Nicolas Leprovost

Eun-Jin Kim

Multi-scale theory of rotating turbulence

Keywords: Turbulence -Stars, interior -Stars, rotation

Aims. To understand the dynamics of stellar interiors, we study the effect of rotation on turbulence. Methods. We consider turbulence induced by an arbitrary forcing and derive turbulence amplitude and turbulent transport coefficients (turbulent viscosity and diffusivity), first by using a quasi-linear theory and then by using a multi-scale renormalisation analysis.

Results. With an isotropic forcing, the quasi-linear theory gives that the turbulent transport coefficients, both parallel and perpendicular to the rotation vector, have the asymptotic scaling Ω -1 for rapid rotation (i.e. when the rotation rate Ω is larger than the inverse of the correlation time of the forcing and the diffusion time), while the renormalisation analysis suggests a weaker dependence on Ω, with Ω -1/2 scaling. The turbulence amplitude is found to scale as Ω 0 -Ω -1 in the rapid rotation limit depending on the property of the forcing. In the case of an anisotropic forcing with inhibited motion in the vertical direction, as should be relevant in a strongly stratified medium, we find that non-diffusive fluxes of angular momentum scale as Ω -2 -Ω -1 for rapid rotation, depending on the temporal correlation of the forcing. We discuss the implications of our result for the dynamics of stellar interiors.

Introduction

Rotation and turbulence are ubiquitous features of many astrophysical bodies and play a crucial role in the dynamics of these bodies. For instance, turbulence is very often invoked as a means to increase the transport of angular momentum [e.g. the accretion rate in disks (Balbus & Hawley 1998)] or the mixing of light elements in stars [e.g. to explain the surface depletion (see Pinsonneault 1997, and references therein)]. Since the rotation rate of stars varies with their mass (spectral types) and age, it is important to develop a general theory of turbulent transport which is valid for arbitrary rotation rate and other stellar parameters (such as molecular viscosity, particle diffusivity, etc).

In the solar context, many authors have investigated the structure of turbulence in rotating bodies to explain the occurrence of differential rotation in the convective zone. The main feature of this type of turbulence is the appearance of non-diffusive terms in the transport of angular momentum, which prevents a solid body rotation from being a solution of the Reynolds equation (Lebedinsky 1941;Kippenhahn 1963). Starting from the Navier-Stokes equation, it is possible to show that these fluxes arise when there is a cause of anisotropy in the system, due to anisotropic background turbulence (see Rüdiger 1989, and references therein) or inhomogeneities such as an underlying stratification (Kichatinov 1987). To explain the observed differential rotation of the sun, it is also of prime importance to understand the influence of rotation on the turbulent transport coefficients such as the turbulent diffusivity of particles and the turbulent conductivity of heat. Kichatinov et al. (1994) have shown that for an isotropic original turbulence with long correlation time, all these coefficients scale as Ω -1 (where Ω is the rotation rate of the sun) in the large rotation limit.

In our recent publications, we have studied the dynamics of the Sun by taking into account the presence of a strong shear (Kim 2005;Leprovost & Kim 2006) and the interaction of this sheared turbulence with different types of waves that can be excited in the Sun due to magnetic fields (Leprovost & Kim 2007a;Kim & Leprovost 2007a), stratification (Kim & Leprovost 2007b) or global rotation (Leprovost & Kim 2007b). The main purpose of these studies was to understand the dynamics of the thin radial shear layer in the Sun, namely the solar tachocline (Spiegel & Zahn 1992), where a sharp transition between latitudinal differential rotation in the convective envelope and nearly uniform rotation in the radiative interior takes place.

In this paper, we consider the opposite limit where the differential rotation is negligible compared to the global rotation. This is relevant for the convection zone and also for rapidly rotating stars which are more massive than the Sun (Küker & Rüdiger 2005). Specifically, we investigate the effect of rotation on the properties of turbulence; i.e. its intensity and its transport. First, using a quasi-linear theory of turbulence with an isotropic arbitrary forcing, we show that all the turbulent transport coefficients, both parallel and perpendicular to the ro-tation vector, have the asymptotic scaling Ω -1 when the rotation rate is sufficiently large. The turbulence amplitude is however found to scale as Ω 0 -Ω -1 in the rapid rotation limit depending on the property of the forcing. We also consider the alternative case of an anisotropic turbulence with inhibited motion in the vertical direction, as should be relevant, for example, in a strongly stratified medium. In this case, we find that the Reynolds stress does not vanish (unlike in the isotropic case where it vanishes) and that the transport of angular momentum scales as Ω -2 -Ω -1 in the rapid rotation limit. Secondly, we perform a multi-scale renormalisation group analysis to improve on the quasi-linear theory. The result suggests that the transport scales as Ω -1/2 , with a weaker dependence on Ω compared to that found by the quasi-linear theory.

The rest of the paper is organised as follows: in Sect. 2, we solve the quasi-linear equations for the fluctuating velocity and density of particles in a rotating frame with an arbitrary external forcing. We then calculate the turbulent intensity and turbulent transport in the case of a homogeneous forcing with arbitrary temporal correlations in Sect. 3. In Sect. 4 and 5, we present results in the case of an isotropic and anisotropic forcing, respectively. Section 6 is devoted to the multi-scale renormalisation group analysis. Finally, we summarise and discuss our results in Sect. 7 and provide the implications for the turbulence in stellar interiors in Sect. 8.

Model (governing equations)

Our starting point is the Navier-Stokes equation with a forcing term f combined with an advection-diffusion equation for the transport of chemical species, in a rotating frame with angular velocity Ω:

∂ t v + v • ∇v = -∇p + ν∇ 2 v + f -2 Ω × v , (1) ∇ • v = 0 , ∂ t N + v • ∇N = D∇ 2 N ,
where ν is the viscosity of the fluid and D the diffusivity of particle. Note that, in the incompressible case considered here, the heat transport can be described by an advection-diffusion equation (e.g. see [START_REF] Chandrasekhar | Hydrodynamic and hydromagnetic stability[END_REF]. Thus, our result for the turbulent diffusivity also holds for the turbulent conductivity, provided that D is replaced by the thermal conductivity. In the following, we let Ω = 2 Ω and assume (without loss of generality) that the rotation is around the x-axis with no large-scale velocity field (note that the combined effect of rotation and large-scale shear is investigated in Leprovost & Kim 2007b). We then express the velocity field and the concentration as the sum of a large-scale field and small-scale fluctuations: v = U 0 + u = u and N = N 0 + n. In the quasi-linear theory [START_REF] Moffatt | Magnetic field generation in fluids[END_REF], Eq. ( 1) can be linearised for the fluctuating fields and then Fourier-transformed to yield equations for the Fourier component X(k, t). To consider arbitrary Prandtl number P r = ν/D, we introduce the following new variables X = exp[νk 2 t] X and X = exp[Dk 2 t] X to absorb the viscosity and diffusivity terms, respectively. Eliminating the pressure variable, the equations for the velocity field and concentration can be written:

∂ 2 t ûx + ω 2 0 ûx = a γ + a 2 ∂ t ĥ1 (t) a -Ω ĥ2 (t) , (2)
∂ t ûz = - βa γ ∂ t ûx + Ωa γ ûx + ĥ2 (t) γ , ûy = -(a ûx + βû z ) , ∂ t n = -ȗ • ∇N 0 , where a = k x /k y , β = k z /k y , γ = 1 + β 2 and ω 2 0 = Ω 2 a 2 /(γ + a 2 ); ĥ1 (t) = γ fx (t) -a fy (t) -βa fz (t) and ĥ2 (t) = -β fy (t) + fz (t).
The homogeneous solution of the first equation can easily be found in terms of trigonometric functions. Using these solutions and the method of variation of constants, the solutions to the first three equations of Eq. ( 2) can then be derived:

ûx = t 0 dt ′ ĥ1 (t ′ ) γ + a 2 cos[ω 0 (t ′ -t)] + θ ĥ2 (t ′ ) γ + a 2 sin[ω 0 (t ′ -t)] , ûy = t 0 dt ′ ĥ1 (t ′ )a γ(γ + a 2 ) -cos[ω 0 (t ′ -t)] (3) + θβ γ + a 2 a sin[ω 0 (t ′ -t)] + ĥ2 (t ′ )a γ γ + a 2 × -θ sin[ω 0 (t ′ -t)] - γ + a 2 β a cos[ω 0 (t ′ -t)] , ûz = - t 0 dt ′ ĥ1 (t ′ )a γ(γ + a 2 ) β cos[ω 0 (t ′ -t)] + θ γ + a 2 a sin[ω 0 (t ′ -t)] + ĥ2 (t ′ )a γ γ + a 2 × θβ sin[ω 0 (t ′ -t)] - γ + a 2 a cos[ω 0 (t ′ -t)] ,
where θ is the sign of (Ωa).

In the following sections, we use Eq. (3) to calculate both the turbulet amplitude and transport (of chemicals or heat). To this end, we prescribe the correlation function of the forcing to be spatially homogeneous with a temporal correlation C(τ):

ĥi (k 1 , t 1 ) ĥ j (k 2 , t 2 ) = (2π) 3 δ(k 1 + k 2 ) C(| t 1 -t 2 |) φ i j (k 2 ) , (4)
for i and j = 1 or 2. The functions φ i j are the power spectra of the forcing.

Turbulent amplitude and transport

Turbulent amplitude

Using Eqs. (3) and (4), it is straightforward to obtain the intensity of turbulence as:

u 2 x = 1 (2π) 3 d 3 k A x 11 φ 11 (k) (γ + a 2 ) 2
(5)

+A x 12 2 φ 12 (k) (γ + a 2 ) 3/2 + A x 22 φ 22 (k) (γ + a 2 )
,

u 2 y = 1 (2π) 3 d 3 k A y 11 a 2 φ 11 (k) γ 2 (γ + a 2 ) 2 +A y 12 2a 2 φ 12 (k) γ 2 (γ + a 2 ) 3/2 + A y 22 a 2 φ 22 (k) γ 2 (γ + a 2 )
.

Here, the functions A i jk (k, ω 0 ) are defined in Eq. (A.1) of Appendix A: they depend only on k and Ω 0 and involve the temporal correlation function of the forcing C(τ) only through the functions κ c and σ c , defined by:

κ c = +∞ 0 e -νk 2 τ cos(ω 0 τ) C(τ) dτ , (6)
σ c = +∞ 0 e -νk 2 τ sin(ω 0 τ) C(τ) dτ .
The turbulent amplitude in the z direction u 2 z can easily be obtained from u 2 y , due to the symmetry of the problem, by making the following replacements:

β → β -1 , a → a β -1 , γ → γ β -2 , (7) φ 11 → φ 11 β -4 , φ 12 → φ 12 β -3 , φ 22 → φ 22 β -2 .

Transport of angular momentum

From Eqs. ( 3) and ( 4), we can also calculate the non-diagonal part of the correlation tensors u i u j to obtain the transport of angular momentum by turbulence:

u x u y = 1 (2π) 3 d 3 k M x 11 aφ 11 (k) γ(γ + a 2 ) 2 (8) +M x 12 2a φ 12 (k) γ(γ + a 2 ) 3/2 + M x 22 aφ 22 (k) γ(γ + a 2 )
,

u y u z = 1 (2π) 3 d 3 k M z 11 a 2 φ 11 (k) γ 2 (γ + a 2 ) 2 +M z 12 2a 2 φ 12 (k) γ 2 (γ + a 2 ) 3/2 + M z 22 a 2 φ 22 (k) γ 2 (γ + a 2 )
.

Here, the functions M x i j and M z i j are given in Eq. (A.1) of appendix A and depend only on k, Ω 0 and the functions κ c and σ c defined in Eq. (6).

Transport of particles

Integrating the last equation of Eq. (3), we obtain the density of chemical species:

ni = (-∂ i N 0 ) t 0 ȗi (t ′ ) dt ′ . ( 9
)
Note that the velocity to be integrated in Eq. ( 9) differs by an exponential factor from that given by Eq. ( 3). However it is more convenient to express the turbulent diffusivity of chemical species D i j T , defined by u i n = -D i j T ∂ j N 0 , in terms of ûi . Specifically, we can obtain the following:

D i j T = 1 (2π) 6 d 3 k 1 d 3 k 2 e -(Dk 2 1 +νk 2 2 )t × (10) t 0 dt ′ e (D-ν)k 2 1 t ′ û j (k 1 , t ′ ) ûi (k 2 , t) .
Using Eqs. ( 3)-( 4) and following a calculation similar to that in Sect. 3.1, we can calculate all the components of the turbulent diffusivity D i j T . The results turn out to depend on the temporal correlation function C(τ) through a new function:

ζ c = +∞ 0 e -Dk 2 τ C(τ) dτ . ( 11
)
These results in a general form are not shown here for the sake of brevity and shall be presented in the case of isotropic and anisotropic forcing in Sect. 4 and 5, respectively.

Isotropic forcing

When the forcing f is isotropic, the correlation function of the forcing (with no helicity) can be written as:

f i (k, t) f j (k ′ , t ′ ) = (2π) 3 C(|t-t ′ |) F(k) (δ i j -k i k j /k 2 ) δ(k+k ′ ) .(12)
The functions φ i j in Eq. ( 4) are related to F(k) in ( 12) by:

φ 11 = γ(γ + a 2 )F(k) , φ 12 = 0 and φ 22 = γF(k) . (13)
Using Eq. ( 13) in Eqs. ( 5), ( 8) and ( 10), we can obtain the turbulence amplitude and transport as follows:

u 2 x = 1 (2π) 3 d 3 k γF(k) (γ + a 2 ) κ c νk 2 , ( 14
)
u 2 y = 1 (2π) 3 d 3 k a 2 F(k) γ(γ + a 2 )        γ(β 2 + a 2 ) νk 2 a 2 κ c + ω 0 ν 2 k 4 + ω 2 0 σ c        , u x u y = - 1 (2π) 3 d 3 k aF(k) (γ + a 2 ) κ c νk 2 , u y u z = - 1 (2π) 3 d 3 k F(k) (γ + a 2 ) β νk 2 κ c , D xx T = 1 (2π) 3 d 3 k γF(k) (γ + a 2 ) I(k, ω 0 ) , D yy T = 1 (2π) 3 d 3 k F(k)(β 2 + a 2 ) (γ + a 2 ) I(k, ω 0 ) .
Here, the functions χ c , κ c and σ c [defined in Eqs. ( 6) and ( 11)] depend only on the modulus of the wave number; The function I(k, ω 0 ) is defined as:

I(k, ω 0 ) = 1 (D -ν) 2 k 4 + ω 2 0 D -ν ν κ c (15) - (D 2 -ν 2 )k 4 -ω 2 0 (D + ν) 2 k 4 + ω 2 0 (κ c + ζ c ) + 2Dk 2 ω 0 (D + ν) 2 k 4 + ω 2 0 σ c .
By performing the integration in the y-direction and changing the colatitude variable to ω 0 = |Ω cos θ|, we obtain the following result:

u 2 x = 2 (2π) 2 |Ω| +∞ 0 dk k 2 F(k) |Ω| 0 dω 0       1 - ω 2 0 Ω 2       κ c (k, ω 0 ) νk 2 , u 2 y = 1 (2π) 2 |Ω| +∞ 0 dk k 2 F(k) |Ω| 0 dω 0 × (16)              1 + ω 2 0 Ω 2       κ c (k, ω 0 ) νk 2 + ω 2 0 Ω 2 ω 0 ν 2 k 4 + ω 2 0 σ c (k, ω 0 )        , D xx T = 2 (2π) 2 |Ω| +∞ 0 dk k 2 F(k) |Ω| 0 dω 0       1 - ω 2 0 Ω 2       I(k, ω 0 ) , D yy T = 1 (2π) 2 |Ω| +∞ 0 dk k 2 F(k) |Ω| 0 dω 0       1 + ω 2 0 Ω 2       I(k, ω 0 ) ,
and vanishing transport of angular momentum: u x u y = u x u y = 0. We now examine the dependence of the turbulence amplitude and transport of particles in Eq. ( 16) on the correlation time of the forcing C(τ).

Infinitely correlated turbulence

When C(τ) = 1, the correlation functions in Eqs. ( 6) and ( 11) take the following forms:

ζ c = 1 Dk 2 , κ c = νk 2 ν 2 k 4 + ω 2 0 and σ c = ω 0 ν 2 k 4 + ω 2 0 . (17)
Using Eq. ( 17) in Eq. ( 16) and performing the integration on the ω 0 variable, we obtain the following equations:

u 2 x = 2 (2π) 2 +∞ 0 dk F(k) ν 2 k 2 1 Ω 2 * -1 + Ω 2 * + 1 Ω * arctan Ω * , u 2 y = 1 (2π) 2 +∞ 0 dk F(k) ν 2 k 2 1 Ω 2 * × (18) 2 + 1 2(1 + Ω 2 * ) + 2Ω 2 * -5 2Ω * arctan Ω * , D xx T = 2 (2π) 2 +∞ 0 dk F(k) Dν 2 k 4 1 Ω 2 * -1 + Ω 2 * + 1 Ω * arctan Ω * , D yy T = 1 (2π) 2 +∞ 0 dk F(k) Dν 2 k 4 1 Ω 2 * 1 + Ω 2 * -1 Ω * arctan Ω * ,
where Ω * = Ω * (k) = |Ω|/(νk 2 ). In the limit Ω * → ∞, the turbulent transport and intensity, both for the parallel and perpendicular components, all tend to zero as Ω -1 . These agree with Kichatinov et al. (1994). The effect of rotation on turbulence amplitude appears with Ω * , which becomes large on large scales. In other words, turbulence quenching is more severe on large scales, leading to effectively strong turbulence on small scales. This is consistent with the reduction of scale of motion as evidenced in numerical simulations [START_REF] Bender | Advanced mathematical methods for scientists and engineers[END_REF].

Interestingly, the turbulent transport of particles is proportional to D -1 , suggesting that the turbulent transport of heat scales as κ -1 with the molecular heat diffusivity κ. Thus, in the Sun where κ ≫ D, the transport of particles is expected to be faster than that of heat by a factor κ/D ∼ 10 7 /10 2 ∼ 10 5 . However, this holds only for an incompressible fluid, as the equation for the transport of heat is not the same as that of particles for a compressible fluid (Spiegel & Veronis 1960). The relevance of this result for the solar context may thus be questionnable.

δ-correlated turbulence

We now consider a short correlated turbulence modelled by C(τ) = τ f δ(τ). In this case, the temporal correlations in Eqs. ( 6) and (11) become:

ζ c = κ c = τ f 2 and σ c = 0 , (19)
leading to the following turbulent intensity and transport:

u 2 x = u 2 y = 2 3(2π) 2 ∞ 0 F(k)τ f ν dk , ( 20
)
D xx T = 1 (2π) 2 ∞ 0 F(k)τ f ν 2 k 2 1 Ω 2 * × -(1 + b) + Ω 2 * + (1 + b) 2 2Ω * arctan Ω * 2 dk , D yy T = 1 2(2π) 2 ∞ 0 F(k)τ f ν 2 k 2 1 Ω 2 * × 1 + b + Ω 2 * -(1 + b) 2 2Ω * arctan Ω * 2 dk .
Here, b = 1/Pr. Eq. ( 20) shows that, in the case of a short correlated forcing, only the turbulent transport is suppressed (by a factor Ω -1 ) whereas the turbulent intensity is of the same order as that without rotation. This is because, in this case, inertial waves only have an effect on the phase of the velocity field and thus do not modify its amplitude. Alternatively, this is because the flow driven by a forcing with a short correlation time τ f < Ω -1 has no coherent motion to be affected by inertial waves (or rotation). This is to be compared with the result obtained in the case of an infinitely correlated forcing (Sect. 4.1) where the turbulence amplitude was reduced as Ω -1 due to rotation. Furthermore, the dependence of the turbulent diffusivity on the molecular diffusivity is not as simple as in the infinitely correlated case. However, in the large rotation limit (Ω * ≫ 1), we can easily see from Eq. ( 20) that the turbulent diffusivity depends neither on P r nor on D. Consequently, the transport of heat and particle is the same for rapid rotation.

Finite correlation time with exponential correlation function

In Sect. 4.1 and 4.2, we considered two extreme limits where the correlation time was infinite or zero, respectively. Here, we discuss a more realistic case where the turbulence has a finite correlation time τ f by using an exponential correlation function

C(τ) = exp[-2τ/τ f ].
In this case, the functions in Eqs. ( 6) and ( 11) can be simplified as:

κ c = νk 2 + 2/τ f (νk 2 + 2/τ f ) 2 + ω 2 0 , (21)
σ c = ω 0 (νk 2 + 2/τ f ) 2 + ω 2 0 , ζ c = 1 Dk 2 + 2/τ f .
Putting Eq. ( 21) in Eq. ( 16), we can calculate the turbulent intensity and transport as previously. We omit, however, the general form of the results which are too complex, and only provide a result that is valid in the limit Ω → ∞. To this end, we can simply take the limit Ω → ∞ in Eq. ( 16) and obtain the following result:

u 2 x ∼ 2 (2π) 2 |Ω| +∞ 0 dk k 2 F(k) ∞ 0 dω 0 κ c (k, ω 0 ) νk 2 , ( 22
)
u 2 y ∼ 1 (2π) 2 |Ω| +∞ 0 dk k 2 F(k) ∞ 0 dω 0 κ c (k, ω 0 ) νk 2 , D xx T ∼ 2 (2π) 2 |Ω| +∞ 0 dk k 2 F(k) ∞ 0 dω 0 I(k, ω 0 ) , D yy T ∼ 1 (2π) 2 |Ω| +∞ 0 dk k 2 F(k) ∞ 0 dω 0 I(k, ω 0 ) .
We note that, in order to obtain Eq. ( 22), the limit Ω → ∞ was taken inside the integral as the resulting integrals are all convergent. Eq. ( 22) shows that both turbulent amplitude and transport scale as Ω -1 for large rotation rate. This is similar to the case of the infinite correlation time case. Furthermore, Eq. ( 22) shows that the ratio between the parallel and perpendicular transport of particles (or heat) is given by:

u 2 x u 2 y = D xx T D yy T = 2 for Ω → ∞ , (23)
again similar to the case of C(τ) = 1. We note that the large rotation limit Ω → ∞ taken above requires that not only Ω ≫ νk 2 but also Ω ≫ τ -1 f . In other words, the correlation time of the forcing should be larger than Ω -1 . In the solar context, Ω ∼ 3 × 10 -6 s. Thus τ f must be larger than a few days. When Ω < τ -1 f , it can easily be shown that the results become similar to those in the case of δ-correlated forcing studied in Sect. 4.2 (i.e. u 2 ∼ Ω 0 and D T ∼ Ω -1 ).

Power law correlation

In view of the highly intermittent nature of stellar activity, it is interesting to consider the case with a correlation function of the forcing as a power-law. For simplicity, we consider a correlation function given by:

C(τ) = τ f τ µ , (24)
with 0 < µ < 1 (note that it is also possible to consider a function with exponent µ > 1 by introducing a cutoff frequency).

To calculate the various correlation functions, we need the following integral:

I = +∞ 0 e (iω 0 -νk 2 )τ τ -µ dτ = i νk 2 1-µ +∞ 0 t -µ e -it-ω * 0 t dt , ( 25
)
where the last equation is obtained using steepest descent method. Using Watson's lemma [START_REF] Bender | Advanced mathematical methods for scientists and engineers[END_REF], the asymptotic behaviour of this integral for ω * 0 = ω 0 /νk 2 ≫ 1 can be found as:

I ∼ Γ(1 -µ) i ω 0 1-µ . ( 26
)
Thus the correlation functions in Eqs. ( 6) and ( 11) become:

ζ c = Γ(1 -µ) Dk 2 τ µ f * , (27)
κ c ∼ Γ(1 -µ)τ µ f * νk 2 ω µ-1 0 * cos (1 -µ) π 2 , σ c ∼ Γ(1 -µ)τ µ f * νk 2 ω µ-1 0 * sin (1 -µ) π 2 ,
where τ f * = τ f νk 2 and ω 0 * = ω 0 /νk 2 are the correlation time and the oscillation frequency of the noise scaled by the diffusion time, respectively. Putting Eq. ( 27) in Eq. ( 16), we then obtain the turbulence amplitude and turbulent transport of chemicals for an isotropic forcing with a power-law correlation:

u 2 x = 2 (2π) 2 +∞ 0 dk F(k)τ µ f * ν 2 k 2 Ω * cos (1 -µ) π 2 2Ω µ * µ(µ + 2) , ( 28
)
u 2 y = 1 (2π) 2 +∞ 0 dk F(k)τ µ f * ν 2 k 2 Ω * cos (1 -µ) π 2 2(µ + 1)Ω µ * µ(µ + 2) , D xx T = 2 (2π) 2 +∞ 0 dk F(k)τ µ f * Dν 2 k 4 Ω * T , D yy T = 1 (2π) 2 +∞ 0 dk F(k)τ µ f * Dν 2 k 4 Ω * T ,
in the rapid rotation limit Ω * = |Ω|/(νk 2 ) ≫ 1. T is a constant which is independent of Ω. From Eq. ( 28), we can see that the turbulence intensity scales as |Ω| µ-1 for rapid rotation. This shows that the scaling of the turbulent intensity depends sensitively on the exponent of the power law. Note that the result of infinitely correlated turbulence is recovered for µ = 0. In comparison, the turbulent diffusivity in both directions scales as |Ω| -1 independent of the power exponent µ. Furthermore, we can see that the ratio between the turbulence in the x-and y-direction depends only on the power exponent:

u 2 x u 2 y ∼ 2 µ + 1 . ( 29
)
Consequently, the turbulence is always more vigorous in the xdirection (along the rotation axis) than in the y-direction. We recover the factor 2 for µ = 0, corresponding to the exponentially correlated case, see Eq. ( 23). As the power exponent is increased, the turbulence becomes more isotropic. Note that the ratio between the transport of species in the x-and y-direction is always 2, as in the case of an exponential correlation function.

To summarise, for an isotropic forcing and rapid rotation, we showed that the turbulent diffusivity is reduced as Ω -1 , irrespective of the correlation time of the forcing, with the transport in the direction parallel to the rotation vector being twice as fast as than in the direction perpendicular to the rotation. This would lead to a warmer pole than equator (Kichatinov & Rüdiger 1995). In comparison, the turbulent intensity is found to be reduced with different scaling with Ω depending on the properties of the correlation function of the forcing. Specifically, in the case of a finite correlation time, the turbulence intensity scales as Ω -1 for rapid rotation, while it is independent of the rotation rate for a short-correlated forcing (Sect. 4.2). Furthermore, for a temporal correlation function given by a power-law, turbulence amplitude scales as Ω n with -1 < n < 0.

Anisotropic forcing

We now consider the case where the forcing f is anisotropic as should be relevant for the turbulence in the convection zone (for instance, due to the underlying stratification). We are particularly interested in the Reynolds stress as the combination of anisotropy and rotation can lead to non-diffusive fluxes of angular momentum. In the case where g is a unit vector in the direction of anisotropy, the correlation function of the forcing can be written [START_REF] Rüdiger | Differential rotation and stellar convection (Gordon and Breach)[END_REF]:

f i (k, t) f j (k ′ , t ′ ) = (2π) 3 C(|t -t ′ |) δ(k + k ′ ) G(k) × (30) δ i j - k i k k k 2 - (g • k) 2 k 2 δ i j -g i g j + g • k k 2 (g i k j + g j k i ) .
This forcing in Eq. ( 30) is highly anisotropic with only a nonvanishing component perpendicular to g:

(f • g) 2 = 0 , (31)
(f × g) 2 = 1 2π 2 ∞ 0 G(k)k 2 dk .
In the following, we assume g to be in the vertical direction.

In general, one can consider a less anisotropic forcing by combining the isotopic part of the forcing (12) and anisotropic part (30).

Rotation parallel to the vertical direction

When the rotation vector is parallel to the vertical direction i.e.

g Ω (as in the case of the turbulence near the pole), Eq. ( 30) translates to:

φ 11 = φ 12 = 0 and φ 22 = γ 2 γ + a 2 G(k) . ( 32
)
Using Eq. ( 32) in Eqs. ( 5) and ( 8), we can show that the turbulence amplitude becomes

u 2 x = 2 (2π) 2 |Ω| ∞ 0 dk k 2 G(k) |Ω| 0 dω 0       1 - ω 2 0 Ω 2       2 ×             1 2νk 2 - νk 2 2(ν 2 k 4 + ω 2 0 )       κ c + ω 0 2(ν 2 k 4 + ω 2 0 ) σ c       , u 2 y = 1 (2π) 2 |Ω| ∞ 0 dk k 2 G(k) |Ω| 0 dω 0       1 - ω 2 0 Ω 2       × (33)             1 + ω 2 0 Ω 2       κ c 2νk 2 + νk 2 κ c -ω 0 σ c 2(ν 2 k 4 + ω 2 0 )       ,
while the transport of angular momentum vanishes. Here again, we can see by using Eq. ( 21) that all the integrals in Eq. (33) converge in the limit Ω → ∞. Thus, turbulence amplitudes are reduced as Ω -1 for large rotation rate. We do not provide here the results for the turbulent diffusivities as they are very similar to those derived in the isotopic case. Thus, in the rapid rotation limit, turbulent diffusivities scale as Ω -1 for all the different types of correlation function of the forcing that were considered previously.

Rotation perpendicular to the vertical direction

Near the equator, the rotation vector is perpendicular to the vertical direction. Since the rotation is assumed to be in the x-direction, the vertical direction for anisotropy can be chosen to be in the z-direction without loss of generality. Eq. ( 30) then translates to:

φ 11 = (γ + a 2 )G(k) , φ 12 = aβG(k) ( 34
)
and

φ 22 = a 2 β 2 γ + a 2 G(k) .
In this case, the turbulent amplitude and diffusivities can be shown to be very similar to those derived in the isotopic case. Consequently, the scalings are the same as those derived in Sect. 4. However, in this case, the transport of angular momentum does not vanish as discussed below. Putting Eq. ( 34) into Eq. ( 8) and keeping only the even terms in β and a (as the odd terms vanish after angular integration), straightforward but cumbersome calculations lead to the following expressions for the correlation function of the y and z components of the velocity:

u y u z = - 1 2(2π) 3 d 3 k θa(1 + a 2 )G(k) (γ + a 2 ) 3/2 ω 0 κ c + νk 2 σ c ν 2 k 4 + ω 2 0 , (35)
while the other correlation functions vanish. Performing the integration over the angular variables, Eq. ( 35) can be simplified as:

u y u z = - χ 2(2π) 2 Ω 2 ∞ 0 dk k 2 G(k) |Ω| 0 dω 0 × (36) ω 0       1 + ω 2 0 Ω 2       ω 0 κ c + νk 2 σ c ν 2 k 4 + ω 2 0 ,
where χ is the sign of Ω. The Reynolds stress in Eq. ( 35) depends on the direction of the rotation, representing nondiffusive momentum transport, i.e., Λ effect [START_REF] Rüdiger | Differential rotation and stellar convection (Gordon and Breach)[END_REF]). This effect favours the creation of velocity gradient rather than smoothing it out and can thus be responsible for the differential rotation of stars.

We now examine the behaviour of Eq. ( 35) for different temporal correlation functions. In the case of an infinitely correlated forcing, Eq. ( 36) can be shown to become:

u y u z = - χ 2(2π) 2 ∞ 0 dk G(k) ν 2 k 2 Ω 3 * × (37) 3 + Ω 2 * 1 + Ω 2 * + Ω 2 * -3 Ω * arctan Ω * .
Eq. ( 37) shows that the non-diffusive fluxes of angular momentum scale as Ω -2 in the large rotation limit. This agrees with Kichatinov (1986). Similarly, we can show that for a finite correlation time (with exponential form), the limit Ω * → ∞ can be taken inside the integral of Eq. ( 36) and consequently, the non-diffusive fluxes of angular momentum scale also as Ω -2 in the large rotation limit.

In comparison, when the forcing is δ-correlated, Eq. ( 36) becomes:

u y u z = - χ 2(2π) 2 ∞ 0 dk τ f G(k) 6νΩ 4 * × (38) Ω * (4Ω 2 * -3) + 3(1 -Ω 2 * ) arctan Ω * ,
which shows that the non-diffusive fluxes of angular momentum scale as Ω -1 in the rapid rotation limit. Finally, for a power-law correlation function, by putting Eq. ( 27) into Eq. ( 36), we obtain the non-diffusive fluxes of angular momentum as:

u y u z = - χ 2(2π) 2 ∞ 0 dk τ µ f * G(k) νk 2 Ω 2 * Γ(1 -µ) × (39)
Ω µ * cos (1 -µ)π/2 2(µ + 2) cos (1 + µ)π/2 cos (1 + µ) π 2 - 1 µ ,
in the rapid rotation limit. In this limit, the Λ-effect scales as Ω µ-2 . Thus, for a power-law correlation function, the Λ-effect scales as Ω -n with 1 < n < 2. In summary, we show that non-diffusive fluxes of angular momentum do not vanish only when the rotation is perpendicular to the inhomogeneity (such as stratification in the radial direction or shear in the azimuthal direction). This is in agreement with previous results. Furthermore, we show that these non-diffusive fluxes scale as Ω -2 -Ω -1 , in the rapid rotation limit, depending on the property of the temporal correlation function of the forcing.

Multi-scale renormalisation group analysis

The results presented in the previous sections are derived using the quasi-linear theory. This is strictly valid only for twoscale turbulence, with a spatial gap between a large-scale for the mean-flow and a small scale for the fluctuations, and for weak turbulence. Therefore, the applicability of this approximation is limited. In this section, we utilise a renormalisation group technique (see [START_REF] Moffatt | [END_REF], and references therein for details) to consider a more realistic situation of a strong turbulence with a no-clear scale separation between turbulence and mean fields. This method is based on the computation of the turbulent diffusivity iteratively by considering a range of scales l ′ n s with:

l n > l n-1 > . . . > l 0 , (40)
and l 0 is the scale at which the molecular diffusion becomes dominant. The effective turbulent diffusivity D n+1 for scale n+1 is then calculated by using the value of the turbulent diffusivity at the preceding step, i.e., D n via quasi-linear theory. Since the turbulent diffusivity scales as D -1 [see Eq. ( 18)], we can calculate D 1 in terms of D 0 as:

D 1 = 1 D 0 1/l 0 1/l 1 P(k, ν, Ω) dk , (41)
where P(k, ν, Ω) is a function depending only on modulus of the wave number, molecular viscosity, and rotation. Note that the integration range is over the scale l such that l 0 < l < l 1 . Repeating this procedure iteratively, one can compute the turbulent diffusivity at scale n + 1 as follows:

D n+1 = 1 D 0 + D n 1/l n 1/l n+1 P(k, ν, Ω) dk .
(42)

If we take the limit n → ∞, we obtain a continuous range of scale and turbulent diffusivity. Thus, writing D(k) as the turbulent diffusivity acting on scales smaller than k -1 , Eq. ( 42) can be rewritten

dD(k) = 1 D 0 + D(k) P(k) dk . (43)
The turbulent diffusivity D T is obtained by integrating over all scales in the inertial range, starting from n = 0 with the value of molecular diffusivity D 0 = D, with the result:

D T ∼ 1 D ∞ 0 P(k, ν, Ω) dk 1/2 . ( 44
)
Eq. ( 44) shows that if the quasi-linear theory gives a turbulent diffusivity as Ω -n , then the renormalisation analysis predicts a scaling Ω -n/2 . Consequently, as the quasi-linear theory predicted a scaling as Ω -1 (irrespective of the form of temporal correlations), the turbulent transport of particles (or heat) scales as Ω -1/2 in the large rotation limit. Interestingly, this result has a weaker dependence on Ω compared to that according to quasi-linear theory. Furthermore D xx T /D zz T = √ 2 in the rapid rotation limit, with a weaker anisotropy compared to the quasilinear result [see Eq. ( 23)].

Finally, we note that quasi-linear or multi-scale renormalisation group analysis cannot capture a strong anisotropy in the rapid rotation limit with a tendency towards twodimensionalisation (Cambon et al. 1997). In order to capture the highly anisotropic nature of rotating turbulence, one needs to use more sophisticated closure, such as wave-turbulence which leads to anisotropic spectra [START_REF] Galtier | [END_REF]). In the extreme limit of rapid rotation, this rotation-induced anisotropy could be captured by assuming geostrophic balance in the equilibrium state [START_REF] Pedlovsky | Geophysical fluid dynamics[END_REF].

Summary and discussion

We studied the structure of turbulence in a rotating medium with an arbitrary external forcing. Using the quasi-linear theory, we first performed a thorough study of the turbulence intensity and transport (of chemical species or heat) in the case of an isotropic forcing by considering different types of temporal correlation functions. We also examined the transport of angular momentum in the case of an anisotropic forcing.

Specifically, we showed that for an isotropic forcing, the transport parallel and perpendicular to the rotation vector has the same scaling with the rotation frequency Ω: they are both reduced by a factor Ω -1 compared to the case without rotation irrespective of the temporal correlation function of the forcing. Furthermore, the transport in the direction parallel to the rotation is twice as fast than the one in the perpendicular direction [see Eq. ( 23)]. This result was shown to be robust independent of the specific form of temporal correlation C(τ) (exponential and power law) or correlation time (finite, short or infinite). Note that this result was shown by Kichatinov et al. (1994) in the case of an infinitely correlated forcing.

In comparison, the scaling of the turbulence intensity depends on the property of the forcing. For exponential correlation function with infinite memory, all the components of the velocity amplitude scale as Ω -1 while their intensity in the direction parallel to the rotation is twice that in the perpendicular direction (similar to the transport). However, for δcorrelated turbulence, the turbulence amplitude becomes independent of the rotation rate. Thus, in this case, inertial waves affect only the cross-phase of the velocity field and not its amplitude. Furthermore, in the case of a finite correlation time with a power-law, the turbulence intensity scales as Ω n with -1 < n < 0 while the ratio of the turbulence amplitude in the direction parallel to that perpendicular to the rotation can be any number between 1 and 2 depending on the exponent of the power-law.

In the case where the driving force is highly anisotropic, it was shown to give rise to non-diffusive fluxes when the rotation is perpendicular to the anisotropy. In the large rotation limit, the transport of angular momentum scales as Ω -2 for a forcing with a finite correlation time. However, for a δcorrelated turbulence, we found that the (quenching) effect of rotation was reduced with non-diffusive fluxes now scaling as Ω -1 . Furthermore, for a power-law correlation function, the momentum flux was found to scale as Ω p with -2 < p < -1.

Finally, we performed a multi-scale renormalisation analysis which permitted us to go beyond the quasi-linear theory in the calculation of the turbulent diffusivity. We found that, in comparison with the quasi-linear theory result of a turbulent diffusivity proportional to Ω -1 , the renormalisation theory predicts a scaling Ω -1/2 . Consequently, we expect the turbulent diffusivity to depend weakly on Ω. It also gave us an anisotropy in transport weaker than the quasi-linear prediction, the parallel component being √ 2 faster than that of the perpendicular. We should however note that the scaling of our results can be modified in the case of a bounded domain of finite size. In the calculation of all the turbulent transport coefficients, we obtained a result proportional to the following type of integral:

I(k, Ω) = H(k) ν 2 k 4 + ω 2 0 d 3 k , (45)
where ω 0 = (Ω • k)/k is the projection of the unit vector in the direction of the wave number on the rotation axis. When the domain of integration is unbounded, the integration over the angular variable makes this integral proportional to Ω -1 , when the rotation rate Ω is sufficiently large [e.g. Eq. ( 18)]. This is because this integral involves some contribution of order 1 (when Ω • k = 0) and others of order Ω -2 . However, in a realistic situation, the domain of integration (in Fourier space) is bounded. That is, there is a minimal wavenumber (corresponding to a maximum length, for instance the size of the box) in the direction of the rotation, for example, k m = min(k x ). The aforementioned scaling of Ω -1 is valid only when ν 2 k 6 ≫ Ω 2 k 2 m . In the opposite case (ν 2 k 6 ≪ Ω 2 k 2 m ), the term ω 2 0 in Eq. ( 45) is always dominant, giving the scaling of Ω -2 for large rotation rate. In this case, the multi-scale analysis would give a scaling of Ω -1 instead of Ω -1/2 .

Implications for stellar convection zone

We discuss here some of the important implications of our results for the Sun and other stars. For parameters typical of the Sun, Ω ∼ 2.3 × 10 -6 s -1 and ν ∼ 10 2 cm 2 s -1 , the effect of a finite-size domain discussed in Sect. 7 can be ignored if the characteristic length scale is larger than (ν/Ω) 1/2 ∼ 1.7×10 4 cm. This is likely to be the case in the solar convection zone as convective motions can extend to a distance of approximately 4 × 10 9 cm. In this case, the quasi-linear theory predicts that the turbulent diffusivity scales as Ω -1 for rapid rotation while the turbulence amplitude scales as Ω 0 -Ω -1 depending on the property of the forcing. In comparison, the renormalisation analysis predicts a turbulent diffusivity scaling as Ω -1/2 . This result thus suggests that the effect of rotation on particle or heat transport is weaker than previously thought (Kichatinov et al. 1994). If the effect of rotation had been much stronger in reducing transport, it could have inhibited the mixing of light elements in the convection zone, thereby leading to a weak mixing as required (Barnes et al. 1999). Our results have confirmed that this cannot possibly be true and that rotation cannot play an important role in reducing the mixing of light elements in the solar convection zone to explain the observed (rather modest) depletion of light elements on the Sun [START_REF] Rüdiger | Differential rotation and stellar convection (Gordon and Breach)[END_REF].

Traditional stellar modelling does not involve any anisotropy in the turbulent diffusivity of particles (or equivalently in the turbulent conductivity of temperature) due to rotation. We have shown here that such an anisotropy can be induced by rotation. This is however not a significant effect as the transport in the direction parallel to the rotation is only twice as fast as that in the perpendicular direction, according to the quasi-linear theory. This anisotropy is quite robust against the temporal correlations used and has been shown to give rise to a warmer pole than the equator (Kichatinov & Rüdiger 1995). This anisotropy is however shown to be weaker according to the multi-scale renormalisation analysis, the ratio being only √ 2. Consequently, rotation may not be sufficient for inducing anisotropic transport in stars, for instance, in order to cause a sufficient latitudinal temperature gradient in the convective envelopes as required by numerical models (Kichatinov & Rüdiger 1995;Küker & Rüdiger 2005). As we have shown in our previous studies, an anisotropic transport can easily be induced by a strong shear layer (such as the solar tachocline, see Kim 2005;Leprovost & Kim 2006), magnetic fields (Kim 2006;Leprovost & Kim 2007a;Kim & Leprovost 2007a) or stratification (Kim & Leprovost 2007b). In these works, shear flows are also shown to quench turbulent transport significantly leading to weak mixing.

Finally, we note that our results depend on the magnitude of Ω, the diffusion rate νk 2 (or Dk 2 ) and the characteristic frequency of the forcing 1/τ f . Obviously, stars of different spectral types have different values for Ω, ν, D and τ f . It is thus important to understand the origin of τ f and to perform a systematic study on the development of turbulence theory as a function of masses and ages. This will be addressed in future studies.

      1 2νk 2 + νk 2 2(ν 2 k 4 + ω 2 0 )       κ c - ω 0 2(ν 2 k 4 + ω 2 0 )
σ c , (A.1)

A x 12 = -θ       ω 0 2(ν 2 k 4 + ω 2 0 ) κ c + νk 2 2(ν 2 k 4 + ω 2 0 ) σ c       , A x 22 =       1 2νk 2 - νk 2 2(ν 2 k 4 + ω 2 0 )       κ c + ω 0 2(ν 2 k 4 + ω 2 0 ) σ c , A y 11 = γ(β 2 + a 2 ) 2νk 2 a 2 + -γ(β 2 + a 2 ) + 2a 2 a 2 νk 2 2(ν 2 k 4 + ω 2 0 ) + 2θβ γ + a 2 a νk 2 2(ν 2 k 4 + ω 2 0 ) κ c +        γ(β 2 + a 2 ) a 2 ω 0 2(ν 2 k 4 + ω 2 0 ) + θβ γ + a 2 a νk 2 ν 2 k 4 + ω 2 0        σ c , A y 12 = θ γ(β 2 + a 2 ) -2a 2 a 2 ω 0 2(ν 2 k 4 + ω 2 0 ) + 2β γ + a 2 a νk 2 2(ν 2 k 4 + ω 2 0 ) κ c -θ γ(β 2 + a 2 ) -2a 2 a 2 νk 2 2(ν 2 k 4 + ω 2 0 ) - 2β γ + a 2 a ω 0 2(ν 2 k 4 + ω 2 0 ) σ c , A y 22 = γ(β 2 + a 2 ) 2νk 2 a 2 + γ(β 2 + a 2 ) -2a 2 a 2 νk 2 2(ν 2 k 4 + ω 2 0 ) - 2θβ γ + a 2 a νk 2 2(ν 2 k 4 + ω 2 0 ) κ c + -γ(β 2 + a 2 ) + 2a 2 a 2 ω 0 2(ν 2 k 4 + ω 2 0 ) - θβ γ + a 2 a νk 2 ν 2 k 4 + ω 2 βγ 2a 2 νk 2 + β(γ + 2a 2 )νk 2 2a 2 (ν 2 k 4 + ω 2 0 ) - (1 -β 2 )θ γ + a 2 a ω 0 2(ν 2 k 4 + ω 2 0 ) κ c -β (γ + 2a 2 )ω 0 2a 2 (ν 2 k 4 + ω 2 0 ) + (1 -β 2 )θ γ + a 2 a νk 2 2(ν 2 k 4 + ω 2 0 ) σ c , M z 12 = (β 2 -1)
γ + a 2 a νk 2 2(ν 2 k 4 + ω 2 0 )

-θβ γ + 2a 2 a 2 ω 0 2(ν 2 k 4 + ω 2 0 )

κ c + -θβ γ + 2a 2 a 2 νk 2 2(ν 2 k 4 + ω 2 0 ) -(β 2 -1) γ + a 2 a ω 0 2(ν 2 k 4 + ω 2 0 )
σ c , M z 22 = -βγ 2a 2 νk 2 -β(γ + 2a 2 )νk 2 2a 2 (ν 2 k 4 + ω 2 0 )

+ (1 -β 2 )θ γ + a 2 a ω 0 2(ν 2 k 4 + ω 2 0 ) κ c + β (γ + 2a 2 )ω 0 2a 2 (ν 2 k 4 + ω 2 0 ) + (1 -β 2 )θ γ + a 2 a νk 2 2(ν 2 k 4 + ω 2 0 ) σ c .
Here again, the functions κ c , σ c and ζ c characterising the influence of the temporal correlation of the forcing on the turbulence are defined by Eqs. ( 6) and ( 11).

Acknowledgements. This work was supported by U.K. PPARC Grant No. PP/B501512/1.

Appendix A: Turbulence amplitude and transport

In this appendix, we provide the form of A i jk (k, ω 0 ) and M i jk (k, ω 0 ) used in Eqs. ( 5) and (8).

A x 11 =


Laura Rampazzi

Cristina Corti
email: cristina.corti@uninsubria.it

Roberto Bugini

Antonio Sansonetti

Marco Biraghi

Lanfredo Castelletti

Isabella Nobile

Clelia Orsenigo

Roberto Bugini B

Antonio Sansonetti B

Marco Biraghi C

Lanfredo Castelletti C

Isabella Nobile C

Thermal analysis and archaeological chronology: the ancient mortars of the site of Baradello (Como, Italy)

Keywords: TG-DSC, medieval mortar, chronology, Baradello, binder/aggregate ratio, hydraulicity index

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

Mortar is an artificial material, commonly used since ancient times and widely conserved, in many cases in good conditions, to the present day. The ease of preparation, the effortless availability of raw materials, the features of durability and versatility, have contributed to the spreading of mortar, both as a joint element of bricks or stones, and as a coating (plaster). All these factors have diffused the use of mortar in civil and military structures. The first rationalization of knowledge on mortars, containing 'recipes' and technical information, dates back to the 'De Architectura' by Vitruvius (I century BC); other evidences are reported in the 'Naturalis Historia' of Pliny (I century AD) and 'De Agri Cultura' of Cato (II century BC). The information gathered in these works have been used for centuries by the experts: the manufacturers of the Renaissance, but still even those of the XIX century, admired the perfection and endurance especially of ancient Roman mortars and tried to copy techniques and materials. The historical sources contain valuable information on the techniques of production and implementation of the mortars, obviously in a general view. The use of chemical and mineralogical analysis techniques, instead, allows collecting information on specific cases, in terms of technology and raw materials. Despite their widespread use in ancient buildings, in fact, it is generally difficult to obtain historical information on the origin of the raw materials or on the production technologies, because mortars are generally considered as 'poor and common' materials: this is a challenge for their characterization.

Due to their largely variable nature, depending on the period and place of preparation, mortars are traces of the progress in building technology and economical status of a population; this information can be obtained on the basis of the refinement of techniques and materials used. Even in their 'simple' function, mortars are a very complex material, consisting of at least two phases, a binder and an aggregate, with possible additives. This complexity can be the clue to understand their production technology, to characterise a broad spectrum of raw materials and to identify stratigraphies: all these aspects can be used to define different constructive histories, observing possible variations in mixtures, in additives, in provenance of the stones used to prepare them [START_REF] Miriello | Characterisation of archaeological mortars and plasters from Kyme (Turkey)[END_REF][START_REF] Riccardi | Thermal, microscopic and X-ray diffraction studies on some ancient mortars[END_REF][START_REF] Elsen | Microscopy of historic mortars -a review[END_REF][START_REF] Vendrell-Saz | Dating ancient lime mortars by geochemical and mineralogical analysis[END_REF][START_REF] Bruno | Chemicalphysical and mineralogical investigation on ancient mortars from the archaeological site of Monte Sannace (Bari-Southern Italy)[END_REF][START_REF] Rampazzi | A chemometric approach to the characterisation of historical mortars[END_REF][START_REF] Rampazzi | A new microwave-assisted approach to the chemical analysis of mortars[END_REF][START_REF] Bugini | Classification of multi-layered plaster in St. Abbondio Cloister at Como, Italy: an analytical tool for architectural archaeology[END_REF]. Finally, information about the use (civil, military, etc.) of the structures can be inferred from the characteristics of their mortars, i.e. refinement, care in the choice of materials, etc. The analyses of mortars are also important in something which is possible to call 'reverse engineering' process, in order to find the best compatible materials when restoring the original ones.

Ancient mortars are composite materials, generally mixtures of lime (as a binder), sand (as an aggregate) and water, sometimes with organic or inorganic additives. The binder and the water give workability to the material, while the aggregate adds volume and influences its mechanical properties. Mud was probably the first binder used, then replaced, since VI millennium BC, by gypsum and then by lime; the latter became the basic component of most of ancient mortars [START_REF] Elsen | Microscopy of historic mortars -a review[END_REF].

Due to this complex composition, different analytical techniques are generally required to characterize mortars: the most relevant are Fourier transform infrared spectroscopy (FTIR) [START_REF] Genestar | Analytical characterisation of ancient mortars from the archaeological Roman city of Pollentia (Balearic Islands, Spain)[END_REF][START_REF] Duran | Study by thermal analysis of mortars belonging to wall paintings corresponding to some historical buildings of Sevillian art[END_REF][START_REF] Bakolas | Thermoanalytical research on traditional mortars in Venice[END_REF], X-ray diffraction (XRD) [START_REF] Miriello | Characterisation of archaeological mortars and plasters from Kyme (Turkey)[END_REF][START_REF] Riccardi | Thermal, microscopic and X-ray diffraction studies on some ancient mortars[END_REF][START_REF] Vendrell-Saz | Dating ancient lime mortars by geochemical and mineralogical analysis[END_REF][START_REF] Genestar | Analytical characterisation of ancient mortars from the archaeological Roman city of Pollentia (Balearic Islands, Spain)[END_REF][START_REF] Duran | Study by thermal analysis of mortars belonging to wall paintings corresponding to some historical buildings of Sevillian art[END_REF][START_REF] Iordanidis | Thermal, mineralogical and spectroscopic study of plasters from three post-Byzantine churches from Kastoria (northern Greece)[END_REF][START_REF] Bartz | Mineralogical characterization of rendering mortars from decorative details of a baroque building in Kozuchow (SW Poland)[END_REF][START_REF] Cardiano | Investigations on ancient mortars from the Basilian monastery of Fragala[END_REF][START_REF] Alvarez | Thermal, mineralogical and chemical studies of the mortars used in the cathedral of Pamplona (Spain)[END_REF][START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Ingo | Thermal and microchemical investigation of Phoenician-Punic mortars used for lining cisterns at Tharros (western Sardinia, Italy)[END_REF][START_REF] Sanjurjo-Sanchez | Chemical and mineralogical characterization of historic mortars from the Santa Eulalia de Boveda temple, NW Spain[END_REF][START_REF] Vecchio | The use of thermoanalytical techniques in the characterization of ancient mortars[END_REF], thermal analysis (TG, DTG, DTA, DSC) [START_REF] Riccardi | Thermal, microscopic and X-ray diffraction studies on some ancient mortars[END_REF][START_REF] Genestar | Analytical characterisation of ancient mortars from the archaeological Roman city of Pollentia (Balearic Islands, Spain)[END_REF][START_REF] Duran | Study by thermal analysis of mortars belonging to wall paintings corresponding to some historical buildings of Sevillian art[END_REF][START_REF] Bakolas | Thermoanalytical research on traditional mortars in Venice[END_REF][START_REF] Iordanidis | Thermal, mineralogical and spectroscopic study of plasters from three post-Byzantine churches from Kastoria (northern Greece)[END_REF][START_REF] Bartz | Mineralogical characterization of rendering mortars from decorative details of a baroque building in Kozuchow (SW Poland)[END_REF][START_REF] Cardiano | Investigations on ancient mortars from the Basilian monastery of Fragala[END_REF][START_REF] Alvarez | Thermal, mineralogical and chemical studies of the mortars used in the cathedral of Pamplona (Spain)[END_REF][START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Ingo | Thermal and microchemical investigation of Phoenician-Punic mortars used for lining cisterns at Tharros (western Sardinia, Italy)[END_REF][START_REF] Vecchio | The use of thermoanalytical techniques in the characterization of ancient mortars[END_REF][START_REF] Paama | Thermal and infrared spectroscopic characterization of historical mortars[END_REF][START_REF] Adams | Thermal analysis (TA) of lime-and gypsum-based medieval mortars[END_REF] sometimes coupled with FTIR [START_REF] Paama | Thermal and infrared spectroscopic characterization of historical mortars[END_REF], electron microscopy with microprobe for elemental analysis (SEM-EDX) [1-3, 10, 12, 13, 17], Raman spectroscopy [START_REF] Iordanidis | Thermal, mineralogical and spectroscopic study of plasters from three post-Byzantine churches from Kastoria (northern Greece)[END_REF], optical microscopy on thin sections [START_REF] Miriello | Characterisation of archaeological mortars and plasters from Kyme (Turkey)[END_REF][START_REF] Elsen | Microscopy of historic mortars -a review[END_REF][START_REF] Bartz | Mineralogical characterization of rendering mortars from decorative details of a baroque building in Kozuchow (SW Poland)[END_REF][START_REF] Cardiano | Investigations on ancient mortars from the Basilian monastery of Fragala[END_REF][START_REF] Ingo | Thermal and microchemical investigation of Phoenician-Punic mortars used for lining cisterns at Tharros (western Sardinia, Italy)[END_REF], inductively coupled plasma atomic emission spectrometry (ICP-AES) [START_REF] Vendrell-Saz | Dating ancient lime mortars by geochemical and mineralogical analysis[END_REF][START_REF] Cardiano | Investigations on ancient mortars from the Basilian monastery of Fragala[END_REF][START_REF] Alvarez | Thermal, mineralogical and chemical studies of the mortars used in the cathedral of Pamplona (Spain)[END_REF][START_REF] Paama | Thermal and infrared spectroscopic characterization of historical mortars[END_REF], ionic chromatography [START_REF] Cardiano | Investigations on ancient mortars from the Basilian monastery of Fragala[END_REF][START_REF] Alvarez | Thermal, mineralogical and chemical studies of the mortars used in the cathedral of Pamplona (Spain)[END_REF], x-ray fluorescence (XRF) [START_REF] Miriello | Characterisation of archaeological mortars and plasters from Kyme (Turkey)[END_REF][START_REF] Duran | Study by thermal analysis of mortars belonging to wall paintings corresponding to some historical buildings of Sevillian art[END_REF], flame emission atomic spectroscopy [START_REF] Alvarez | Thermal, mineralogical and chemical studies of the mortars used in the cathedral of Pamplona (Spain)[END_REF], mercury intrusion porosimetry [START_REF] Duran | Study by thermal analysis of mortars belonging to wall paintings corresponding to some historical buildings of Sevillian art[END_REF], neutron activation analysis (NAA) [START_REF] Sanjurjo-Sanchez | Chemical and mineralogical characterization of historic mortars from the Santa Eulalia de Boveda temple, NW Spain[END_REF]. Many critical issues are connected with the analysis of mortars. For example, given their particular composition and, often, inhomogeneity, it could be difficult to prepare aliquots for analysis truly representative of the entire sample, or to obtain reproducible results. The complex visual identification of the phases binder and aggregate, often impossible to distinguish without observation in thin sections (not always available in a short time), must also be added. Every mortar constitutes a specific case of analysis, according to its age, location, technology, background and, therefore, features.

This study deals with the mortars of the archaeological site surrounding the Baradello Castle (Como, Italy). The input was given by the team of archaeologists that, during a recent excavation, has brought to light several masonry structures. Their history was only partially defined by means of the visual analysis of buildings and the examination of historical resources that attest their construction. Therefore, the purpose of this study is to trace a constructive history by means of chemical and mineralogical analysis of the mortars used. Sampling was conducted on all accessible masonry structures and has been designed to obtain the maximum representativeness. The mineralogical analysis of thin sections was followed by X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FTIR), thermal analysis (TG-DSC) (which proved to be particularly informative) and, finally, observation with a scanning electron microscope equipped with X-ray microprobe for the elemental analysis (SEM-EDX). Baradello Castle (Figure 1) was built in a dominant position (432 m above mean sea level) as it was meant to be an important observation point within the defense system around the city of Como (Figure 2).

MATERIALS AND METHODS

Archaeological background

The fortification includes many structures, such as two concentric walls around the tower, and it was erected in three different periods: the first framework dates to Early Middle Ages, the second to XII century and the third to Viscontis Age (XIV-XV century).

The two last phases were already examined more than fifty years ago [START_REF] Belloni | Il castello Baradello. Zona A[END_REF], therefore the new survey focused on the earliest stage of the fortification. The new project, which began in 2008, is still in progress [START_REF] Biraghi | Le nuove scoperte archeologiche[END_REF] and is being carried out with a multidisciplinary approach that involves archival researches, reconnaissance (on land, LiDar application), archaeological excavations (Figure 3), stratigraphic studies of walls and lab-analysis of the castle's mortars (chemical,

C).

The study has achieved important results that allowed us to identify a third circle of the walls (Figure 3) which dates back to the beginning of Early Middle Ages.

The outer facades of the walls are made of pebbles of gonfolite (a sedimentary rock of Oligocene and Miocene epoch) arranged in sub-horizontal rows and reemployed material (such as fragments of bricks and carved stones of Roman age) bound by strong mortar, while the emplekton (the rubble stone fill) is composed of plentiful mortar and small stones.

Sampling

The samples were taken from the places that, after a macroscopic assessment and an historical classification by the archaeologists, were considered as representative of the different typologies of mortars. Since the purpose of this study was the classification of mortars and not a survey on their state of preservation, only macroscopically non-degraded samples were taken. For every masonry structure, a number of representative samples was taken, choosing, when possible, semidetached fragments and using chisels only when necessary.

Once in the laboratory, the outermost parts of the samples were removed, in order to favour the analysis of the core of the mortars. The samples did not present layers and could be considered as homogeneous.

A site map is shown in Figure 3.

Twenty-five samples were collected, as shown in Figure 4 and described in Table 1, where the structure of provenance and the age of construction, hypothesized by the archaeologists on the basis of historical documents or stylistic observations, are reported.

Samples coming from the tank showed at naked eye the presence of cocciopesto, which gives them a peculiar pinkishreddish colour (Figure 5). Cocciopesto is composed by crushed terracotta or bricks and was widely used since Roman age in order to induce a hydraulic setting to the material for the coating of tanks, wells, aqueducts, etc., when volcanic materials such as pozzolana were not available [START_REF] Elsen | Microscopy of historic mortars -a review[END_REF][START_REF] Bakolas | Characterization of structural byzantine mortars by thermogravimetric analysis[END_REF]. Hence these mortars are considered hydraulic, due to the formation of calcium silicates at the interface between the brick fragments and lime ('pozzolanic reactions'), giving particular characteristics of adhesion, strength, impermeability to water and resistance to continuous contact with damp [START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Silva | Characterisation of roman mortars from the archaeological site of Tróia (Portugal)[END_REF][START_REF] Velosa | Characterisation of roman mortars from Conímbriga with respect to their repair[END_REF]. As already said, these materials were commonly used in structures such as cisterns, baths, aqueducts, tanks to catch fishes, etc., in permanent or frequent contact with water. The use of crushed bricks in lime mortars is described in Vitruvius (book II, chapter VII, paragraphs XVII-XVIII [START_REF] Vitruvius | On Architecture[END_REF]), which quotes this composite material as opus testaceum.

Representative aliquots were collected from every sample, discarding the surfaces that were exposed to the atmosphere or to the soil. The fragments were then grinded and homogenized in a stainless steel mechanical ball mill Tac 400 MS and kept in clean Kartell LPDE vials until analysis. paper-based databases were used to identify the mineral phases in each X-ray powder spectrum, by comparing experimental peaks with reference patterns. The semi-quantitative mineralogical composition of each mineralogical phase detected in the XRD spectrum was calculated, too.

Thermal Analysis

Thermal analysis (TGA/DSC) was performed by means of a NETZSCH STA 409 PC instrument. Samples were placed in alumina crucibles, the temperature program ranged from room temperature to 900°C, at a heating rate of 10°C min -1 under pure nitrogen atmosphere. TG and DSC raw data were corrected on the basis of a background curve.

Fourier Transform Infrared Spectroscopy

FTIR spectra were recorded on KBr pellets (Sigma-Aldrich FTIR Grade) in transmission mode, by a BioRad Excalibur Series FTS 3000 spectrometer (detector DTGS) in the 4000-400cm -1 range, with a resolution of 4cm -1 , 16 scans.

Pellets were prepared by mixing samples and KBr in an agate mortar, pouring the mixture in a press and then applying a pressure of 6ton/cm 2 for 1 minute.

Scanning Electron Microscopy and Energy Dispersive X-ray spectroscopy

The morphology of the different components of the mortars was observed with a FEI/Philips XL30 ESEM. The elemental analyses were performed using an X-ray energy dispersive spectrometer EDX Quantax 400 coupled to SEM.

Every sample was observed 'as is' in low vacuum mode (1torr) at 20kV, by using GSE, BSE and X-Ray detectors.

Thin Sections

Thin sections were prepared following the standard method. A Nikon Eclipse E400 Pol microscope with Nikon Pol objectives was used for the observation.

RESULTS AND DISCUSSION

All the samples were analyzed in FTIR, TG-DSC and XRD. Some of them, chosen on the basis of their particular features, were observed by means of optical microscopy of thin sections and SEM-EDX.

Thin sections

The observation of thin sections was performed on only one sample for each structure, because it was considered representative of the whole.

The results of the observations are reported in Table 2 The analysis of binder using optical microscopy on thin sections does not allow us to distinguish the presence of a magnesian fraction in the lime. On the contrary, the texture of the binder allows us to differentiate the homogeneous mortars from those containing lumps (always made of calcium carbonate, but clearly different from the surrounding binder).

The aggregate is clearly discriminated by optical microscopy allowing us to separate the lime used as binder from the limestone clasts used as aggregate. In this case the aggregate is always made of natural sand, too. It is quite difficult to understand the provenance of the sand in this region, where the Quaternary glaciers carried out detrital materials from the Alpine range containing igneous and metamorphic silicate rocks from the Basement and sedimentary rocks from the Mesozoic series.

The presence of the cavities, their shape and size, is due to some features of the mix design (binder/aggregate ratio, aggregate shape, amount of water) but could be even strongly affected by the weathering of the mortars during the period of burial.

Sample C2 includes a hydraulic fraction made of bricks mechanically crushed. Samples C7 and C8 contain a silicate aggregate made of quartz coming from igneous rocks and quartz coming from metamorphic rocks (undulose extinction); clasts of metamorphic rocks contain quartz, plagioclase, feldspar or muscovite. Sample C20 (Figure 6b)

(wall E downhill) is different from the other ones featuring high percentage of coarse grained aggregate. Samples C8, C14 (Figure 6a), C16 contain small fragments of wood (size circa 1.0 mm), possibly coming from the addition of charcoal to the inorganic aggregate. Samples C14 (Figure 6a), C16, C20 (Figure 6b) contain limestone in the aggregate, marking a different supply of sand. Samples C14 (Figure 6a) and C16 are quite similar, but the aggregate percentage is different.

XRD

In

FTIR

The species that have been identified are reported in Table 5.

The presence of calcite and silicates in all the samples, as was evidenced from XRD, was confirmed by FTIR, too. All the spectra exhibit bands between 1430 and 1440cm -1 , characteristic of carbonates and corresponding to asymmetric C-O stretching in carbonate ion. Peaks ascribed to calcite are those at 874 (out-of-plane bending vibration), 713 (in-plane bending vibration), 1798 and 2514cm -1 (combination modes). It can be seen that some samples (C3, C4, C5, C23 and C24) present another peak near 856cm -1 (out-of-plane bending vibration), signalling the presence of aragonite, too, partially in agreement with XRD [START_REF] Chakrabarty | Aragonite crystals with unconventional morphologies[END_REF][START_REF] Derrick | Infrared Spectroscopy in Conservation Science[END_REF][START_REF] Gunasekaran | Raman and infrared spectra of carbonates of calcite structure[END_REF]. Bands falling in the regions 1200-900 and 500-400cm -1 can be generally attributed to the SiO 4 tetrahedra of silicates. In most of the samples the peak at 1080cm -1 can be recognized and attributed to quartz, whose most diagnostic signal is the doublet near 780 and 795cm -1 . Quartz cannot be clearly seen in every sample, but this could probably be due to the fact that its doublet can be overlapped by the peaks of other silicates. Gypsum can be detected in sample C6, which presents its characteristic peaks at 3453, 3408, 1681, 1621

(hydroxyl stretching and bending absorption bands), 1147, 1116 (S-O asymmetric stretching vibration bands), 671 and 600cm -1 (S-O bending bands) [START_REF] Derrick | Infrared Spectroscopy in Conservation Science[END_REF]. This sample shows the highest abundance of gypsum in XRD, too. Finally, the presence of traces of organic matter can be hypothesized in samples C1, C2, C3, C4, C8, C10, C12, C13, C14, C20, where marked peaks in the regions around 2800-2900cm -1 and 1630cm In Figure 7 and Figure 8 infrared spectra for samples C4 and C6 are shown for example. The spectrum of C4 (Figure 7)

presents the typical signals detected in all the samples, i.e. those ascribed to calcite, quartz and silicates. In addition to these, it is possible to note, however, one of the characteristic peaks of aragonite (858cm -1 ) and one (1634cm -1 ) attributable to organic substances in trace amounts, probably due to the presence of a carbonyl or a carbon double bond group. The spectrum of the sample C6 (Figure 8) is unique in that it has clear signals of gypsum (3543, 3408, 1681, 1621, 1147, 1116, 671, 600cm -1 ) and a peak (1385cm -1 ) generally associated with the presence of nitrates.

TG-DSC

All samples have been analyzed by thermal analysis (TG and DSC simultaneously).

All of them show the TG signals characteristic of the loss of CO 2 , due to the decomposition of carbonates. Calcium carbonate generally decomposes between 750 and 850°C, but its signal falls in different positions depending on the state of crystallization, the presence of other salts or organic additives, the grain size and the atmosphere used for the analysis [START_REF] Bakolas | Thermoanalytical research on traditional mortars in Venice[END_REF][START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Ingo | Thermal and microchemical investigation of Phoenician-Punic mortars used for lining cisterns at Tharros (western Sardinia, Italy)[END_REF][START_REF] Paama | Thermal and infrared spectroscopic characterization of historical mortars[END_REF]. The temperature of decomposition of calcium carbonate, calculated from DTG (first derivative of TG) curves, can be seen in Table 6. The position of these peaks shifts from about 650°C for polycrystalline calcite, up to about 850°C for the monocrystalline form [START_REF] Reyes-Zamudio | Clay minerals in historic buildings[END_REF][START_REF] Duran | A thermal study approach to Roman age wall painting mortars[END_REF]. In most of the analyzed samples, calcite decomposes at temperatures lower than 850°C, pointing out the presence of recarbonated lime or impurities, for example magnesian lime or clays. If dolomite is present, a double peak appears at 780 and 860°C [START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Paama | Thermal and infrared spectroscopic characterization of historical mortars[END_REF]. Quartz can be identified from the DSC plot, using an endothermic peak at about 573°C, corresponding to the transition phase α → β, without loss of mass [START_REF] Bartz | Mineralogical characterization of rendering mortars from decorative details of a baroque building in Kozuchow (SW Poland)[END_REF][START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Ingo | Thermal and microchemical investigation of Phoenician-Punic mortars used for lining cisterns at Tharros (western Sardinia, Italy)[END_REF]. This characteristic peak is present in almost all samples except those coming from the tank (C1, C2, C3 and C4). Gypsum has two DTG and DSC peaks in the range 120-200°C: signals of this type are present in the samples C1, C6, C9, C12 and C24. Other samples have less clear peaks, for example, C3, C5 and C22. Sometimes it can be difficult to distinguish the two peaks, because they can be overlapped, especially depending on the heating rate set (which also influences the temperature where they fall) [START_REF] Borrachero | The use of thermogravimetric analysis technique for the characterization of construction materials[END_REF]. The two endothermic DSC peaks correspond, respectively, to the loss of 1.5 molecules of water (at about 138°C) and to the one of the remaining 0.5 molecules [START_REF] Bartz | Mineralogical characterization of rendering mortars from decorative details of a baroque building in Kozuchow (SW Poland)[END_REF][START_REF] Adams | Thermal analysis (TA) of lime-and gypsum-based medieval mortars[END_REF]. Gypsum may be a product of the sulphation caused by acid rain due to the presence of H 2 SO 4 , or it could have been added directly as a constituent of the binder to improve the features of compactness, hardness and time of setting [START_REF] Bartz | Mineralogical characterization of rendering mortars from decorative details of a baroque building in Kozuchow (SW Poland)[END_REF][START_REF] Adams | Thermal analysis (TA) of lime-and gypsum-based medieval mortars[END_REF].

Samples C3, C12, C20, C22 and C23 show DTG peaks in the range 300-350°C, due to dehydroxylation of micaceous minerals (in general, the peaks between 200 and 600°C are attributed to the loss of structural OH -from micaceous minerals and phyllosilicates) [START_REF] Iordanidis | Thermal, mineralogical and spectroscopic study of plasters from three post-Byzantine churches from Kastoria (northern Greece)[END_REF]. The peaks associated with micas can also be found around 630°C, such as those visible in the samples C2, C6, C9, C10, C11, C25 and particularly strong in C3 and C4.

The data obtained from TG were subsequently analysed, using the conventional pattern of temperature ranges reported in the literature [START_REF] Genestar | Analytical characterisation of ancient mortars from the archaeological Roman city of Pollentia (Balearic Islands, Spain)[END_REF][START_REF] Bartz | Mineralogical characterization of rendering mortars from decorative details of a baroque building in Kozuchow (SW Poland)[END_REF][START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Ingo | Thermal and microchemical investigation of Phoenician-Punic mortars used for lining cisterns at Tharros (western Sardinia, Italy)[END_REF][START_REF] Bakolas | Characterization of structural byzantine mortars by thermogravimetric analysis[END_REF]:

-room temperature to 120°C: loss of adsorbed water from the mortar;

-120 to 200°C: loss of water from hydrated salts;

-200 to 600°C: loss of structurally bound water in hydraulic compounds (aluminosilicates hydrates, minerals CSH, micas, etc.); ->600°C: loss of CO 2 from the decomposition of carbonates.

The resulting data can be seen in Table 7. It can be seen that the total mass loss (%) ranges from 12.6 (C20) to 51.6 (C1).

The observation of thin sections showed that all samples have a carbonatic binder, but those from wall C, wall D and wall E have a carbonatic aggregate, too. This characteristic makes it impossible to calculate, for these samples, the binder/aggregate, defined as the ratio between the calcite (determined by the loss of CO 2 ) and the rest of the sample.

B/A was calculated, and reported in Table 8, only for samples coming from tank, wall A and wall B.

The index of hydraulicity of the mortar, defined as the ratio between the amount of CO 2 lost above 600°C from carbonates and the amount of H 2 O linked to hydraulic compounds and lost between 200 and 600°C [START_REF] Duran | Study by thermal analysis of mortars belonging to wall paintings corresponding to some historical buildings of Sevillian art[END_REF][START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Paama | Thermal and infrared spectroscopic characterization of historical mortars[END_REF][START_REF] Bakolas | Characterization of the lumps in the mortars of historic masonry[END_REF], was calculated for all the samples. Consequently, higher CO 2 /H 2 O ratios correspond to less hydraulic mortars, and vice versa. The results are shown in Table 8. Some authors classified 'true' lime mortars as having CO 2 /H 2 O ratios higher than 10, hydraulic lime mortars between 4 and 10 and pozzolanic mortars <3 [START_REF] Duran | Study by thermal analysis of mortars belonging to wall paintings corresponding to some historical buildings of Sevillian art[END_REF][START_REF] Iordanidis | Thermal, mineralogical and spectroscopic study of plasters from three post-Byzantine churches from Kastoria (northern Greece)[END_REF][START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Ingo | Thermal and microchemical investigation of Phoenician-Punic mortars used for lining cisterns at Tharros (western Sardinia, Italy)[END_REF]. In our study, samples C7, C15, C16 and C17 have CO 2 /H 2 O ratios exceeding 10, so they can be considered as true lime mortars. Most of the others range between 3 and 5, so they can be classified as hydraulic. It is generally difficult to recognise hydraulic lime, but its presence is compatible with an attested use in this region. Quarries of hydraulic lime can be found about 20 km East from the Baradello Castle, in the area near the small lakes of Brianza region where the Upper Cretaceous marly limestone formation called 'Scaglia Lombarda' outcrops.

The particularity of the sample C6 is outstanding, with its very low ratio (1.41), probably because of its content in gypsum, as evidenced also by XRD and FTIR analysis. Its particular composition, in fact, generates a loss of CO 2 equal to only 5.62% by weight, giving a percentage of calcite in the sample of 13.04% and, consequently, B/A and CO 2 /H 2 O ratios particularly low. The structure where C6 was taken has been buried for centuries, so it can be excluded that the presence of gypsum is due to sulphation phenomena for exposition to air pollutants, but rather to a deliberate addition to the binder. Archaeologists did not observe macroscopic differences in this part of wall, so they cannot assert the presence of a work of conservation during centuries, but they cannot exclude this hypothesis a priori. Its peculiar composition, so different from those of the other samples taken from the same structure, could suggest that C6 was a replacement done during a small work of conservation. The use of gypsum has never been attested before in samples of masonry mortars from Lombardy, whilst it was found in renderings or decorative plasters [START_REF] Bugini | Plaster technologies in Northern Italy from 15th to 19th century[END_REF].

Samples C3, C4 and C12 have ratios <3. The first two samples come from the tank and present a high loss of mass between 200 and 600°C, maybe related to the presence of visible cocciopesto. For what concerns sample C12, it just seems quite poor in calcite.

The numerical results can be better seen in Figure 9, Figure 10 and Figure 11. For what concerns the tank, it can be seen that samples C1 and C2, taken in the corner near to the Hexagonal Tower, are nearly overlapped, differently from samples C3 and C4, taken on the side toward the Tower of the Castle.

In Figure 11 (see Figure 11a for the whole graph and Figure 11b for a zoom on the central area, where many samples were overlapped) the hydraulicity index is shown, vs. the % amount of CO 2 , making the classification of samples immediately clear, according both to their coefficient of hydraulicity and their content in carbonates [START_REF] Moropoulou | Characterization of ancient, Byzantine and later historic mortars by thermal and X-ray diffraction techniques[END_REF][START_REF] Ingo | Thermal and microchemical investigation of Phoenician-Punic mortars used for lining cisterns at Tharros (western Sardinia, Italy)[END_REF]. The peculiarity of the structure named wall D is to have mortars with a marked content in carbonates. It is now possible to see how the sample C6 has the most hydraulic composition, i.e. the lowest hydraulic index. The samples coming from the tank tend to group according to the side to which they belonged (C1-C2 vs. C3-C4), as was seen from B/A ratios.

Finally, it is interesting to note how the samples from wall E are distributed: C18 and C19, taken uphill, are much more aerial than those taken downhill. The archaeologists attributed this structure to one only construction campaign, but these results point towards the hypothesis of distinct workdays, or the simultaneous work of two groups with different experiences and procedures of building.

The same considerations are evident in Figure 10, where the hydraulicity vs. the structure of provenance is shown.

Finally, the presence of peculiar peaks of gypsum, quartz and calcite in DSC curves is summed up in Table 9. A SEM image and EDX map of sample C3 (tank) is shown in Figure 12, where fragments of cocciopesto, rich of Al, Si, K and Fe, can be seen. For what concerns sample C6 (wall A), where gypsum was detected by all analytical techniques, its presence is confirmed by SEM-EDX, too. Gypsum aggregates have 'cloudy' (Figure 13) or needle-like (Figure 14) shapes, as

shown by the EDX map, where sulphur is clearly accumulated and associated with calcium. All the matrices of the mortars are, obviously, rich of calcium and magnesium, as can be seen from XRD, too, which pointed out the presence of pure calcitic and calcitic-magnesian limes. Most of the samples present a considerable amount of silicon, aluminium, and potassium, coming from the silicates constituting the aggregate dispersed in the binder. Some samples contain particles rich of copper and titanium, too.

CONCLUSIONS

The 'traditional' techniques showed, in this work, the existence of differences between groups of structures, but the thermal analysis was actually discriminating. The various structures were, in fact, characterized by the index of hydraulicity and, except for the mortars with carbonate aggregate, by the binder/aggregate ratio. In particular, the coupling of index of hydraulicity and %CO2 revealed the presence of patterns bound to the different structures. All these values were obtained by thermal analysis, which proved to be the most meaningful techniques to evidence a possible building chronology.

It is, in fact, generally difficult to obtain comparable results when it comes to mortars, because of their specific heterogeneity and variability, and the use of different analytical techniques is often the only way to have a global vision of mortars, to classify them and to recognize groups.

The analyses of samples from Baradello site generally confirmed the hypotheses of the archaeologists, showing a common homogeneity between the materials used in the same structure, while some differences emerged between the castle walls. The results stressed the peculiarity of the tank, whose mortars contain mechanically crushed bricks known as 'cocciopesto': some small differences emerged between the samples, confirming the hypotheses made by the archaeologists concerning subsequent integrations. In addition to the results of thermal analysis, it can also be observed that samples C3 and C4, related to a recent (19th or 20th century) maintenance work contain traces of aragonite, which is not present in C1 and C2.

Wall A, B, C, D and E were classified by archaeologist as ascribable to the same building epoch, between 6th and 7th century AD, since they all presented similar characteristics at a macroscopic analysis. Wall A is, by far, the most heterogeneous structure. Sample C5 can be considered as an original ancient mortar; C6 is the only one sample in which all the techniques evidenced the presence of gypsum added to lime and can be identified as a repair mortar. Local sources of gypsum can be found on the Lake Como shores (Nobiallo, near Menaggio) nearly 35km north from the Baradello site.

Concerning sample C7, thermal analysis showed its different characteristics respect to the other samples in the same structure. The presence of white lumps inside, as seen in thin sections, suggested a different dating, probably due to another work of integration following the construction. In addition, this is one of the only two samples showing traces of dolomite.

Wall B is more homogeneous and can be considered as coeval with wall A from an archaeological evaluation, because they share many characteristics. In particular, thermal analysis evidenced that samples C8 and C11 have a composition similar to C5, which can be identified as an original material of wall A. All the mortars of wall B can be considered as hydraulic according to the index calculated by means of thermal analysis. It is generally not easy to recognise hydraulic lime, but its use is attested as widespread in this area. Sources of hydraulic lime can be found in the area of the small lakes of Brianza, between Como and Lecco.

Wall C belongs to the second group of structures, together with wall D and E. As already said, they contain both silicatic and carbonatic aggregate: the first can probably come from the surroundings of the Castle, the second from the area of Cernobbio (5 km northern). A slight difference between C12 and C13-C14 can be seen in thermal analysis: this can be related to the presence of traces of added gypsum in C12 and the use of magnesian lime (as seen from XRD analysis) in C13 and C14, instead of calcitic lime as in C12.

Wall D also showed in TG-DSC a low hydraulicity and a particular richness in carbonatic components, both in the binder and, as already said, in the aggregate (as seen from thin sections). It seems to have been built with a different technique, as observed from the different kind of raw materials used, even if there no differenced were evidenced by the archaeological point of view.

Coming to wall E, thermal analysis seemed to support that the samples taken uphill are less hydraulic than those taken downhill. The archaeologists classified this whole structure as built in the same period but, considering the results of these analyses, different phases of construction cannot be excluded.

Summing up the obtained results, a general distinction can be observed between the group constituted by wall A and B and that of wall C, D and E. Within this second group, thermal analysis also highlighted that wall D is clearly different from C and E. The same technique contributed to show that some structures, substantially homogeneous, were reshaped with some integration after their construction.

The different binders and aggregates used, coming from different quarries, can thus suggest the existence of various moments of construction, with an expansion of the built structures. From the archaeological point of view, it was common in Early Middle Ages to observe a technique of construction with different teams of builders, working simultaneously at different structures under the guidance of a supervisor, using different techniques and raw materials coming from different areas or materials of reuse.

Figure 1 -Figure 2 -

 12 Figure 1 -The Tower of the Baradello Castle, Como, Italy

Figure 3 -

 3 Figure 3 -Site map

Figure 4 -Figure 5

 45 Figure 4 -Provenance of samples

Figure 6 -

 6 Figure 6 -a: sample C14 -Thin section, cross-polarized light. A lime binder, with irregularly shaped cavities, contains clasts of quartz and metamorphic rock together with a wood fragment (top centre). b: sample C20 -Thin section, cross-polarized light. A lime binder, with cracks coated of secondary calcite crystals, contains rounded clasts of limestone (right), metamorphic rock clasts (centre) and angular crystals of quartz (left)

Figure 7 -Figure 8 -

 78 Figure 7 -FTIR spectrum of sample C4 in KBr (Absorption peaks: calcite: 2517, 1797, 1448, 874, 713cm -1 ; aragonite: 858cm -1 ; quartz: 801, 778cm -1 ;silicates: 3590, 3457, 3397, 1037, 977, 478, 455cm -1 ; organic compounds: 1634cm -1 )

Figure 9 -Figure 10 -

 910 Figure 9 -B/A ratio relative to the different structures

9 -

 9 Presence of peculiar peaks in DSC curves3.5 SEM-EDXA selection of samples was observed and analysed using SEM-EDX.

Figure 12 -

 12 Figure 12 -SEM images and EDX distribution maps of Al, Si, Fe and K (clockwise from upper left) of cocciopesto fragments in sample C3

Figure 13 -Figure 14 -

 1314 Figure 13 -SEM images and EDX elemental maps of Ca and S of 'cloudy' gypsum crystals in sample C6

Table 1

 1

	-Description of the mortar samples
	2.3 X-ray Powder Diffraction Analysis
	X-ray diffraction analysis was conducted using a Bruker AXS D8 ADVANCE diffractometer operating at 30kV and
	40mA, Cu-Kα radiation (λ=1.5418Å) monochromated with a graphite monochromator. Scans were collected on powder
	samples in the range 5-55° 2θ; step scan Δ2θ = 0.

02°, counting time t = 1s. EVA software (DIFFRACplus EVA) and

Table 3 -

 3 and Table3.

			binder		cavities
	structure sample
			composition	texture	shape	size (mm)
	tank	C2	calcite (lime) lump	rounded	0.5
	wall A	C7	calcite (lime) lump	irregular	0.5
	wall B	C8	calcite (lime) lump	rounded	0.2 -0.6
	wall C	C14	calcite (lime) lump	irregular	0.2 -0.8
	wall D	C16	calcite (lime) lump	irreg. cracks 0.1 -0.2
	wall E	C20	calcite (lime) homogeneous irregular	0.2 -0.8
	Table 2 -Results of petrographic analysis
			aggregate features
	structure sample
			clasts		size (mm)	shape	corners	sorting	aggregate %
			quartz (metam.)	0.8	equant	subrounded -
	tank	C2
			brick		0.4	equant	subangular
			quartz		0.1 -0.5	prismatic	subangular scarce
	wall A	C7
			metam		1.5	prismatic	rounded
			quartz (metam.)	0.1 -0.8	equant -prism. angular	medium	low
	wall B	C8
			muscovite		0.4 -0.6	lamellar	angular
			quartz		0.1 -1.0	equant	subangular very scarce medium
	wall C	C14	metamorphic	0.5 -4.1	prismatic	subrounded
			limestone		0.8 -2.0	equant	rounded
			quartz		0.1 -0.3	equant	subangular very scarce very low
	wall D	C16	metamorphic	0.2 -0.8	equant	subrounded
			limestone		2.0 -5.0	equant	rounded
			quartz		0.1 -0.3	equant	subangular scarce	high
	wall E	C20	metamorphic	0.2 -9.0	equant	subrounded
			limestone		0.5 -2.2	equant	rounded

Results of petrographic analysis

Legend: qz = quartz, met. = metamorphic rock fragments, lim. = limestone, mus. = muscovite

Table 4

 4 KAlSi 3 O 8 ] and orthoclase [monoclinic KAlSi 3 O 8 ]. Fillosilicates as serpentine [(Mg, Fe) 3 Si 2 O 5 (OH) 4 ] and micas (muscovite [KAl 3 Si 3 O 10 (OH,F) 2 ]) are also common. Six samples (C1, C3, C4, C6, C12, C20), especially those coming from the tank, contain little amount of gypsum (CaSO 4 •2H 2 O) that could have been added to the raw materials of the mortars, or could be a degradation product[START_REF] Iordanidis | Thermal, mineralogical and spectroscopic study of plasters from three post-Byzantine churches from Kastoria (northern Greece)[END_REF]. Gypsum was not highlighted by the observation of thin sections because it is difficult to be seen, unless it is greatly abundant. Three samples (C3, C4, C5) include a little amount of aragonite [CaCO 3 ], the metastable orthorhombic form of calcium carbonate, less common than calcite and generally found in the shells of molluscs[START_REF] Edwards | The conservational heritage of wall paintings and buildings: an FT-Raman spectroscopic study of prehistoric, Roman, mediaeval and Renaissance lime substrates and mortars[END_REF][START_REF] Moropoulou | Investigation of the technology of historic mortars[END_REF][START_REF] Genestar | Ancient covering plaster mortars from several convents and Islamic and Gothic palaces in Palma de Mallorca (Spain). Analytical characterisation[END_REF][START_REF] Prieto-Taboada | Spectroscopic evaluation of the environmental impact on black crusted modern mortars in urban-industrial areas[END_REF]. Aragonite is not a common mineral in the area of Baradello, but it could be found in the raw materials that constitute the mortar, due to the presence of shells of

						magnesian
	structure		sample calcite aragonite	dolomite quartz gypsum feldspars phyllosilicates pyroxenes other
						calcite
			C1	+++			++	+	+
			C2			+++	+++	+
	tank
			C3		+	+++	+	+
			C4	+++	+		+++	+	+	+
			C5	++	+		+++	+
	wall A		C6	++			+++	+	++	+
			C7	+++		+	+
			C8			+++	+++	+	+
			C9	+++			+++	++	+	+
	wall B
			C10	+++			+++	+++	+	++
			C11	+++			++	+	+
			C12	+++			+++	+	++	+
	wall C		C13			+++	+++	+	+
			C14			+++	+++	+	+
			C15			+++	+++	+/-
	wall D		C16			+++	+	+	+
			C17	+++			+	+	+/-
	wall	E	C18			+++	+++	+	++
	(uphill)		C19	++			+++	+
			C20			+++	+++	+/-	+	+	+/-
			C21			+++	+++	++	+	+
	wall	E	C22			++	+++	++	+
	(downhill)		C23	+++		+
			C24	+++			+++	+++	+	+
			C25	+++			+++	+++	+

X-ray diffraction results are reported. The relative abundance is also shown, on the basis of the intensity of peaks. Quartz [SiO 2 ] and calcite [CaCO 3 ] (either magnesian calcite, [(Ca, Mg)CO 3 ]) are the most common mineral phases found in each mortar, even if in different relative amounts. For example, samples C3, C7 and C17 are constituted almost exclusively of calcite, while the opposite situation is present in the sample C6. It can also be noted that the sample C23 does not contain quartz. Most of the samples, except for C3, C7, C15 and C23, contain the feldspar albite [NaAlSi 3 O 8 ]: a peculiar situation can be seen in C24 and C25, where it is particularly abundant. Other feldspars can be recognized, as microcline [triclinic gastropods coming from the near Lake Como. Two samples (C7 and C23) have little amounts of dolomite [CaMg(CO 3 ) 2 ]. All the samples coming from wall E (except C23) show the peculiarity of quartz being the most abundant mineral phase. The presence of muscovite, a mineral of the family of micas, in many samples may indicate

that the burning of the calcareous stones to prepare the quick lime was done at temperatures lower than 800-900°C

[START_REF] Sanjurjo-Sanchez | Chemical and mineralogical characterization of historic mortars from the Santa Eulalia de Boveda temple, NW Spain[END_REF]

.

Table 4 -

 4 XRD semiquantitative analysis data.

Legend: +++ very abundant (>40%); ++ abundant (15-40%); + present (3-15%); +/-scarce (<3%)

Table 5 -

 5 -1 can be observed, due to the presence of C-H and carbonyl groups. FTIR data

	structure	sample calcite aragonite quartz silicates gypsum nitrates organic compounds (traces)
		C1	*		*	*		*
		C2	*		*	*		*
	tank
		C3	*	*	*	*		*
		C4	*	*	*	*		*
		C5	*	*		*
	wall A	C6	*		*	*	*	*
		C7	*			*
		C8	*		*	*		*
		C9	*		*	*
	wall B
		C10	*		*	*		*
		C11	*		*	*
		C12	*		*	*		*
	wall C	C13	*		*	*		*
		C14	*		*	*		*
		C15	*		*	*
	wall D	C16	*		*	*
		C17	*		*	*
		C18	*		*	*
	wall E (uphill)
		C19	*		*	*
		C20	*		*	*		*
		C21	*		*	*
		C22	*		*	*
	wall E (downhill)
		C23	*	*	*	*
		C24	*	*	*	*
		C25	*		*	*

Table 6 -

 6 Temperature of decomposition of calcium carbonate

			temperature of CO2 loss
	structure	sample
			in DTG curves (°C)
		C1	806
		C2	792
	tank
		C3	780
		C4	770
		C5	804
	wall A	C6	794, 878
		C7	823
		C8	783
		C9	778
	wall B
		C10	786
		C11	809
		C12	786
	wall C	C13	793
		C14	786
		C15	849
	wall D	C16	817
		C17	818
		C18	778
	wall E (uphill)
		C19	814
		C20	783
		C21	820
		C22	782
	wall E (downhill)
		C23	781
		C24	787
		C25	793

Table 7 -

 7 Mass loss according to the conventional pattern of temperature ranges

Table 8 -

 8

	structure	sample CO2/H2O B/A % calcite
		C1	4.96	1.08 51.91
		C2	6.11	1.16 53.66
	tank
		C3	2.67	0.75 42.96
		C4	2.06	0.41 29.31
		C5	3.99	0.86 46.36
	wall A	C6	1.41	0.15 13.04
		C7	10.58	4.79 82.71
		C8	3.28	0.61 38.06
		C9	4.27	0.41 29.07
	wall B
		C10	3.57	0.39 28.21
		C11	4.67	0.97 49.32
		C12	2.74		28.58
	wall C	C13	4.67	n.d.*	45.33
		C14	4.11		33.94
		C15	11.99		74.18
	wall D	C16	10.56	n.d.*	75.63
		C17	11.47		71.31
		C18	8.55		31.30
	wall E (uphill)
		C19	8.95		42.70
		C20	4.34		19.59
		C21	4.00		38.52
				n.d.*
		C22	3.49		22.44
	wall E (downhill)
		C23	3.56		34.06
		C24	3.88		31.25
		C25	3.44		25.25

index of hydraulicity, binder/aggregate ratio and calcite % obtained by TA results Legend: n.d. = value not determined, *see text for details

  The analyses showed that they all have an aggregate consisting of natural sand, but A and B contain only silicate sands, while C, D and E include also carbonate sands, suggesting a different source of supply for raw materials. It is worth to remember that the Baradello Castle site lies on a geological overthrust connecting Miocene quartz sandstones and conglomerates (Gonfolite) southbound, and Lower Jurassic limestones (Calcari selciferi lombardi) northbound: the sand available in these two geological areas is quite dissimilar. Thus, the different composition of raw materials used for the aggregate is completely coherent with the geology of the region and point towards two different areas of material supply and two different moments of construction, at least.

It can also be noted that magnesian lime is nearly absent from wall A and B (except for sample C8), whilst it is more diffuse in wall C, D and E. The nearest source of magnesian lime can be found in the area of Valmadrera, in the socalled 'Triangolo Lariano', the region between the two branches of the Lake Como (about 25 km eastward). A close relationship between Como and this region in Roman and Early Medieval ages is well documented and was eased by the presence of a pedemountain road connecting Como to Erba and Lecco.

Acknowledgements

The research has been conducted by Musei Civici di Como in collaboration with Università di Padova, Università di Milano Bicocca, Università dell'Insubria -Como. Under ministerial concession, the direction of archaeological excavation is entrusted to G.P. Brogiolo, L. Castelletti, I. Nobile. The project has been financed by Regione Lombardia and Comune di Como.