L Arevalo

V Cooray

Preliminary study on the modeling of negative leader discharges

HAL is

Introduction

A great deal of scientific work has been dedicated to the study of discharges in long spark gaps under different voltage stresses and polarities with the aim of understanding the physics of the discharge and of being able to predict the behavior of natural discharges like the lightning discharge. These experimental studies have been used to develop empirical and physical models capable of assisting in the design of protection systems against lightning, of the insulation coordination and of protecting high voltage electrical equipment in general.

However, because of the complexity of the negative discharge, most of the modeling work has been dedicated to positive discharges. In nature, most of the lightning discharges are of negative polarity. Given this, the development of numerical models capable of reproducing negative discharges will improve the design of protection systems and the design of equipment that can withstand overvoltages originating from direct and/or indirect strikes.

The first step towards attaining the goal of modeling negative lightning discharges, is to model laboratory gap discharges developed in a controlled 1-2 environment, such as laboratory discharges produced under switching impulses. Thus, the research presented in this contribution represents an effort to reproduce discharges in long laboratory spark gaps under different kinds of switching impulses.

The first lightning attachment model incorporating assessments of real physical parameters was proposed by Dellera and Garbagnati [START_REF] Dellera | Lightning stroke simulation by means of the leader progression model. Part I: Description of the model and evaluation of free-standing structures[END_REF]. This model, called the leader progression model, includes a simulation of the propagation of both negative and positive leaders achieved using electrostatic considerations. Later, a model that simulates positive leader channels, and which incorporated more of the physics of the discharge, was constructed by Rizk [START_REF] Rizk | Modeling of lightning incidence to tall structures. Part I: Theory[END_REF][START_REF] Rizk | Modeling of lightning incidence to tall structures Part II[END_REF]. Then, based on Les Renardieres' experimental work [4,[START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF], which gave a clear description of the leader behavior under laboratory conditions, Gallimberti [START_REF] Gallimberti | The mechanism of long spark formation[END_REF] developed the physics pertinent to understanding the development of leaders. Subsequently, Gallimberti and Bondiou [START_REF] Bondiou | Theoretical modelling of the development of the positive spark in long gaps[END_REF] built up a physical methodology that simulates positive upward leaders under switching impulses. Based on the work of Bondiou and Gallimberti [START_REF] Bondiou | Theoretical modelling of the development of the positive spark in long gaps[END_REF], Becerra and Cooray [START_REF] Becerra | A self consistent upward leader propagation model[END_REF] developed a model to study the attachment of lightning flashes to grounded structures. However, none of the researchers investigating this, namely, Dellera and Garbagnati [START_REF] Dellera | Lightning stroke simulation by means of the leader progression model. Part I: Description of the model and evaluation of free-standing structures[END_REF], Rizk [START_REF] Rizk | Modeling of lightning incidence to tall structures. Part I: Theory[END_REF][START_REF] Rizk | Modeling of lightning incidence to tall structures Part II[END_REF] and Becerra and Cooray [START_REF] Becerra | A self consistent upward leader propagation model[END_REF] modeled the negative stepped leader from first principles.

In 1994, Bacchiega et al [START_REF] Bacchiega | Theoretical modelling of the laboratory negative stepped leader[END_REF] developed the first theoretical model of the propagation of a negative stepped leader across long air gaps. By 1998, Castellani et al [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF] had conducted experimental measurements in such long gaps to identify the bileader process, and thereby providing a great amount of detail on the negative leader development.

In 2000, Mazur et al [START_REF] Mazur | Computer simulation of a downward negative stepped leader and its interaction with ground structures[END_REF] presented results of a physical model of negative leaders under the effect of a downward-coming positive leader. In 2002, Gallimberti and Lalande et al [START_REF] Lalande | Observations and modeling of lightning leaders[END_REF][START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF] presented a bipolar model to reproduce triggered lightning, in which they included positive and negative lightning based on the measurements of Les Renardieres [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF] and Castellani [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF] and the model of Bacchiega [START_REF] Bacchiega | Theoretical modelling of the laboratory negative stepped leader[END_REF].

The other available models are stochastical that make use of fractals, and the electrical breakdown models of Niemeyer et al. [START_REF] Niemeyer | Fractal dimension of dielectric breakdown[END_REF], Tsonis and Elsnet [START_REF] Tsonis | Fractal characterization and simulation of lightning[END_REF], and Sanudo et al. [START_REF] Sanudo | Fractal dimensions of lightning discharge[END_REF], and some of them use the electrical characteristics of the leader channel published by Petrova [START_REF] Petrova | Lightning stroke simulation by means of the fractal approach: Attractive and protective zones for structures[END_REF]. The latest available model by Beroual et al [START_REF] Beroual | Predictive dynamic model of the negative lightning discharge based on the similarity with long laboratory sparks. Part 1: physical process and modeling[END_REF][START_REF] Beroual | Predictive dynamic model of the negative lightning discharge based on the similarity with long laboratory sparks. Part 2: Validation[END_REF] that calculates the evolution of a lightning discharge is based on electrical network parameters derived from electromagnetism, wave propagation and gas discharge theory.

All of the above-mentioned models are approximations, however, which, in one way or other, cannot reproduce all the different stages of the negative discharge. Thus, to reproduce processes such as the relaxation time or the positive leader propagation or space leader, they tune parameters or assume circuital arrangements.

Here, a new physical model is proposed based on the stages and characteristic measurements identified by Castellani [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF] and the subsequent phases of the physics of the negative discharge identified and used by Castellani, Mazur and their co-workers [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF][START_REF] Mazur | Computer simulation of a downward negative stepped leader and its interaction with ground structures[END_REF][START_REF] Lalande | Observations and modeling of lightning leaders[END_REF]. The methodology considers processes such as the negative leader channel, composed of negative and positive streamers and the negative leader stem. For the positive and negative streamers, we used the streamer criterion of Gallimberti [START_REF] Gallimberti | The mechanism of long spark formation[END_REF].

Once the first corona has been incepted, the charge in the streamer zone is calculated, assuming that the streamer zone is characterized by a minimum propagation electric field of 450 kV/m and 750 kV/m for positive and negative corona, respectively [START_REF] Gallimberti | The mechanism of long spark formation[END_REF][START_REF] Bondiou | Theoretical modelling of the development of the positive spark in long gaps[END_REF][START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF][START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF].

The voltage drop in the leader channels is evaluated using the "local thermodynamic equilibrium" (LTE) equations of Gallimberti [START_REF] Gallimberti | The mechanism of long spark formation[END_REF]. Once the radius of the leader channel has been given as an input parameter, reported values from experimental measurements [4], the equations can predict the development of the electric field inside the leader channel.

Physical processes underlying the negative discharge

The evolution of a negative discharge is determined by environmental conditions, like the humidity, temperature and pressure, and by the electric field characteristics. The physical description of the negative discharge assumed here is based on the characteristics presented by Castellani [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF], Les Renardieres 1978 [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF], Lalande 2002 [START_REF] Lalande | Observations and modeling of lightning leaders[END_REF] and Gallimberti et al [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF]. This description corresponds to observations of experiments performed in a rod-plane arrangement to avoid interaction between positive and negative discharges. The stimuli used to analyze the electrical discharge were switching impulses, with different front and tail times and peak magnitudes. The electrical response of these configurations under negative polarity is of great interest from an engineering point of view because the majority of stresses caused by lightning in electrical networks are of negative polarity, and even when they do not dominate the breakdown process, they can play a significant role in multielectrode and phase-to-phase arrangements. Furthermore, the study of negative discharges will improve the knowledge of the lightning mechanism, which might help to clarify shielding problems and predict outage rates of power lines.

The first phenomenon involved in a negative discharge is the formation of a negative streamer corona (NC) at time t i . Then, after at time t 1 , a discharge process called a "pilot" appears (PL), in which the simultaneous development of a positive streamer propagating towards the cathode and a negative streamer propagating towards the anode occurs.

At a time t 2 , a space leader (SL) develops from a space stem, and at t 3 , the conditions are suitable for a negative leader (NL) to propagate towards the anode. On the basis of the published results [START_REF] Bondiou | Theoretical modelling of the development of the positive spark in long gaps[END_REF][START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF], one can conclude that the thermodynamic conditions for the formation of positive leaders ought to be very similar to those for negative leaders.

At time t 4 , a stem is formed in the space where the two pilots are located. However, this stage is not observed for gaps of less than 4 m [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF][START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF]. From this stem, a negative streamer and a leader crossing the space (SL) are formed. The leader possesses two heads, one of which is a positive leader and propagates towards the cathode, the other being a negative leader that propagates towards the anode.

When the space leader reaches the negative electrode, a negative corona is formed and the leader channel is illuminated brightly. This step is called a step discharge. Briefly, the negative discharge propagates discontinuously, hence the name, and with complex processes taking place between the negative and positive streamers. Figure 1 was adapted from [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF] to illustrate the various stages of a negative discharge. Gallimberti [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF] with the permission of Elsevier.

3

The principles of the model

The methodology

The complexity of the electric field calculations and the need to implement all the different physical equations required made it necessary to employ software for a finite element method. The calculation procedure was divided up into a number of different physical stages as follows:

3.1.1
The negative streamer region First, the background electric field was calculated for the applied switching impulse voltage source to find out the voltage level of the source at which the streamer inception criterion is satisfied.

(

)

stab x N dx ≥ ⋅ - ∆ η α exp (1) 1-4
where α is the first ionization coefficient, η is the attachment coefficient and x is the size of the active region where (α -η) > 0 and N stab is the minimum charge that produces a space charge field high enough to reproduce the streamer tip. The ionization and attachment coefficients are calculated as a function of the local fields.

For the calculation of the streamer region, the methodology presented in [START_REF] Arevalo | A new static calculation of the streamer region for long spark gaps: numerical simulations[END_REF] is used. The calculation of the streamer zone divides up naturally into four main stages:

1. As the streamer region is characterized for an almost constant electric field between 700 -1000 kV/m [START_REF] Gallimberti | The mechanism of long spark formation[END_REF][START_REF] Bondiou | Theoretical modelling of the development of the positive spark in long gaps[END_REF][START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF][START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF], the volumetric region between the tip of the leader channel/or the high voltage electrode and the ground point is divided into several layers. For each layer, the area where the field is equal to or higher than the stabilization electric field (of 750 kV/m) is identified.

2. In this streamer region, it is assumed that the streamers propagate along the electric field lines and that the drop in the potential along the electric field line to the boundary of the region is equal to 750 kV/m.

3. The streamer region is then divided into several layers and the charge is calculated for each layer. This calculation requires that the direction of the electric field vector be taken into consideration, including the direction at the edge of the region, as illustrated in Figure 2. 4. Using Gauss's theorem, the streamer charge located in each layer Q i is calculated, and the total charge accumulated in the volume for each time step Q total is computed.

⋅ ⋅ = s i S d E Q r r 0 ε (2) = = ∆ n i i total Q Q 1 (3)
Then the negative streamer charge is calculated, using Equations ( 2) and ( 3), where 0 is the permittivity of the air, E corresponds to the electric field vector and S is the vector of the surface over which the integration is performed.

The next stage in the discharge process is the socalled "pilot system", and the calculation for this is made as follows.

3.1.2

The "Pilot Streamer System"

Even though, there is no clear explanation of the physics underlying the discharge process because of its complexity and the lack of experimental data, some experimental results have indicated that each pilot streamer discharge starts at the lower tip of the preceding one and that this form of initiation is associated with every current pulse [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF][START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF]. A consistent explanation was, however, proposed by Bacchiega [START_REF] Bacchiega | Theoretical modelling of the laboratory negative stepped leader[END_REF].

For the calculation of the potential profile and the reproduction of a pilot system, the following assumptions should be made:

1. It has been stated in the literature that the electric field necessary for the propagation of a negative streamer is almost constant and has a value of 750 kV/m [START_REF] Gallimberti | The mechanism of long spark formation[END_REF]. Thus, the region in front of the tip that fulfils this electric field criterion has been identified, and the maximum axial length of the region has been calculated (see Figure 2). 2. Once the negative streamer is formed, a positive leader discharge will develop in the direction of the high voltage electrode. The simulation of this leader follows the methodology presented in the literature [START_REF] Arevalo | Modeling of positive discharges in laboratory gaps under switching impulses Gas Discharge Conference GD[END_REF][START_REF] Arevalo | Numerical Simulation of Long Laboratory Sparks Generated by Positive Switching Impulses[END_REF][START_REF] Arevalo | Laboratory long gaps simulation considering a variable corona region International Conference on Lightning Protection[END_REF]. The inception and the location of the 1-5

propagation of this leader is assumed to be at the central axis of the arrangement and, as mentioned, it propagates towards the high voltage electrode.

3.

Laboratory measurements made by Les Renardieres' Group have shown that, before the formation of the first pilot system, there is an enhancement of the electric field between the negative streamer and the grounded electrode, as a result of this, the electric field in the local region close to the tip of the negative streamer has a magnitude of 1•10 6 V/m [START_REF] Rizk | Modeling of lightning incidence to tall structures. Part I: Theory[END_REF]. The existence of this electric field and the requirement that a positive discharge can propagate towards the high voltage electrode makes it necessary to assume that, from the tip of the negative streamer region, a first stem leader develops towards the high voltage electrode with an electric field magnitude of 1•10 6 V/m. In front of this stem leader, a positive streamer is located. 4. For the simulation of the positive and negative leader channel, Gallimberti's LTE equation is used [START_REF] Gallimberti | The mechanism of long spark formation[END_REF].

The leader is decomposed into elementary segments with length dl, temperature T, pressure P and a molecular density n that is assumed to be uniform along the channel.

The potential drop U L in the segment i will then be: U Li = E Li • dl i where dli is the length of the segment i and E Li is the potential gradient of the segment of length i.

Gallimberti's model gives the evolution of the internal electric field E L as a function of the injected current, supposing that the conductivity of the leader channel is essentially controlled by electronic collisions between neutral molecules and accelerated electrons in the electric field E L . This estimation supposes that all the injected current I L is used to dilate the leader, and that the mass remains constant as the expansion takes place. The formulation describing this hypothesis results in the following set of equations:

i i i i i i dl n a dl n a ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ 2 0 2 0 π π (4)
( )

L L I E dt a d p ⋅ = ⋅ ⋅ - ⋅ 2 0 1 π γ γ (5)
where a i is the radius of the leader segment at a defined instant in time, n i is the neutral molecules' density at the same time, and a 0i and n 0i are the initial conditions for the leader formation. P 0 is the atmospheric pressure, is the ratio between the specific heat at constant pressure and the volume constant, d( a 2 ) is the variation in the crosssectional area of the leader, and E L I L is the power injected into the channel during the time step, dt.

From equation [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF], it is possible to calculate the channel section for the next time step from the function by considering the function at the time t, and the internal electric field and the charge

I L dt. ( ) () dt I E p t a dt t a L L ⋅ ⋅ ⋅ - + ⋅ = + ⋅ ⋅ 0 2 2 1 γ γ π π (6)
As the mass is constant, the molecules' density enables us to write:

( ) () ( ) ( ) dt t a t a t n dt t n + ⋅ ⋅ = + 2 2 π π (7)
And, using the hypothesis that E L /n is constant, the internal electric field at that time will be equal to:

( ) ( ) ( ) ( ) t E t n dt t n dt t E L L + = + (8)
From this it is possible to calculate the time evolution of the internal electric field for each segment and the potential drop along the leader channel:

= ⋅ = ∆ k i i Li L dl E U 1 ( 9
)
where k is the total number of segments.

Assuming that a first leader segment is incepted and that its initial characteristics are given by the parameters described in table 1, the electric field calculation is performed using finite element methods. In 1-6 front of this leader channel, a positive streamer region is located, as presented in figure 3.

Figure 3. Illustration of the initial propagation of the pilot system. Note that it is assumed that the positive leader discharge is propagating along the main axis of the electrode arrangement. 5. To consider the positive streamer region, the region that satisfies the condition for a stable positive streamer of 450 kV/m, close to the high voltage electrode is determined using the same methodology described to calculate the negative streamer region in section 3.1.1, using equations ( 2) and (3). For major details, please refer to the reference [START_REF] Arevalo | A new static calculation of the streamer region for long spark gaps: numerical simulations[END_REF].

The potential distribution of every pilot streamer discharge is composed of a series of potential drops over: the positive streamer region, the positive leader channel and the negative streamer region located in front of the grounded electrode.

Several different phenomena need to be considered, including: the positive leader channel, the positive and the negative streamers. The relative impact of the different phenomena gives the total charge, which leads to the characteristic pulsing of the pilot system, defined by the so-called "relaxation time", which has been tuned to RLC electrical circuits in other models.

The electrical charge per unit length that is required to sustain the progression of the pilot streamer discharge was assumed to be 13.8 µC/m, as Castellani estimated in [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF].

In Figure 4, one can observe the potential profile distribution before and after the streamer development, and after the formation of the first pilot streamer discharge. The advance of the pilot is calculated from the minimum electrical charge per unit length required to sustain the pilot and the charge in the streamer region using equations [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF] and [START_REF] Mazur | Computer simulation of a downward negative stepped leader and its interaction with ground structures[END_REF]. The advance of the positive leader l L (i) can be determined by integrating the velocity of the leader, which is described in equation [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF]; where Q total is the corresponding total charge of the pilot system, q L is the charge per unit length required to sustain a pilot system, L(t+dt) is the length of the next leader segment and L L (t) is the length of the present leader segment.

L i total i q Q l L ) ( ) ( ∆ = ∆ (10) l L l t L dt t L ∆ + = + ∆ ) ( ) ( (11)

"Negative Leader Phase"

The measured results extracted from the literature [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF][START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF] reveal that the charge necessary to incept a

Positive streamer region

Positive leader

Negative Streamer leader channel is of the order of 5.4 µC. This value is used in the present calculation to initiate the simulation of the leader stage. This charge includes all of the charge associated with the different streamers and the pilot systems developed up to the inception of the leader.

We assumed that the inception mechanism for the negative leader is similar to the one for the positive leader [START_REF] Arevalo | Modeling of positive discharges in laboratory gaps under switching impulses Gas Discharge Conference GD[END_REF][START_REF] Arevalo | Laboratory long gaps simulation considering a variable corona region International Conference on Lightning Protection[END_REF]. This assumption is valid because the threshold electrical charge corresponds to the charge necessary to heat the "stem" to a critical temperature, which will lead to the formation of the first section of the leader channel. This negative leader is calculated using the thermo-equilibrium equations of Gallimberti [START_REF] Gallimberti | The mechanism of long spark formation[END_REF], described in equations ( 4) to [START_REF] Bacchiega | Theoretical modelling of the laboratory negative stepped leader[END_REF]. The potential gradient immediately before a new leader segment is created is assumed to be 7.5 x 10 5 V/m, from experimental evidence [START_REF] Gallimberti | The mechanism of long spark formation[END_REF][START_REF] Bondiou | Theoretical modelling of the development of the positive spark in long gaps[END_REF][START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF][START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF].

The first incepted negative leader is assumed to propagate from the tip of the high voltage electrode to the tip of the negative streamer region, as presented in figure 5.

Figure 5. Illustration of the initial propagation of the negative leader. Note that it is assumed that the negative leader is propagating straight along the main axis of the electrode arrangement.

For the next stages of the simulation of the advancement of the negative leader, equations [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF] and ( 11) are used and the charge per unit length required to sustain the negative leader propagation, q L , was 108 µC/m based on the values published in the literature [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF].

The new negative streamer region is found in front of this negative leader, then a new pilot system is calculated. 4

Application of the methodology

Two different rod-plane configurations were tested with different switching impulses and gap distances to check the validity of the model. First, the methodology was applied to the same geometrical set-up as that used by the Les Renardieres Group [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF], which consisted of a gap between a conical rod and a plane. The conical electrode had a tip radius of 10 mm. The simulation was made assuming a gap distance of 2 m with an applied impulse peak voltage of -1550 kV and a 20/1600 µs impulse waveshape.

The results of the simulations presented in Figures 6 and7 showed that the trend exhibited by the simulated current agrees with the measured signatures determined by the Les Renardieres' Group [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF]. The maximum magnitude of the current was 10 A for both the measurements and simulations. The simulation of the current pulsates, which is because of the restarting process and because of the exchange of charge in the channel arising from the presence of both a negative leader and a positive leader moving towards the high voltage electrode. Overall, the results from the model including the path of the discharge are in good agreement with the experimental results of Les Renardieres [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF]. However, it is important to note that, even though the shape of the pulses correspond to those of the measurements, slight differences are evident in the timing. All pulsation behavior corresponds to the pilot streamer system representation and the initiation of pilot streamers is dependent on the exchange of positive and negative charges. It is likely, therefore, that the exchange of charges happens faster in the experiment because of external conditions in the arrangement that are not taken into account in the calculation, such as imperfections on the high voltage electrode, or external corona sources. The final parameters available for comparison are the time to breakdown and the breakdown voltage, which were calculated and are summarized in Table 2. The time to breakdown and the breakdown voltages exhibited errors of 35.9% and 4.5%, respectively, for the particular case shown. The high error in the time to breakdown can be attributed to the path taken by the discharge. As was demonstrated by Arevalo and co-workers [START_REF] Arevalo | Numerical Simulation of Long Laboratory Sparks Generated by Positive Switching Impulses[END_REF][START_REF] Arevalo | Laboratory long gaps simulation considering a variable corona region International Conference on Lightning Protection[END_REF], the path followed by the discharge determines the time to the breakdown, but it does not have a notable influence on the calculation of the breakdown voltage.

Negative leader

Negative streamer region

Owing to a lack of detailed experimental information in the literature, it is not possible to extract more parameters for the same configuration, but data for positive and negative streamer charges and for the velocity of the positive and negative leaders can be obtained from the calculation.

In another example of the usefulness of the simulation method, a rod-plane arrangement, 7 m gap distance, 2.8 MV peak voltage and waveform of 6/3000 µs configuration studied by Les Renardieres [START_REF] Renardières | Negative discharges in long air gaps at les Renardieres[END_REF] was reproduced. The only available data from the configuration is the streak photograph. However, simulations of the current and the streak image are available in the literature [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF]. Therefore, comparisons of our results with the published ones [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF] are presented here. Current simulated by Gallimberti [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF] Simulated current -method presented here Figure 8. Computed currents of a negative discharge in a rod -plane configuration (Gap = 7m, Vmax = 2.8MV, waveform 6/3000 µs). The continuous line corresponds to the methodology described in this paper; the line with dots corresponds to the simulation of [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF] The simulation of the current using the methodology of this paper shows an additional peak in the current between 10 and 12 µs because of the differences between the methodologies used to calculate the pilot system. However, with the exception of the aforementioned peak, one can claim that both methodologies reproduce similar peak current magnitudes and rise times. The maximum error in the current peak magnitudes is 26% for the second peak. Simulated current -method presented here Gallimberti et al [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF] Results Figure 9. Computed and photographic streak image of the discharge in a rod-plane configuration (Gap = 7 m, Vmax = 2.8 MV, waveform 6/3000 µs). The continuous line corresponds to the methodology presented in the paper; the line with dots corresponds to the simulation of [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF]. The photographic image has been reproduced from [START_REF] Gallimberti | Fundamental process in long air gap discharges[END_REF] with copyright of Elsevier.

It is difficult to make a meaningful comparison between the photographic results and the simulation; however one can see that both simulations follow a path that apparently agrees with the photograph, and both have similar tendency.

Analysis of the parameters that affect the calculation

The proposed model includes a number of input parameters and, it is necessary to check their sensitivity on the computational procedure and to examine their variability. From all the input parameters, the ones that make the most notable change to the results are the electrical charge required to incept a negative leader, the electrical charge per unit length required to sustain the pilot progression and the electric field of the positive and negative streamer zone. Figure 10 The charge per unit length required to sustain the pilot progression is a parameter used to calculate the advance of the pilot system in the gap space. This parameter directly affects the velocity and the current of the negative leader. The magnitude is taken from laboratory gap measurements performed by Castellani [START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF], since of this value was obtained for long spark gap experiments similar to the ones produced in this paper. The parameter was varied between +60% and -60% of the reference value taken to verify its variability, see figure 10 (a). The analysis shows that the lower the magnitude required to sustain the pilot progression, the longer the delay before the first peak of the current; the first peak occurs between 0 and 0.5 µs. If the parameter is changed by 60%, the magnitude of the first peak current changes by 66%; for moderate variations of 20% in the parameter, a change of 16% in the first peak current is observed. Other that these changes, the general trend does not change.

As far as the charge required to incept a negative leader is concerned, the lower the magnitude, the shorter the inception time of the negative leader. For example, when the parameter is increased or decreased by 20%, the inception time of the negative leader changes by ±3%, respectively. With respect to the peak current, magnitude changes of 20% in the required charge will produce an increment in the peak current of 3.5%. Therefore, one can conclude that this parameter affects the calculation, but only weakly.

According to Gallimberti [START_REF] Gallimberti | The mechanism of long spark formation[END_REF], the potential gradient of the streamer depends on the gas pressure and water vapor content. We have taken values of between 400 and 500 kV/m for positive and 700 to 1000 kV/m for the stabilization electric field of the negative streamer. As can be seen in Figure 10 (c) and (d), the breakdown voltage is not deeply affected by this factor: variations in the breakdown voltage of less than 1% were obtained for the positive streamer and of 2% for the negative one. Consequently, for the calculation the values chosen were 450 and 750 kV/m for positive and negative streamer inception, respectively.

Conclusion

A first approximation to a self-consistent numerical simulation based on the physical process underlying the negative leader discharge has been presented here. It includes a novel way of calculating the positive streamer region for the pilot streamer system and enables the current and voltage characteristics of the complete negative discharge to be predicted. The experimental results are compared with simulations and show good agreement for the representation of the magnitudes of the current and path of the discharge.

Figure 1 .

 1 Figure 1. Negative leader development process. This figure has been reproduced from a manuscript by Gallimberti [13] with the permission of Elsevier.

Figure 2 .

 2 Figure 2. Delimitation of the streamer region the arrows indicate the direction of the electric field, which is also the direction of propagation of the streamers

Figure 4 .

 4 Figure 4. Potential distribution before and after the negative streamer development and after the formation of the first pilot streamer discharge.Figure (a) is an overall view and (b) an enlargement of the potential incepted leader and the streamer channel.

Figure 6 .

 6 Figure 6. Current pulses during the stem-space propagation in a 2 m wide gap between a conical rod and a plane with a switching impulse of 20/1600 µs.

Figure 7 .

 7 Figure 7. Streak simulated path of the discharge compared with the photographic evidence of the path extracted from the literature for a 2 m conical rod-plane gap during a switching impulse of 20/1600 µs.

Figure 10 .

 10 Figure 10. Variability of the input parameters: (a) the charge per unit length required to sustain the pilot progression; (b) the charge required to incept a negative leader; (c) the electric field of the positive streamer zone; and (d) the electric field of the negative streamer zone.

Table I .

 I Input parameters used for the calculation of the negative leader discharge based on measurements made by Castellani and coworkers[START_REF] Castellani | Laboratory study of the bi-leader process from an electrically floating conductor Part 2: Bileader properties[END_REF]

	Process	Condition	Magnitude	Ref.
	Positive Electric Field	Positive streamer Stabilization electric field	450 kV/m	[6]
	Negative Electric Field	Negative streamer stabilization electric field	750 kV/m	[6]
	Initial channel
	radius of a newly created	Positive and negative leader	1•10 -3 m	[6]
	leader segment
	Pilot	Charge contained in the first pilot	0.5 µC	[10]
	progression	Charge per unit length required to sustain the PC	13.8 µC/m	[10]
	Space leader	Minimum charge per unit length required to incept a	53.1 µC/m	[10]
		space leader
		Charge required to incept a negative leader	5.4 µC	[10]
	Negative	Charge required to sustain the negative leader	108 µC/m	[10]
	Leader Phase	Potential gradient before a
		new leader segment is	750 kV/m	[6]
		created
		Initial length of the leader	2 x 10 -2 m	[6]
		Charge required to sustain a positive leader channel	40 -55 µC/m	[4]
	Positive	Potential gradient before a
	Leader Phase	new leader segment is	450 kV/m	[6]
		created
		Initial length of the leader	2 x 10 -2 m	[6]

Table 2 .

 2 Comparison between the measurements and the results of the simulation for a 20/1600 µs impulse across a gap of 2 m between a conical rod and a plane.

		Breakdown Voltage [kV]	Time to Breakdown [µs]
	Les Renardieres	1481	20.6
	Simulations	1546	28
	% error	4.5	35.9

Acknowledgements

The author would like to thank ABB/Power Systems HVDC Ludvika, Sweden for the economic support of the PhD candidate (L.A.).


Diego Ramón Medina-Muñoz

Dr Rita-Dolores D Medina-Muñoz

Dr Diego R Medina-Muñoz

Determinants of expenditures on wellness services: The case of Gran Canaria

Keywords: tourism, tourist destination, tourist demand, wellness tourism, tourist expenditure tourism, tourist destination, tourist demand, wellness tourism, tourist expenditure L83, M10, M20, M30

come

Introduction

The tourism and travel sector has become one of the most important economic activities and has enjoyed one of the highest growth rates in the world economy (e.g., WORLD TOURISM ORGANIZATION, 2009). International travel can be grouped into five basic categories according to the principal motivations (WORLD TOURISM ORGANIZATION, 2001): leisure, recreation and holidays, business and professional, health, religious and pilgrimage, and visits to family and friends. Each of those motivations generates a significant tourist demand comprising travellers who visit destinations with tourism offers that meet their requirements. Thus, a tourism destination may develop and market one or more tourism products in order to satisfy better the different motives for the trips, one of those motives being health and wellness tourism. 4 sand, volcanic lava, landscape, certain plant species, nature in general, religion, and the possibility of meditation (KAPCZYNSKI and SZROMEK, 2007).

According to MUELLER and KAUFMANN (2001), it is particularly necessary to generate information about the wellness vacation demand and the preferences of tourists who travel for wellness motives. This would help those tourist destinations interested in developing and marketing health and wellness tourism to design their wellness offer successfully. That type of information would also facilitate the adaptation of the wellness facilities and services currently offered by the destinations and tourism enterprises interested in this form of tourism. However, no empirical research designed to describe and analyse the health and wellness tourism demand was found in the literature. Therefore, the first objective of this work refers to the description of the tourists who visit wellness centres on Gran Canaria, one of the islands forming the Spanish archipelago of the Canary Islands.

In addition, the study of the determinants of expenditure on wellness services during the stay in the destination would be of great interest for the design of marketing strategies that permit the destination to attract "quality tourists" with a high expenditure on this component of the leisure offer (e.g., NICOLAU and MÁS, 2005). It would also contribute to the improvement of the destination's competitiveness, understood as "its ability to increase tourism expenditure, to increasingly attract visitors while providing them with satisfying, memorable experiences, and to do so in a profitable way, while enhancing the well-being of destination residents and preserving the natural capital of the destination for future generations" (RITCHIE and CROUCH, 2003:2). Moreover, it could increase the An exploratory empirical research was carried out in order to achieve those two objectives. The work first describes the sociodemographic attributes of visitors to the wellness centres on Gran Canaria, the principal characteristics of the trip to the island, traveller behaviour for wellness motives and the expenditure on wellness services during the stay in the destination. That is followed by an evaluation of the influence that those characteristics exercise on the probability of higher tourist expenditure on wellness services during the stay in the destination.

Individual tourist expenditure on wellness services

Research on tourists' expenditure during their travels is of great interest to those in charge of the destinations and tourism firms interested in improving or maximising tourism revenues. However, most research on tourist expenditure has focused on tourism demand as a whole rather than on individual tourist expenditure (e.g., DWYER and FORSYTH, 2008;LAESSER and CROUCH, 2006). That macroeconomic approach does not permit the identification of the tourists with the highest expenditure during their travels. Moreover, the few empirical works that have been designed to describe individual expenditure have not always understood it in the same way. In general, empirical works have measured the total expenditure by a tourist during one trip, which includes expenditure on transport, accommodation, food and beverages, and other leisure services (e.g., LAESSER and CROUCH, 2006;NICOLAU and MÁS, 2005;MOK and IVERSON, 2000;PÉREZ and SAMPOL, 2000). However, few studies have been interested in the expenditure on certain components of the destination's tourism offer, such as local food products (e.g., SKURAS et al., 2006) or park fees (MMOPELWA et al., 2007), and only during the stay in the destination (e.g., MEHMETOGLU, 2007;VAUGHAN et al., 2000).

This study of wellness tourist expenditure aims to contribute to the emergent field of research on tourist expenditure by products, such as rural and nature tourism (e.g., [START_REF] Fredman | Determinants of visitor expenditures in mountain tourism[END_REF][START_REF] Kastenholz | Analysing determinants of visitor spending for the rural tourist market in North Portugal[END_REF]FELSENSTEIN and FLEISCHER, 2003;LEONES et al., 1998), nautical tourism (e.g., LEE, 2001), visits to National Parks (e.g., WHITE and STYNES, 2008;DOWNWARD and LUMSDON, 2004;SPOTTS and MAHONEY, 1991), and historic site and festival tourism (e.g., TAYLOR et al., 1993;THRANE, 2002;LONG and PERDUE, 1990). This work evaluates, in an exploratory fashion, the possible effects that the sociodemographic characteristics, behaviour as a wellness tourist and certain characteristics of the trip may have on expenditure on wellness services. and CROUCH, 2006;SKURAS et al., 2006;NICOLAU and MÁS, 2005;MMOPELWA et al., 2007). The most studied characteristics include: geographical origin, gender, age, marital status, having children, educational level and occupation, which are also the variables selected in this research.

Various authors have offered empirical evidence suggesting that the geographical origin of the visitors is a determinant that favours tourist expenditure, in that the greater the physical distance between the place of residence and the destination, the higher the transport costs seem to be (e.g., LEONES et al., 1998;LEE, 2001).

Other authors propose that, given the fixed cost of the transport, the greater distance to the destination also favours a longer stay and higher expenditure in the destination (e.g., SILBERMAN, 1985;NICOLAU and MÁS, 2005).

The gender of the tourist has received scant attention in the literature as a possible determinant of tourist expenditure (e.g., THRANE, 2002). However, the literature on wellness indicates that females have traditionally displayed greater concern for health, beauty and mind although a growing interest among males has also been noticed (e.g., MUELLER and KAUFMANN, 2001). Therefore, it seems logical to propose that women spend more on wellness services and treatments.

Unlike gender, the age of the tourist is one of the determinants of tourist expenditure most cited in the literature although there is no consensus on its possible impact (e.g., NICOLAU and MÁS, 2005). In general, it seems that the young and the middle-aged especially value the possibility of tourism and allocate part of their budgets to tourist expenditure, while that expenditure decreases in advanced Similarly, the wellness literature suggests a stronger demand for wellness treatments among young and middle-aged individuals (e.g., MUELLER and KAUFMANN, 2001).

With regard to marital status, it is considered that married couples and opposite sex couples could exhibit a higher tourist expenditure (e.g., CAI, 1998). The principal argument for that association refers to the different purchasing behaviour of men and women in that, when they travel together, they may incur higher expenditure than two members of the same sex travelling together. That argument is not valid in this research since it measures the expenditure on wellness services by each individual and not by family, couples or friends travelling together.

However, partnered individuals might also exhibit higher individual tourism expenditure because a couple may consider leisure and consumption options that perhaps they would not have considered as single travellers. Moreover, the relationship with a partner may increase interest in wellness services and treatments that may be positive to their relationship as a couple. Therefore, it is opportune to evaluate, in an exploratory fashion, whether partnered individuals have higher individual expenditure on wellness services. In addition, individuals with children usually exhibit higher tourist expenditure (e.g., [START_REF] Agarwal | Tourist spending and race of visitors[END_REF] and may also exhibit a greater need to consume wellness services and treatments as a consequence of having less time available for personal attention (MUELLER and KAUFMANN, 2001).

The empirical literature also suggests a positive relationship between the tourist's income and his/her expenditure on tourism products (e.g., NICOLAU and MÁS, , 2002;LEE, 2001), although it is not conclusive in the case of occupation (e.g., PÉREZ and SAMPOL, 2000) or that of educational level (e.g., NICOLAU and MÁS, 2005) as determinants of tourist expenditure. However, it seems logical to propose that tourists with higher incomes and educational levels display greater expenditure on wellness services.

The review of the literature in the previous paragraphs permits the following hypotheses to be put forward: H1. Daily expenditure on wellness services is higher when the tourist takes a longdistance trip.

H2. Daily expenditure on wellness services is higher among women than men.

H3. Daily expenditure on wellness services is higher when the tourist is young or middle-aged. H4. Daily expenditure on wellness services is higher when the tourist is married or living with a partner. H5. Daily expenditure on wellness services is higher when the tourist has children.

H6. Daily expenditure on wellness services is higher when the tourist has a higher educational level. H7. Daily expenditure on wellness services is higher when the tourist has an occupation that offers a high income.

Characteristics of the trip to Gran Canaria. Some empirical works have provided results that indicate that the length of the stay in the destination is one of the principal determinants of tourist expenditure. In his research on the expenditure of the Swiss on their international travels, LAESSER (2004) observed that twice as long (LAESSER and CROUCH, 2006). Similarly, in his study of nature tourism in Northern Norway, MEHMETOGLU (2007) concludes that tourists who stay longer in the destination are more likely to be considered "light spenders" in terms of expenditure per day in the destination. The main argument to justify those results refers to longer stay tourists having less urgency to enjoy the leisure offer in the destination and, therefore, distribute their budget for tourist expenditure over a longer holiday (e.g., LAESSER and CROUCH, 2006;THRANE, 2002).

The literature also suggests that the principal motive for the trip may be a determinant of tourist expenditure. The few empirical works to analyse that relationship have found no significant differences in tourist expenditure depending on the purpose of the trip (e.g., CAI et al., 2001;THRANE, 2002), although some authors have found lower relative expenditure on some tourism products such as sun, sand and sea (e.g., LAESSER and CROUCH, 2006). However, the main purpose of the trip may have a positive influence on spending on certain concepts of expenditure or on tourist services that may be more required, depending on the motive for the trip (e.g., LAESSER and CROUCH, 2006). In that respect, it seems logical to propose that tourists who travel for wellness motives spends more on wellness services and treatments than those who travel for other motives, such as sun, sand and sea, golf, rural, cultural or business tourism.

Moreover, the factors that influence the choice of destination may also affect tourist expenditure (e.g., THRANE, 2002;LAESSER and CROUCH, 2006). More 2002) is one of the few authors to directly analyse the association between tourist expenditure and interest in certain tourist attractions and concludes that tourists who display more interest in music spend more during a music festival. Following that line of argument, it could be proposed that expenditure on wellness services is higher among tourists who consider the offer of spas and wellness centres important when choosing the destination for their holidays.

In light of all the above, the following hypotheses can be made: H8. Daily expenditure on wellness services is lower when the stay in the destination is longer.

H9. Daily expenditure on wellness services is greater when the principal motive for the trip is related to wellness tourism. H10. Daily expenditure on wellness services is greater when the tourist considers the offer of wellness centres important when choosing the destination.

Behaviour as a wellness tourist. Another criterion that may be used as an alternative to the sociodemographic characteristics of the tourist or his/her behaviour on a single trip to segment tourist demand is the general travel behaviour of the tourist. Thus, MUELLER and KAUFMANN (2001) indicate that wellness behaviour on a single trip depends on conditions such as the time available, climatic conditions, the companions or physical and mental state of the tourist at the time of the trip. Therefore, those authors recommend the analysis of the general behaviour as a wellness tourist in any attempt to segment the wellness This work analyses, in an exploratory fashion, the effect that the frequency of wellness-motivated international trips has on expenditure on wellness services.

Tourists who regularly travel for wellness motives give a higher evaluation of the wellness resources and services offered by destinations and therefore it may seem reasonable to think that they probably spend more on wellness services during their stay in a destination. Moreover, their experience of wellness-motivated international travel suggests a greater knowledge of the different wellness treatments, which, in turn, facilitates the decision to purchase wellness services.

Thus, the following hypothesis is proposed: H11: Daily expenditure on wellness services is higher when the tourist regularly travels internationally for wellness motives.

Research design and methodology

Tourism on Gran Canaria

Covering an area of 1,560.10 km 2 and with a population of 807,049, Gran Canaria is one of the seven islands comprising the Canary Islands, a Spanish archipelago in the Atlantic Ocean. In 2008, the island received 2,685,267 foreign tourists, who stayed in its 144,954 tourist beds (CANARIAN GOVERNMENT DEPARTMENT OF TOURISM, 2009). Practically all the visitors were from Europe, in particular Germany (27.9%), the United Kingdom (22.4%) and Central and Northern Europe, where the principal countries of origin are Norway (9.0%), Sweden (7.7%), Holland (6.4%), Denmark (4.7%), Finland (4.5%), Ireland (4.1%), Switzerland (3.1%), Belgium (2.5%), Austria The average winter shade temperatures range from 18.6°C in March to 24.6°C in October and November. Those favourable climatic conditions, together with the attractiveness of its beaches and coastline lead almost 80% of Gran Canaria's tourists to mention sun, sand and sea tourism as the main motive for their visit.

The island also boasts tourist attractions and an important business offer linked to other tourism products, such as rural tourism, golf or wellness; therefore, the island's government and tourism sector have set the objective of increasing the number of visitors with motives other than sun, sand and sea. In that context, wellness tourism has been chosen as a tourism product around which the tourist demand can diversify. By the end of the 19 th Century, British health tourists were already being drawn to Gran Canaria by its climate, the quality of its underground waters and its other natural resources. Apart from the favourable natural conditions, the island has also developed a significant offer of spa and wellness centres and hotels that offer wellness services.

Research approach and survey

The research method chosen for this work was the survey and the type of questionnaire used was the "face-to-face" personal interview, which was conducted on visitors to the 15 wellness centres that form part of the "Gran Canaria Spa&Wellness" association. That association comprises tourism entrepreneurs providing wellness facilities and services and its objective is to The design of the questionnaire on the basis of previous works and expert opinions reinforces the validity of content and validates the measurement instrument used in this research [START_REF] Babbie | The practice of social research[END_REF][START_REF] Hambrick | Strategic awareness within top management teams[END_REF]. The process of designing the questionnaire began by preparing a list of possible determinants of tourist expenditure on wellness services, based on a review of the existing literature (e.g., MEHMETOGLU, 2007;MMOPELWA et al., 2007;LAESSER and CROUCH, 2006;NICOLAU and MÁS, 2005). The few academic and professional publications on wellness tourism were also taken into consideration (e.g., MUELLER and KAUFMANN, 2001;[START_REF] García-Altés | The development of health tourism services[END_REF]SMITH and KELLY, 2006;[START_REF] Kapczynski | Hypotheses concerning the development of Polish spas in the years 1949-2006[END_REF]. Then, a focus group was organised, in which six experts, who were representatives of the "Gran Canaria Spa&Wellness" association, participated with the aim of analysing the concept of wellness tourism, describing the profile of wellness tourists and, in particular, identifying the characteristics of the tourist and the trip that could affect the level of expenditure on wellness services. The conclusions of that meeting were considered in the preparation of the second draft of the questionnaire.

In order to perfect the second draft of the questionnaire for wellness centre customers and reduce its length, and to design the fieldwork in the wellness centres, a specific questionnaire for the managers of the 15 wellness centres in the association was drawn up. The responses of all those association members not only job status. The second section asked about the following characteristics of the trip to Gran Canaria: the length of stay in the destination, the principal motive for the trip, the importance of wellness centres in the choice of destination, and the approximate expenditure on wellness services during the stay on the island. The third and final section asked about the frequency with which the tourist makes international trips for wellness purposes.

Data collection and sample selection

A team of 18 interviewers conducted the surveys in the island's 15 wellness centres in October and November 2007. Two interviewers, who between them spoke at least Spanish, English and German, were assigned to each centre. A total of 989 wellness centre users completed the questionnaire correctly and the sample error was 3.1% for a reliability level of 95%.

Results

Sociodemographic characteristics of the customers of the wellness centers

This section presents the results of the descriptive analysis of the sociodemographic characteristics of the tourists participating in this research; thus, the general characteristics of the demand for wellness on Gran Canaria are described (see Spain and Continental Europe to the Canary Islands means that the cost of air transport is much higher than for residents of the other Canary Islands. Therefore, in later analyses, the spending behaviour of residents of other Canary Islands will be compared with that of other Spanish and European visitors.

INSERT TABLE 1

With regard to the gender of the respondents, women represented 53.7% of the customers and men 46.3%. However, women represent only 33.64% of the visitors to the island (CANARIAN GOVERNMENT DEPARTMENT OF TOURISM, 2009), which leads to the conclusion that female tourists make more use of the wellness and spa centres than male tourists. The average age of visitors to the wellness centres was 42 years. An analysis of the age groups revealed that almost 40% of the customers were younger than 36 years old (39.87%). The percentage was similar (41.59%) in the case of the market segment comprising tourists of between 36 and 55 years old. The oldest customers represented the remaining 18.54%, with the over-65s making up 6.02% of the tourists who make use of wellness facilities and services.

In the case of marital status, 72.39% of the customers of spa and wellness centres on Gran Canaria were married (55.74%) or living with their partners (16.65%).

Unmarried individuals not living with a partner comprised the next largest group (20.91%), followed by the divorced or separated (4.77%) and widows/widowers (1.93%). Moreover, 64.35% of the respondents have at least one child, which could The educational levels stated by the respondents indicate that the wellness centre demand has higher or intermediate levels of education. More specifically, it is notable that half of the respondents have finished higher university degree courses while 31.10% have completed intermediate studies and 16.77% elementary studies.

Only 1.63% of wellness tourists have no completed official studies. The professional and labour situations permit the potential income and tourism expenditure to be estimated. The results obtained in this study reveal that the most common situation is one of a non-managerial employee of a firm, which is the case of 35.79% of the respondents. However, the high presence of managerial employees (18.10%) and, especially, of entrepreneurs and self-employed individuals (22.49%) should be highlighted. Those two segments of tourist demand represent 40.59% of the total wellness demand. The significant presence of pensioners and retirees deserves mention since they represent 9.3% of wellness customers.

Characteristics of the trip and frequency of wellness-motivated trips

The customers of wellness centres spend an average of almost 8 nights on the island and their principal motive for the trip was the possibility of enjoying sun and beach: 58.82% of the surveyed tourists. However, that percentage is lower than the 80% obtained for all the tourists that visit the island (CANARIAN GOVERNMENT DEPARTMENT OF TOURISM, 2009). Those results suggest a greater diversification of the wellness centre demand than of the island's total tourist demand according to the principal motive for the trip. The second most important motive for visiting the 19 island refers to wellness tourism and the possibility of enjoying wellness and spa centres: 24.59%. That is followed by professional and business tourism (6.31%), attending a conference, seminar or similar event (1.42%) and visiting family members (2.19%). However, health tourism linked to surgery or the treatment of an illness was the least important motive among visitors to the wellness centres on Gran Canaria: 0.77%. The other motives (rural, cultural, nautical, cruise, golf and sport tourism, and special events) were mentioned by less than 1.2% of the surveyed tourists.

Moreover, the results obtained in this research suggest that the availability of wellness-related leisure tourism is an attractive complement to a visit to the island for any other primary purpose. More specifically, 42.7% of the interviewees indicated the medium or high importance of the possibility of enjoying wellness centres at the time of choosing the destination for their holidays.

The current low level of knowledge about the profile of wellness tourists has led some authors to recommend the analysis of the general behaviour of the traveller for wellness motives rather than that of behaviour on a single trip (e.g., MUELLER and KAUFMANN, 2001). With regard to international travel for wellness purposes, 62.92% of the interviewees had never travelled for that reason. That high percentage supports the recentness of the idea of holidays for wellness purposes although all predictions point to a significant increase in wellness-motivated international travel in coming years (e.g., WORLD TOURISM ORGANIZATION, 2009).

An explanatory model of expenditure on wellness services 2009), that amount is slightly lower than the mean daily expenditure of the total tourists who visit Gran Canaria (38.72 euros). Thus, if we add the other expenditure in the destination (e.g., transport, food, and other leisure services) to that spent on wellness services, it is highly probable that the total daily expenditure in the destination by the customers of the island's wellness centres is greater than the average expenditure of all visitors to Gran Canaria.

In line with previous empirical studies (e.g., KOZAK et al., 2008;[START_REF] Jang | Understanding travel expenditure patterns: a study of Japanese pleasure travelers to the United States by income level[END_REF], an ordinary least squares (OLS) regression analysis was conducted in order to evaluate the hypotheses proposed in this work regarding possible determinants of expenditure on wellness services. According to NICOLAU and MÁS (2005), the majority of the studies on holiday expenses are based on the classical regression methodology, which seems to be appropriate when the dependent variable is quantitative and the number of null observations (nil tourist expenditure) is not large. The use of truncated regression models on the sub-sample resulting from the elimination of nil values would be an alternative whenever the number of nil values is large and the size of the sample does not allow the gathering of a wide range of possible values. However, in this study, the dependent variable referred to the amount of euros spent on wellness services per day; the tourists were interviewed as they exited a wellness centre after receiving wellness services, which minimises the number of tourists who reported zero; and a total of 989 wellness centre users participated in this research. Specifically, the OLS regression analysis was conducted by using the log value of the wellness expenditure per tourist per day with a view to achieving a better fit of the model (e.g., [START_REF] Jang | Understanding travel expenditure patterns: a study of Japanese pleasure travelers to the United States by income level[END_REF]. The independent variables included in the model referred to the 11 possible determinants of daily wellness expenditure reflected in the hypotheses proposed in this work. Age (X3= number of years) and stay in the destination (X8= number of nights) are quantitative variables.

Geographical origin (X1) was split into two different dummies in order to identify possible different structural behaviours based on the geographical origin: "tourists from European countries" (1= yes, 0= no) and "tourists from Spain" (1= yes, 0= no); the base case (zero for both dummies) is represented by tourists from the Canary Islands. Gender (X2) was measured as 1 if male, and 0 if female. Similarly, having children (X5) had two different values (1= one or more children, 0= childless); tourists were not asked about the specific number of children.

The other independent variables included in the model were recodified based on the theoretical foundation of the hypotheses. Marital status (X4) achieved two different values: 1= married or living with partner, and 0= other situations; this option was chosen because of the theoretical interest of an analysis of possible differences in the daily wellness expenditure of tourists with partners and that of the tourists in other situations (H4). Educational level (X6, where 1= higher education, 0= intermediate education or lower) and occupation (X7, where 1= businessperson or self-employed, 0= other job situations) were selected as proxies of income. As stated in H6 and H7, daily expenditure on wellness services is expected to be higher when the tourist has a higher education level and his/her occupation offers a high income. With regard to the principal motive for the trip was measured with value of 1 if it is moderately or very important, and 0 in the case of unimportant or moderately unimportant. Finally, frequency of international trips for wellness (X11) was measured with two values: 1= some frequency, 0= never.

Table 1 displays the values of the mean and standard deviation obtained for the daily wellness spending of each group of tourists identified in the 9 qualitative independent variables. The correlation analysis between the different independent variables generated correlations values below 0.333 and no significant association was found [START_REF] Bachaus | Multivariate analysemethoden[END_REF].

The results of the OLS regression analysis are shown in Table 2. The model diagnostics could be considered adequate since the model F value was 11.195 at an observed significance level (p< 0.01). Moreover, the adjusted R 2 achieved a value of 0.318, suggesting that the independent variables explained about 32% of the observed variance of the daily wellness expenditure. Specifically, seven of the 11 independent variables in the model affected the dependent variable.

INSERT TABLE 2

With regard to the sociodemographic characteristics, only three of the seven possible determinants of daily expenditure on wellness were significant: geographical origin, age and having children. As can be seen in , "age" and "having children" are significantly and positively associated with "daily expenditure on wellness". As expected, the average daily spending on wellness services is significantly higher if the geographical origin of the tourist is a long distance from the destination and if the tourist has children. However, the daily expenditure on wellness significantly increases with the age, while the wellness literature suggests a stronger demand for wellness treatments among young and middle-aged individuals. Beta values also suggest that "geographical origin -tourists from European countries" has a stronger association with "daily expenditure on wellness services" in comparison with "age" and "having children". Additionally, "gender", "marital status", "educational level" and "occupation" have no significant influence on "daily expenditure on wellness". Therefore, hypotheses H1 and H5 are not rejected by the results while H2, H3, H4, H6 and H7 are not supported.

The three characteristics of the trip to Gran Canaria that were evaluated as possible determinants of "daily expenditure on wellness" were significant. As Table 2 shows, Beta values also suggest a strong association. More specifically, expenditure on wellness services increases when the tourist considers the presence of wellness centres important and the principal purpose of the trip is to practice wellness tourism. With regard to "stay in the destination", measured as the number of nights in the destination, the findings suggest the content of H8, to the extent that "stay in the destination" had a significant negative association with "daily expenditure on wellness services", meaning that the lower the stay in the The frequency of international trips for wellness motives is another variable that enables significant differences in wellness spending to be identified. More specifically, tourists that have previously taken trips for wellness purposes spend significantly more money on wellness services per day in comparison with those who have never travelled abroad for wellness motives. The Beta value suggests a strong positive association (see Table 2). Therefore, exploratory hypothesis H11 is not rejected.

Conclusions

One of the main trends in tourism is the appearance and rapid spread of new motives, such as wellness tourism, to take international trips. This work argues that it is necessary to know the characteristics of that new demand so that regions interested in tourism development can successfully design the tourism offer associated with those new tourism products. The first objective of this research was to describe the tourists who visit the wellness centres on Gran Canaria, an important Spanish sun and beach destination that has committed to the diversification of its tourism offer by developing the wellness tourism product.

In that regard, one pioneering academic and professional contribution of this work is that it provides empirical evidence about the wellness tourism demand. In particular, it describes the sociodemographic profile of the tourist, the characteristics of his/her trip to Gran Canaria and the frequency of his/her 25 international trips for wellness purposes. Moreover, it shows that such tourists display a high level of spending on wellness services during their stay, which may increase their total expenditure in the destination to above the average of the island's visitors as a whole.

The second objective of this study was to identify the tourist characteristics that contribute to an explanation of the higher expenditure on wellness services during the stay in the destination. That objective has its theoretical basis in the literature on the tourism competitiveness of a region and in the importance of the maximisation of tourism revenues by tourist regions or by regions interested in developing tourism. In that respect, the destination's marketing policy and strategy should focus not only on capturing and satisfying the market segments that are most attractive in terms of and tourist spending during the stay in the destination but also on promoting sustainable development.

The main academic contribution of this study refers to the generation, for the first time in the literature, of an explanatory model of tourist spending on wellness services during the stay in the destination. Moreover, the results also reveal the importance of analysing traveller behaviour in general as a principal criterion to identify the tourists with greatest spending potential during their stay in the destination. Also important are the characteristics of the trip, such as the length of the stay in the destination, the principal motive of the visit and the importance that the tourist attaches to each component of the leisure offer. Thus, while the sociodemographic characteristics of the tourists are the most used, they are not sufficient to give an understanding of spending in the destination. With regard to the professional implications, the results of the work suggest that the managers of the destination and the wellness centres should direct their marketing and communication strategies toward the market segment comprising tourists who regularly take trips for wellness reasons. They should also concentrate their marketing efforts on capturing tourists who visit the destination for wellness motives and who consider the offer of wellness centres important when choosing the destination for their holidays. With a view to reaching these market segments, direct promotions and communications actions (e.g., specific sales promotions and advertising campaigns) aimed at customers of wellness centres could be implemented in major origin markets. These actions could be extended to the members of sports clubs and associations, and any other organisation that might be interested in promoting health and wellbeing amongst their members, including those against some types of disease. Additionally, it would be valuable to work with intermediaries specialising in wellness tourism, including tour operators, travel agencies and on-line intermediaries selling wellness travel.

Moreover, a shorter stay and a longer distance between the place of residence and the destination also contribute to explaining higher expenditure in the destination, possibly because the fixed cost of travelling to the destination leads the tourist to increase his/her spending efforts in order to obtain the greatest benefit from the high fixed cost of transport services. However, tourists who incur lower transport costs and stay a longer time in the destination distribute their spending over the entire holiday and, therefore, their daily spending decreases. For that reason, it would be interesting to design promotions for short stays and trips although that On the other hand, the sociodemographic characteristics, such as gender, marital status, educational level and occupation, do not explain expenditure on wellness services. However, the literature on tourist spending emphasises those variables as possible determinants of spending and, moreover, they constitute criteria for tourism demand segmentation that are quite frequently used by the heads of tourist destinations and the marketing managers of tourism firms. The results obtained in this research confirm that geographical origin, age, and having children are the only sociodemographic characteristics that may explain spending on wellness services. Therefore, another professional implication refers to the appropriateness of designing promotions and communications actions aimed specifically at tourists coming from major international markets, particularly those who are middle-aged or senior citizens and those with children. Wellness packages could be further designed to better satisfy the needs of those market segments. For example, a complementary leisure offer for children could be offered to parents interested in health and wellbeing, who would then be able to take advantage of the wellness services while the children are entertained in a safe environment; moreover, innovative wellness activities for parents and children to enjoy together could be developed. With regard to age, special packages of wellness treatments and With the aim of improving the explanatory model generated in this work, it is recommendable that future research analyses other possible determinants of tourist spending, such as the type of accommodation and how the trip was organised. It would also be interesting to design new studies of wellness expenditure by nationality in order to analyse possible cultural differences that may help explain the higher expenditure on wellness. Moreover, a more detailed analysis could identify the wellness treatments and services with greater potential demand and on which spending could be increased. Finally, similar empirical studies could be designed and applied to other regions and tourist destinations as well as to tourism products other than wellness tourism, such as health tourism, sun, sand and sea tourism, cultural tourism, rural tourism, golf tourism, sports tourism or business and events tourism.

  social benefits of the investments in the development of infrastructures and facilities linked to the wellness tourism product, which sometimes entail the consumption of certain scarce natural resources as well as possible environmental degradation. Therefore, the second objective of this research is to identify the determinants of expenditure on wellness services during the stay in the destination.

  Sociodemographic characteristics. Most works that study the determinants of tourist expenditure refer to sociodemographic characteristics as possible factors that explain the differences in tourist expenditure (e.g.,MEHMETOGLU, 2007;

  the destination involves an increase of only 45.9% in the total tourist expenditure per person. That percentage rises to 62.9% in the case of international visitors to Australia

  However, no empirical works were found that have considered this criterion in the study of tourist expenditure.

  5%) and Italy (0.8%). Other nationalities comprised 5.4% of the foreign visitors to the island.The climate of Gran Canaria is characterised by being spring-like all year round.

  island's wellness offer in the principal tourist origin markets of Europe in coordination with the Tourism Office, which is the organisation responsible for marketing Gran Canaria as a tourism destination.

  to be reduced by eliminating the items that were given the lowest rating by those experts, but also provided an estimation of the capacity and average occupancy of each wellness centre and knowledge of some characteristics of their customers. The new version of the wellness centre customer questionnaire was prepared taking the results of a pretest on 20 visitors to wellness centres into consideration. The final version of the questionnaire comprised three clearly differentiated sections. The first section contained open and closed questions to identify the sociodemographic and classification characteristics of the respondents: place of origin, gender, age, marital status, having children, level of education, and present

  was represented in the total number of completed questionnaire, in accordance with criteria such as the maximum daily capacity of each centre and the average daily number of customers visiting each centre during October and November of the previous two years. Furthermore, in order to guarantee randomness in the selection of customers, as a general guideline, each pair of interviewers interviewed each and every customer of a wellness centre as they exited the centre after receiving wellness services. Interviews were conducted every day of the week in order to avoid any possible related bias. When the data gathering stage had ended, the variables were codified and the data contained in each valid questionnaire were tabulated. The data were recorded directly in the editor of the SPSS program for Windows, the statistical package used in the analyses.

  behaviour as tourists. The high presence of that market segment suggests the importance of offering complementary activities for children.

  a daily average of 29.39 euros on wellness services during their stay on the island. According to figures published by the CANARIAN GOVERNMENT DEPARTMENT OF TOURISM (

  , a dummy variable was used with value of 1 if the motive is wellness tourism, and 0 otherwise. Importance of wellness centres in the choice of destination (X10)

  higher is the daily expenditure on wellness. Thus, hypotheses H8, H9 and H10 are not rejected.

  risk in terms of maintaining the same level of occupancy in the destination's accommodation establishments. In any case, what is important is the maximisation of revenues and margins rather than of the number of tourists.

  offered based on the most common ailments and health problems among the elderly.

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Table 1 )

 1 . The tourists living on other islands in the Canarian Archipelago represented 19.37% of the 989 tourists who visited wellness centres while those from the rest of Spain constituted 16.5% and the remaining 64.13% comprised residents of other European countries. More specifically, the most represented nationalities of the foreign tourists were those of the two large tourist origin markets in Europe: Germany (38.99%) and the United Kingdom (27.52%). The remaining 33.49% came from various countries, particularly from Holland, Ireland, Belgium, Austria and France. The distance from Mainland

	Switzerland,
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Table 2

 2

	, the

Table 1 .

 1 Characteristics of the tourists and the daily expenditure on wellness

	World Tourism Organization, Madrid.

a Parenthesis refers to a variable used as a reference group.

b

The variable is quantitative.

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