinen, Fabian Mauchle and Bruno Wilhelm especially for discussions on lake sedimentation processes and flood history. Particular thanks go to Daniela Fischer for essential support in the field and the lab. We would also like to thank Mathias Trachsel (University of Bergen) for statistical discussions, and Krystyna Saunders, Rixt de Jong and Iván Hernandez-Almeida (University of Bern) for comments on this manuscript.

Supplementary material

Fig. S1 Visible Reflectance Spectroscopy (VIS-RS) properties with a) the first derivative spectra characteristic of three samples with different calcite/siliciclastics ratios [typical end-members], and b) the proxy-proxy calibration model between the mean first derivative and LOI-based calcite/siliciclatics ratios The fresh sediment cores were scanned for Visible Reflectance Spectroscopy (VIS-RS) using a portable Gretag-Spectrolino (GretagMcBeth, Switzerland). This instrument provides reflectance spectra in the visible range (380 -730 nm) with a spatial resolution of 2 mm (Trachsel et al. 2010). We calibrated the VIS-RS reflectance properties of 44 individual samples with known proportions of calcite/siliciclastics as derived from LOI. We used a simple linear model (r 2 = 0.932) between the mean value of each first-derivative spectrum (overall spectral slope) and conventional LOI950 measurements and the residuals, respectively (Fig. S1). The first-derivative spectrum enables eliminating the influence of water content in the sediments [START_REF] Debret | Spectrocolorimetric interpretation of sedimentary dynamics: The new "Q7/4 diagram[END_REF]). The principle is the following: the higher the calcite(bright)/siliciclastics(dark) ratio is, the brighter is the sample and the higher is the mean reflectance spectrum and its 1 st derivative. This allows us to rapidly infer the sediment composition (mainly calcite/siliciclastics) of the entire core section at a very high (2 mm) resolution.

References (supplementary material S1)

Debret, M., Sebag, D., Desmet, M., Balsam, W., Copard, Y., Mourier, B., Susperrigui, A.-S., Arnaud, F., Bentaleb, I., Chapron, E., Lallier-Vergès, E., Winiarski, T., 2011. Spectrocolorimetric interpretation of sedimentary dynamics: The new "Q7/4 diagram". Earth-Science Reviews 109, 1-19.

Trachsel, M., Grosjean, M., Schnyder, D., Kamenik, C., Rein, B., 2010. Scanning reflectance spectroscopy (380-730 nm): a novel method for quantitative high-resolution climate reconstructions from minerogenic lake sediments. Journal of Paleolimnology 44, 979-994.

Fig. S2 Properties of the relation between warm season precipitation (this study) and summer

European Alps temperature [START_REF] Büntgen | Summer Temperature Variations in the European Alps, A.D. 755-2004[END_REF]) with a) the 31-year running correlations plotted with time, b) the continuous wavelet transform analysis of this running correlation (Morlet wavelet) with contour lines representing the 95 % significance level, and c) the corresponding power spectrum of the time series (REDFIT; Welch method) (black line), theoretical red-noise spectrum (red line) and falsealarm levels at 95% and 99% (green and blue lines respectively). Spectral peaks at periods of [58 -73] years are inconsistent with AR(1) origin.

Our data show that the relation between precipitation and temperature (PP-TT) in the NW Alps is not stationary over the past millennium (Fig. S2a). On multidecadal time scales, phases with strong positive correlations (up to r = 0.5) alternate with phases of strong negative correlations (up to r = -0.6). A Continuous Wavelet Transform (CWT) analysis was performed on this 31-year running correlation time series to detect possible periodicities along the record (Fig. S2b). Highest squared correlation strength (power) can be detected for periods >60 years, while noise remains principally for lower periods. Then, we tested in a REDFIT spectral analysis [START_REF] Schulz | REDFIT: estimating red-noise spectra directly from unevenly spaced paleoclimatic time series[END_REF] whether non-AR1 components can be identified in the time series (Fig. S2c). Results show that spectral peaks with periods of [58 -73] years exceed the false-alarm levels (χ 2 95% and 99%). This suggests that the 31-yr running correlations between warm-season precipitation and temperature oscillated with a highly significant periodicity between 58 -73 years throughout the last millennium.