Vaughan T J Phillips

Jun-Ichi Yano

Ice Multiplication by Fragmentation of Spherical Freezing of Drops: A Theoretical Investigation

The ice multiplication by fragmentation associated with collision-freezing of supercooled drops is investigated. A zero-dimensional dynamical system describing the time evolution of the number density of the supercooled drops and the ice crystals in a mixed-phase cloud is developed. The analytical solutions of this system are derived for various asymptotic limits. Especially, when the ice fragmentation by freezing supercooled drops is considered in its isolation, a complete analytical solution is available, which shows that the originally-existed supercooled drops are simply all converted into the ice crystals multiplied by a fragmentation number per freezing. When constant generation of both supercooled drops and ice crystals is considered, the icecrystal number tends to linearly increase with time with the rate given by the sum of the primary-ice and the the supercooled-drop sources with the latter multiplied by the fragmentation number per freezing. The study concludes that the ice multiplication by fragmentation of freezing drops can be a crucial process in certain situations. Yet its multiplication tendency is limited by availability of supercooled rain or drizzle and by whether the freezing temperature allows more than one fragment to be emitted per frozen drop so as to allow a positive feedback with collisional raindrop-freezing. When there is no more rain or drizzle, the multiplication stops. When there is copious supercooled rain initially and temperatures are optimal, then this fragmentation can be more prolific than the other multiplication processes such as the Hallett-Mossop process and the ice-ice collision process.

Introduction

Coincident measurements of concentrations ice particles in clouds and of active ice nucleus (IN) aerosols in the environment have shown a discrepancy by orders of magnitude, in clouds too warm for homogeneous freezing [START_REF] Hobbs | Ice multiplication in clouds[END_REF][START_REF] Auer | Observations of ice crystal an dice nuclei concentrations in stable cap clouds[END_REF]. This has been explained in terms of fragmentation of ice. Several pathways of fragmentation have been observed in the laboratory, but their relative importance has been unclear.

One possible type of fragmentation is shattering during collision-freezing of supercooled drops.

Generally the rate of freezing of a drop is limited by the rate of dissipation of latent heat. Initially, the latent heat from freezing is converted to thermal energy of the drop, raising its temperature close to 0 • C almost instantaneously. In this first stage, needles of ice permeate the drop. In the subsequent stage, the latent heat flows down the gradient of temperature to the surroundings, once a steady state has been reached. The rate of dissipation to the air governs the freezing rate. The fastest rate of freezing occurs for freezing as near as possible to the drop surface. Hence, the freezing occurs from the outside inwards, with a shell of ice encasing the liquid core.

It is experimentally observed that as the ice shell thickens, the expansion on freezing can sometimes cause the shell to shatter [START_REF] Wildeman | Fast dynamics of water droplets freezing from the outside in[END_REF]. Splinters of ice can then be emitted into the air. Many ealier laboratory experiments [START_REF] Latham | Generation of electric charge associated with the formation of soft hail in thunderclouds[END_REF][START_REF] Brownscombe | Freezing and shattering of water drop sin free fall[END_REF][START_REF] Dye | The influence of environmental parameters on the freezing and fragmentation of suspended water drops[END_REF][START_REF] Hobbs | The fragmentation of freezing water droplets in free fall[END_REF][START_REF] Takahashi | Deformation and fragmentation of freezing water drops in free fall[END_REF]) measured this fragmentation for drops in free-fall: See [START_REF] Mossop | Concentration of ice crystals in clouds[END_REF] as a review. Only a few splinters are typically emitted per frozen drops (usually less than about 1-10 depending on temperature and size). On the other hand, there is a positive feedback that can lead to all supercooled drops being frozen quite quickly: as ice splinters are emitted they can grow and after a minute or so are large enough to collide efficiently with supercooled drops, freezing them. splintering for a New Mexican convective cloud. It was found that it was less prolific than the [START_REF] Hallett | Production of secondary ice particles during the riming process[END_REF] process of rime-splintering between -3 and -8 • C when cloud-droplets freeze on impact with graupel. However, [START_REF] Lawson | The microphysics of ice and precipitation development in tropical cumulus clouds[END_REF] simulated another observed case with a spectral bin microphysics parcel model and found the positive feedback to account for the observed ice. Phillips et al. (2018) provided a comprehensive formulation to treat rain-or drizzle-drop freezing fragmentation by pooling published data from past laboratory studies in the literature. Two modes were represented:

• Mode 1: spherical drop-freezing when incident ice particle is less massive than the drop or due to immersed IN activating, with an outer ice shell growing inwards and breaking;

• Mode 2: non-spherical drop-freezing when incident ice particle is more massive than the drop and ice particles may be present in the splash.

Only mode 1 has been observed in the lab comprehensively. For mode 2 a theoretical approach for the physics of the collision was invoked. After creating an empirical formulation for mode 1 alone, [START_REF] Phillips | Secondary ice production by fragmentation of freezing of drops: formulation and theory[END_REF] applied it to a simulation of a composite case of deep convection from the tropical Atlantic in ICE-T (Ice in Clouds Experiment-Tropical). They found observed ice concentrations were predicted in a bin microphysical parcel model with the new schemes for an assumed probability of 50% (-10 • C) for the chance of any drop in the splash containing ice due to partial freezing on impact.

The empirical scheme for numbers of secondary fragments per drop, N, from spherical dropfreezing of mode 1 was expressed by [START_REF] Phillips | Secondary ice production by fragmentation of freezing of drops: formulation and theory[END_REF] with a 2D phase-space of drop diameter and freezing temperature. A peak of N near -15 • C was observed. Construction of a zerodimensional (0D) model of monodisperse populations of crystals, graupel (from drop-freezing) and supercooled drops without fallout showed that the positive feedback with exponential growth of ice concentrations only occurs for N > 1, which is found at diameters of 0.1 mm between about -13 and -17 • C, 0.2 mm (or 0.4 mm) between about -10 • C and -20 • C and 1.6 mm when colder than -7 • C. When mode 2 was included in the 0D model, the realm of instability was extended to a wider range of temperature.

The purpose of the present paper is to extend this investigation of the efficiency of the ice fragmentation by spherical drop-freezing from such a theoretical point of view. A similar 0D dynamical system is applied for this purpose, generalized to include a source of raindrops from coalescence, as presented in the next section. By taking the similar methodologies, [START_REF] Yano | Ice-ice collisions: An ice multiplication process in atmospheric clouds[END_REF] and Yano et al. (2016) have investigated the efficiency of the Hallett-Mossop and the iceice collision ice-multiplication processes. Yano et al. (2016) further list the various methodologies applicable for interpreting the behavior of a dynamical system, and in that manner, the basic nature of a given microphysical process described by the given dynamical system can be elucidated.

The most basic strategy is to develop an asymptotic expansion by assuming a certain physical parameter to be small. We emphasize that a given physical parameter may not be necessarily small arithmetically, but it can still be introduced as a small parameter in order to obtain insights to the basic behavior of the system. This most basic strategy is systematically applied in the present study. After introducing a formulation of the problem in the next section, theoretical analysis considers first the ice fragmentation process of mode 1 in its isolation in Sec. 3 and the feedbacks are quantified. General analyses are presented in Sec. 4. Obtained results are further discussed in Sec. 5.

0D Model

a. Basic Assumptions

We consider a cloud at subzero temperatures consisting of supercooled drops, ice crystals, and graupel. Their number densities are designated by n r , n i , and n g , respectively. Collisions between ice crystals and supercooled drops cause freezing to the latter leading to formation of graupel, associated with N (up to 10-100) secondary fragments of ice emitted by each frozen drop. The rate that supercooled drops freezes by collision of ice particles may be given by Kn r n i , where K is a coefficient describing the product of the collision efficiency and rate of geometric sweep-out of air volume per second (collection kernel, m 3 sec -1 ).

We further assume constant generations, c 0 and c i , of supercooled rain and ice crystals, respectively. We also assume that the supercooled rain is lost by fall with a characteristic time scale of τ f . Some of those additional processes may be turned off in the following as required.

Here, being consistent with our earlier studies [START_REF] Yano | Ice-ice collisions: An ice multiplication process in atmospheric clouds[END_REF], Yano et al. 2016[START_REF] Phillips | Secondary ice production by fragmentation of freezing of drops: formulation and theory[END_REF], no spatial dimension is considered.

b. Mathematical Formulation

From the physical assumptions just introduced above, the mathematical description of this microphysical system is presented by

ṅr = c 0 -Kn r n i -n r /τ f (2.1a) ṅi = c i + KN ′ n r n i (2.1b)
Note that the drop-freezing is initiated by capture (loss) of an ice particle, and leads to a gain of N ice particles by fragmentation so that N ′ = N -1 more ice particles are found after drop-freezing in total per drop frozen. This means that for some conditions of drop size and freezing temperature (as reviewed by Pruppacher and Klett 1997), there can be N ′ < 0, such as at diameters less than about 0.1 mm near the optimum of -15 • C and 0.5-1 mm at temperatures several degrees warmer or colder than this. For a monodisperse population of drops, when N ′ < 0, then there is capture of splinters causing a decrease with time of the crystal concentration with incomplete freezing of the population of drops. This corresponds to a situation in reality with a continuum of drop sizes in any cloudy volume but with insufficiently numerous large drops to cause any explosion of ice concentration (Sec. 1).

Additionally, the time evolution of the graupel number, n g , is presented by ṅg = Kn r n in g /τ g also assuming that the graupel is lost from the cloud-microphysical system by fall-out with a characteristic time-scale, τ g . However, the graupel number, n g , does not affect the subsequent evolution of the remaining part of the system. Thus, in the following, the evolution of n g will not be considered.

c. Nondimensionalization

Nondimensionalization is crucial to ensure a systematic investigation of a given system, because it reduces the parameters characterizing the system to a smaller number of nondimensional parameters, thus a parameter space for the investigation is much reduced.

In the following, we add the subscript * to the nondimensional variables in order to distinguish them from the dimensional variables. Here, we introduce an unspecified time scale, τ, for nondimensionalizing the time. Although the fall time-scale, τ f , of rain is a natural choice for the scale, we retain a certain freedom for the model analysis in this manner. Thus,

d dt = 1 τ d dt * .
(2.2)

The particle number densities may be nondimensionalized by (2.4b)

n i = n i * /Kτ, (2.3a) n r = n r * /KN ′ τ. ( 2
Here, we have introduced the following nondimensional parameters:

ĉ0 = KN ′ τ 2 c 0 , (2.5a) ĉi = Kτ 2 c i , (2.5b) ε s = τ/τ f .
(2.5c)

Note that ĉ0 and ĉi are the nondimensional generation rates of liquid and ice; ε s measures the magnitude of the rain sedimentation rate. Furthermore, the dimensionless tag, K, is introduced, multiplying the freezing-fragmentation terms so as to indicate a contribution of this process to the solutions. Formally K = 1 by nondimensionalization when this process is present and may be set zero otherwise. Note that the time scale, τ, is defined by Eq. (3.5) below in retrospect.

Basic Analysis: Freezing-Drop Fragmentation Process in Isolation

As the simplest case, the shattering of ice particles by freezing drops is considered in isolation away from the other processes. We obtain this situation by setting, ĉ0 = ĉi = ε s = 0 in the above general system (2.4):

ṅr * = -Kn r * n i * , (3.1a) ṅi * = Kn r * n i * . (3.1a)
Here, the tag, K, for the ice fragmentation process is still retained. Note that there is no nondimensional parameter controlling this system. Thus, even in numerical terms, the interest of the investigation of this system only depends on the initial condition. As seen below, the initial condition can further be normalized, because the characteristic time-scale, τ, of the system remains arbitrary due to the absence of rain sedimentation.

a. Full Solution

We identify two equilibrium solutions in this system: either n r * = 0 or n i * = 0 and the other number density remains an arbitrary finite value. As suggested by [START_REF] Yano | Ice-ice collisions: An ice multiplication process in atmospheric clouds[END_REF], as well as Yano et al. (2016: See especially their Sec. 6.2), the standard procedure would be to perform the linear perturbation analysis against these equilibrium states for inferring the tendency of the system away from these equilibrium states.

However, in the present case, a full analytical solution is available for Eqs. (3.1a, b). This is realized by noting that the sum of the two particle number densities is conserved, as seen by taking the sum of Eqs. (3.1a, b):

d dt * (n r * + n i * ) = 0. (3.2)
In dimensional terms, the above conservation law is

d dt (N ′ n r + n i ) = 0.
Thus, more precisely, the sum of the number densities, weighted by the ice-fragmentation number, N ′ , on the rain water number density, n r , is conserved with time.

From Eq. (3.2), n i * can be written in terms of n r * by

n i * = n 0 -n r * , (3.3)
where n 0 = n r * (0) + n i * (0) is the initial total particle number. By substituting the above expression into Eq. (3.1a), we obtain a differential equation solely in terms of n r * . It can be readily solved to obtain:

n r * = (n r * (0) + n i * (0))(1 + n i * (0) n r * (0) e Kn 0 t * ) -1 . (3.4a)
Further substitution of the above solution into Eq. ( 3.3) leads to a solution for n i * :

n i * = (n r * (0) + n i * (0))(1 + n r * (0) n i * (0) e -Kn 0 t * ) -1 . (3.4b)
Note that the time-scale, τ, introduced above for non-dimensionalising the system remains arbitrary. Thus without loss of generality, we may set the initial condition to n 0 = 1, which implies that

τ = K -1 (N ′ n r (0) + n i (0)) -1 (3.5)
by referring to Eq. (2.3a, b). Note that the dimensionless characteristic time-scale for the exponentials of both n r * and n i * is just 1/n 0 in Eq (3.4a,b). This characteristic time-scale becomes τ when dimensionalised. In other words, the initial particle number dictates the characteristic timescale of the system, as shown by [START_REF] Phillips | Secondary ice production by fragmentation of freezing of drops: formulation and theory[END_REF]. Furthermore, the behaviour of the system can be investigated systematically by solely changing the initial condition for n r * as a result. The examples of time evolution for those normalizations are shown in Fig. 1.

It may also be worthwhile to note that the solutions (3.4a, b) are given in dimensional terms by:

n r = (n r (0) + n i (0)/N ′ )[1 + n i (0) N ′ n r (0) e K(N ′ n r (0)+n i (0))t ] -1 (3.6a
)

n i = (N ′ n r (0) + n i (0))[1 + N ′ n r (0) n i (0) e -K(N ′ n r (0)+n i (0))t ] -1 (3.6b)
The result shows that regardless of the initial condition, the system exponentially approaches to a state only with the ice crystals, n i = N ′ n r (0) + n i (0) by converting all the supercooled drops into the ice. As a result, we also see that the state with n r = 0 and n i = 0 is unstable, and that with n r = 0 and n i = 0 is stable against small perturbations.

Finally, the IE factor, f (t) = n i (t)/n i (0), evolves as

f (t) = 1 + N ′ n r (0)/n i (0) 1 + N ′ n r (0)/n i (0)e -K(N ′ n r (0)+n i (0))t
and the eventual ice enhancement reaches the maximum

f = 1 + N ′ n r (0)/n i (0) as t → ∞.

b. Perturbation Analysis

A particular interest here is how the ice-number evolution is modified by slightly modifying its value by n ′ i * . As a result, the rain-water number is also perturbed by, say, n ′ r * . In this isolated setting, the total number density is conserved, thus we may set

n ′ r * = -n ′ i * . (3.7)
The pertubation equation for n ′ i * is given by

ṅ′ i * = K(n i * n ′ r * + n r * n ′ i * )
or by substituting the number-concentration constraint (3.7),

ṅ′ i * = λ n ′ i * ,
where

λ = K(n r * -n i * ) (3.8)
may be considered a feedback parameter characterizing this perturbation evolution. We see that at the initial stage when the rain-water is relatively abundant, n r *n i * > 0, and the feedback is positive (splintering feedback), whereas when the ice number has substantially multiplied, n r *n i * < 0, and the feedback becomes negative (drop-depletion feedback)

For any system in general, a feedback parameter may be defined as the rate of change of a forcing, Q, of system evolution with respect to its response (a change in its state variable, x) with λ = dQ/dx. Here, λ > 0 for a net positive feedback and λ < 0 for a net negative feedback. The forcing may be a function of variables quantifying several processes

(Q = Q(x, X 1 , X 2 , ...X n ))
. By performing this derivative as a partial derivative with respect to only one process variable, then the derivative quantifies the feedback from that process (

λ i = (∂ Q/∂ X i )(dX i /dx)).
The total feedback parameter is the sum of partial derivatives corresponding to all the processes by the chain rule:

λ = ∑ i λ i .
Thus the net feedback parameter has contributions from all the positive and negative feedbacks of the system.

One can regard the above evolution equation ( ṅi * = Kn r * n i * ) for ice crystal number as arising from two separate processes: splintering of freezing drops, controlled by n r * , and depletion of drops by collision with crystals, controlled by n i * . The evolution equation of the system (Eq. 3.8) may be viewed as a special case of a more general equation, ṅi

* = Q where Q = Q(n i * , n r * ),
with rain and ice concentrations being hypothetically 'independent' contributions to the forcing Q causing a response in ice concentration. The feedback strength has contributions from both pro-

cesses: λ = dQ/dn i * = ∂ Q/∂ n i * + ∂ Q/∂ n
λ 2 = ∂ Q/∂ n r * dn r * /dn i * = Kn i * × (-1
) < 0 measuring the feedback from drop depletion (a negative feedback). Thus, we arrive again at the same equation as above,

λ = λ 1 + λ 2 = K(n r * -n i * ).
The identified characteristics of the system is summarized in the phase-space of (n r * , n i * ) in Fig. 2. Positive and negative feedbacks (λ > 0 and λ < 0 ) are realized over the regions of instability and stability (separated by the thick line, n i * = n r * ), where splintering and drop-depletion feedbacks prevail respectively. The system travels along the trajectory (line with arrow, in which n r * + n i * = n 0 ) from one equilibrium point that is unstable (n r * = n 0 ) to the other that is stable (n i * = n 0 ). While travelling along the trajectory, the system goes from the realm of instability (n r * > n i * ) to that of stability (n r * < n i * ).

General Analysis

When the general case with ĉ0 = 0, ĉi = 0, ε s = 0 is considered, a full analytical solution is no longer available, thus various asymptotic expansion methods are called for.

Probably the most useful starting point is to note that both the supercooled-drop and ice-particle number densities, n r * and n i * , can evolve independently in absence of the drop freezing and the subsequent ice fragmentation. We may consider this as a leading-order behaviour of the system (Eq 2.4) by setting K = 0:

n (0) r * = (n r * (0) - ĉ0 ε s )e -ε s t * + ĉ0 ε s , (4.1a) n (0) i * = n i * (0) + ĉi t * . (4.1b)
The ice-fragmentation process is introduced as a process at O( K). Here, taking K as a perturbation parameter for the asymptotic expansion is merely a device for analytically inferring the behavior of the system. Recall that in fact, K = 1 by definition. However, in this manner, we can investigate how the evolution of the system is modified in the presence of ice fragmentation.

Physically, this amounts for investigating the feedback to the system due to a presence of ice fragmentation.

Thus, a full solution may be approximately obtained by adding an O( K)-contribution to the above leading-order solution:

n r * = n (0) r * + Kn (1)
r * , (4.2a)

n i * = n (0) i * + Kn (1) i * . (4.2b)
The O( K)-equations are given by ṅ( 1)

r * = -n (0) r * n (0) i * -ε s n (1) r * , (4.3a) ṅ(1) i * = n (0) r * n (0) i * . (4.3b)
Here, note that the first term, n

(0) r * n (0)
i * , in the left-hand side of both equations represents the feedback to the system due to the ice-fragmentation process. It may also be important to note that this term is nonlinear, but sicne this term is given, the above equations can readily be transformed into an integral form n

(1)

r * = -e -ε s t * t * 0 n (0) r * n (0)
i * e ε s t dt * , (4.4a)

n (1) i * = t * 0 n (0) r * n (0) i * dt. (4.4b)
By substituting the leading-order solutions (4.1a, b) into the above, and performing the integrals, we obtain

n (1) r * = [ ĉ0 ε 2 s (n i * (0) - ĉi ε s ) -(n r * (0) - ĉ0 ε s )(n i * (0)t * + ĉi 2 t 2 * )]e -ε s t * - ĉ0 ε 2 s [(n i * (0) - ĉi ε s ) + ĉi t * ], (4.5a) n (1) i * = (n r * (0) - ĉ0 ε s ) n i * (0) ε s (1 -e -ε s t * ) + ĉi ε s (n r * (0) - ĉ0 ε s )[ 1 ε s -( 1 ε s + t * )e -ε s t * ] + ĉ0 ε s (n i * (0)t * + ĉi 2 t 2 * ). (4.5b)
We see that apart from the exponentially decaying tendency associated with the sedimentation, the feedback is merely algebraic with no obvious destabilization tendency to the system.

The solutions may be further simplified by setting the initial condition to n r * (0) = ĉ0 /ε s and

n i * (0) = 0: n (1) r * = - ĉ0 ĉi ε 2 s [ 1 ε s (e -ε s t * -1) + t * ], (4.6a) n (1) i * = 1 2 ĉi ĉ0 ε s t 2 * . (4.6b)
The solution by this feedback analysis would be indicative of the evolution of the system at onset of ice fragmentation. Note that the solution for n r * may further be approximated to

n (1) r * ≃ - 1 2 ĉi ĉ0 ε s t 2 * (4.7)
to the limit of t → 0. We see that the dimensionless concentrations of rain water and ice crystals decrease and increase at the same rate, initially. This is consistent with the splintering positive feedback of ice multiplication (noted above for the simplest system) being boosted by the rates of primary generation of crystals and drops, as well as by the fall-out time of drops being long (compared to the characteristic time for glaciation). Fall-out of drops has a damping effect on the splintering positive feedback.

b. Taylor-Expansion Solution

The feedback analysis of the last subsection suggests that the initial tendency of the system due to the ice fragmentation is merely algebraic. It further suggests that an exact initial tendency of the system may be obtained by directly applying the Taylor expansion to the solution. Thus, we set

n r * = ∞ ∑ j=0 n r * , j t j * , (4.8a
) With j = 0, we obtain n r * ,1 = ĉ0 -Kn r * ,0 n i * ,0 -ε s n r * ,0 , (4.9a)

n i * = ∞ ∑ j=0 n i * , j t j * . ( 4
n i * ,1 = ĉi + Kn r * ,0 n i * ,0 . (4.9b)
Here, n r * ,0 and n i * ,0 are defined by the initial conditions. With j = 1, we further obtain

n r * ,2 = -K(n r * ,0 n i * ,1 + n r * ,1 n i * ,0 ) -ε s n r * ,1 , (4.10a
)

n i * ,2 = K(n r * ,0 n i * ,1 + n r * ,1 n i * ,0 ). (4.10b) n r * ,1 = 0, n i * ,1 = ĉi , n r * ,2 = - K ĉi ĉ0 ε s , n i * ,2 = K ĉi ĉ0 ε s .
Putting them together, the initial tendency of the solution is given by

n r * = ĉ0 ε s (1 -K ĉi t 2 * ) + • • • , (4.11a
)

n i * = ĉi (t * + K ĉ0 ε s t 2 * ) + • • • . (4.11b)
This solution (4.11a, b) may be compared with (4.7) and (4.6b) obtained in the last subsection. We see that the feedback analysis underestimates the effect of the ice fragmentation by half due to its perturbation treatment.

c. Feedback Analysis with Distorted Physics

A variation to the feedback analysis in Sec. 4.a is to differentiate the strength of the ice freezingfragmentation to the hydrometeor types. Thus, we re-write Eqs. (2.4a, b) as

ṅr * = ĉ0 -K1 n i * n r * -ε s n r * , (4.12a) ṅi * = ĉi + K2 n r * n i * . (4.12b)
Here, we may artificially assume that the freezing fragmentation is negligible for the supercooled drops, thus K1 ≪ 1, and the effect of freezing-fragmentation is only felt by the ice particles to the leading order, assuming K2 = O(1). Thus, we solve the problem with the following asymptotic expansion:

n r * = n (0) r * + K1 n (1) r * + • • • n i * = n (0) i * + K1 n (1) i * + • • •
Though such an assumption is hardly justified from a physical basis, this procedure provides a useful insight into the evolution of the system during ice fragmentation. For this reason, we shall call this method the feedback analysis with distorted physics.

The leading-order solution for n r * is, thus, given by Eq. (4.1a), whereas we solve the full problem for n i0 * to the leading order:

( d dt * -K2 n (0) r * )n (0) i * = ĉi *
This equation is linear in respect to n (0)

i * , and its solution can easily be written in an integral form:

n (0) i * = [n i * (0) + t * 0 ĉi exp(- t ′ * 0 K2 n (0) r * dt ′′ * )dt ′ * ] exp( K2 t * 0 n (0) r * dt * )
The second integral is readily performed and we obtain

t * 0 n (0) r * dt * = 1 ε s (n r * (0) - ĉ0 ε s )(1 -e -ε s t * ) + ĉ0 ε s t *
However, the first integral

t * 0 exp(-K2 t ′ * 0 n (0) r * dt ′′ * )dt * = t * 0 exp[- K2 ε s (n (0) r * (0) - ĉ0 ε s )(1 -e -ε s t * ) - K2 ĉ0 ε s t * ]dt *
is not readily integrable, thus we focus on the two limits, t * ∼ 0 and t * → ∞.

(i) When t * ∼ 0, noting 1e -ε s t * ≃ ε s t * , the integral becomes

t * 0 exp(-K2 t ′ * 0 n (0) r * dt ′′ * )dt * ≃ 1 K2 n (0) r * (0) (1 -e -K2 n (0) r * (0)t * ),
and the solution is

n (0) i * ≃ (n (0) i * (0) + ĉi k2 n (0) r * (0) )e K2 n (0) r * (0)t - ĉi k2 n (0) r * (0)
.

(4.13a)

(ii) When t * → ∞, the integral asymptotically approaches to:

t * 0 exp(-K2 t ′ * 0 n (0) r * dt ′′ * )dt * → - ε s K2 ĉ0 exp[- K2 ε s (n (0) r * (0) - ĉ0 ε s ) - K2 ĉ0 ε s t * ] + 1 K2 n (0) r * (0) , thus n (0) i * → (n (0) i * (0) + ĉi K2 n (0) r * (0) ) exp[ K2 ε s (n (0) r * (0) - ĉ0 ε s ) + K2 ĉ0 ε s t * ] - ε s ĉi K2 ĉ0 . (4.13b)
It is seen that in both limits, the ice-crystal number increases exponentially with time when the decrease of supercooled drops by freezing is neglected.

We now turn to O( K1 ) in order to see the modifications of the both particles numbers by the exponential ice-fragmentational multiplication:

( d dt * + ε s )n (1) r * = -n (0) r * n (0) i * , (4.14a) ( d dt * - K2 K1 n (0) r * )n (1) i * = K2 K1 n (1) r * n (0) i * . (4.14b)
Though it is possible to explicitly write down the full asymptotic solutions for the same two limits as considered for the leading order, the expressions are lengthy, and not particularly illuminating. For this reason, we focus on the case with n r * (0) = ĉ0 /ε s and n i * (0) = 0. Note that coincidentally, in this case, the two leading-order solutions (4.13a, b) agree. We also focus only on the most dominant exponential term in the solution of Eqs. (4.14a, b), neglecting the various exponentially-decaying terms that follow.

After these simplifications, the first-order modification to the supercooled rain density, n r * , due to the freezing is given by:

n (1) r * = - ε s ĉi K2 ( K2 ĉ0 + ε 2 s ) e ( K2 ĉ0 /ε s )t * + • • • (4.15a)
The result shows that as the first-order effect, collision freezing exponentially depletes the supercooled water due to the exponential multiplication of ice. If this result is taken literally, the supercooled water would be completely depleted over a finite time. In reality, as the supercooled water begins to deplete, the ice fragmentation also slows down, thus it is more likely that supercooled water will be depleted in slower rate.

In turn, the first-order modification to the ice-particle number is:

n (1) i * = - ĉ0 ĉi K2 ĉ0 + ε 2 s [ K2 2 ĉ2 0 + ε 2 s K2 ĉ0 + ε 4 s ε s K2 ĉ0 ( K2 ĉ0 + ε 2 s ) + ĉ0 ĉi K2 ĉ0 + ε 2 s t * ]e K2 n (0) r * (0)t * + • • • (4.15b)
We see the exponential ice multiplication tendency is first, suppressed by the constant factor (the first term), and further decreases linearly with time (the second term). Thus, the dominant exponential term becomes negative over a finite term, and the only remaining positive contributions are neglected exponentially-decaying terms. We interpret that the result suggests that though the ice multiplies with time, the growth is weaker than exponential.

d. Asymptotic Tendency towards t * → ∞

The analysis of the last subsection suggests (though it does not show) that the supercooled water would deplete with a rate slower than exponential, and also the ice multiplies by fragmentation with a rate slower than exponential. The most likely case is that the supercooled water decreases, and the ice particles increase both algebraically with time. Thus, to the asymptotic limit towards t * → ∞, we may set n r * ∼ n r * 0 t -α * , (4.16a)

n i * ∼ n i * 0 t β * . (4.16b)
Here, both α and β are expected to be positive constants, and the symbol ∼ suggests that we are only concerned with the asymptotic tendency toward t * → ∞. This balance is achieved when -α + β = 0, or α = β . It may be important to note that in asymptotic limit of t * → ∞, the equation for the supercooled rain number becomes quasi-stationary (i.e., the temporal tendency in the left hand side does not contribute to the leading order), and also the sedimentation no longer plays a leading role (i.e., the 3rd term is the right hand side does not contribute to the leading order).

By substituting α = β into Eq. (4.17b), we find

β n i * 0 t β -1 * ∼ ĉi * + Kn r * 0 n i * 0 .
Since the right hand side is positive definite, it must be balanced by a constant growing tendency of the ice number, thus

β = 1,
and also α = 1 as a result.

Constants, n r * 0 and n i * 0 , may be determined in a more straightforward manner by considering the conservation of the total particle number, which is obtained by taking the sum of Eqs. (2.4a, b):

d dt * (n r * + n i * ) = ĉ0 + ĉi -εn r * .
As

n i * ∼ ( ĉ0 + ĉi )t * .
(4.20b)

In the limit of the long time scale, the ice crystals multiply by fragmentation linearly with time at a rate defined by the sum of the supercooled-drop and primary-ice sources.

Finally, the asymptotic solution becomes when dimensions are included:

n r ∼ c 0 KK(N ′ c 0 + c i ) t -1 , (4.20a) n i ∼ (N ′ c 0 + c i )t. (4.20b)
The definition of IE ratio, f ≡ n i * (t * )/n i * (0) (t * ), is the ratio of the total ice concentration to the primary ice concentration. Thus it is a measure of the extent of secondary ice production.

So in the general case, if ĉi is non-zero and t → ∞:

f = (N ′ c 0 /c i + 1)/(n i (t = 0)/(tc i ) + 1) → 1 + N ′ c 0 /c i (1)
The maximum IE ratio is approached on a timescale of n i (t = 0)/c i .

Otherwise without continuous primary ice nucleation, ĉi = 0 implies:

f (t) → N ′ c 0 t/(n i (t = 0))
taking the dimensional expressions for the source terms. Now there is no upper limit on the IE ratio, and it increases with time at a constant rate of N ′ c 0 /n i (t = 0).

quasi-stationarity with the generation rate balancing with the collision freezing rate. The rain sedimentation does not play a leading role.

e. Perturbation Analysis

Finally, the perturbation analysis considered in Sec. 3.b may also be performed for the full problem. However, the procedure is more involved with the fact that the source and the sink terms are involved in the full system, thus the total number density is no longer conserved, and it is no longer possible to invoke a simple constraint as Eq. (3.7). Instead, we have to consider explicitly the two perturbation equations for both n ′ r * and n ′ i * :

ṅ′ r * = -( Kn i * + ε s )n ′ r * -Kn r * n ′ i * , (4.21a)
ṅ′ i * = K(n i * n ′ r * + n r * n ′ i * ). (4.21b)
The analysis of this system is further involved due to the fact that the linear operator in the righthand side also evolves with time. However, the problem can be simplified, as implicitly assumed in Sec. 3.b, when the evolution of the reference state, (n r * , n i * ), is considered much slower than that of the perturbations. In this case, the time derivative in the left-hand side may be replaced by an eigenvalue, λ , and the above problem reduces to that of finding the eigenvalues for the linear operator in the right hand side, which is determined from

λ 2 -( Kn r * -Kn i * -ε s )λ + K2 n i * n r * -( Kn i * + ε s ) Kn r * = 0.
Solving it for λ , we obtain the two eigenvalues (adding the subscripts 1 and 2 for distinction):

λ 1 = ( Kn r * -Kn i * -ε s ) 2 [1 + {1 + 4ε s Kn r * ( Kn r * -Kn i * -ε s ) 2 } 1/2 ], λ 2 = ( Kn r * -Kn i * -ε s ) 2 [1 -{1 + 4ε s Kn r * ( Kn r * -Kn i * -ε s ) 2 } 1/2 ].
λ 2 = 0. The latter is simply interpreted as a manifestation of the conservation of the total particle number number (3.7).

When a finite sedimentation rate (ε s = 0) is considered, the threshold from the positive to negative feedback for λ 1 reduces by ε s (i.e., bias towards a negative "feedback"). Thus, the neutral line shifts from n i * = n r * to n i * = n r * -ε s : the critical ice number density becomes smaller in the presence of sedimentation. Some examples of the neutral lines are plotted in Fig. 3a for selective sedimentation rates, ε s . In the presence of sedimentation, however, the neutral line no longer separates between the stable and the unstable regimes, but the stable regime upper left of the neutral line identified in Fig. 2 also destabilizes due to the second eigenvalue, λ 2 . Most importantly, in these general situations, the perturbation evolution of the system is no longer interpreted in terms of a single "perturbation" parameter, but the actual perturbation evolution is determined by a linear combination of two exponential tendencies characterized by two eigenvalues, sensitively depending on the initial perturbation. Trajectories in the phase-space are shown in Fig. 3a for the special case of ε s = 0 and ĉi = 0 by numerical integration of Eq (2.4), with primary ice only implicitly represented by an initial nonzero value of ice concentration, n i * (t * = 0) = 0.001. The simulations are integrated until t * = 10 3 .

The evolution over time of the IE ratio, f , for the same trajectories is shown in Fig. 3b. Two stages of the glaciation are evident. First, there is the usual exponential growth of the ice concentration by an IE ratio of up to 10 3 in a dimensionless time of t * < 1-10. During this stage, the rain concentration is depleted almost to zero. If the integration of the corresponding dimensional equations were performed, this initial stage would correspond to the time-scale of explosive fragmentation noted above of τ = 1/(K(n i (0) +n r (0)N ′ )). Second, there is the subsequent exponential relaxation to the asymptotic solution when t * > 1-10, coinciding with a balance between supply of fresh drops ( ĉ0 > 0) and their depletion by fragmentation and collisions with ice splinters. In this second stage, the rain concentration decreases steadily, inversely proportional with time.

Discussions a. Efficiency of the Ice Multiplication Processes

Our theoretical investigation reveals that the ice multiplication by fragmentation of the freezing drops is not always as potentially effective as the two other ice multiplication processes examined by [START_REF] Yano | Ice-ice collisions: An ice multiplication process in atmospheric clouds[END_REF], and Yano et al. ( 2016): Hallett-Mossop and ice-ice collision processes. As shown in the Appendix C of [START_REF] Yano | Ice-ice collisions: An ice multiplication process in atmospheric clouds[END_REF], the ice multiplication rate by the Hallett-Mossop process is linearly proportional to the existing graupel number density. This leads to an exponential multiplication of ice with time, assuming that the system is in quasi-equilibrium between the ice particles and graupel, and also the supercooled cloud-droplets are re-adjusted to an equilibrium number as graupel sweeps them through as the latter falls. As carefully analyzed in [START_REF] Yano | Ice-ice collisions: An ice multiplication process in atmospheric clouds[END_REF] see the Appendix especially), the ice multiplication by ice-ice collision is fundamentally nonlinear, being proportional to the square of the ice-number density. In the idealized state with infinite water-vapor supply, this leads to an explosive increase of the ice number to infinity within a finite time.

Compared to those rather strong multiplication tendencies found with these two processes, the ice multiplication by fragmentation of freezing drops often tends to be rather modest. This stems from the fact that this process relies on a number of supercooled drops available in the system, and it essentially limits the possible number of ice fragments that can be generated. When the ice fragmentation by freezing drops is considered in its isolation, as in Sec. 3, this conclusion is especially clear: the final total number of ice fragments found is simply N ′ n r (0) + n i (0) with N = N ′ + 1 the number of secondary ice fragments generated per freezing collision, and n r (0), n i (0) are the initial numbers of supercooled drops and ice crystals, respectively.

When both supercooled-water rain and primary-ice sources are added to the system, as in Sec. 4, the behavior of the system becomes more involved. However, in the limit of the long time-scale, the ice fragmentation number simply increases linearly with time by the rate that both supercooled water and primary ice are supplied. We see that such a multiplication tendency is, generally, much weaker than any of those found in Hallett-Mossop or ice-ice collision process.

However, in some observational situations, fragmentation of freezing drops does become a primary process of ice multiplication, especially when there is copious supercooled rain initially and temperatures are optimal. A case during the ICET is such an example.

b. Comparison with the Observation

A quantitative observational comparison with the present theoretical results is provided by [START_REF] Lawson | The microphysics of ice and precipitation development in tropical cumulus clouds[END_REF], who present the measurements during ICET over U.S. Virgin Islands in July 2011. Their flights measuring cloud properties were performed in ascending order so that a Lagrangian interpretation of data following the convective updrafts can be developed. The Lagrangian interpretation would also be further facilitated by considering the ice-fragmentation process in its isolation as in Sec. 3.

In this framework, the key conserved quantity is the total particle number with the supercooled drops multiplied by the fragmentation number per collision freezing. It also follows that the total water content, as a sum of liquid and ice, is also conserved. Hence, the scatter plots for liquid and ice values, both in terms of the number density and the water content would form a linear line with a negative slope. The scatter plots using the measurements for varying heights (temperature levels) averaged over various convective cores for both quantities in Fig. 4 clearly present a negative correlation both for the number density (a) and the water content (b), being consistent with the theory presented in Sec. 3, though the distributions are hardly aligned to any line.

The strong scatter may be understood by two reasons: 1) the plots are hardly form a single convective event, but a collection of many convective cores, though all of them are in similar regime; 2) the source and the loss terms for both liquid and ice, neglected in the theory, would not be negligible in practice. Considering those two caveats, these two scatter plots would rather be considered a good support for the idealized theory presented in Sec. 3.

A rather surprising result from their data analysis is a fast glaciation process: they found that 3-5 g m -3 supercooled liquid are almost all glaciated over 3 min time at an elevation of 1-1.5 km higher within an updraft core. Here, this rapid glaciation rate is easily explained by the theoretical analysis in Sec. 3. Note that the collision efficiency is defined by

K = πr 2 d v T
in terms of the radius, r d , of the supercooled drops, and the differential vertical velocity, v T , between the supercooled drops and ice crystals. Assuming, r d ∼ 10 2 µm∼ 10 -4 m, v T ∼ 1 m/s, we obtain

K ∼ r 2 d v T ∼ 10 -8 m 3 /s
Also setting n r ∼ 1 cm -3 ∼ 10 6 m -3 , N ′ ∼ 1, the characteristic glaciation rate is estimated by

N ′ Kn r ∼ 10 -1 1/s,
and the characteristic time scale is τ ∼ 10 sec by following the definition (3.5). Noting from Fig. 1 that it typically takes about ten characteristic time scales for completing the glaciation, a time required for completing the glaciation is also only 10 2 sec, being fairly comparable with the observational estimate of 3 min.

c. Asymptotic Expansion Approaches

The present study has employed various asymptotic-expansion methods in order to elucidate the basic behavior of the system for ice fragmentation by collision freezing of supercooled drops.

of them are not precise even in asymptotic sense, but merely for providing qualitative insights.

Nevertheless, they help to identify more rigorous methods for deriving more precise solutions.

For example, the feedback analysis in Sec. 4.a suggests the initial tendency of the system is algebraic in time, that further motivates us to determine the exact initial tendency by the Taylor expansion method (Sec. 4.b). Though the exact Taylor-expansion result demonstrates that the feedback analysis halves the initial parabolic tendency, at the same time, without support of the feedback analysis, the Taylor-expansion solution is justified only in a strong limit of t → 0, because in general, the presence of an exponentially growing tendency invalidates the Taylor-expansion solution very rapidly in time. Another example is an approach of distorted physics considered in Sec. 4.c. Again, though this analysis is not quantitatively accurate, the result suggests to consider an algebraic tendency of the solution in the long time-scale limit, as considered in Sec. 4.d.

Asymptotic analyses presented herein are hardly exhaustive either, because some asymptotic analyses are not particularly illuminating. For example, it is possible to consider the modification of the solution presented in Sec. 3 by adding the sources, ĉ0 and ĉi , as perturbation terms. However, such an analysis hardly elucidates a rather drastic change of the system behavior by adding finite sources, as considered in Sec. 4. Some of the analyses turn out to be rather subtle. For example, an alternative feedback analysis by another assumption of distorted physics, K2 ≪ K1 ∼ 1, may also be considered. In this case, the system exponentially approaches new equilibrium states up to first order. Yet it is rather difficult to draw any physically meaningful conclusions from this solution.

d. Stochasticity of the Ice Fragmentation Process

Finally, we should keep in mind that the fragmentation number, N ′ , of ice per collision freezing is not a fixed number even when a collision between a drop and an ice particle with fixed diameters is considered. One reason is that the number of fragments per spherically frozen drop (mode 1) varies by orders of magnitude depending on temperature. Vertical motions due to sloped incloud drafts combined with sedimentation create variability of the ambient temperature. Equally, even for identical conditions of freezing temperature and drop size, the number of drop fragments varies dramatically from drop to drop. Macroscopically, it may be assumed to be probabilistic. [START_REF] Kolomeychuk | The fragmentation and electrification of freezing drops[END_REF] observed that about 10% of all drops in their experiment (1.6 mm) emitted about 90% of all the detected splinters. Considering only drops freezing between -15 • C and -18 • C, 90% of the splinters were from a third of the drops.

Details of the collision process are complex for the most purposes of cloud microphysical modeling, and the outcome of the fragmentation number by collision freezing may simply be considered a random number (cf., [START_REF] Kolomeychuk | The fragmentation and electrification of freezing drops[END_REF]. As a result, the ice fragmentation may be considered a stochastic process. This is an aspect, where further investigation may be warranted.

Here, we present short general theoretical observations for facilitating further studies.

As a whole, the contribution of stochasticity would also be best understood by adding this component as a perturbation, say, δ N ′ . In the general case considered in Sec. 4, under the asymptotic limit to the long time scale, the ice-particle number, n i , asymptotically evolves solely depending on the strength of the two hydrometeor sources. This asymptotic tendency is not affected, even when the fragmentation number fluctuates randomly. The latter effect only affects the time evolution of the supercooled-drop number, n r . It can be shown that the resulting fluctuation, n ′ r , is dictated by a Brownian motion induced by white noise, δ N ′ , under a linear drag with a drag coefficient linearly increasing with time.

On the other hand, when the ice fragmentation process is considered in isolation, as in Sec. with selective choice of the sedimentation rates: ε s = 0 (solid), 0.25 (long dash), 0.5 (short dash), and 0.75 (chain dash). Note that in the general case, the system is unstable at both sides of the neutral line, with a weak instability to the upper left of the neutral line when the sedimentation effect is small. Some examples of trajectories are also shown based on the asymptotic solution (4.20) with ĉ0 = 10 -1 (blue), 10 -2 (green), and 10 -3 (red): the system evolves in the direction of decreasing n r * and increasing n i * . . . . . . . . 35 The values marked by + symbol are measurements for updraft cores: (a) particle number density (1/cm 3 ), (b) water content (g/m 3 ). Data is from Table 1 of [START_REF] Lawson | The microphysics of ice and precipitation development in tropical cumulus clouds[END_REF]. . . 36 FIG.

1. Time evolution of the particle number densities with various initial conditions when the icefragmentation process is considered in its isolation: for (a) supercooled drops, n r * , and (b) ice crystals, n i * .

All the variables are nondimensionalized as described in the text. choice of the sedimentation rates: ε s = 0 (solid), 0.25 (long dash), 0.5 (short dash), and 0.75 (chain dash). Note that in the general case, the system is unstable at both sides of the neutral line, with a weak instability to the upper left of the neutral line when the sedimentation effect is small. Some examples of trajectories are also shown based on the asymptotic solution (4.20) with ĉ0 = 10 -1 (blue), 10 -2 (green), and 10 -3 (red): the system evolves in the direction of decreasing n r * and increasing n i * . (g/m 3 ). Data is from Table 1 of [START_REF] Lawson | The microphysics of ice and precipitation development in tropical cumulus clouds[END_REF].

Equilibrium point

  r * dn r * /dn i * = λ 1 + λ 2 and with dn r * /dn i * expressing the linkage between rain and ice concentrations by the evolution equation for the drop depletion process. Considering Q = Kn r * n i * , the feedback parameter thus has the two contributions, λ 1 = ∂ Q/∂ n i * = Kn r * > 0 measuring the feedback from splintering (a positive feedback) and

  .8b) In substitution of these Taylor expansions into the original full equations (2.4a, b) leads to a solution. In deriving the solution, we note particularly n r * n i * =

Fig. 2 .Fig. 3 .

 23 Fig. 2. The stability characteristics of the system when no source and sink is considered as in Sec. 3. . . . . . . . . . . . . . . . . . . . . . . . . 34 Fig. 3. The stability characteristics of the system in the general case: shown are the neutral lines

Fig. 4 .

 4 Fig. 4. Scatter plots between the liquid and the ice values observed during the ICE-T campaign.

  FIG.2. The stability characteristics of the system when no source and sink is considered as in Sec. 3.

FIG. 4 .

 4 FIG. 4. Scatter plots between the liquid and the ice values observed during the ICE-T campaign. The values marked by + symbol are measurements for updraft cores: (a) particle number density (1/cm 3 ), (b) water content

  t * → ∞, n r * → 0, thus the above conservation law asymptotically approaches to Substituting Eq. (4.16a) into Eq. (4.19) with β = 1, we immediately obtain n i * 0 = ĉ0 + ĉi . By further substituting this result into Eq. (4.18), we find n r * 0 = ĉ0 [ K( ĉ0 + ĉi )] -1 Thus, the asymptotic solution towards t → ∞ becomes

	n r * ∼	ĉ0 K( ĉ0 + ĉi )	t -1 * ,	(4.20a)
	ṅi * ∼ ĉ0 + ĉi .	(4.19)

Acknowledgment. The first author (VTJP) was supported by three research grants, from Swedish Research Council for Sustainable Development (FORMAS; award number 2018-01795), Swedish Research Council (VR; award number 2015-05104) and US Department of Energy Atmospheric Sciences Research Program (award number: DE-SC0018932). The topics of these awards concern mechanisms for ice production in clouds and their relation of aerosol conditions.


Katerine Romeo
email: katerine.romeo@univ-rouen.fr

Marion Chottin

Pierre Ancet

Christele Lecomte

Edwige Pissaloux

Simplification Of Painting Images For Tactile Perception By Visually Impaired Persons

Keywords: Segmentation, painting, tactile perception, multi-modal perception, blindness, visually impaired

The access to artworks by visually impaired people requires a simplified tactile representation of paintings. This paper presents the difficulties of direct transcription of artworks and the test results of simplification of the paintings done by Australian Aborigines which don't have purely visual elements such as shadows or perspective. The implemented methodology is bottom-up: it starts with tactile representation of basic elements relevant to the understanding of the whole painting, then their association into more complex concepts. The context of associations is explained through audio-description. The results of the tests with visually impaired persons are analyzed and explained.

Introduction: difficulties of a direct transcription

When visually impaired people (VIP) go to a museum, they encounter difficulties with the perception of artworks. To benefit from the beauty, emotion or information inspired by these objects, a tactile representation can be one of the solutions for blind persons. The museums for art and science are traditionally organized prominently for the sense of sight, and this organization has been based for a long time on the prohibition of touching the artwork. However, the sense of touch conveys information and aesthetic pleasure as well as the sense of sight. The cognitive virtues of touching have been recognized since Aristotle, and today they are being rediscovered and measured [11], [START_REF] Hatvelle | Toucher pour connaître. Psychologie cognitive de la perception tactile manuelle[END_REF]. The aesthetic virtues of contact through touch have been known ever since antiquity and they have been honored regularly in the history of art [START_REF]Sculpture and Touch[END_REF], [START_REF] Marinetti | Le tactillisme. Manifeste futuriste[END_REF]. Besides, several blind persons such as Marie Heurtin and Helen Keller testify about their tactile and aesthetic experiences [START_REF] Keller | Histoire de ma vie[END_REF], [START_REF] Villey | Le Monde des aveugles[END_REF].

The direct transcription of a painting into a tactile material is possible with different techniques such as embossing, thermoforming, serigraphy, thermo-graphical printing or 3D printing [START_REF] Krivec | Adapting Artworks for People Who Are Blind or Visually Impaired Using Raised Printing[END_REF]. Examples of realizations with different materials may be touched in the Musée des Beaux Arts of Rouen (France) or the Pinacotheca de Brera of Milan (Italy). Some recommendations for the transcription of documents [START_REF] Bris | Recommandations pour la transcription des documents[END_REF] specify minimum distances between graphical elements of 2.3 mm in order to respect the tactile acuity. This distance is similar to the gap between the dots in Braille letters. Kinesthetic and proprioceptive perception should be taken into account when geometrical characteristics are used in graphics, such as length of segments or angle: distinction of small contours and very small changes in directions in segments could be ignored and not perceived otherwise. Even when respecting these rules and the original page layout, the result of the transcription of painting images present an overabundance of information which can be discouraging to VIP's.

Another problem in the transcription of paintings is the presence of shadows difficult to recognize when included in the contours of an object. The deformation of objects through perspective, which is usually used for depth perception, may not be as intuitive to VIP's who have to be taught ego-centered and exo-centered views of the body, buildings or objects. Some very interesting work has been done in automatic image transcription of very simple educational graphics [START_REF] Haddad | Image Processing and Pattern Recognition Tools for the Automatic Image Transcription[END_REF] [START_REF] Chen | Analyse et Interprétation d'images à l'usage des personnes non-voyantes, Application à la generation automatique d'images en relief à partir d'équipements banalisés[END_REF] that can be applied to tactile transcription of some simple paintings or graphics, but it is necessary to decompose paintings to give access to the meaning of the global picture. Some terms or expressions may mean nothing to congenitally blind persons as mentioned such as "face view" and "profile", or "shadow" and "perspective". However, color names and the semantic field of light should not be banned: these terms are always associated, even with VIP's, to other sensations, (such as the yellow color with heat) and to emotions (such as the red color with anger) and then they can be extremely meaningful [START_REF]Dire le non-visuel. Approches pluridisciplinaires des discours sur les perceptions autres que la vue[END_REF].

Thus, a direct transcription of the painting on a tactile surface is not relevant for VIP's as it does not allow appropriate understanding of its content and appreciation of the esthetic quality of the artwork. A multi-modal approach is necessary, accompanying the graphical representation with an audio description of the shapes of the objects represented in the painting as well as their location and role [START_REF] Fryer | An Introduction to Audio Description: A Practical Guide[END_REF] [START_REF] Ginley | Museums: A Whole New World for Visually Impaired People[END_REF]. The audio description should use the specific words and definitions from the specific vocabulary used by VIP's [8] [9].

This paper proposes a multi-modal representation of paintings in an interactive way. The paper is organized as follows. Section 2 presents the difficult problem of conservation of meaning after a transcription into a tactile representation. Section 3 discusses the proposed multi-modal representation and the tests with visually impaired persons. Finally we present in section 4 our conclusions and some reflections on future work.

Conservation of meaning

First of all, an artwork is dependent on its visual intensity as explained by De Coster and Loots [START_REF] De Coster | Somewhere in between Touch and Vision: In Search of a Meaningful Art Education for Blind Individuals[END_REF]. It is the degree of vision one can use to understand a painting or a sculpture which can be considered as a visual narrative. Blind people are conscious of this important visual concept on art. The Museums often propose audio-guides or art teachers to explain the artworks in the exhibitions but the explanations are often on the context and not the visual appearance. The cognitive effort produced by visually impaired persons to catch the meaning of an artwork is usually made by the associa-tion of the story behind the creation of this artwork and its perceptual features. Of course art is not only about its appearance but also history and philosophy. A meaningful art education would include both explanations and tactile perception. Different details can be presented in an analytical way, including all the eye-catching points of interest. However these details don't have a great significance if the context is not introduced. Only when the global meaning of the painting is understood, can blind persons reach an aesthetic experience [START_REF] Lorin-Etuy | Comment le cerveau réagit-il à l'art? Hors-Série Art[END_REF]. We should also add that aesthetic experiences are not always pleasurable -some are meant to disturb, to perplex or to question and this can also be described and aroused by the narrative experience, or felt by audio-tactile perception.

The tactile perception is a singular approach to feel a painting: this information is not complete as it is. And it is the same incomplete information when we consider solely the visual appearance of an artwork for seeing people. All of the available information is necessary to explore the artwork in a tactile manner, to understand the shape of its different parts and the relation between them, the context. People, visually impaired or not, generally do not have the experience to explore an artwork tactilely. This is why a multi-modal representation of artworks seems to be quite relevant.

Multi-modal representation

To evaluate the relevance of the proposed multi-modal representations of paintings, the bottom-up methodology is implemented. It starts with tactile representation of basic elements relevant to the whole painting understanding, then their associations into more complex concepts and finally into the whole image. The tactile shapes, the context of their associations and the global meaning of the painting are explained with audio-description.

Experimental environment

This bottom-up methodology was implemented while testing the aboriginal painting "Dream of the snake" from the Australian painter Warlimpirrnga Tjapaltjarri which is in the Museum of Quai Branly in Paris, France (see Fig. 1 left). It should be noticed that the objects on aboriginal paintings have no edges: they are composed of elementary dots that form different shapes. There are several important reasons for this choice of painting: first of all, this artwork is composed by simple geometric forms, which can be reproduced on a tablet with haptic feedback and that can be recognized through non-expert touch from a user. The second reason is that this painting does not have purely visual elements like shadows, reliefs or perspectives that congenitally blind persons could not understand. Finally, the third reason is that this artwork has a tactile dimension because the original paintings were done on the soil, with fingers digging in the sand [START_REF] Glowczewski | Du rêve à la loi chez les Aborigènes : mythes, rites et organisation sociale en Australie[END_REF]. These characteristics of aboriginal paintings guided our selection of a touch stimulation device. The University of Lille's touch-stimulating prototype STIMTAC was selected (Fig. 1 right). This device is designed for displaying tactile textures through scanning the STIMTAC surface with a finger. The second set of tests were completed with twelve participants, two of them congenitally blind, six adventitiously blind and four with severe visual impairment. The test was based on the discovery of the aboriginal painting "Dream of the snake". The evaluation was done by VIP's where 58% of the participants were men and 42% women. The majority of the participants were more than fifty years old and some were in their twenties. A great number were retired and 25% were employed (See Fig. 2). All of the participants were expecting a new haptic tool which could help them to discover some artwork through touching.

Results

The participants to the tests appreciated the idea of discovering a painting in a tactile way. The decomposition of the painting "Dream of the snake" into small elements like dots, lines and composite figures (see Fig. 3) was intended to test the ability to recognize simple forms in the different tactile perception of friction. The stimulation of the vibro-tactile device STIMTAC creates the perception of textures and models the interaction between the finger and the surface of the pad. The perception may be the texture of an object or the smoothness/roughness of the explored area comparing to the strength of the finger's contact and the orientation of its movement.

The experiment was carried out in three phases: First of all, a texture detection test was made. Two different textures were shown side by side and the participant had to feel with a finger on the screen of the STIMTAC tablet, the textures and the limit between them. Then, the participant had to indicate which one of the two textures presented a rough surface. The detection results of this experiment with textures are shown in Fig. 4. The majority of VIP's detected the surface with the rough texture but the novel nature of the signal surprised some of the participants who were expecting to touch some hard material, and they could not name what they felt as "surface with friction" or "rough surface" or anything else. The white textures present a smooth surface and the dark textures create a friction-like perception.

Texture Detection Results

Nothing False

Right

The next experiment had a progressive approach in the detection of figures, from simple to complex shapes. Apart from the explanations on the experiment and the test progress, no help was provided.to the participants, no feedback was given. The participants had to describe what they detected. Circular shapes were difficult to recognize, the first figures were dots (See Fig. 5. pictures 1, 2, 3) and the aim of this experiment was to detect one or several textured patches on the surface. The second set of figures was parallel lines and concentric circles (Fig. 5.pictures 4, 5, 6). The aim was to feel the transition from one texture to another between the lines and to feel the white areas between them, as well as the center of the circles. The third set was more complex figures as assembled from simple forms (Fig. 5.pictures 7, 8) where the global perception was asked. Finally the snake (Fig. 5.picture 9) was scanned with one finger and detected partially or as a global shape. It is a difficult exercise to name the perception of a complicated shape without help. However it can be observed that the detection results are improving with experience as shown in Fig. 6, and the participants have some perception of the global shape on the screen. The third phase of the experiment continued with the same figures, but this time with an audio signal describing the painting and the meaning of the symbols (Fig. 3) and it finishes with the legend of the snake in aboriginal Australia. The participants were asked if the audio-description helped the detection of the figures. Only fifty percent of them agreed because they found the haptic signal was not sufficiently in relief or in contrast. The alternation of textures and contours was expected. As the second part of the experiment was done without feedback, the figures could not have been learned. An audio-description during the tactile discovery is necessary. The majority of the participants agreed that the figures become meaningful with the context explained and that the tactile perception enhance the meaning given through the audio-description if a real tactile dialogue is provided.

The decomposition of the painting into small elements proved to be too analytic and the integration of these small details in the whole image was not sufficiently clear. We understand that the notion of detail is very well known by visually impaired persons but it appears that contextualizing these details more clearly is necessary.

The nature of the vibratory signal on the tablet did not allow for adequate recognition of defined figures. We had assumed that these vibrations would be more easily recognizable by VIP's who are experts in touching sensations. In fact, the tests showed us that it was not always the case: visually impaired persons perceive the vibrations as small very sensitive movements, not giving the sensation of a shape or the border between two textures. As a general rule, the non-expert persons in touching had better results in perception of figures.

Conclusion

This paper addresses the evaluation of the tactile representation of an aboriginal painting by visually impaired persons. The original bottom-up evaluation methodology was defined and implemented. The collected results show that the simplification of the tactile representation linked to the audio contextualization help the tactile exploration of the paintings. Inversely, audio-description is a guide to the tactile recognition of shapes. This is a relevant approach to make paintings accessible to visually impaired people and to all persons curious of tactile experiences who want to enhance their sensibility to art. Our future work will explore the possibility of a tablet with tactile pins or points (taxels) that are more adequate for perception of the borders of figures. The participants to our tests asked for a greater size of tablet (A4) and they wish to touch with both hands and more than one finger. For the discovery of a painting it seems essential to explore bi-manually and pluri-digitally. For an efficient tactile transcription of paintings and accompanying audio-description, it is really necessary to work in close collaboration with visually impaired people in order to produce the most relevant results. This participative approach is a guarantee of the appropriation of the tool by the visually impaired people.

Fig. 1 . 2 Participants

 12 Fig. 1. The painting "Dream of the snake" and the Touch stimulation device STIMTAC

Fig. 2 .

 2 Fig. 2. Participants to the evaluation of the painting on the tactile tablet STIMTAC.

Fig. 3 .

 3 Fig. 3. The decomposition of the aboriginal painting "Dreaming snake" in small elements and the description of the context.

Fig. 4 .

 4 Fig. 4. Results on texture detection tests with visually impaired participants.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Simple and complex figures presented for detection by VIP

Acknowledgment

The research on which this work is based is funded by CNRS, TETMOST project, and it is done with the collaboration between University of Rouen Normandy and Department of Seine Maritime and Espace Handicap of the University of Rouen.