Hydraulic and physical properties of stonewool substrates in horticulture

Scientia Horticulturae 104 (2005) 391 405 www.elsevier.com/locate/scihorti Review Hydraulic and physical properties of stonewool substrates in horticulture S. Bougoul a, *, S. Ruy c, F. de Groot d, T. Boulard b a Université de Batna, Laboratoire de Physique Energétique Appliquée, Laboratoire d Etude des systèmes Energétiques Industriels, Rue Chahid Mohamed El Hadi Boukhlouf, Batna, Algeria b INRA-PSH, Domaine St. Paul Site Agroparc, 84914 Avignon Cedex 09, France c INRA-CSE, Domaine St. Paul Site Agroparc, 84914 Avignon Cedex 09, France d Grodania A/S Hovedgaden 501 DK-2460 Hedemusene, Denmark Received 1 August 2004; received in revised form 20 December 2004; accepted 26 January 2005 Abstract Hydraulic conductivity and water content dependence on the substrate suction are certainly among the most crucial physical parameters because they are responsible for water movement and retention in the substrate. Only few data are currently available in the literature and it is the reason why an experimental study was undertaken to characterize more precisely the properties of two slab types manufactured by the Grodan Company, Floriculture 1, a high density rock wool substrate and Expert 1 with a low density one, respectively. Hydraulic conductivity was evaluated by different methods. Saturated conductivity was determined for more than 200 different rock wool substrate types and a model was derived relying on the saturation conductivity and rock wool density for rock wool substrates ranging between 30 and 100 kg/m 3. Saturation conductivity value, measured separately by the INRA (National Institute of Agronomic Research) and manufacturer labs, was found to be much higher than the single value found in the literature for the rock wool type HP. It was also shown that for unsaturated conditions, in the range of the values commonly observed in the cropping conditions (between 0 and 5 cm), the less dense slabs also possess the highest conductivity. For higher suctions, a reverse phenomenon can be observed. Water retention curves, measured with sand tables, were also derived from these measurements for sorption and drying in order to determine the hysteresis properties of the medium. The * Corresponding author. Tel.: +213 33 86 89 75; fax: +213 33 86 89 75. E-mail address: s_bougoul@hotmail.com (S. Bougoul). 0304-4238/$ see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.scienta.2005.01.018

392 S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 experimental results were fitted to the van Genuchten model to identify the model parameters and capillary capacitance from the retention curves. Contrary to the classical description of the hysteresis, a non-closure of the drying sorption cycles could also be observed for the first drying sorption cycles. # 2005 Elsevier B.V. All rights reserved. Keywords: Stonewool; Rockwool; Hydraulic conductivity; Retention curves; Hysteresis Contents 1. Introduction...... 392 2. Material and methods.... 394 2.1. Hydraulic characteristics of stonewool.... 394 2.2. Hydraulic conductivity measurement...... 394 2.3. Water content determination... 395 3. Results...... 395 3.1. Physical properties..... 395 3.2. Saturated hydraulic conductivity determination...... 396 3.3. Saturated hydraulic conductivity variations with respect to stonewool density 396 3.4. Hydraulic conductivity determination..... 396 3.5. Determination of the retention curve...... 398 4. Discussion... 400 4.1. Saturated hydraulic condctivity..... 400 4.2. Hydraulic conductivity in unsaturated conditions..... 400 4.3. Water retention... 401 4.4. The non-closure of the sorption and drying cycles.... 402 5. Conclusion... 403 References... 405 1. Introduction Soil-less culture began to expand rapidly from the 1970s because it helps controlling accurately the environmental factors of the plants and more specifically, meeting its watering and mineral requirements. In addition, it solves the serious pathologic problems caused by soil borne contamination (fusarium, verticillium, nematodes). Soilless cultures are now preponderant in most greenhouse cultures, mainly because they prevent contamination problems and make it possible to obtain substantially higher yields. Greenhouse crop surface in Europe reaches approximately 50,000 ha out of which 20,000 ha are grown on artificial substrates, stonewool being the most popular soil-less medium. The interest for the growing crops on artificial substrate is mainly due to the strong dependence of plant production on environmental conditions, and to the fact that optimum growing conditions are rarely met under open field conditions because of a poor water and mineral availability. In particular, the root system environment being better controlled in soil-less cultures, water and mineral excess and deficiencies

S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 393 (saline stress and root anoxia) can be restricted and more generally, a closer management of water and mineral inputs in time and space is possible with respect to the plant needs during the successive vegetative stages. Most of the soil-less crop substrates are porous and granular media such as perlite and pouzzolane, or fibrous media like rock or stone wools. All are acting as hydraulic and mineral reservoirs and mechanical supports for plants but with physical and chemical characteristics which can strongly vary. The knowledge of the physical properties of these substrates should allow for a better management of water and mineral supply to maintain optimal growth conditions. Stonewool cut into 7.5 cm high slabs is one of the artificial substrates which offer a very large water permeability and the highest water content at low suction. However, the decrease of the substrate water content induces a reduction of the substrate hydraulic conductivity together with an increase of the suction and for frequent uses, its life cycle is limited and its mechanical stability is poor. For avoiding pollution and contamination of the shallow water, the environmental constraints limiting mineral rejects into the environment have led to focus on the recycling of mineral solutions and on artificial substrate in agreement with this technique. Stonewool slabs used as a substrate for ornamental and vegetable crops facilitate the control of the nutrient composition of the solution by means of automated processes and this is why this technique is widely used in combination with recycling systems. As recycling allows for decoupling water and nutrient inputs, it is necessary to know the exact physical and hydraulic properties of stonewool for an accurate supply of nutrients in real time. Despite its increasing use as a soil-less substrate, physical properties of stonewool are poorly documented. The experimental determination of the distribution of nutrients and water in stonewool slabs was investigated by De Rijck and Schrevens (1998) and its modelling was performed by Heinen (1997) and Heinen and de Willigen (1999). However, only one reference is currently available on the determination of the main physical properties of stonewool, a study made by Da Silva et al. (1995) who measured the main physical and hydraulic properties of one type of stonewool (HP type of Grodan). Lack of data on stonewool hydraulic properties is particularly detrimental for the uses of this substrate in the following cases: (i) simulating water and solute flow in the stonewool substrate plant greenhouse atmosphere for managing water and mineral supply; and (ii) characterising new kinds of stonewool with various characteristics (density, isotropy). As stonewool substrate is very permeable and porous and as it is used in nearly unsaturated conditions, classical methods for the measure of hydrodynamic properties may not apply. The aim of this paper is: (i) to investigate the relation between stonewool density (which may easily be controlled during the manufacture of the substrate) and hydraulic conductivity at saturation; and (ii) to test three different methods usually used in soil science (namely Wind evaporation method, Wind infiltration method and DRIP infiltration method) on the two types of stonewool slabs. To provide such information, we characterised the physical and hydraulic properties of two stonewool slab types, Floriculture 1 and Expert 1 manufactured by the Grodan Company and differing mainly in density and orientation of the fibres.

394 S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 2. Material and methods 2.1. Hydraulic characteristics of stonewool Stonewool is a porous medium filled with cavities of various forms in which the fluid is retained by the action of capillarity forces. It is made of fibres of approximately 4 mm diameter and its apparent density varies between 40 and 100 kg/m 3. A large variety of substrates can be produced, differing only by the density of the stonewool slabs produced during the manufacturing process. The Floriculture slab density reaches 0.0675 kg/dm 3 versus 0.0460 kg/dm 3 for the Expert slab. Density can also vary inside the same slab as shown by the density of the top layer of Floriculture slabs which is 30% higher than the lower layers whereas Expert slab density is constant over the height. Because of the fibre orientation, this material is also anisotropic. Thus, the fibre orientation of Floriculture slabs is horizontal whereas it is vertical for Expert slabs. Water availability in the substrate is given by the retention curve relating water content and water suction in the slab. Theoretically, the total potential of the liquid phase in a porous and rigid medium is the sum of not only the retention potential due to the interaction between the liquid and solid phases but also of the gravitational potential with respect to a reference horizon and of an osmotic potential due to the presence of salts and gases in the medium. In this study, we shall only consider the retention and gravitational potentials and neglect the others. Water retention in stonewool is dependent on the hysteresis phenomenon. Thus, for the same suction, the water content of the slab is higher in desorption than in sorption and the van Genuchten model (see Appendix A) is often used to describe this phenomenon. Hydraulic conductivity K can be expressed either with respect to the volume content u or the suction h. As the material shows no detectable volume change in drying, K(u) has a negligible hysteresis whereas K(h) depends strongly on the suction (h). The Mualem van Genuchten model presented in Appendix B is often used to simulate K(h). Likewise the capillary capacity C(h) = du/dh can be deduced from the characteristic water retention curves (see Appendix B). 2.2. Hydraulic conductivity measurement For both stonewool types, the saturated hydraulic conductivity K s is determined by the so-called constant head method (Klute and Dirksen, 1986). In our case, the sample was a 0.15 m diameter cylinder of 0.07 m height. The hydraulic conductivity K(h) was measured using two complementary methods allowing for the determination of the hysteresis properties: the Wind method in evaporation and infiltration, respectively (Wind, 1968). However, as shown by Tamari et al. (1993) and Mohrath et al. (1997), this method does not allow for a precise determination of K(h) near saturation (between 0 and 20 cm of suction) because of its high sensibility to pressure sensor background noise. Complementary measurements in the low suction range were performed in steady state regime by means of the drip infiltration method which imposes both a flux in surface and suction at the base of the sample. The variation of hydraulic potentials within the sample are recorded which makes it possible to calculate the non-saturated hydraulic

S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 395 Fig. 1. Schematic representation used for the determination of hydraulic conductivity in infiltration (nonstationary mode). conductivity K(h). For evaporation, saturated stonewool samples were placed in the cylinders supported by balances (Fig. 1) and the upper surface of the cylinders was submitted to evaporation. For infiltration, we used the same principle as previously, but the sample surface was watered instead of dried. Watering was performed by means of a rain simulator whose flow rate could be precisely adjusted. A variable suction developed in the upper part of the slab whereas a null flux plan was maintained at the base of the substrate. 2.3. Water content determination Water retention in stonewool also depends on hysteresis and must be determined both in sorption and desorption. In the latter case, the retention curve was determined by starting from saturation and by successive measurements of both water content and suction of stonewool samples of 27 cm 3 where increasing suctions were applied. For sorption, the retention curve was determined using the same initially dry samples (hydraulic potential = 100 cm) which were then progressively watered. A sand suction table was used for low suctions and a porous plate extractor coupled with a high pressure gas was used for high suctions. 3. Results 3.1. Physical properties Stonewool is sterile, chemically inert, reacts slightly alkaline (ph 7.5) and contains almost no ions (EC = 50 100 ms/cm) (De Rijck and Schrevens, 1998). Table 1 summarizes the main physical properties (density and porosity) of the two studied stonewool samples, Floriculture and Expert, as well as the density of the stone used as raw material for the substrate.

396 S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 Table 1 Density and porosity of the stone used as basic material and apparent density and porosity of the Floriculture and Expert stonewool slabs Stone Floriculture rockwool slab Expert rockwool slab Density (kg/dm 3 ) 2750 0.0675 0.046 Porosity (%) 0 96.9 97.6 3.2. Saturated hydraulic conductivity determination The variation of water flow rates as a function of hydraulic charge differences for two samples of each type of stonewool can be fitted by a regression line (Table 2). Saturated hydraulic conductivity could be deduced from the Darcy law with K s = 3.06 10 3 and 1.47 10 3 m/s, respectively, for the two samples of Floriculture and 8.38 10 3 and 4.64 10 3 m/s for the two samples of Expert. Only two replications were done and their order of magnitude is similar to the values measured by the manufacturer on more than 200 stonewool samples differing essentially by their density (Fig. 2). On average, the saturated hydraulic conductivity of Expert stonewool was three times higher than that of Floriculture. Yet, it is worth noting that even for a similar stonewool type, saturated hydraulic conductivity can differ by a factor of 2 depending on the sample. These differences can partially be due to the density differences according to the position of the sample in the slab, as it is the case for Floriculture in which the manufacturing process induces a density of the upper part which is theoretically 1.2 times higher than for the lower part, or it can also be due to the measurement method which could overestimate conductivity because of the possible leakages on the circumference of the stonewool cylinder. 3.3. Saturated hydraulic conductivity variations with respect to stonewool density Rock wool density strongly influences the hydraulic properties of the slab. Therefore, saturated hydraulic conductivity variation with respect to stonewool density was systematically studied by means of saturated hydraulic conductivity measurements performed on more than 200 different stonewool samples of different density (40 100 kg/ m 3 ). Fig. 2 shows that the slab density has a remarkable effect on its saturated conductivity, with an important increase of conductivity when the density decreases and conversely. 3.4. Hydraulic conductivity determination For both Floriculture and Expert stonewool types, hydraulic conductivity K(h) was determined with respect to the vertical direction, i.e. following the height of the slab. Table 2 Straight regression line for the experimental determination of hydraulic conductivity at saturation of two stone wool samples First sample Second sample Floriculture y = 0.0023x + 0.146, R 2 = 0.993 y = 0.0059x 0.029, R 2 = 0.993 Expert y = 0.0008x + 0.098, R 2 = 0.993 y = 0.0016x 0.081, R 2 = 0.956

S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 397 Fig. 2. Scatter plot of K s for Grodan products with different densities together with INRA (for two different densities) and Da Silva (for only one density) measurements. Conductivity versus suction for both the steady state and transient regimes in evaporation and watering are given for Floriculture and Expert in Figs. 3 and 4, respectively, together with the corresponding curves fitted to the Mualem van Genuchten model whereas the parameters of the fitted curves are presented in Tables 3 and 4. The results obtained for the different ranges of suction were gathered to get curves with a wider validity (symbol M-VG total in Figs. 3 and 4). The parameters of the curves fitted to the Mualem van Genuchten model are given in Tables 3 and 4. Fig. 3. Hydraulic conductivity K(h) of Floriculture stonewool following the height of the substrate.

398 S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 Fig. 4. Hydraulic conductivity K(h) of Expert stonewool following the height of the substrate. 3.5. Determination of the retention curve Water retention characteristics, u(h), for sorption and drying are given for Floriculture and Expert in Figs. 5 and 6, respectively. These data were fitted to the van Genuchten model (Appendix A) and their fitted parameters (according to Eq. (A.1)) are given in Table 5. Table 3 Floriculture stonewool: hydraulic conductivity parameters adjusted for evaporation, infiltration in transient and steady state regimes Adjustment parameters for the Mualem van Genuchten model Mean K s (m/s) a (m 1 ) n m Wind evaporation 0.00212 36.581 2.1223 0.5288 Wind infiltration 0.00212 0.827 1.0298 0.0288 Drip infiltrometer 0.00212 16.977 2.5202 0.6032 All data 0.00212 15.917 3.0614 0.6733 Table 4 Expert stonewool: hydraulic conductivity parameters adjusted for evaporation, infiltration in transient and steady state regimes Adjustment parameters for the Mualem van Genuchten model Mean K s (m/s) a (m 1 ) n m Wind evaporation 0.006 63.919 1.8664 0.4642 Wind infiltration 0.006 59.656 1.3372 0.2521 Drip infiltrometer 0.006 66.862 2.1120 0.5265 All data 0.006 39.212 2.1811 0.5415

S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 399 Fig. 5. Water retention curves of Floriculture stonewool in sorption and drying. For both the stonewool types, the curves which were determined during sorption are less reliable than those determined during the drying process. This is mainly due to the superposition of the gravity and capillarity forces during sorption and might also be due to the length of time before reaching equilibrium which was much shorter for sorption than for drying. Fig. 6. Water retention curves of Expert stonewool in sorption and drying.

400 S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 Table 5 Expert and Floriculture stonewool slabs: water content parameters adjusted to the van Genuchten model Adjustment parameters of the van Genuchten model u s u r a (cm 1 ) n m Floriculture Sorption 0.975 0.026 1.677 6.029 0.8341 Drying 0.7288 0.0103 0.0726 4.1688 0.760 Expert Sorption 0.983 0.0192 1.94515 3.9857 0.7491 Drying 0.5712 0.0037 0.096 3.2990 0.6968 4. Discussion These results pointed out three main points which deserve to be discussed: (i) the differences between the saturated hydraulic conductivity values of Floriculture and Expert samples and the few values reported in the literature; (ii) the existence of a non-closure of the water content hysteresis curves for a null suction; and (iii) the variations of the hydraulic properties with respect to the apparent stonewool density. 4.1. Saturated hydraulic conductivity The saturated hydraulic conductivity values that we measured for Floriculture and Expert at the INRA (National Institute of Agronomic Research) research center of Avignon, though noticeably smaller, were in the same order of magnitude than those found by the manufacturer Grodan (Fig. 2). Conversely, the average values of 2.25 10 3 m/s measured for Floriculture and 6.5 10 3 m/s for Expert, respectively, were noticeably higher, i.e. 3 times higher for Floriculture and 10 times higher for Expert, than the value found by Da Silva et al. (1995) for the Grodan slab type HP, i.e. 7.14 10 4 m/s. Even if we take into account the effect of density on saturated conductivity (60 kg/m 3 ), the value reported by Da Silva seems to be considerably underestimated compared to INRA and Grodan measurements. If we refer again to Fig. 2, we can also notice the remarkable effect of stonewool density on the saturated conductivity, which is consistent with the other hydraulic properties and can be fitted by a regression line: K s = 24.1 0.45 density + 0.0023 density 2. 4.2. Hydraulic conductivity in unsaturated conditions Several hydraulic conductivity measurement methods were used for sorption and drying. They show (Figs. 3 and 4) that hydraulic conductivity systematically decreases when suction increases. In these figures, no hysteresis can be shown, probably because it is hidden by a strong dispersion of the experimental data in the low suction range due to a high sensor sensibility (Tamari et al., 1993). For low suctions, in the range 0 20 cm, Floriculture and Expert hydraulic conductivities (Fig. 7) are quite lower than that of the HP type measured by Da Silva. This is quite surprising if we

S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 401 Fig. 7. Comparison of the hydraulic conductivity of the three types of stonewool. consider that at saturation, the hydraulic conductivity of HP is much lower than that of Floriculture and Expert. Comparison between Expert and Floriculture also showed that Expert, the less dense slab, logically presents the highest conductivity for the suction values usually observed in the cropping conditions (range: 0 5 cm). Beyond 5 cm, a reverse phenomenon is observed. It is thus likely that near saturation, hydraulic conductivity becomes dependent on the continuity of a few pores presenting a large diameter whereas for dryer conditions, hydraulic conductivity depends on the interconnection of pores which present a much smaller diameter. Thus, the structure of rockwool has a direct effect on the conductivity of the material. Finally, high density slabs like Floriculture have less large pores and much more small pores highly connected than a low density slab like Expert. This explains a very rapid decrease of hydraulic conductivity in the Expert slab with drying in the range of low suctions and a stabilisation of the hydraulic conductivity for high suctions. 4.3. Water retention Water retention curves with respect to suction for the two types of substrate and for the HP type are characterised by a suction rise up to 50 cm, when water content decreases and tends towards 0. HP and Floriculture curves adjusted to the van Genuchten function (Fig. 8) highlight a strong hysteresis characterised by the large differences in water retention for a same suction in drying and sorption, respectively. For drying conditions, except near

402 S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 Fig. 8. Comparison of the water retention curves for Floriculture and Grodan HP stonewool for sorption and drying. saturation, HP water content is much lower than that of Floriculture whereas the opposite is observed for sorption. When comparing Expert and HP slabs (Fig. 9) under drying conditions, both the slab types have a very similar behaviour for suctions higher than 10 cm. However, at low suctions, as those observed under cultural conditions (<10 cm), Expert water content is much lower than that of HP and a strong hysteresis between drying and sorption is observed as previously. For a suction equal to 7.5 cm corresponding to the slab height, water content reaches 70% for Floriculture versus 50% and 47% for HP and Expert, respectively. In summary, Floriculture slabs are denser and consequently present greater water content whereas Expert slabs are less dense and retain less water in the range of the suctions commonly observed in cultural conditions. 4.4. The non-closure of the sorption and drying cycles At saturation (h = 0), Figs. 8 and 9 show clearly that the water content of the HP slab is the same after either sorption or drying whereas for the measurements made by the manufacturer, it can be considerably different for the Expert slab type and slightly different for Floriculture. This is the so-called phenomenon of non-closure of the sorption and drying cycles. This behaviour can partly be attributed to the measurement difficulties with tensiometers in the low suction range. However, as this phenomenon is generally reported by the manufacturer for stonewool slabs with densities lower than 55 kg/m 3 and as it is not stable over time and can disappear after several sorption drying cycles, it can

S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 403 Fig. 9. Comparison of the water retention curves for Expert and Grodan HP stonewool for sorption and drying. be attributed to other causes such as the presence of a wetting agent used to impregnate the stone fibres. 5. Conclusion We studied the physical properties of two stonewool slab types, focusing on their hydraulic properties. Measurements of the saturated hydraulic conductivity performed by two different laboratories (INRA, France; Grodan, Denmark) using different methods give results of the same order of magnitude. However, the found values were not in agreement with the only value reported in the literature without possible explanation for this difference. Our results show that the saturated hydraulic conductivity of the slab strongly depends on its density with a strong decrease of conductivity as density increases. Based on a large number of measurements, an experimental relationship was found between rock wool density and saturated hydraulic conductivity for rock wool densities ranging between 30 and 100 kg/m 3. In non-saturated conditions, hydraulic conductivity decreases more quickly in less dense slabs while suction increases. For the same suction, a strong difference in water content between sorption and drying was also evidenced together with a non-closure of the hysteresis curves for less dense slabs. As stonewool substrate is used very close to the saturation (typically with a suction between 0 and 20 cm), the measurements of the hydraulic conductivity show a large dispersion due to the experimental noise of the pressure transducers. Using the tested methods (Wind evaporation, Wind infiltration, DRIP infiltration), on such a substrate, may be possible only with a very good calibration of the pressure transducers.

404 S. Bougoul et al. / Scientia Horticulturae 104 (2005) 391 405 Determining water content and hydraulic conductivity as a function of suction made it possible to obtain an accurate estimation on water retention by the substrate and on the possibilities for the root system to exploit it. It was shown that when suction increases beyond 10 cm, hydraulic conductivity decreases quickly. This is particularly worrying under sorption conditions because the low water content is then insufficient to allow for a sufficient water supply in response to atmospheric demand. To prevent such stress conditions, excessive drying of stonewool must be avoided to keep both the hydraulic conductivity and water content at rather high levels. In particular, this can be achieved by increasing the irrigation frequency. More generally, a precise knowledge of saturated and non-saturated hydraulic conductivity and retention curves versus suction should facilitate the fertirrigation management. It can also help improve the design of the slabs by combining different stonewool qualities. Such composite substrates should allow for a more complete exploitation of the substrate volume by the roots and for reducing the time constant of the changes of nutrient solution composition at root level. These physical characteristics can also be considered for parameterisation of numerical models used to simulate water and mineral transfers into the substrate and for a modelassisted optimisation of the design of the stonewool slabs. Appendix A The Van Genuchten (1980) model of the retention curve: 8 uðhþ u < 1 r ¼ ð1 þjahj u s u n Þ m ; h 0 (A.1) r : 1; h > 0 where u s, u r are the volume contents at saturation and residuals and a, n, m are the form parameters of the u(h) curve, where m =1 ( 1 n ). Appendix B The Mualem van Genuchten model (Celia et al., 1990; Rathfelder and Abriola, 1994) for hydraulic conductivity: 8 ð1 ðajhjþ n 1 ð1 þðajhjþ n Þ m 2 >< Þ KðhÞ ¼K s ð1 þðajhjþ n Þ m=2 ; h 0 (B.2) >: 1; h > 0 where K s is the saturated hydraulic conductivity with m =1 1/n. The capillary capacity C(h) = du/dh, deduced from the characteristic curve of water, is given by (Heinen, 1997): CðhÞ ¼ ðu s u r Þnma n jhj n 1 ð1 þjahj n Þ 1 m ; h 0 (B.3) 0; h > 0

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