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, ^{3} 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 m^{3} s^{−1}. However, such a large amount of dense water could sink fast to aphotic layers due to vertical static instability (the

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 ^{3} 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 m^{3} s^{−1}. Sin embargo, una cantidad tan grande de agua densa podría hundirse rápidamente a las capas afóticas debido a la inestabilidad estática vertical (

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 (

Scientists are now looking for ways to avoid these limiting factors and fertilize the ocean artificially at a large spatial scale (

Several artificial upwelling mechanisms have already been proposed. One of them is to cause upwelling using surface wave energy (^{3} s^{−}^{1} from a depth of 300 m employing a 1.2-m diameter rigid tube. Using a “Bristol Cylinder” (^{3} s^{−}^{1} of water from a depth of 500 m. A recent attempt (^{3} s^{−}^{1}.

An upwelling mechanism based on the particular temperature and salinity vertical gradients of some tropical and subtropical ocean regions was proposed by ^{−}^{4} m^{3} s^{−}^{1} ≃ 45 m^{3} day^{−}^{1}.

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 (^{3} s^{−}^{1} (^{3} s^{−}^{1}. A third approach to force the upwelling of the unstable buoyant plume (^{3} s^{−}^{1} 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 m^{3} s^{−}^{1} 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 (^{5} m^{3} day^{−1}≃1.2 m^{3} s^{−1} (

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 (

The design proposals for offshore floating wind turbines depend on the way the turbine is stabilized (

Dense deep water (with density _{b}) 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 _{w} of the wind turbine and the upward volume transport _{v} are therefore related through _{w}=_{v}_{b}gh, or _{v}=_{w}/(_{b}gh), which provides the upwelling transport through the tube as a function of the turbine’s power _{w} and the surface hydraulic jump h. In the next section we calculate the surface jump h, using several background density vertical profiles _{w}=2 MW) the volume transport could be very large (of the order of 10^{3} m^{3} s^{−}^{1}). 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 _{f} 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 m^{3} s^{−}^{1}. 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.

In this section we calculate the free surface height difference, or hydraulic head,

$\frac{\partial p}{\partial z}=-g\rho$, | (1) |

where _{i}(_{ib} equals the exterior pressure _{e}(_{eb} below the pump. At the free surface (top, subscript t) interior and exterior pressures are equal to the atmospheric pressure _{a}, that is, _{e}(0)=_{i}(_{a}. We assume that in the steady state the water inside the tube is homogeneous and has a density equal to the bottom density

$\rho \left(H\right)\left(h-H\right)={\int}_{H}^{0}\rho \left(z\right)dz$ , | (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

$h=H+\frac{1}{\rho \left(H\right)}{\int}_{H}^{0}\rho \left(z\right)dz$ , | (3) |

which depends on the length of the tube and on the vertical distribution of background density,

The simplest case is that of a two-layer model (see _{t} and _{b}, respectively. In this case (see

$h}_{2}=H\frac{{\rho}_{\text{b}}-{\rho}_{\text{t}}}{{\rho}_{\text{b}}$. | (4) |

Thus, in the simplest case _{2} depends linearly both on H and on the density difference _{b}−_{t}. We note that, as expected, _{2}=0 when _{b}=_{t}. Assuming values of H=−300 m, _{b}=1029 kg m^{−}^{3}, and _{t}=1026 kg m^{−}^{3}, typical of the western Mediterranean Sea in the summer season, we obtain _{2}≃−87 cm. Notice that this profile is an extreme case in the sense that it maximizes the jump |

This is the simplest case of a continuously stratified ocean, in which density _{l} is assumed to be linearly dependent with depth, that is, _{l}(z)≡_{t}+_{0}_{t}=_{l}(0) is the surface density, and _{0}<0 is a constant density stratification (see _{l},

$h}_{1}=\frac{{\vartheta}_{0}{H}^{2}}{2\left({\rho}_{\text{t}}+{\vartheta}_{0}H\right)}=\frac{H}{2}\frac{{\rho}_{\text{b}}-{\rho}_{\text{t}}}{{\rho}_{\text{b}}}=\frac{{h}_{2}}{2$ . | (5) |

The jump _{l}=_{2}/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 _{l}=_{2}/2≃−44 cm.

Here we consider a density profile _{a}(

_{a}(_{1} + _{2} arctan(_{3} + _{4} |
(6) |

where _{1}, _{2}, _{3}, and _{4} are given constants used to match the _{a} asymptotes to the boundary values _{t} and _{b}, and to adjust the pycnocline depth (see

$\begin{array}{c}{\rho}_{\text{b}}\left({h}_{\text{a}}-H\right)=\frac{{c}_{2}}{{c}_{4}}\left[{c}_{3}\mathrm{arctan}\left({c}_{3}\right)-\frac{1}{2}\mathrm{ln}\left({c}_{3}^{2}+1\right)\right]-\\ -{c}_{1}H-\frac{{c}_{2}}{{c}_{4}}\left({c}_{3}+{c}_{4}H\right)\mathrm{arctan}\left({c}_{3}+{c}_{4}H\right)+\\ +\frac{{c}_{2}}{2{c}_{4}}\mathrm{ln}\left[{\left({c}_{3}+{c}_{4}H\right)}^{2}+1\right]\equiv \xi \left(H\right)\text{\hspace{0.17em}},\end{array}$ | (7) |

where _{b}(_{a}−_{a} we get

$h}_{\text{a}}=H+\frac{\xi \left(H\right)}{{\rho}_{\text{b}}$. | (8) |

For the same typical values in the western Mediterranean used previously we obtain _{a}≃−18 cm, which has a magnitude smaller than the previous |_{2}|=87 cm and |_{l}|=44 cm.

Finally, we obtain the water jump _{e} using experimental in situ density profiles _{j}_{i}_{i}_{e} for the four mean density profiles (subscript e={

$h}_{\text{e}}=H+\frac{1}{{\overline{\rho}}_{\text{e}}\left(N\right)}{\displaystyle \sum _{i=1}^{N-1}\frac{{\overline{\rho}}_{\text{e}}\left({z}_{i}\right)+{\overline{\rho}}_{\text{e}}\left({z}_{i}+1\right)}{2}}\Delta {z}_{i$, | (9) |

where the vertical grid size is non-uniform ∆_{i}_{i}_{i}_{e} (_{A}_{B}_{C}_{D}

In winter the reduced solar radiation causes a cooling of the upper ocean and therefore a vertical homogenization of the water (_{A}_{A}_{B}_{C}_{a} (_{C}_{a}|≃ 18 cm. In autumn, surface cooling decreases vertical stratification and therefore the surface jump decreases to |_{D}^{3} s^{−}^{1}, 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

The loss of power due to the increase in kinetic energy is simply related to the velocity head _{u}=^{2}/(2g)= F_{v}^{2}/(2gπ^{2}R^{4}). We obtain the loss of power due to friction in the pipe, related to the _{f} in hydraulic engineering, using the Darcy-Weisbach equation

$h}_{\text{f}}={f}_{\text{D}}\frac{L}{D}\frac{{u}^{2}}{2g$, | (10) |

where _{D} is the Darcy-Weisbach friction factor,

${h}_{\text{u}}+{h}_{\text{f}}=\frac{{u}^{2}}{2g}\left(1+{f}_{\text{D}}\frac{L}{D}\right)$. | (11) |

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

$\frac{1}{\sqrt{{f}_{\text{D}}}}=-2{\mathrm{log}}_{10}\left(\frac{\epsilon}{3.7D}+\frac{2.51}{\mathrm{Re}\sqrt{{f}_{\text{D}}}}\right)$, | (12) |

where ^{8}, corresponding to the expected highly turbulent flow inside the pipe, (12) simplifies to

$f}_{\text{D}}={\left[-2{\mathrm{log}}_{10}\left(\frac{1}{3.7}\frac{\epsilon}{D}\right)\right]}^{-2$. | (13) |

As a conservative value we take _{D} = 0.03, which approximately corresponds to a very large ratio

When changes in kinetic energy and friction are considered, the power provided by the turbine _{w} is used to save the hydraulic head _{u}, and the head loss _{f}, so the volume flux is

$F}_{\text{V}}=\frac{{P}_{\text{W}}}{g{\rho}_{\text{b}}\left(h+{h}_{\text{u}}+{h}_{\text{f}}\right)$ | (14) |

Replacing _{u}+_{f} given by (11) in (14) and using _{v} = π^{2}_{v},

${F}_{\text{V}}^{3}\frac{{\rho}_{\text{b}}}{2{\pi}^{2}{R}^{4}}\left(1+{f}_{\text{D}}\frac{L}{2R}\right)+{F}_{\text{V}}g{\rho}_{\text{b}}h-{P}_{\text{W}}=0$. | (15) |

For large pipe radius R, the first term in (15) may be neglected, and _{v}(_{v}=_{w}/(_{b}_{f}). The solution of _{v}(

As expected, the increase in kinetic energy and the loss of energy due to friction can only be neglected for a large radius _{W}≃2 MW, the resulting volume transport is _{v}≃547 m^{3} s^{−}^{1}, which is still a very large volumetric flow. In this case the velocity head _{u}≃0.15 cm is similar to the hydraulic head _{f}≃0.06 cm is about

The horizontal pressure jump ∆

$\Delta p\left(z\right)=\{\begin{array}{cc}-{p}_{\text{e}}(z),& z\ge {h}_{\text{a}};\\ -{p}_{\text{e}}(z)+{p}_{\text{i}}(z),& z<{h}_{\text{a}},\end{array}$ | (16) |

where _{e}_{i}(

$\begin{array}{c}{\displaystyle {p}_{\text{e}}(z)=g\frac{{c}_{2}}{{c}_{4}}\left[{c}_{3}\mathrm{arctan}({c}_{3})-\frac{1}{2}\mathrm{ln}\left({c}_{3}^{2}-1\right)\right]-g{c}_{1}z-}\\ {\displaystyle -g\frac{{c}_{2}}{{c}_{4}}({c}_{3}+{c}_{4}z)\mathrm{arctan}({c}_{3}+{c}_{4}z)+}\\ {\displaystyle +g\frac{{c}_{2}}{2{c}_{4}}\mathrm{ln}\left[{({c}_{3}+{c}_{4}z)}^{2}-1\right]}\end{array}$ | (17) |

and

_{i}(_{b}(_{a} – |
(18) |

The resulting pressure jump ∆_{a}, ∆_{a}|=18 cm. At _{a}=−18 cm we have ∆_{a})≃18×10^{2}_{a}=0.18 dbar. One dbar approximately corresponds to the hydrostatic pressure of a 1-m-depth water column. At depths _{a} the pressure jump increases due to the larger weight per unit depth of the interior water column, and reaches ∆_{a}. Note that ∆_{a} to _{a}(

For a rigid vessel such as that proposed in this study, the circumferential hoop stress _{θ}

$\sigma}_{\theta}=\frac{\Delta p*R}{t$, | (19) |

where above ∆_{max} required for a stainless steel pipe (maximum allowable stess _{θ}^{2} MPa) subject to a pressure difference ∆^{2}Pa, is (_{max}∼_{θmax}/∆^{3}. For a circular pipe of radius _{min}∼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.

As we have seen in the previous sections, the surface water jump _{w}, is used to uplift water from _{v} obtained is

$F}_{\text{v}}=\frac{{P}_{\text{w}}}{g{\rho}_{\text{b}}h$. | (20) |

For typical values _{w}=2 MW, _{b}=1029 kg m^{−}^{3} and _{v}≃10^{3} m^{3} s^{−}^{1}. Obviously this is a huge volume flux. Such a volume of water per unit of time with density _{b} brought to the sea surface, where density ratio _{b}/_{t}≃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 _{N} beyond which no further sinking occurs. Following the classical results of (_{N} is given by

$H}_{\text{N}}=2.1\sqrt[4]{\frac{\pi {f}_{0}}{{N}^{3}}$ , | (21) |

where _{0} is the surface buoyancy flux and

$N}^{2}\equiv \frac{-g}{{\rho}_{0}}\frac{d{\rho}_{\text{B}}}{dz$ , | (22) |

where _{0} is a constant reference density and _{B}(_{0} is related to the volume flux _{v} by

$f}_{\text{0}}={g}^{\prime}\frac{{F}_{\text{v}}}{\pi$ , | (23) |

where

$g}^{\prime}\equiv g\frac{{\rho}_{\text{b}}-{\rho}_{\text{t}}}{{\rho}_{\text{0}}$. | (24) |

Finally, the neutral height _{N} is related to the volume flux _{v} through

$H}_{\text{N}}=2.1\sqrt[4]{\frac{{g}^{\prime}{F}_{\text{v}}}{{N}^{3}}$ . | (25) |

For the values g^{−}^{2}, _{v}=10^{3} m^{3} s^{−}^{1}, and ^{−}^{1}, we obtain a neutral height _{N}≃90 m, or _{N}≃77 m for _{v}=500 m^{3} s^{−}^{1}. Thus, due to the very large volume flux _{v} 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 (_{v} of the point source by

$\mathrm{tan}\frac{\theta}{2}=\frac{D}{2{z}_{\text{v}}}$. | (26) |

Experimental results show that the total angle _{v}≃_{N}. Thus, the upwelled water must be dispersed to the surface over a circle of radius r∼_{N} tan(

In this case the wind power _{w} must be employed both for water upwelling (_{u}) and water spreading (_{s}). The larger the water volume transport _{v}, the larger the amount of energy required for water spreading _{s }and the smaller the amount of energy for water upwelling _{u}, which means the smaller the amount of water transport _{v} achieved. An estimate of the power _{s }(_{v}) consumed for homogeneously sprinkling, over a circle of radius R, a volume transport of water _{v} leaving a point source located at the circle’s centre is given in Appendix D. The basic result is that P_{s}(R, _{v})=(_{v}. 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 _{v} over a radius _{s }(_{v}) might be an order of magnitude larger. In order to take this uncertainty into account, we include a factor κ_{0} in _{s}, that is,

$P}_{\text{s}}\left(R,{F}_{\text{v}}\right)={\kappa}_{0}\frac{{\rho}_{\text{b}}g}{3}R{F}_{\text{v}$ , | (27) |

and assume that typically κ_{0}∼10. Thus, the variables _{u}, _{s }, _{v} and

_{w} = _{u} + _{s} , |
(28) |

$F}_{\text{v}}=\frac{{P}_{\text{u}}}{g{\rho}_{\text{b}}h$, | (29) |

$R=2.1\sqrt[4]{\frac{{g}^{\prime}{F}_{\text{v}}}{{N}^{3}}}\mathrm{tan}\theta$ , | (30) |

$P}_{\text{s}}={\kappa}_{0}\frac{{\rho}_{\text{b}}g}{3}R{F}_{\text{v}$ , | (31) |

where the unknowns are (_{u}, _{s}, _{v}, _{v},

$F}_{\text{v}}\left({\kappa}_{0}\frac{2.1}{3}\sqrt[4]{\frac{{g}^{\prime}}{{N}^{3}}}\mathrm{tan}\theta {F}_{\text{v}}^{1/4}+h\right)=\frac{{P}_{\text{w}}}{{\rho}_{\text{b}}g$ , | (32) |

which can be solved numerically noticing that the second term on the left hand side of (32), _{v}_{w}=2 MW, ^{−}^{2}, ^{−}^{1}, and _{v}=6.3 m^{3} s^{−}^{1}, _{s}=1.99 MW, and _{u}=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 m^{3} s^{−}^{1}. 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.

Once we have an estimation of the order of magnitude of the upwelling volume transport (say _{v}∼500 m^{3} s^{−}^{1}), 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 _{N}=7 µ mol l^{–1}, _{P}=0.4 µ mol l^{–1} and _{S}=5 µ mol l^{–1}. The transport of nutrients to the photic layer is therefore _{N}= _{N}_{v} ∼3.5 mol s^{−}^{1}, _{P}=_{P}_{v}∼0.2 mol s^{−}^{1} and _{S}=_{S}_{v}∼2.5 mol s^{−}^{1}. This means mass transports _{N}∼217 g s^{−}^{1}=19 t day^{−}^{1}, _{P}∼19 g s^{−}^{1}=1.6 t day^{−}^{1} and _{S}∼150 g s^{−}^{1}=13 t day^{–1}, where we have used the molecular masses _{N}≃62 g mol^{–1}, _{P}≃95 g mol^{–1} and _{S}≃60 g mol^{–1}. Typical values of offshore water transport caused by wind stress in natural upwelling coastal areas are of the order of 10^{3} m^{3} s^{−}^{1} per km of coastline (upwelling index ^{3} m^{3} s^{−}^{1} km^{−}^{1}). Thus, an upwelling _{v}∼500 m^{3} s^{−}^{1} produced by a wind turbine is roughly equivalent to the natural upwelling along a 500 m coastline.

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 (

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 m^{3} s^{−}^{1}, 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 (

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 (

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 _{0} exp(−_{0}), where _{0}<0 is the height at the vortex centre, and _{0}≃20 km is the horizontal length scale of the vortex. The volume integral of _{0}_{0}^{2}. The mass anomaly of this vortex is therefore M_{0}V_{m}=_{v} will take a time _{m}= _{0}_{v}) to complete the vortex mass anomaly M_{0}=10^{3} kg m^{−}^{3}, ^{−}^{3}, _{0}=1 cm, _{0}=20 km and _{v}=500 m^{3} s^{−}^{1}, we obtain an order of magnitude of the time period ^{7} s∼10^{2} 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 (

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.

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).

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

${\int}_{H}^{0}\frac{\partial {p}_{1}}{\partial z}}\partial z=-{\displaystyle {\int}_{H}^{0}{\rho}_{\text{t}}gdz$, | (A1) |

whose integration is

_{1}(0) − _{1}(_{t} |
(A2) |

or _{1}(_{a}−g_{t}_{a} is the atmospheric pressure. Inside the tube,

${\int}_{H}^{h}\frac{\partial {p}_{2}}{\partial z}}\partial z=-{\displaystyle {\int}_{H}^{h}{\rho}_{\text{b}}gdz$, | (A3) |

whose integration is

_{2}(_{2}(_{t}( |
(A4) |

or _{2}(_{a}−_{b}(_{1}(_{2}(

$h=H\frac{{\rho}_{\text{b}}-{\rho}_{\text{t}}}{{\rho}_{\text{b}}}$. | (A5) |

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

${\int}_{H}^{0}\frac{\partial {p}_{1}}{\partial z}}dz=-g{\displaystyle {\int}_{H}^{0}\left[{\rho}_{\text{t}}+{\vartheta}_{0}z\right]dz$ , | (B1) |

whose integration is

$p}_{1}(0)-{p}_{1}(H)=g{\rho}_{\text{t}}H+g{\vartheta}_{0}\frac{{H}^{2}}{2$ , | (B2) |

or

${p}_{1}(H)={p}_{\text{a}}-g{\rho}_{\text{t}}H-g{\vartheta}_{0}\frac{{H}^{2}}{2}$, |

where _{a} is the atmospheric pressure. Inside the tube,

${\int}_{H}^{h}\frac{\partial {p}_{2}}{\partial z}}dz=-g{\displaystyle {\int}_{H}^{h}{\rho}_{\text{b}}dz$ , | (B3) |

and integrating,

_{2}(_{a} − _{b}( |
(B4) |

Assuming that in the steady state there is no horizontal pressure gradient at _{1}(_{2}(

$-{p}_{\text{t}}-{\vartheta}_{0}\frac{{H}^{2}}{2}={\rho}_{\text{t}}h-{\rho}_{\text{t}}H+g{\vartheta}_{0}Hh-{\vartheta}_{0}{H}^{2}$, | (B5) |

and solving for

$h=\frac{{\vartheta}_{0}{H}^{2}}{2\left({\rho}_{\text{t}}+{\vartheta}_{0}H\right)}$ . | (B6) |

The objective here is to find an arctangent density profile

_{1} + _{2} arctan(_{2} + _{3} |
(C1) |

where _{1}, _{2}, _{3} and _{4} are constants. To do so, we first linearly map the depth range [_{min}, _{max}] into the range of the tangent function [_{min}, _{max}] = [−tan(_{max}), −tan(_{min})], where [_{max}, _{min}] =[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

Therefore,

$Z(z)={Z}_{\mathrm{min}}+(z-{z}_{\mathrm{min}})\frac{{Z}_{\mathrm{max}}-{Z}_{\mathrm{min}}}{{z}_{\mathrm{max}}-{z}_{\mathrm{min}}}$ , | (C2) |

Finally, the arctangent profile arctan[

$\frac{{\rho}_{\text{t}}-\rho (z)}{{\rho}_{\text{t}}-{\rho}_{\text{b}}}=\frac{\mathrm{arctan}({Z}_{\mathrm{max}})-\mathrm{arctan}\left[Z(z)\right]}{\mathrm{arctan}({Z}_{\mathrm{max}})-\mathrm{arctan}({Z}_{\mathrm{min}})}$ , | (C3) |

Thus, we obtain the constants,

$c}_{1}={\rho}_{\text{t}}-\frac{\left({\rho}_{\text{t}}-{\rho}_{\text{b}}\right)\mathrm{arctan}({Z}_{\mathrm{max}})}{\mathrm{arctan}({Z}_{\mathrm{max}})-\mathrm{arctan}({Z}_{\mathrm{min}})$ , | (C4) |

$c}_{2}=\frac{{\rho}_{\text{t}}-{\rho}_{\text{b}}}{\mathrm{arctan}({Z}_{\mathrm{max}})-\mathrm{arctan}({Z}_{\mathrm{min}})$ , | (C5) |

$$c}_{3}={Z}_{\mathrm{min}}-\frac{{z}_{\mathrm{min}}\left({Z}_{\mathrm{max}}-{Z}_{\mathrm{min}}\right)}{{z}_{\mathrm{max}}-{z}_{\mathrm{min}}$$ , | (C6) |

$c}_{4}=\frac{{Z}_{\mathrm{max}}-{Z}_{\mathrm{min}}}{{z}_{\mathrm{max}}-{z}_{\mathrm{min}}$ . | (C7) |

Given the depth range [_{min}, _{max}]=[−300, 0] m, and the density range [_{b}, _{t}] = [1029.1, 1025.6] kg m^{−}^{3}, we obtain {_{1}_{2}_{3}_{4}} = {1027.0, −0.66190, 9.9666, 0.19987}.

Once the arctangent density profile

$${\int}_{{z}_{\mathrm{min}}}^{{z}_{\mathrm{max}}}\rho (z)dz={\int}_{{z}_{\mathrm{min}}}^{{z}_{\mathrm{max}}}\left[{c}_{1}+{c}_{2}\mathrm{arctan}\left({c}_{3}+{c}_{4}z\right)\right]dz$$ . | (C8) |

The arctangent integral

${c}_{2}{\int}_{{z}_{\mathrm{min}}}^{{z}_{\mathrm{max}}}\mathrm{arctan}\left({c}_{3}+{c}_{4}z\right)dz$ , | (C9) |

is done by a change of variable, _{3} + _{4}_{4}

Therefore

$\begin{array}{c}{\displaystyle {c}_{2}\int \mathrm{arctan}(x)\frac{dx}{{c}_{4}}=}\\ {\displaystyle =\frac{{c}_{2}}{{c}_{4}}\int \mathrm{arctan}(x)=}\\ {\displaystyle =\frac{{c}_{2}}{{c}_{4}}\left[x\mathrm{arctan}(x)-\frac{1}{2}\mathrm{ln}\left({x}^{2}+1\right)\right]=}\\ {\displaystyle =\frac{{c}_{2}}{{c}_{4}}{\left[\left({c}_{3}+{c}_{4}z\right)\mathrm{arctan}\left({c}_{3}+{c}_{4}z\right)-\frac{1}{2}\mathrm{ln}\left\{{\left({c}_{3}+{c}_{4}z\right)}^{2}+1\right\}\right]}_{{z}_{\mathrm{min}}}^{{z}_{\mathrm{max}}}}\end{array}$ . | (C10) |

And hence, the pressure outside the tube,

$\begin{array}{c}{\displaystyle {p}_{1}(0)-{p}_{1}(H)=}\\ {\displaystyle =-g{\left[{c}_{1}z+\frac{{c}_{2}}{{c}_{4}}\left({c}_{3}+{c}_{4}z\right)\mathrm{arctan}\left({c}_{3}+{c}_{4}z\right)-\frac{1}{2}\mathrm{ln}\left\{{\left({c}_{3}+{c}_{4}z\right)}^{2}+1\right\}\right]}_{H}^{0}}\end{array}$ , |

and finally,

$\begin{array}{c}{\displaystyle {p}_{1}(0)-{p}_{1}(H)=}\\ {\displaystyle =-g\frac{{c}_{2}}{{c}_{4}}\left[{c}_{3}\mathrm{arctan}\left({c}_{3}\right)-\frac{1}{2}\mathrm{ln}\left({c}_{3}^{2}+1\right)\right]+}\\ {\displaystyle +g{c}_{1}H+\frac{g{c}_{2}}{{c}_{4}}\left({c}_{3}+{c}_{4}H\right)\mathrm{arctan}\left({c}_{3}+{c}_{4}H\right)+}\\ {\displaystyle +\frac{g{c}_{2}}{2{c}_{4}}\mathrm{ln}\left[{\left({c}_{3}+{c}_{4}H\right)}^{2}+1\right]}\end{array}$ . | (C11) |

Inside the tube

_{2}(0) − _{2}(_{b}( |
(C12) |

Equating _{1}(0)=_{2}(0)=_{a}, and _{1}(_{2}(

In this appendix we estimate the order of magnitude of the power _{s}(_{v}) required for sprinkling a water volume transport _{v} over a circle of radius _{0} of a water particle of density _{0} and vertical _{0} velocity components is

${E}_{0}=\frac{1}{2}\rho \left({u}_{0}^{2}+{w}_{0}^{2}\right)$, | (D1) |

From the equations of uniformly accelerated motion _{0}_{0}^{2}, we see that the initial kinetic energy required to throw a water particle a horizontal distance

_{0} = _{0} = $\sqrt{rg/2}$
, and hence the kinetic energy density

${E}_{0}(r)=\frac{\rho g}{2}r$ , | (D2) |

The area differential is _{0}(^{2} ^{3}. As a function of the upwelling volume transport _{v}≡^{2}^{2}_{s}(_{v})≡

$P}_{\text{s}}(R,{F}_{\text{v}})={\kappa}_{0}\frac{\rho g}{3}R{F}_{\text{v}$ . | (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 _{v} over a radius _{0} in _{s}, assuming that typically _{0}∼10.