sm80s1-4297

Planet Ocean
J.L. Pelegrí and D. Vaqué (eds)

Artificial upwelling using offshore wind energy for mariculture applications

Álvaro Viúdez, Marta Fernández-Pedrera Balsells, Rocío Rodríguez-Marroyo

Department of Physical Oceanography and Technology, Institut de Ciències del Mar, CSIC, P. Marítim de la Barceloneta 37-49, Barcelona, Spain. E-mail: aviudez@cmima.csic.es, martafpb@gmail.com, rrodriguez@icm.csic.es

Summary: Offshore wind is proposed as an energy source to upwell nutrient-rich deep water to the ocean photic layers. A spar-buoy wind turbine with a rigid tube about 300 m long is proposed as a pipe to drive deep water up to the surface. The minimum energy required to uplift the water is the potential energy difference between surface waters inside and outside the pipe, which depends on the background density profile. The corresponding surface jump or hydraulic head, h, calculated for several analytical and experimental density profiles, is of the order of 10 cm. If the complete turbine power (of the order of several MW) is used for raising the water (assuming a 100% pump efficiency), in a frictionless flow, very large water volumes, of the order of thousands of m3 s−1, will be transported to the photic layers. In a more realistic case, taking into account pipe friction in wide pipes, of the order of 10 m radius, and a power delivered to the fluid of 1 MW, the volume transport is still very large, about 500 m3 s−1. However, such a large amount of dense water could sink fast to aphotic layers due to vertical static instability (the fountain effect), ruining the enhancement of primary production. Hence, some ways to increase the turbulent entrainment and avoid the fountain effect are proposed. From the energetic viewpoint, artificial upwelling using offshore wind energy is a promising way to fertilize large open sea regions. This mariculture application is, however, severely subjected to atmosphere and ocean climatology, as well as to ecological dynamics. The general problem is multidisciplinary, and some important physical, engineering and ecological questions need to be seriously addressed to improve our confidence in the approach presented here.

Keywords: artificial upwelling; mariculture applications; ocean fertilization; offshore wind energy; spar-buoy wind turbine.

Afloramiento artificial producido con energía eólica con aplicación a la maricultura

Resumen: Analizamos el uso de la energía eólica marina como fuente de energía para aflorar aguas profundas ricas en nutrientes a las capas fóticas del océano. Una turbina de viento tipo boya-pértiga, con un tubo rígido de unos 300 m de largo, se propone para transportar las aguas profundas hasta la superficie. La energía mínima necesaria para elevar el agua es la diferencia de energa potencial entre las aguas superficiales dentro y fuera de la tubería, que depende del perfil de densidad de fondo. El salto superficial de agua, o cabezal hidráulico h, calculado para varios perfiles analíticos y experimentales de densidad, resulta ser del orden de 10 cm. Si la potencia total de la turbina (del orden de varios MW) se utiliza para elevar el agua (suponiendo una eficiencia de la bomba del 100%), en un flujo sin fricción, el transporte de volumen de agua transportado a las capas fóticas es muy elevado, del orden de miles de m3 s−1. En un caso más realista, teniendo en cuenta la fricción en tuberías de un ancho del orden de 10 m radio, y una potencia proporcionada al fluido de 1 MW, el transporte volumen sigue siendo muy grande, de alrededor de 500 m3 s−1. Sin embargo, una cantidad tan grande de agua densa podría hundirse rápidamente a las capas afóticas debido a la inestabilidad estática vertical (efecto fuente) arruinando la mejora de la producción primaria. Por lo tanto se proponen algunas maneras de aumentar el arrastre turbulento y evitar el efecto fuente. Desde el punto de vista energético, el afloramiento artificial utilizando energía eólica marina parece una manera prometedora de fertilización de grandes regiones del mar abierto. Esta aplicación de maricultura, sin embargo, depende severamente de la climatología atmosférica y océanica, así como de la dinámica ecológica. El problema global es multidisciplinar, y algunos aspectos importantes de física, ingeniería, y ecología tienen que ser mejor estudiados para poder aumentar nuestra confianza en el método aquí presentado.

Palabras clave: afloramiento artificial; aplicaciones a la maricultura; fertilización oceánica; energía eólica marina; turbina eólica tipo boya-pértiga.

Citation/Como citar este artículo: Viúdez A., Fernández-Pedrera Balsells M., Rodríguez-Marroyo R. 2016. Artificial upwelling using offshore wind energy for mariculture applications. Sci. Mar. 80S1: 235-248. doi: http://dx.doi.org/10.3989/scimar.04297.06B

Editor: J.L. Pelegrí.

Received: June 16, 2015. Accepted: October 14, 2015. Published: September 30, 2016.

Copyright: © 2016 CSIC. This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-by) Spain 3.0 License.

Contents

Summary
Resumen
Introduction
The surface jump h
Loss of power due to the increase in kinetic energy and energy dissipation in the pipe
Horizontal pressure jump
Power budget considering power consumption required for water sprinkling
Nutrients transport
Discussion and concluding remarks
Acknowledgements
References

INTRODUCTIONTop

Upwelling of deep water, caused by surface wind stress along the coast, occurs naturally in some oceanic regions and favours high primary production. High primary production regions require both large solar radiation input and a high inorganic nutrient content. The above two requirements seldom occur naturally, and either solar radiation or nutrient concentration becomes a limiting factor inhibiting primary production. High solar radiation and nutrient availability occur simultaneously in surface tropical and subtropical ocean regions where natural upwelling along coastal waters occurs. The scarcity of these natural upwelling regions in the whole ocean and the need to obtain fish resources led to the start of aquaculture activities, first on land and, more recently, in coastal marine environments (Costa-Pierce 2002Costa-Pierce B.A. 2002. Ecological Aquaculture: The Evolution of the Blue Revolution. Blackwell Science, 382 pp.). However, these activities do not seem to be sufficient to satisfy the fish requirements of an increasing world population (FAO 2010FAO. 2010. The state of the world fisheries and aquaculture. Rome: Food and Agriculture Organization of the United Nations.). The message that comes out most clearly from FAO (2010)FAO. 2010. The state of the world fisheries and aquaculture. Rome: Food and Agriculture Organization of the United Nations. is the concern that the world fisheries are on a dangerous course (Pauly and Froese 2012Pauly D., Froese R. 2012. Comments on FAO’s State of Fisheries and Aquaculture, or ‘SOFIA 2010’. Mar. Pol. 36: 746-752.).

Scientists are now looking for ways to avoid these limiting factors and fertilize the ocean artificially at a large spatial scale (UNESCO 2010UNESCO 2010. Ocean fertilization. A scientific summary for policy makers. 17 pp.). The activity is known as mariculture (e.g. McKinley and Takahashi 1991McKinley K.R., Takahashi P.K. 1991. Deep ocean water, artificial upwelling, and open ocean mariculture: A promise for the future. IEEE 1: 195-199.), in close relation to aquaculture, which refers to similar fish production activities carried out at smaller spatial scales in land factories. Several attempts at ocean fertilization have been tested. Experiments based on the addition of iron or phosphorus to ocean sites seem to have been concluded negatively (de Baar et al. 2008de Baar H.J.W., Gerringa L.J.A., Laan P., et al. 2008. Efficiency of carbon removal per added iron in ocean iron fertilization. Mar. Ecol. Prog. Ser. 364: 269-282. , UNESCO 2010UNESCO 2010. Ocean fertilization. A scientific summary for policy makers. 17 pp.). Alternative ways to fertilize the ocean are based on the artificial upwelling of nutrient-rich deep waters. This paper proposes an energetically efficient mechanism to upwell deep waters using offshore wind energy.

Several artificial upwelling mechanisms have already been proposed. One of them is to cause upwelling using surface wave energy (Kirke 2003Kirke B. 2003. Enhancing fish stocks with wave-powered artificial upwelling. Ocean Coast. Manag. 46: 901–915.). Wave-powered inertia pump theory (Liu 1999Liu C. 1999. Research on artificial upwelling and mixing at the University of Hawaii at Manoa. IOA Newslett. 10: 1-8.), using a simple pump design originally conceived by Isaacs et al. (1976)Isaacs J.D., Castel D., Wick G.L., 1976. Utilization of the energy in ocean waves. Ocean Eng. 3: 175-187., predicts a volume transport of 0.45 m3 s1 from a depth of 300 m employing a 1.2-m diameter rigid tube. Using a “Bristol Cylinder” (Davis 1990Davis J. 1990. Wave energy absorption by the Bristol Cylinder-linear and non-linear effects. Proc. Inst. Civil Eng. 89: 317-340.) of 21.5 m diameter and 40 m length, Kirke (2003)Kirke B. 2003. Enhancing fish stocks with wave-powered artificial upwelling. Ocean Coast. Manag. 46: 901–915. estimated that a power of 1328 kW could be obtained in 1.9-m Hawaiian swells and pointed out that such a power would be able to raise 50 m3 s1 of water from a depth of 500 m. A recent attempt (White et al. 2010White A., Björkman K., Grabowski E., et al. 2010. An open ocean trial of controlled upwelling using wave pump technology. J. Atmos. Ocean. Tech. 27: 385-396.) used an Isaacs wave pump mechanism consisting of a 300-m-long flexible tube with a one-way valve at the deeper end and the upper end attached by a 15-m cable to a surface buoy. However, this experiment was interrupted early due to material failure caused by the vertical shear and stress of water currents on the tube’s flexible material (Kithil 2011Kithil P.W. 2011. Comments on “An open ocean trial of controlled upwelling using wave pump technology”. J. Atmos. Ocea. Tech., 28: 847-849. , White et al. 2011White A.E., Letelier R.M., Björkman K.M., et al. 2011. Reply. J. Atmos. Ocean. Tech. 28: 850-851.). Horizontal pressure differences inside and outside the tube may also have played a role in the material failure. A more complex and robust design was proposed by Salter (2009)Salter S.H. 2009. A 20 GW thermal 300-metre3/sec wave-energised, surge-mode nutrient pump for removing atmospheric carbon dioxide, increasing fish stocks and suppressing hurricanes. Proceedings of the 8th European Wave and Tidal Energy Conference, Uppsala, Sweden.. This device uses sea wave energy to pump surface warm water downwards through a down-tube reaching the thermocline. The warm water will mix with cold, nutrient-rich water, rise to the neutral density layers, and spread sideways. For a wave climate of typical trade-wind sites, and a design including a 100-m-diameter, 200-m-long tube could generate a volumetric flow rate of about 300 m3 s1.

An upwelling mechanism based on the particular temperature and salinity vertical gradients of some tropical and subtropical ocean regions was proposed by Stommel et al. (1956)Stommel H., Arons A.B., Blanchard D. 1956. An oceanographical curiosity: the perpetual salt fountain. Deep-Sea Res. 3: 152-153.. In these regions the temperature and salinity of sea water are higher in surface layers than in deep layers. The density profile remains statically stable due to the large temperature differences, thus counteracting the effect of salinity gradients on the density profile. The mechanism, named the perpetual salt fountain, consists in initially raising cold and low-salinity water along a vertical tube that allows heat exchange. As water raises diabatically, its temperature increases due to the warmer environmental water outside the tube. Since the low salinity remains unchanged during the ascent, the water becomes lighter than the environmental water at every depth z, resulting in a perpetual flux of deep water at the upper end of the tube, close to the sea surface. Due to its low efficiency, the perpetual salt fountain was initially considered only an “oceanographic curiosity” (Stommel et al. 1956Stommel H., Arons A.B., Blanchard D. 1956. An oceanographical curiosity: the perpetual salt fountain. Deep-Sea Res. 3: 152-153., Groves 1958Groves G.W. 1958. Flow estimate for the perpetual salt fountain. Deep Sea Res. 5: 209-214.). However, this mechanism was recently tested in the ocean (Tsubaki et al. 2007Tsubaki K., Maruyama S., Komiya A., et al. 2007. Continuous measurement of an artificial upwelling of deep sea water induced by the perpetual salt fountain. Deep-Sea Res. I 54: 75-84., Maruyama et al. 2011Maruyama S., Yabuki T., Sato T., et al. 2011. Evidences of increasing primary production in the ocean by Stommel’s perpetual salt fountain. Deep-Sea Res. I 58: 567-574.) following an earlier experiment by Maruyama et al. (2004)Maruyama S., Tsubaki K., Taira K., et al. 2004. Artificial upwelling of deep seawater using the perpetual salt fountain for cultivation of ocean desert. J. Oceanogr. 60: 563-568. using a 300-m-long, 0.3-m-diameter tube whose upper end was located at about 60 m depth. The estimated volume transport was approximately 5.2×104 m3 s1 ≃ 45 m3 day1.

Artificial upwelling has also been proposed and developed in Norwegian fjords. Upwelling in fjords has been used to prevent the development of toxic dinoflagellates in the upper brackish layer during summer and to favour toxin-free, high-quality mussels (Berntsen et al. 2002Berntsen J., Aksnes D.L., Foldvik A. 2002. Production enhancement by artificial upwelling: a simulation study. Hydrobiology 484: 177-190., Aure et al. 2007Aure J., Strand Ø., Erga S.R., et al. 2007. Primary production enhancement by artificial upwelling in a western Norwegian fjord. Mar. Ecol. Prog. Ser. 352: 39-52., McClimans et al. 2010McClimans T.A., Handå H., Fredheim A., et al., 2010. Controlled artificial upwelling in a fjord to stimulate non-toxic algae. Aquac. Eng. 42: 140-147. ). In one case, artificial upwelling was caused by forcing down through a vertical pipe (27 m long and 1.25 m in diameter) surface brackish water using an electrical pump (60 kW), which produced a downward volume transport of about 2 m3 s1 (Aure et al. 2007Aure J., Strand Ø., Erga S.R., et al. 2007. Primary production enhancement by artificial upwelling in a western Norwegian fjord. Mar. Ecol. Prog. Ser. 352: 39-52.). Artificial upwelling starts outside the pipe due to the turbulent entrainment of the ascending buoyant plume with the environmental nutrient-rich deep water. In another case, artificial upwelling was caused by a similar vertical plume but forced this time by a 100-m-long air bubble curtain submerged at about 40 m depth (McClimans et al. 2010McClimans T.A., Handå H., Fredheim A., et al., 2010. Controlled artificial upwelling in a fjord to stimulate non-toxic algae. Aquac. Eng. 42: 140-147. ). The estimated vertical flux of deep seawater was 65 m3 s1. A third approach to force the upwelling of the unstable buoyant plume (McClimans et al. 2010McClimans T.A., Handå H., Fredheim A., et al., 2010. Controlled artificial upwelling in a fjord to stimulate non-toxic algae. Aquac. Eng. 42: 140-147. ) used a 26 m3 s1 discharged of freshwater at 40 m depth from a hydropower plant. A diffuser plate at the lower end of the pipe was employed to increase turbulent entrainment. In this case the field experiment showed an entrainment of 117 m3 s1 of nutrient-rich deep seawater. These experiments show an increase in nutrients in the euphotic zone resulting in an increase in phytoplankton and a decrease in toxic algae.

Related also to the upwelling mechanism proposed here is the Japanese TAKUMI project, which was set up in Sagami Bay, Japan in 2003, a prototype of the Ocean Nutrient Enhancer (Ouchi et al. 1998Ouchi K., Yamatogi T., Kobayashi K., et al. 1998. Density Current Generator—A new concept machine for agitating and upwelling a stratified water. Ocean Community Conference, 1, Baltimore, MD, USA, The Marine Technology Society, 129-136.). This device consists of a submersed spar-type floating structure and a steel riser pipe that upwells dense water from 200 m depth and mixes it with light surface water so that the mixed water spreads at its neutral buoyancy depth within the euphotic layer. The resulting volume transport is about 105 m3 day1≃1.2 m3 s1 (Ouchi 2007Ouchi K. 2007. Three years operation of ocean nutrient enhancer TAKUMI in Sagami Bay. Seventh ISOPE Ocean Mining Symposium, 1-6 July, Lisbon, Portugal, International Society of Offshore and Polar Engineers.). The source of energy of this Ocean Nutrient Enhancer is a diesel generator of about 100 Kw, which makes it unsustainable, though other design possibilities based on ocean thermal energy could overcome this problem (Otsuka 2011Otsuka K. 2011. Inclusive Impact Index Triple I and its application for Ocean Nutrient Enhancer. Proceedings of the Twenty-first International Offshore and Polar Engineering Conference, International Society of Offshore and Polar Engineers.).

In this article we propose the use of offshore wind as the power source for artificial upwelling. Wind turbine farms are stably operating on land and coastal areas. However, offshore wind energy (Henderson et al. 2003Henderson A.R., Morgan C., Smith B., et al. 2003. Offshore wind energy in Europe – a review of the state-of-the-art. Wind Energ. 6: 35-52.) is being seriously considered as an emerging sustainable complementary source of energy. From the first offshore wind turbine deployed in 1990, cumulative installed power reached 1500 MW in 2009 mainly in northwest European waters (Zaaijer 2009Zaaijer M.B. 2009. Review of knowledge development for the design of offshore wind energy technology. Wind Energ. 12: 411-430.). In the first six months of 2015, 584 commercial offshore wind turbines with a combined capacity totalling 2343 MW were fully grid connected in Europe. Once the 15 commercial wind farms under construction have been completed, the total capacity will be about 4268 MW (EWEA 2015EWEA. 2015. The European offshore wind industry - key trends and statistics 1st half 2015. Tech. rep. European Wind Energy Association.). Offshore wind farms have a higher cost in construction, deployment and maintenance than those on land or in coastal areas. However, the possibility of installing a large number of turbines with a minimum visual impact and the persistence of strong winds in some ocean regions make offshore wind farms an attractive solution to the electric energy demand. Interest in offshore wind farms may increase considering the possibility of using these turbines to favour ocean primary production.

The design proposals for offshore floating wind turbines depend on the way the turbine is stabilized (Jonkman 2009Jonkman J.M. 2009. Dynamics of offshore floating wind turbines – model development and verification. Wind Energ. 12: 459-492., Jonkman and Matha 2011Jonkman J.M., Matha D. 2011. Dynamics of offshore floating wind turbines – analysis of three concepts. Wind Energ. 14: 557-569. ). One of them, the spar-buoy floating wind turbine, incorporates a long vertical tube (typically 150 m long) submerged below the turbine’s tower that deepens the centre of buoyancy and stabilizes the turbine structure (Roddier and Weinstein 2010Roddier D., Weinstein J. 2010. Floating wind turbines. Mech. Eng. 132: 28-32., Viré 2012Viré A. 2012. How to float a wind turbine. Rev. Environ. Sci. Biotechnol. 11: 223-226.). It seems possible to combine a spar-buoy floating wind turbine with a wave energy converter, with the shape of a torus attached at the basis of the turbine’s tower so that wind and wave energy are simultaneously obtained at the same location (Muliawan et al. 2013Muliawan M.J., Karimirad M., Moan T. 2013. Dynamic response and power performance of a combined Spar-type floating wind turbine and coaxial floating wave energy converter. Renew. Energy 50: 47-57.). The idea that we explore in this work is to use the power of a spar-buoy turbine to raise deep water through an extended spar converted into a long (say 300 m deep) hollow tube (Fig. 1).

sm4297fig1.jpg

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Fig. 1. – Schematic of the spar-buoy turbine upwelling device illustrating its essential (not to scale) parts. Labels indicate the negative buoyant plume (p), the pipe’s bottom depth (H), neutral height (HN), and diffuser platform (d). White arrows indicate water flow and the grey shade qualitatively indicates nutrient concentration.

Dense deep water (with density ρb) inside a vertical submerged hollow tube open at the bottom and top ends will sink and, in the state of rest, will cause a negative hydraulic jump (say h) at the free surface, where pressure equals the atmospheric pressure, because the water column inside the tube is heavier than that outside. The amount of energy E required to pump a mass M of deep water to the surface is therefore the potential energy E=Mgh, where g is the acceleration due to gravity. The power Pw of the wind turbine and the upward volume transport Fv are therefore related through Pw=Fvρbgh, or Fv=Pw/(ρbgh), which provides the upwelling transport through the tube as a function of the turbine’s power Pw and the surface hydraulic jump h. In the next section we calculate the surface jump h, using several background density vertical profiles ρ(z) and therefore provide an estimate of the upwelling transport. It is found that the order of magnitude of h is only 0.1 m, so, due to the large power of wind turbines (typically Pw=2 MW) the volume transport could be very large (of the order of 103 m3 s1). With such a large volume transport, pipe friction can only be neglected if very wide pipes, of the order of 15 m radius, are used. For thinner pipes, the head loss hf becomes as important as the hydraulic head and needs to be taken into account. Still, considering pipe friction, the volume transport is very large, of the order of 500 m3 s1. Using the turbulent entrainment theory of buoyant plumes it is found that once the dense water has upwelled to the sea surface, it will sink to a neutral height which, depending on the relative depths of the photic layer and thermocline layers, will usually be located below the photic layer, thus making the mechanism useless for primary production enhancement. To avoid this problem, some form of water dilution at the surface must be introduced. Water sprinkling, as a way to increase turbulent entrainment and water aeration (as a fountain) is considered, but this process is very energy-consuming. Other mechanical ways requiring no energy consumption, for example introducing a diffuser plate or multiport diffusers, as used for ocean outfall dilution in reverse osmosis desalination plants to reject brine, are also possible. We conclude that, though several important physical, ecological, and engineering questions must be addressed before its effectiveness is proven, artificial upwelling using offshore wind power is a promising approach for enhancing ocean primary production on a large scale.

THE SURFACE JUMP hTop

In this section we calculate the free surface height difference, or hydraulic head, h, between the water column inside and outside the tube for a number of density profiles. To do so we assume hydrostatic balance,

p z =gρ ,

(1)

where p(z) is the pressure field, g>0 is the acceleration due to gravity, ρ(z) is the density field and z is the vertical coordinate. We further assume that, in the steady state, the horizontal pressure gradient at the lower base of the tube is zero, so there is no horizontal flow acceleration in the exterior fluid as it enters the tube. Therefore, at the tube’s bottom depth, z=H (subscript b), the interior pressure pi(H)=pib equals the exterior pressure pe(H)=peb below the pump. At the free surface (top, subscript t) interior and exterior pressures are equal to the atmospheric pressure pa, that is, pe(0)=pi(h)=pa. We assume that in the steady state the water inside the tube is homogeneous and has a density equal to the bottom density ρ(H). We therefore neglect density changes as the fluid ascends along the tube. Under these assumptions integration of the hydrostatic condition (1) leads to

ρ( H )( hH )= H 0 ρ( z ) dz ,

(2)

which means that water masses per unit of horizontal area of the water column inside and outside the tube are equal. From (2) we obtain the solution for the jump h,

h=H+ 1 ρ( H ) H 0 ρ( z ) dz ,

(3)

which depends on the length of the tube and on the vertical distribution of background density, ρ(z). Notice that H<0, and since 0<ρ(z)≤ρ(H) for z∈[H, 0], we have Hh<0. We describe next three different solutions for h depending on three different simplified vertical density profiles, ρ(z), and as a fourth case we consider an experimental density profile.

Theoretical models of stratification

Two-layer model

The simplest case is that of a two-layer model (see Fig. 2). In the upper layer, where the tube is located, and in the lower layer, below the tube, densities are homogeneous and equal to ρt and ρb, respectively. In this case (see Appendix A) the solution for h is:

h 2 =H ρ b ρ t ρ b .

(4)

Thus, in the simplest case h2 depends linearly both on H and on the density difference ρbρt. We note that, as expected, h2=0 when H=0 and/or ρb=ρt. Assuming values of H=−300 m, ρb=1029 kg m3, and ρt=1026 kg m3, typical of the western Mediterranean Sea in the summer season, we obtain h2≃−87 cm. Notice that this profile is an extreme case in the sense that it maximizes the jump |h|. An upper density layer deeper than the tube’s depth will cause a zero jump, while a shallower layer will cause the weight difference between exterior and interior water column decrease, and therefore will cause a smaller |h|.

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Fig. 2. – The different theoretical density profiles used to compute the surface jump h: the two-layer density profile ρ2, the linear density profile ρl, and the arctangent density profile ρa as a function of depth z. In this schematic the density boundary values are ρt= 1025.5 kg m3 and ρb= 1029.1 kg m3.

Linear density profile

This is the simplest case of a continuously stratified ocean, in which density ρl is assumed to be linearly dependent with depth, that is, ρl(z)≡ρt+ϑ0z, where ρt=ρl(0) is the surface density, and ϑ0<0 is a constant density stratification (see Fig. 2). Following the approach described above (see Appendix B) we obtain the surface jump hl,

h 1 = ϑ 0 H 2 2( ρ t + ϑ 0 H ) = H 2 ρ b ρ t ρ b = h 2 2 .

(5)

The jump hl=h2/2 because the mass of the background water column (per unit area) in the linear density profile is half that in the two-layer profile. Thus, for the same typical values used previously, we obtain the jump hl=h2/2≃−44 cm.

Arctan density profile

Here we consider a density profile ρa(z) that is more realistic than the previous ones in the sense that it reproduces the ocean pycnocline. This profile may be modelled using the arctangent function

ρa(z)= c1 + c2 arctan(c3 + c4z) ,

(6)

where c1, c2, c3, and c4 are given constants used to match the ρa asymptotes to the boundary values ρt and ρb, and to adjust the pycnocline depth (see Fig. 2 and Appendix C). Integrating (3) we obtain

ρ b ( h a H )= c 2 c 4 [ c 3 arctan( c 3 ) 1 2 ln( c 3 2 +1 ) ] c 1 H c 2 c 4 ( c 3 + c 4 H )arctan( c 3 + c 4 H )+ + c 2 2 c 4 ln[ ( c 3 + c 4 H ) 2 +1 ]ξ( H ),

(7)

where ρb(haH) and ξ(H) stand for the masses per unit area of the water column inside and outside the tube, respectively. Solving for ha we get

h a =H+ ξ( H ) ρ b .

(8)

For the same typical values in the western Mediterranean used previously we obtain ha≃−18 cm, which has a magnitude smaller than the previous |h2|=87 cm and |hl|=44 cm.

Experimental density data

Finally, we obtain the water jump he using experimental in situ density profiles ρj (t, zi) representative of the Catalan Sea (western Mediterranean). No offshore wind farm has yet been erected in the Mediterranean, but some studies have already addressed the suitability of climatological Mediterranean offshore winds for profitable energy applications (Lavagnini et al. 2006Lavagnini A., Sempreviva A.M., Transerici C., et al. 2006. Offshore wind climatology over the Mediterranean Basin. Wind Energ. 9: 251-266.). The profiles were spatially averaged to produce a representative profile of the region, and time-averaged to produce a representative profile of each season, ρ ¯ A ( z i ) from January to March, ρ ¯ B ( z i ) from April to June, ρ ¯ C ( z i ) from July to September, and ρ ¯ D ( z i ) from October to December. The sampling depths are zi={−0, −10, −20, −30, −50, −75, −100, −125, −150, −200, −250, −300} m , i = 1, . . . , N = 12. These mean density profiles are plotted in Figure 3. In order to compute he for the four mean density profiles (subscript e={A, B, C, D}), we use the trapezoidal rule to numerically integrate (3), that is,

h e =H+ 1 ρ ¯ e ( N ) i=1 N1 ρ ¯ e ( z i )+ ρ ¯ e ( z i +1 ) 2 Δ z i ,

(9)

where the vertical grid size is non-uniform ∆zizizi+1. Using (9) and the data ρe (Fig. 3), we obtain the jumps hA=−4.9 cm, hB=−8.1 cm, hC=−16 cm, and hD=−12 cm, corresponding approximately to the four seasons.

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Fig. 3. – Seasonal mean density profiles σ(z) (thick lines) in the Catalan Sea, where sigma σ(z) ≡ρ(z)−103 kg m−3. Labels stand for the time periods January-March (A), April-June (B), July-September (C), and October-December (D). The spatial average comprises a total of five hydrographic stations while the time average covers a time period of about three years. Thin lines represent σ(z) plus one standard deviation.

In winter the reduced solar radiation causes a cooling of the upper ocean and therefore a vertical homogenization of the water (ρA in Fig. 3). The surface jump |hA|=4.9 cm therefore reaches a minimum value. In spring the surface layer becomes warmer and therefore lighter, so vertical stratification increases. As a consequence, the surface jump increases to |hB|=8.1 cm. In summer, with the pycnocline completely developed, vertical stratification reaches maximum values, and ρC becomes more similar to the arctangent profile ρa (Fig. 2). The surface jump increases to |hC|=16 cm and approaches the jump size of the arctangent profile |ha|≃ 18 cm. In autumn, surface cooling decreases vertical stratification and therefore the surface jump decreases to |hD| = 12 cm. The cases described above show that the surface water jump h is only of the order of a few centimetres. This means that very large water volumes inside the tube, of the order of thousands of m3 s1, will be transported up if the complete turbine power (also assuming a 100% pump efficiency) is used to raise the water. Such a huge amount of dense water on the surface runs a risk of sinking fast to aphotic layers due to vertical static instability (the so called fountain effect), ruining the enhancement of primary production. This possibility is analysed in a section below. We notice that the power required to raise the deep water is the potential energy associated with the water surface jump h only in the steady state, in which the vertical velocity inside the tube is steady. However, the water volume is initially forced upward from a state of rest, and therefore part of the wind energy must be used to increase the kinetic energy of the water inside the tube. This is a transient effect that, though interesting from the energy balance, is not addressed in this study, which only considers steady state energy balances. We have also assumed that the increase in kinetic energy is negligible and that the water withdrawn within the vertical tube is instantaneously replaced with deep water to keep the hydrostatic balance. This assumes that friction effects in the pipe are negligible, which in turn assumes, given the large volume flux involved, a very large pipe radius, R, so the vertical velocity inside the pipe is small. In the next section we consider the loss of power due to the increase in kinetic energy and the loss of power due to friction, which depend on the size of the pipe radius R.

LOSS OF POWER DUE TO THE INCREASE IN KINETIC ENERGY AND ENERGY DISSIPATION IN THE PIPETop

The loss of power due to the increase in kinetic energy is simply related to the velocity head hu=u2/(2g)= Fv2/(2gπ2R4). We obtain the loss of power due to friction in the pipe, related to the head loss, hf in hydraulic engineering, using the Darcy-Weisbach equation

h f = f D L D u 2 2g ,

(10)

where fD is the Darcy-Weisbach friction factor, u is the fluid velocity, and L and D are the length and diameter of the pipe, respectively. Velocity head and head loss can be combined

h u + h f = u 2 2g ( 1+ f D L D ) .

(11)

The Darcy-Weisbach friction factor fD is obtained from the Colebrook-White equation for Reynolds numbers Re>4000,

1 f D =2 log 10 ( ε 3.7D + 2.51 Re f D ) ,

(12)

where ε is the roughness height of the pipe. For very large Reynolds numbers, Re>108, corresponding to the expected highly turbulent flow inside the pipe, (12) simplifies to

f D = [ 2 log 10 ( 1 3.7 ε D ) ] 2 .

(13)

As a conservative value we take fD = 0.03, which approximately corresponds to a very large ratio ε/D= 0.005, or a roughness height ε=5 cm for a diameter D=10 m. Typical roughness height of clean materials are two orders of magnitude smaller than this magnitude, but macro fouling caused by coarse matter, such as algae or mussels, will substantially increase ε.

When changes in kinetic energy and friction are considered, the power provided by the turbine Pw is used to save the hydraulic head h (now taken as a positive quantity), the velocity head hu, and the head loss hf, so the volume flux is

F V = P W g ρ b ( h+ h u + h f )

(14)

Replacing hu+hf given by (11) in (14) and using Fv = πR2u, we obtain the depressed cubic equation for the volume flux Fv,

F V 3 ρ b 2 π 2 R 4 ( 1+ f D L 2R )+ F V g ρ b h P W =0 .

(15)

For large pipe radius R, the first term in (15) may be neglected, and Fv(R) tends to the frictionless solution for slow flows Fv=Pw/(gρbhf). The solution of Fv(R) is plotted in Figure 4.

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Fig. 4. – The volume flux Fv(R) (in m3 s1, thick continuous lines), as a function of the pipe radius R (m), solution of (15) for three different values of power Pw delivered by the wind turbine. The horizontal dashed lines correspond to the inviscid slow flow limit Fv=Pw/(gρbh)

As expected, the increase in kinetic energy and the loss of energy due to friction can only be neglected for a large radius R. Given a hydraulic jump h=0.15 m, a pipe radius R≃10 m, and a wind turbine providing a power FW≃2 MW, the resulting volume transport is Fv≃547 m3 s1, which is still a very large volumetric flow. In this case the velocity head hu≃0.15 cm is similar to the hydraulic head h, and the head loss hf≃0.06 cm is about h/2. Given the large values of R required to reduce the velocity head and pipe resistance head, we may instead think of the vertical tube or pipe as a wide cylindric container, with open bottom and surface ends, filled with dense deep water, the upper end located above the ocean free surface to avoid entrance of light water. The energy delivered by the wind turbine is used to withdraw the heavy water from this container. In the next section we calculate the horizontal pressure jump across the tube in order to have an insight into the resistance requirements of the tube’s materials.

HORIZONTAL PRESSURE JUMPTop

The horizontal pressure jump ∆p(z) across the tube is

Δp( z )={ p e (z), z h a ; p e (z)+ p i (z), z< h a ,

(16)

where pe(z) and pi(z) are the hydrostatic pressure of the water column at depth z outside and inside the tube, respectively. In the case of the arctangent density profile, which is the most realistic profile, ∆p(z) is computed from the integration of (6) from z to z=0 for the tube’s exterior, and from z to z=h for the tube’s interior, giving

p e (z)=g c 2 c 4 [ c 3 arctan( c 3 ) 1 2 ln( c 3 2 1 ) ]g c 1 z g c 2 c 4 ( c 3 + c 4 z)arctan( c 3 + c 4 z)+ +g c 2 2 c 4 ln[ ( c 3 + c 4 z) 2 1 ]

(17)

and

pi(z) = gρb(haz).

(18)

The resulting pressure jump ∆p(z) is shown in Figure 5. At the ocean surface ∆p(0)=0 since both pressures are equal to the atmospheric pressure. At zha, ∆p(z) equals the (minus) weight (per unit area) of the exterior water column and therefore decreases almost linearly (density differences are very small) along the first |ha|=18 cm. At ha=−18 cm we have ∆p(ha)≃18×102Pa=0.18 dbar. One dbar approximately corresponds to the hydrostatic pressure of a 1-m-depth water column. At depths z<ha the pressure jump increases due to the larger weight per unit depth of the interior water column, and reaches ∆p=0 at z=H, which was the boundary condition chosen to compute ha. Note that ∆p increases rapidly from z=ha to z≃−50 m since is in this layer where density ρa(z) increases fast with depth (Fig. 2). Though this pressure seems to be small, it may be enough to deform flexible material tubes made of very thin (0.5 mm thickness) polyvinyl chloride or polyethylene, such as those used in some upwelling experiments. Thus, some periodic reinforcement of a flexible tube like that done in Maruyama et al. (2011)Maruyama S., Yabuki T., Sato T., et al. 2011. Evidences of increasing primary production in the ocean by Stommel’s perpetual salt fountain. Deep-Sea Res. I 58: 567-574. seems to be necessary. Deployment of a flexible tube in the open ocean, assuring that is initially full of water, is therefore essential for the correct operation of the device. Deformation, including torsion, in the tube caused by vertical shear currents seems to be a more serious problem to the operation of flexible tubes. The structure supporting the pipe will presumably also have an important effect on the pipe. Strain measurements taken in the riser pipe of TAKUMI (Maeda et al. 2007Maeda K., Takahashi I., Miyabe H., et al. 2007. Strain measurement on riser pipe of TAKUMI. Proceedings of the Seventh ISOPE Ocean Mining Symposium, Lisbon, Portugal, The International Society of Offshore and Polar Engineers.) indicated that the main factor affecting the riser pipe behaviour was not the direct effect of wave and current fields on the pipe, but their indirect effects through the motion of the floating structure.

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Fig. 5. – Horizontal pressure jump ∆p(z) across the tube’s wall.

For a rigid vessel such as that proposed in this study, the circumferential hoop stress σθ can be obtained using the Barlow thin wall method,

σ θ = Δp*R t ,

(19)

where above ∆p* is the pressure difference due to the hydraulic head and head loss, R is the pipe radius, and t is the wall thickness. Thus, an order of magnitude of the maximum permissible radius-to-wall thickness (R/t )max required for a stainless steel pipe (maximum allowable stess σθ max of the order of 102 MPa) subject to a pressure difference ∆p*≃2×∆p≈40×102Pa, is (R/t)maxσθmax/∆p*∼25×103. For a circular pipe of radius R=10 m, this implies a minimum wall thickness tmin∼4 mm. Furthermore, horizontal currents will cause a shear force on the tube, which will in turn tend to bend it. Weighting the bottom may help to reduce this problem. Thus, as mentioned before, deformation, including torsion, in the tube by bending moments caused by vertical shear currents seems to be more serious a problem for the operation of the container than the circumferential hoop stress caused by the hydraulic head and head loss.

Neutral height of the sinking plume

As we have seen in the previous sections, the surface water jump h is only of a few centimetres. The power generated by offshore wind turbines is of the order of several MW. If the complete wind power, say Pw, is used to uplift water from z=h to the surface z=0, that is, any other energy consuming processes is neglected, the volume flux Fv obtained is

F v = P w g ρ b h .

(20)

For typical values Pw=2 MW, ρb=1029 kg m3 and h=0.16 m, we have a water volume transport estimate Fv≃103 m3 s1. Obviously this is a huge volume flux. Such a volume of water per unit of time with density ρb brought to the sea surface, where density ratio ρb/ρt≃1.003>1, will have no time for a complete mixing, and will inevitably lead to a fast sinking of the dense deep water developing a downwelling plume, leaving little time for primary production to develop. Due to the horizontal turbulent entrainment, this vertical plume will later reach a neutral height HN beyond which no further sinking occurs. Following the classical results of (Morton et al. 1956Morton B.R., Taylor G.I., Turner J.S. 1956. Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. 234: 1-23.) for point buoyancy sources, an order of magnitude of HN is given by

H N =2.1 π f 0 N 3 4 ,

(21)

where f0 is the surface buoyancy flux and N is the environment Brunt-Väisälä or buoyancy frequency,

N 2 g ρ 0 d ρ B dz ,

(22)

where ρ0 is a constant reference density and ρB(z) is the environment (background) density profile. In our case, f0 is related to the volume flux Fv by

f 0 = g F v π ,

(23)

where gʹ is the reduced gravity

g g ρ b ρ t ρ 0 .

(24)

Finally, the neutral height HN is related to the volume flux Fv through

H N =2.1 g F v N 3 4 .

(25)

For the values gʹ=10(1029−1026)/1028 m s2, Fv=103 m3 s1, and N=1/50 s1, we obtain a neutral height HN≃90 m, or HN≃77 m for Fv=500 m3 s1. Thus, due to the very large volume flux Fv the nutrient-rich deep water will probably sink to the bottom limit of the photic layer, and primary production will not be considerably enhanced.

To avoid this large vertical sinking, the upwelled water must be dispersed in some way at the surface to increase turbulent entrainment. The point source theory for vertical plumes used above must therefore be modified to include finite area sources. A number of corrections have been developed (Hunt and Kaye 2001Hunt G.R., Kaye N.G. 2001. Virtual origin correction for lazy turbulent plumes. J. Fluid Mech. 435: 377-396.) to take into account finite area sources in the theory of turbulent entrainment. A simple estimate may be given by noticing that the angle θ of the plume’s cone (Fig. 6) is related to the diameter D of the source and to the virtual height zv of the point source by

tan θ 2 = D 2 z v .

(26)

Experimental results show that the total angle θ enclosed is approximately 18°-20°. If the mixed upwelled water is to remain close to the surface, we must impose zvHN. Thus, the upwelled water must be dispersed to the surface over a circle of radius r∼HN tan(θ/2)≃15 m. This water dispersion could be achieved by a number of mechanical, non-power-consuming ways. For example, a horizontal perforated disc platform of 15 m radius located at the surface but below the deep water outflow could work well as a diffuser plate, or multiport diffusers such as those used in ocean outfall dilution could be installed. Note also that this radius r≃15 m has an order of magnitude similar to the pipe radius R required to keep power losses due to friction below reasonable low values. However, if oxygen enrichment, or aeration, of the deep water is desirable, a portion of the turbine’s energy could be used for water spreading (a fountain aerator) and only the remaining portion will be consumed for water raising. A gross estimation of the order of magnitude of the power required for water sprinkling is given in the next section.

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Fig. 6. – Schematic explaining the finite area source symbols.

POWER BUDGET CONSIDERING POWER CONSUMPTION REQUIRED FOR WATER SPRINKLINGTop

In this case the wind power Pw must be employed both for water upwelling (Pu) and water spreading (Ps). The larger the water volume transport Fv, the larger the amount of energy required for water spreading Ps and the smaller the amount of energy for water upwelling Pu, which means the smaller the amount of water transport Fv achieved. An estimate of the power Ps (R, Fv) consumed for homogeneously sprinkling, over a circle of radius R, a volume transport of water Fv leaving a point source located at the circle’s centre is given in Appendix D. The basic result is that Ps(R, Fv)=(ρg/3)RFv. We stress that the above relation is only a rough estimate of the minimum bound for the ideal power required to spread a volume flux of water Fv over a radius R. The practical power Ps (R, Fv) might be an order of magnitude larger. In order to take this uncertainty into account, we include a factor κ0 in Ps, that is,

P s ( R, F v )= κ 0 ρ b g 3 R F v ,

(27)

and assume that typically κ0∼10. Thus, the variables Pu, Ps , Fv and R are coupled through the following system of equations

Pw = Pu + Ps ,

(28)

F v = P u g ρ b h ,

(29)

R=2.1 g F v N 3 4 tanθ ,

(30)

P s = κ 0 ρ b g 3 R F v ,

(31)

where the unknowns are (Pu, Ps, Fv, R). By simple substitution we arrive at a single algebraic equation for Fv,

F v ( κ 0 2.1 3 g N 3 4 tanθ F v 1/4 +h )= P w ρ b g ,

(32)

which can be solved numerically noticing that the second term on the left hand side of (32), Fvh, is small compared to the first one. For typical values Pw=2 MW, gʹ=0.028 m s2, N=1/50 s1, and θ=20°, we obtain Fv=6.3 m3 s1, R≃9 m, Ps=1.99 MW, and Pu=0.01 MW. This solution implies that 99.4% of the wind power will be employed for water sprinkling while only the remaining 0.6% will be used for raising water at a very reduced volume flux of about 6 m3 s1. These results show that water sprinkling, as a way to favour both the turbulent entrainment and oxygen enrichment of the deep water, is very power consuming and, as stated above, use of non-power-consuming diffusion devices such as multiport diffusers or a perforated disc platform is more advantageous.

NUTRIENTS TRANSPORTTop

Once we have an estimation of the order of magnitude of the upwelling volume transport (say Fv∼500 m3 s1), we can also provide an estimation for the transport of nutrients to the photic layer. Time and space averages of vertical profiles of nitrates, phosphates and silicates, characteristics of the Catalan Sea, are given in Figure 7. As a first approximation we can assume that at a depth z=−300 m the mass concentration of nitrates, phosphates and silicates is, respectively, CN=7 µ mol l–1, CP=0.4 µ mol l–1 and CS=5 µ mol l–1. The transport of nutrients to the photic layer is therefore FN= CNFv ∼3.5 mol s1, FP=CPFv∼0.2 mol s1 and FS=CSFv∼2.5 mol s1. This means mass transports TN∼217 g s1=19 t day1, TP∼19 g s1=1.6 t day1 and TS∼150 g s1=13 t day–1, where we have used the molecular masses MN≃62 g mol–1, MP≃95 g mol–1 and MS≃60 g mol–1. Typical values of offshore water transport caused by wind stress in natural upwelling coastal areas are of the order of 103 m3 s1 per km of coastline (upwelling index U=103 m3 s1 km1). Thus, an upwelling Fv∼500 m3 s1 produced by a wind turbine is roughly equivalent to the natural upwelling along a 500 m coastline.

sm4297fig7.jpg

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Fig. 7. – Seasonal mean nitrate, phosphate, and silicate profiles for the same time periods as in Figure 3: January-March (solid line), April-June (dotted line), July-September (dashed line), and October-December (dash-dotted line). Thin lines represent the corresponding value plus one standard deviation.

We notice that large artificial upwelling in an ocean region with poor biological productivity can change the trophic cycle positively or negatively. The sudden growth of primary producers can generate the expected positive response, i.e. a higher productivity with the consequent increase of economically important marine resources. It can also generate a negative response, favouring proliferation of damaging or simply non-autochthonous species that will put an end to the flora and fauna of the region. Short, medium, and large time scale responses of the ecosystem to large artificial upwelling are therefore a very important issue that has been addressed little (McAndrew et al. 2007McAndrew P.M., Björkman K.M., Church M.J., et al. 2007. Metabolic response of oligotrophic plankton communities to deep water nutrient enrichment. Mar. Ecol. Prog. Ser. 332: 63-75., Williamson et al. 2009Williamson N., Komiya A., Maruyama S., et al. 2009. Nutrient transport from an artificial upwelling of deep sea water. J. Oceanogr. 65: 349-359.) and needs to be seriously taken into account. The relation between edible fish production and upward flow rate of deep nutrient-rich water also needs to be studied (Groves 1958Groves G.W. 1958. Flow estimate for the perpetual salt fountain. Deep Sea Res. 5: 209-214., Thorson 1961Thorson J. 1961. Flow estimate for an artificial upwelling in the sea. Deep Sea Res. (1953) 7(4): 296-297., Matsuda et al. 1999Matsuda F., Szyper J., Takahashi P., et al. 1999. The ultimate ocean ranch. Sea Technol. 40(8): 17-26.). The ecosystem response is a problem beyond the scope of this work, which only develops, in terms of power requirements, the idea of artificial upwelling using wind power. Artificial upwelling using wind power is, however adjustable, and can be easily reduced, or turned off, if there is any risk of adverse effects to the ecosystem.

DISCUSSION AND CONCLUDING REMARKSTop

In this work we have put forward the idea of taking advantage of in situ offshore wind energy to cause upwelling of nutrient-rich deep water to favour primary production in the ocean. With a wind turbine power of several MW, the order of magnitude of the upward water transport is several hundreds of m3 s1, a transport much larger than that achieved using surface wave energy or the perpetual salt fountain mechanism. Here we propose enlarging the spar of a spar-buoy wind turbine to channel the deep water up to the sea surface. This long rigid tube also acts as a ballast, providing additional stability to spar-buoy offshore wind turbines. This work is, however, only a first approximation to the general problem. We basically have found that the upward volume transport could be so large that turbulent water entrainment must be enhanced in some way to avoid sinking of the dense water to a neutral depth below the photic layer. An energetically cheap way to dilute the dense upwelled water is therefore required.

There are still a large number of basic questions that need to be properly addressed. One of them is the ecological response. Though a water rich in nutrients is in theory beneficial to primary production, a large and concentrated source of nutrients may be harmful in an oceanic region where the ecosystem is not yet well adapted to a large availability of nutrients. For example, at the Mississippi mouth excess nutrients carried down the river in floods cause massive algal blooms. A slow startup of the fertilization could therefore be desirable. Since it is planned to obtain offshore wind energy in wind farms comprising a large number of wind turbines, it could be possible to use only a part of the electric power of several wind turbines to raise deep water. This will reduce the upward volume transport assigned to each wind turbine, therefore favouring turbulent entrainment, though it would require an expensive 300-m spar-buoy on each wind turbine. Thus, the benefits of artificial upwelling might largely counteract minor potential negative environmental impacts of offshore wind farms (Wilson et al. 2010Wilson J.C., Elliott M., Cutts N.D., et al. 2010. Coastal and offshore wind energy generation: Is it environmentally benign? Energies 3(7): 1383-1422.).

Another important problem concerns engineering issues. Nutrient-rich water below the photic layer is located in many ocean regions at a minimum depth of about 300 m. This implies the design, manufacturing, testing, and deployment of vertical spars almost twice as large as the ones currently designed for spar-buoy offshore wind turbines (EWEA 2013EWEA. 2013. Deep water - The next step for offshore wind energy. Tech. rep. European Wind Energy Association.). Also, since energy losses due to friction in the pipe become large for a reasonable pipe radius, a compromise between wall thickness, pipe radius and upward volume transport must be reached through very careful engineering analysis. Fatigue, stress and load analysis of these new structures need to be carried out.

Some physical oceanographic questions also need to be addressed with great care. Point-like localized upwelling of large water volume transport cannot be maintained for long periods of time unless ocean currents advect the surface mixed water far from the wind turbines and replace it with new, lighter water. Otherwise the complete water column will homogenize into dense water and deep sinking of the upwelled water will occur despite turbulent entrainment. However, while some amount of ocean currents is beneficial to maintain the nutrients in the photic layer, large amplitude currents, and in particular the large vertical shear associated with baroclinic currents, might be detrimental to the stability of the long tube and the turbine’s tower. It is therefore important for mariculture applications to select the offshore wind farm location taking into account the wind, wave and ocean current climatology.

The influence of surface waves and shear currents favouring turbulent entrainment needs to be properly investigated. Also, in the absence of significant background ocean shear currents, the upwelled water mixed with the surface water could form large bowls of dense water which, on a time scale of a few days, will start rotating cyclonically, due to the centripetal and Coriolis accelerations, and will approach the cyclogeostrophic balance that is characteristic of mesoscale and submesoscale dynamics. An order of magnitude of the time required to generate a cyclogeostrophic cyclone from a source of deep water can be roughly estimated assuming that in a completely mixed and cyclogeostrophic balanced state the cyclone develops a Gaussian radial (negative) free surface height anomaly h(r)=h0 exp(−r/r0), where r is the radial distance, h0<0 is the height at the vortex centre, and r0≃20 km is the horizontal length scale of the vortex. The volume integral of h(r) between r=0 and r=∞ is the volume anomaly Vʹh0r02. The mass anomaly of this vortex is therefore =−ρ0Vʹ>0. A constant flux of mass anomaly m=ρʹFv will take a time t=Mʹ/Fʹm= ρ0Vʹ/(ρʹFv) to complete the vortex mass anomaly . For ρ0=103 kg m3, ρʹ=3 kg m3, h0=1 cm, r0=20 km and Fv=500 m3 s1, we obtain an order of magnitude of the time period t∼107 s∼102 days. This is still too long time a period to neglect the effect of background ocean currents, but if exceptionally quiescent conditions are found, or if several wind turbines are employed, the high volume transports achieved open up the possibility of artificial submesoscale eddy formation. Mesoscale and submesoscale eddies are very coherent stable structures in which mixing with surrounding waters is inhibited. When these vortices leave the upwelling site, they retain the nutrients and other sea water properties for a long time period. In the extreme case of very quiescent waters, an anticyclone may be artificially generated by downwelling light water using a nearby wind turbine. If a cyclone-anticyclone pair is generated close enough, the two vortices will form a dipole vortex which will start translating coherently along the dipole axis. That scenario seems quite simplistic because the upwelling produced by a turbine only occurs when wind is present, so wind stress on the surface ocean layer will likely provide, via turbulent momentum diffusion, enough horizontal momentum to the upper layer waters to avoid generation of shallow eddies, or at least to strongly modify its surface structure. However, we mention this artificial dipole formation because the large vertical transport of water that wind turbines might cause opens up the opportunity to modify the regional submesoscale ocean circulation. On the other hand, large wind farms may produce a significant disturbance on the surface wind stress, generating upwelling/downwelling velocities in the wind wake that might affect the local ecosystem (Broström 2008Broström G. 2008. On the influence of large wind farms on the upper circulation. J. Mar. Syst. 74: 585-591.).

As an overall conclusion we might say that, from the energetic point of view, artificial upwelling using offshore wind energy seems to be a promising way to enhance primary production in the ocean. Mariculture application of this approach implies the fertilization of large regions in the open ocean, and is therefore severely subjected to atmosphere and ocean climatology, as well as to ecological dynamics. The political issues also need to be addressed: if productivity is enhanced in the open ocean, costs and benefits must be properly distributed among the interested countries. The general problem is a multidisciplinary one, and we have noticed that some important physical, engineering, and ecological questions need to be seriously addressed to obtain a more complete confidence in the approach presented here.

ACKNOWLEDGEMENTSTop

We thank two anonymous reviewers for their very useful comments. Partial support for this study was obtained through projects CTM2011-28867 and CTM2014-56987-P (Spanish Ministry of Science and Innovation).

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APPENDICES

Appendix A. – Surface jump in the two-layer model

Using the hydrostatic approximation (1), the pressure outside the tube is

H 0 p 1 z z= H 0 ρ t gdz ,

(A1)

whose integration is

p1(0) − p1(H) = gρtH ,

(A2)

or p1(H)=pa−gρtH, where pa is the atmospheric pressure. Inside the tube,

H h p 2 z z= H h ρ b gdz ,

(A3)

whose integration is

p2(h) − p2(H) = −gρt(hH) ,

(A4)

or p2(H)=pagρb(hH). The height h is obtained assuming that in the steady state there is no horizontal pressure gradient at z=H, i.e., p1(H) = p2(H), so

h=H ρ b ρ t ρ b .

(A5)

Appendix B. – Surface jump in the linear density profile

Using the hydrostatic approximation (1), the pressure outside the tube is

H 0 p 1 z dz=g H 0 [ ρ t + ϑ 0 z ]dz ,

(B1)

whose integration is

p 1 (0) p 1 (H)=g ρ t H+g ϑ 0 H 2 2 ,

(B2)

or

p 1 (H)= p a g ρ t Hg ϑ 0 H 2 2 ,

where pa is the atmospheric pressure. Inside the tube,

H h p 2 z dz=g H h ρ b dz ,

(B3)

and integrating,

p2(H) = pagρb(H)(hH) .

(B4)

Assuming that in the steady state there is no horizontal pressure gradient at z=H, i.e., p1(H)=p2(H), we obtain

p t ϑ 0 H 2 2 = ρ t h ρ t H+g ϑ 0 Hh ϑ 0 H 2 ,

(B5)

and solving for h,

h= ϑ 0 H 2 2( ρ t + ϑ 0 H ) .

(B6)

Appendix C.Surface jump in the arctangent density profile

The objective here is to find an arctangent density profile ρ(z) of the form

ρ(z) = c1 + c2 arctan(c2 + c3z),

(C1)

where c1, c2, c3 and c4 are constants. To do so, we first linearly map the depth range [zmin, zmax] into the range of the tangent function [Zmin, Zmax] = [−tan(Xmax), −tan(Xmin)], where [Xmax, Xmin] =[1.55, −1.47] are values slightly [smaller, larger] than [π/2, −π/2]. These values are chosen so that the vertical arctangent asymptotes adjust to the upper and lower density values, as well as to match the pycnocline depth to about z=50 m.

Therefore,

Z(z)= Z min +(z z min ) Z max Z min z max z min ,

(C2)

Finally, the arctangent profile arctan[Z(z)] is linearly mapped into the density profile ρ(z), that is,

ρ t ρ(z) ρ t ρ b = arctan( Z max )arctan[ Z(z) ] arctan( Z max )arctan( Z min ) ,

(C3)

Thus, we obtain the constants,

c 1 = ρ t ( ρ t ρ b )arctan( Z max ) arctan( Z max )arctan( Z min ) ,

(C4)

c 2 = ρ t ρ b arctan( Z max )arctan( Z min ) ,

(C5)

c 3 = Z min z min ( Z max Z min ) z max z min ,

(C6)

c 4 = Z max Z min z max z min .

(C7)

Given the depth range [zmin, zmax]=[−300, 0] m, and the density range [ρb, ρt] = [1029.1, 1025.6] kg m3, we obtain {c1, c2, c3, c4} = {1027.0, −0.66190, 9.9666, 0.19987}.

Once the arctangent density profile ρ(z) has been found, we need to integrate it in order to obtain the pressure field outside the tube,

z min z max ρ(z)dz = z min z max [ c 1 + c 2 arctan( c 3 + c 4 z ) ]dz .

(C8)

The arctangent integral

c 2 z min z max arctan( c 3 + c 4 z )dz ,

(C9)

is done by a change of variable, xc3 + c4z , dx = c4 dz .

Therefore

c 2 arctan(x) dx c 4 = = c 2 c 4 arctan(x) = = c 2 c 4 [ xarctan(x) 1 2 ln( x 2 +1 ) ]= = c 2 c 4 [ ( c 3 + c 4 z )arctan( c 3 + c 4 z ) 1 2 ln{ ( c 3 + c 4 z ) 2 +1 } ] z min z max .

(C10)

And hence, the pressure outside the tube,

p 1 (0) p 1 (H)= =g [ c 1 z+ c 2 c 4 ( c 3 + c 4 z )arctan( c 3 + c 4 z ) 1 2 ln{ ( c 3 + c 4 z ) 2 +1 } ] H 0 ,

 

and finally,

p 1 (0) p 1 (H)= =g c 2 c 4 [ c 3 arctan( c 3 ) 1 2 ln( c 3 2 +1 ) ]+ +g c 1 H+ g c 2 c 4 ( c 3 + c 4 H )arctan( c 3 + c 4 H )+ + g c 2 2 c 4 ln[ ( c 3 + c 4 H ) 2 +1 ] .

(C11)

Inside the tube

p2(0) − p2(H) = −gρb(hH).

(C12)

Equating p1(0)=p2(0)=pa, and p1(H)=p2(H), we obtain (8).

Appendix D. – Power required for water sprinkling

In this appendix we estimate the order of magnitude of the power Ps(R, Fv) required for sprinkling a water volume transport Fv over a circle of radius R. The initial kinetic energy density E0 of a water particle of density ρ leaving the circle centre with horizontal u0 and vertical w0 velocity components is

E 0 = 1 2 ρ( u 0 2 + w 0 2 ) ,

(D1)

From the equations of uniformly accelerated motion r(t) = u0t and z(t) = w0t − 1/2gt2, we see that the initial kinetic energy required to throw a water particle a horizontal distance r reaches a minimum when

u0 = w0 = rg /2 , and hence the kinetic energy density

E 0 (r)= ρg 2 r ,

(D2)

The area differential is dA(r)=2π r dr and the energy differential is dE(r)=E0(r) H dA(r) =ρ gπ H r2 dr, where H is a unit of vertical length. Therefore, the energy E(R) = (ρ gπ H /3)R3. As a function of the upwelling volume transport FvπR2W=πR2H/T , where TH/W is the time interval taken to advance a length H at a vertical velocity W. Finally the power consumed for water spreading Ps(R, Fv)≡E(R)/T is

P s (R, F v )= κ 0 ρg 3 R F v .

(D3)

We stress that the above relation is only a rough estimate of a minimum bound for the ideal power required to spread a volume flux of water Fv over a radius R. The practical power might be an order of magnitude larger. In order to take this uncertainty into account, we have included a factor κ0 in Ps, assuming that typically κ0∼10.



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