# An ecologically aware modification of the Morison's equation for long term marine growth effects

Cian Warby, Frederic Dias, Franck Schoefs, Vikram Pakrashi

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Cian Warby, Frederic Dias, Franck Schoefs, Vikram Pakrashi. An ecologically aware modification of the Morison's equation for long term marine growth effects. Mechanics Research Communications, 2024, 139, pp.104293. 10.1016/j.mechrescom.2024.104293. hal-04934448

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# An ecologically aware modification of the Morison's equation for long term marine growth effects

Cian Warby<sup>a</sup>, Frederic Dias<sup>b,c</sup>, Franck Schoefs<sup>d</sup>, Vikram Pakrashi<sup>a,*</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

### Keywords:

Morison's equation

Marine growth

Ecology

Hydrodynamic force

## ABSTRACT

The Morison's equation continues to be an important approximation of the load effect of waves on structures. However, such forces can evolve with time due to marine growth and the evolution of long term growth within the Morison equation is challenging. Incorporation of long term marine growth gives rise to two timescales associated with the Morison equation. A short time scale is associated with the period of a wave, while a long time scale is associated with the change in geometry of the structure due to marine growth. This paper proposes a new modification of the Morison's equation where these multiple time scales of change are addressed. The proposed method allows for a better understanding of how the force predicted by the Morison equation changes over time. The approach allows considering marine growth as an integral part of the Morison equation, expanding the capability to handle a significantly wider range of conditions related to lifetime performance of marine infrastructure, like wind turbines. The proposed approach is compatible with ecological processes and consequently the sampling of randomness over time is physically and biologically viable. The proposed approach allows incorporation of changes in the geometry of these structures through simple biological sampling and subsequently provides a way of updating Morison's equation to provide more accurate force estimates. This ecologically compatible geometry allows for forces estimates closer to reality, departing from the existing and simplistic smooth and rough regimes which are typically used as binary parameters.

## 1. Introduction

Understanding the hydrodynamic forces on marine structures has long been a difficult problem. This is due to a number of uncertainties that must be predicted, the sea state around the structure, the vertical profile of water particle velocity and after a period of deployment even the geometry of the structure. This final uncertainty is what this paper will focus on. Once a marine structure is deployed marine growth will begin to accumulate on the surface of the structure, changing the geometry of the structure. This in turn results in a limited number of methods that can be used to predict the hydrodynamic forces on the structure. The most accurate method of calculating hydrodynamic force would be computational fluid dynamics (CFD) but this requires a detailed description of the geometry of the structure. The proposed solution is to use the Morison equation which can be used to calculate the hydrodynamic force and adapting it as marine growth accumulates.

Morison's equation is an empirical formula widely used to approximate the forces on cylinders or piles placed in the ocean [1,2]. With the rise of offshore renewable energy, the relevance and importance of this equation is more significant than ever in the marine environment [3]. Morison's equation requires that the diameter of the cylinder be less than 20% of the wavelength in this marine environment. The approximation of forces on a cylinder calculated from this equation is typically done due to a lack of accurate and adequate measurements from sources such as pressure sensors. This equation is widely popular in industry, and followed by the American Petroleum Institute, to assess forces expected on equipment or platforms in the ocean [4]. However, with renewable energy aspirations becoming increasingly important globally, these offshore structures (e.g. wind turbines, wave devices) are often slender, flexible and exposed to high natural forces, thereby requiring a close assessment of the expected forces for their lifetime

<footnote><sup>a</sup> UCD Centre for Mechanics, Dynamical System and Risk Laboratory, UCD School of Mechanical and Materials Engineering, University College Dublin, Ireland</footnote>

<footnote><sup>b</sup> School of Mathematics and Statistics, University College Dublin, Ireland</footnote>

<footnote><sup>c</sup> Centre Borelli, ENS Paris-Saclay, France</footnote>

<footnote><sup>d</sup> Research Institute of Civil Engineering and Mechanics (GeM), Sea and Littoral Research Institute (IUML), CNRS, Nantes Université, UMR 6183/FR, 44300 Nantes, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: cian.warby@ucdconnect.ie (C. Warby), frederic.dias@ucd.ie, frederic.dias@ens-paris-saclay.fr (F. Dias), franck.schoefs@univ-nantes.fr (F. Schoefs), Vikram.Pakrashi@ucd.ie (V. Pakrashi).</footnote>
performance. A particularly important aspect in this regard is the need to better understand the effect of marine growth, which can be more pronounced on such structures. Offshore wind turbines are particularly slender and flexible and the impact of such effects has been found to be significant [5]. Effects of marine growth are also considered in normative documents [6]. Despite the importance of marine growth on structures and efforts around design or assessment guidance, there is a fundamental knowledge gap in terms of inability to accommodate such an effect integrally within the Morison's equation [7,8].

Adapting Morison's equation to long term marine growth effects can have significant positive impact in terms of technological readiness levels of offshore renewable solutions and existing coastal or marine infrastructure in terms of conceptual design, scaled wave basin tests, construction, lifetime performance, operations and maintenance, decommissioning and even repowering [9–11].

This paper addresses this knowledge gap directly and provides a new formulation which allows to adapt the Morison equation to multiple timescales and stochasticity coming from marine growth, while respecting ecological and biological processes. Such an adaptation of Morison's equation can lead to rapid design and assessment of structures, while ensuring safety and performance in a realistic manner, in line with natural processes that guide the evolution of these structures over their lifetime. A core challenge behind such adaptation lies in the different timescales over which hydrodynamics and the marine growth operate, along with the randomness associated with them. Often, marine growth effects are assessed by considering simple measurements for biological growth, derived from samples of growth taken from site [12]. Even with modern inspection methods including underwater techniques, the fundamental approach remains the same whereby the changed geometry is input to Morison's equation and the roughness is approximated by considering the samples obtained from a diver with an inspection probe or with underwater cameras. Solving Morison's equation repeatedly for such snapshots of change can be inefficient and also inadequate for a continuous and long term assessment. Also, this approach cannot consider biological or ecological processes that may be available for a certain site, thereby making the estimates unrealistic in certain circumstances. To address these limitations, this paper provides a modification of the Morison's equation which allows creating a framework that can be generally adapted to both estimated and measured conditions of marine growth and can also be calibrated in a site-specific manner.

Apart from fundamentally expanding the capability of the current form of the Morison equation, an additional and practical benefit of using this proposed model lies in the fact that the force changes gradually. This is different from the traditional smooth and rough regime approach, where a sudden regime switch between smooth and rough forces is considered based on changing the values of the Morison equation coefficients [13–15]. Consequently, the proposed model allowing for a gradual change avoids unrealistic effects from sudden changes and brings together physical processes of the ocean along with its ecology and biology.

The work also has an influence on rehabilitation possibilities of coastal structures, including the possible creation of an artificial reef due to the location of several offshore wind platforms and especially in the context of floating energy islands [16].

This paper limits the adaptation of ecology and biology only to hard shells. In particular, we look primarily at the growth of hard bodied organisms such as the blue mussel (Mytilus Edulis) and at the impact this growth has due to an increase in surface roughness as well as the diameter of the cylinder [17]. Soft body growth involves large deformations, which come with a very different challenge in terms of fluid structure interaction, especially due to the large non-commutative rotations [18]. As such, any biological growth that is mentioned in this paper is hard bodied biological growth and is treated as a change in the geometry of the structure rather than soft bodied growth such as seaweeds which have the ability to move and deform. The turbulence

generated by soft bodied growth is also different and will require a separate treatment [19]. The growth considered in this paper is also assumed to be a complete coverage of growth. Therefore effects of coverage while being important will not be considered [20]. Despite being focused on hard growth, the modification remains very widely useful since it is the hard growth for offshore structures around the world that tends to cause higher impacts for increased hydrodynamic forces due to marine growth.

## 2. Theory

The Morison's equation is used to approximate forces on a cylinder due to wave particle motion. This is essentially a semi empirical formula that attempts to summarize the force experienced by a cylinder in two parts: an inertial component and a drag component. This formula is applicable while the diameter of the cylinder is up to 20% of the wavelength impeding on the cylinder or $D/\lambda < 0.2$:

$$F = \frac{1}{2} \rho D C_D \|u(t)\| u(t) + \frac{\pi}{4} \rho D^2 C_M \dot{u}(t). \quad (1)$$

In Eq. (1) D refers to the diameter of the structure and called equivalent diameter in presence of marine growth, $\rho$ refers to the density of water, $C_D$ refers to the drag coefficient, $C_M$ refers to the inertial coefficient and $u$ is the velocity of the wave particle motion. From now on, when referring to the velocity and the acceleration, we will consider the component of the velocity and acceleration in the direction of the wave, assuming that the flow is quasi-one dimensional.

The time dependence here is based on the variation of the velocity and acceleration. This is what will be referred to as the hydrodynamic timescale from now on. The hydrodynamic timescale is of the order of 1 s or less. This is however not the only timescale: we also have the timescale of biological growth. The biological timescale is of the order of tens of hours or days. This timescale exists as a few factors in the Morison equation change as marine growth begins to accumulate on the cylinder in question.

The most important parameters that impact the force from waves are $C_D$ and $C_M$ corresponding to the drag coefficient and the inertial coefficient respectively. Consequently, the change in diameter and the roughness on the cylinder are the two most important aspects to look into and both evolve over time due to marine growth or biofouling, while guided by the ecology and biology of the process. Both $C_D$ and $C_M$ are normally assumed to depend on the same three numbers, Reynolds number (Re), Keulegan–Carpenter number (KC) and relative roughness ($\epsilon$). The Reynolds number is given by $Re = \frac{\mu\lambda}{\nu}$, where $u$ is the wave particle velocity, $\lambda$ is the wavelength of the associated wave and $\nu$ is the kinematic viscosity. The Keulegan–Carpenter number is given by $KC = \frac{uT}{D}$, where $T$ is the wave period and $D$ is the diameter of the cylinder. As biological growth accumulates on the cylinder the roughness increases from a smooth state to a rough state. In this case roughness is defined as the average peak to valley growth ($k$) (as displayed in Fig. 1) of the cylinder over its diameter $e = \frac{k}{D}$. We will refer to this biological long time scale using $\tau$ with $\tau$ defined such that $\frac{l}{\tau} = \epsilon$ and $\epsilon \ll 1$. Using this framework we will now look at the long term change of the Morison equation:

$$F = \frac{1}{2} \rho D(\tau) C_D (Re, KC, e(\tau)) u(t) |u(t)| + \frac{\pi}{4} \rho D(\tau)^2 C_M (Re, KC, e(\tau)) \dot{u}(t). \quad (2)$$

The next step is to examine how the force changes on the biological time scale and hydrodynamic effects are considered locally within the time step. In order to examine the change in force with biological growth the long time scale derivative is taken and the assumption is made that the short time scale effects are independent i.e. that wave statistics do not change over the time scale. These changes are the ones resulting from a changed geometry of the structure, a change in
<figure><img src="image_4.png" /><figcaption>Fig. 1. Illustration of the definition of k from [21] in order to explain how relative roughness is calculated.</figcaption></figure>

diameter and a change in the Morison coefficients with roughness. This change in force is given by the following equation:

$$ \frac{\partial F}{\partial \tau} = \frac{1}{2} \rho \frac{\partial D}{\partial \tau} C_D u(t) |u(t)| + \frac{1}{2} \rho D(\tau) \frac{\partial C_D}{\partial e} \frac{\partial e}{\partial \tau} u(t) |u(t)| + \frac{\pi}{2} \rho D(\tau) \frac{\partial D}{\partial \tau} C_M \dot{u}(t) + \frac{\pi}{4} \rho D(\tau)^2 \frac{\partial C_M}{\partial e} \frac{\partial e}{\partial \tau} \dot{u}(t) + \mathcal{O}(\epsilon). \quad (3) $$

The $\mathcal{O}(\epsilon)$ terms refer to variations in the velocity and acceleration over the timescale of biological growth and the assumption is that these changes are significantly less than those associated with the four main terms in eq (3).

Taking a closer look at the four terms, we see that in the first order difference two of the terms contain $\frac{\partial D}{\partial \tau}$, which is the rate of change of diameter: $\frac{1}{2} \rho \frac{\partial D}{\partial \tau} C_D u(t) |u(t)|$ and $\frac{\pi}{2} \rho D(\tau) \frac{\partial D}{\partial \tau} C_M \dot{u}(t)$. In other words, this is twice the rate of growth as we presume equal growth on both sides of the cylinder. This figure can be found through site inspection as well as estimated from databases that contain growth data from similar locations using resources such as [22,23].

Looking at the second and fourth terms in eq (3) we see that both contain the $\frac{\partial e}{\partial \tau}$ term which is the rate of change of relative roughness with time: $\frac{1}{2} \rho D(\tau) \frac{\partial C_D}{\partial e} \frac{\partial e}{\partial \tau} u(t) |u(t)|$ and $\frac{\pi}{4} \rho D(\tau)^2 \frac{\partial C_M}{\partial e} \frac{\partial e}{\partial \tau} \dot{u}(t)$. Since relative roughness is defined as the average peak to valley distance over the diameter of the cylinder, we can use the geometry of the organism to estimate this roughness. The approach used here for example is to take half the maximum organism length over the diameter of the cylinder as an approximation of this roughness. It is important to note that this relative roughness is an approximation around the entire cylinder at any given depth but this is the convention as most tests which calculate the Morison coefficients using roughness do so with a homogenized roughness [1,13,24].

Using the well known wave basin tests conducted originally by Sarpkaya [2], functions have been interpolated to give the Morison equation coefficients for various Reynolds numbers at different values for roughness and Keulegan-Carpenter number KC = 20 Figs. 2 and 3. Keulegan-Carpenter number KC = 20 was chosen as for the upper percentile of forces KC ≈ 20 for all time. Fig. 2 is a direct adaptation of the experiments performed by Sarpkaya [2]. The Morison force can then be calculated using a growth profile developed from the local area as well as wave heights reported from the local area. The local growth in particular can be used to calculate the change in Morison force as defined in eq (3) allowing calculation of the evolution of the force over time for a given area. Using wave height statistics and forecasting a force probability model can be generated [25]. This is done by taking a wave height spectral distribution and calculating the associated velocities and acceleration and subsequently calculating the force while taking into account the accumulated growth on the cylinder as well as account for the associated change in the Morison coefficients. Using this method it is possible to create a force probability distribution and this can then be used to inform fatigue studies as well as lifetime estimates for the site.

<figure><img src="image_5.png" /><figcaption>Fig. 2. Morison coefficient $C_D$ versus Reynolds number for various levels of roughness.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Morison coefficient $C_M$ versus Reynolds number for various levels of roughness.</figcaption></figure>

## 2.1. Higher-order expansion

Until this point we have only looked at the first order change in force on the long time scale and assumed that higher order terms are negligible. To assess whether higher order terms are of consequence or not, we will look at the second order term and evaluate its magnitude:

$$ \frac{4}{\rho} \frac{\partial^2 F}{\partial \tau^2} = \pi \dot{u}(t) \left[ \frac{\partial^2 C_M}{\partial e^2} \left( \frac{\partial e}{\partial \tau} \right)^2 D^2 + \frac{\partial C_M}{\partial e} \frac{\partial^2 e}{\partial \tau^2} D^2 + 4D \frac{\partial C_M}{\partial e} \frac{\partial e}{\partial \tau} \frac{\partial D}{\partial \tau} + 2C_M D \frac{\partial^2 D}{\partial \tau^2} + 2C_M \left( \frac{\partial D}{\partial \tau} \right)^2 \right] + 2u(t) |u(t)| \left[ \frac{\partial^2 C_D}{\partial e^2} \left( \frac{\partial e}{\partial \tau} \right)^2 D + \frac{\partial C_D}{\partial e} \frac{\partial^2 e}{\partial \tau^2} D + 2 \frac{\partial C_D}{\partial e} \frac{\partial e}{\partial \tau} \frac{\partial D}{\partial \tau} + C_D \frac{\partial^2 D}{\partial \tau^2} \right]. \quad (4) $$

In order for the higher order terms to be negligible we require that $|\frac{\partial F}{\partial \tau}| \gg \Delta\tau |\frac{\partial^2 F}{\partial \tau^2}|$. In order to understand the magnitude of $\frac{\partial^2 F}{\partial \tau^2}$ the force will be calculated using both the first and second changes and compared. Further justification for not including the second order terms will be included in the following section where the magnitude of the second order term will be calculated. Another issue with the higher order terms is that they can be quite difficult to calculate since many of the terms rely on the acceleration of growth. Demonstrating these terms to be negligible makes it easier to adapt the modifications proposed in this paper to Morison's equation. Growth measurements are not
taken very regularly in marine settings due to the cost and difficulties involved with such measurements [26]. Due to this, acceleration of growth will have to be inferred from some type of growth model and currently marine growth models are often basic in nature [27]. However, with the proposed approach it is possible to accommodate current models and more complex models in the future, as well as monitoring and data driven model updates [28,29]. One such model will be outlined and an example of how to apply this theory will be shown in the following section.

Changes in acceleration and velocity are excluded but can be considered as the seasonal changes in average wave particle velocity and acceleration resulting from the seasonal changes in wave height. The change in such a velocity can be calculated using the time averaged value of each term over the long time step.

$$ \frac{\partial}{\partial \tau} \left( \frac{\partial u}{\partial t} \right) = \frac{\partial}{\partial t} \left( \frac{\partial u}{\partial \tau} \right) = \frac{\partial}{\partial t} \left( \frac{\lambda}{T} \right) \\ = \frac{\partial}{\partial t} \left( \frac{1}{T} \frac{\partial \lambda}{\partial \tau} \right) + \frac{\partial}{\partial t} \left( \lambda \frac{\partial}{\partial \tau} \left( \frac{1}{T} \right) \right). \qquad (5) $$

Neglecting changes in acceleration and velocity is not necessary but simplified changes in velocity and acceleration can be calculated using either on site measurements such as satellite measurements or else using forecasting models if attempting to predict forces for the future or as mentioned before using data collecting buoys.

### 3. Application

Using growth parameters developed using test sites in the North Atlantic [12] a growth model in both time and space was developed. This growth model relies on three fundamental behaviours of our model hard growth organism the blue mussel (Mytulis Edulis). These behaviours are growth through feeding on organisms in the water column, a negative growth or death term through crowding and a diffusion term:

$$ \frac{\partial G(z, \tau)}{\partial \tau} = \left( p(z)G(z, \tau) - dG(z, \tau)^2 + \nu \frac{\partial^2 G(z, \tau)}{\partial z^2} \right) DB. \qquad (6) $$

In the above equation G refers to the extent of growth, p refers to an availability of nutrients at a given depth, d refers to a death constant associated with crowding, ν is the diffusion coefficient of the mussels and DB refers to a random sampled increment that is comprised of a constant term and a Gaussian sampled term. This generates different extents of growth for the same conditions. This random term is included to help capture some of the unpredictability of growth as two different organisms of the same species can grow at different rates under the same conditions due to genetic differences. Figs. 4 and 5 show two realizations of this randomly generated mature growth. Fig. 6 shows the progress of growth over time where the impact of seasonality can be seen on the growth once it is mature in the fluctuations in extent of growth. In reality, real world growth is significantly more complex but when sampled over long time intervals, the model considered here can be adequate to represent the need for hydrodynamic force estimates to be a function of biological processes. More realistic growths, which consider several environmental factors like chlorophyll content and temperature in the surrounding waters, can also be accommodated in the approach proposed in this paper without modification [21]. To demonstrate the biological and ecological compatibility of the modified Morison's equation considered in this paper, multiple growth cycles were introduced at different time steps but follow the same growth rate. This is done in order to simulate the multiple spawning events that give rise to growth in Mytulis Edulis, which is the example species considered in this paper [21]. This delay is the result of multiple fixation events from the larval form of Mytulis Edulis present in the water column.

In order to create an accurate estimate for the force on the cylinder at a given depth an accurate depth dependent velocity and acceleration profile is required. The velocity and acceleration used are generated

<figure><img src="image_7.png" /><figcaption>Fig. 4. An example of randomly generated mature growth with depth using the death coefficient of 0.006 and a diffusion coefficient of 5e-6 (Growth 1).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Another example of randomly generated mature growth with depth using the death coefficient of 0.004 and a diffusion coefficient of 8e-6 (Growth 2).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. The progression growth over time at the surface of the structure over a 5 year period (Growth 1).</figcaption></figure>

using the reported or forecast wave height. This is useful as forecast data usually reports in parameters such as significant wave height
<figure><img src="image_10.png" /><figcaption>Fig. 7. Morison force profile on a smooth cylinder with the upper and lower bounds referring to two standard deviations of wave height above and below the mean respectively.</figcaption></figure>

which then provides information on the distribution of wave height which subsequently provides the distribution of velocity, acceleration, Reynolds number etc. In this paper wave height data and statistics were sampled for a location off the west coast of Ireland at coordinates 53.018, -9.782 [30]. For sites where real world data is not available significant wave height forecasting can be used instead. In this case, the probability distribution of wave height can be defined using (7)

$$P(H) = \frac{2H}{H_{rms}^2} \exp\left(-\left(\frac{H^2}{H_{rms}^2}\right)\right) \quad (7)$$

where H refers to the wave height and $H_{rms}$ is the root mean square of the wave height. This wave height is then used to generate the water particle velocity and acceleration which is then used in the Morison equation. According to the first order Stokes model the water particle velocity and acceleration are given by the following depth dependent equations:

$$u = \pi H \frac{\cosh(2\pi(d-z)/\lambda)}{T \sinh(2\pi d/\lambda)}, \quad (8)$$

$$\dot{u} = 2\pi^2 H \frac{\cosh(2\pi(d-z)/\lambda)}{T^2 \sinh(2\pi d/\lambda)}, \quad (9)$$

where z refers to the depth, with the mean surface level at z = 0 and the bottom located at z = d. The period T, the wavelength λ and the wave heights used are the averaged values of all wave statistics over the long time step. It is important to note that these velocity and acceleration equations contain no sinusoidal elements which would normally be present. We are however only interested in the magnitude of the force not the fluctuation over the short time scale. This is the reason for the exclusion of this term. This does create the assumption that the drag and inertial forces are in phase. This can be solved by performing an evaluation of the force within each timestep. However the purpose of this paper is to investigate the evolution of force: this is the reason for this simplification.

Using the velocity and acceleration generated from wave height and the geometry of the structure generated using the growth model we can investigate the change in force with time. The force generated in this application is for a cylinder with a 2 m diameter and 50 m depth. The lower and upper percentiles refer to two standard deviations in wave height above and below the average wave height. The increase in force in the cylinder due to roughness between the smooth cylinder Fig. 7 and the rough cylinders Figs. 8 and 10 is clearly observed. The jump behaviour in the force measurements is due to the Morison equation falling into different roughness bands giving different Morison

<figure><img src="image_11.png" /><figcaption>Fig. 8. Morison force profile on a rough cylinder with the upper and lower bounds referring to two standard deviations of wave height above and below the mean respectively.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Morison force profile on a rough cylinder by only considering Morison coefficient and change in diameter, with the upper and lower bounds referring to two standard deviations of wave height above and below the mean respectively.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Morison force profile for growth 2 on a rough cylinder with the upper and lower bounds referring to two standard deviations of wave height above and below the mean respectively.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Morison force profile on a rough cylinder by only considering Morison coefficient and change in diameter, with the upper and lower bounds referring to two standard deviations of wave height above and below the mean respectively.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Second order difference in force over time calculated as the cylinder accumulates growth with the upper and lower percentiles referring to two standard deviations of wave height above and below the mean respectively.</figcaption></figure>

coefficients which can be seen in Figs. 2 and 3. These calculations are done using the first order change in force only. An example of the force increase in force using the conventional Morison equation can be seen in Figs. 9 and 11. Fig. 12 shows the second order difference in force for growth 1. As can be seen the second order term has a balanced contribution. The sum total contribution from the second order term is $4N$. This is less than 0.1% of the overall force and excluding this contribution will not have a significant impact on the force calculation while significantly speeding up calculations. Fig. 13 shows the fast Fourier transform of the force time series produced by growth 1. This Fourier transform is rough as the frequency has no effect on the second order change in force, these jumps come from the transition between roughness regimes such as those that can be seen in Figs. 2 and 3.

Using the proposed theory of biological growth along with wave height distributions, a probability distribution of hydrodynamic force over time can be obtained. This force distribution can be used to inform failure models and provide estimates on the lifetime of the equipment. This model can also be improved in a number of ways such as providing a more accurate wave height distribution than the one provided, improving the accuracy of the growth model with information gathered from the experiment site [12], as well as using more complicated growth models such as those described in [21].

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** The fast Fourier transform of the second order difference in force over time calculated as the cylinder accumulates growth with the upper and lower percentiles referring to two standard deviations of wave height above and below the mean respectively.</figcaption></figure>

The distribution of force on the cylinder is also significantly changed due to the presence of marine growth. Since marine growth is concentrated towards the surface the force experienced increases at a faster rate than the force at greater depths. This is due to an increase in both diameter and roughness. This will impact long term maintenance fatigue and also stress cycles which will tell us whether repowering of a location due to geotechnical fatigue is possible or not. The changes in the distribution pattern due to marine growth will have an effect on fatigue life and even types of inspection that should be done to get better values [31]. Finally it is important to note that the presence of marine growth itself also has an effect in corrosion further reducing expected lifetime of the structure [32,33].

Even with all these complexities present, the modification presented in this paper allows for a natural incorporation of several of these processes in a reasonably manner, allowing for both current and future inspection and monitoring aspects.

## 4. Conclusion

This paper presents a modification of the Morison's equation to accommodate marine growth and biofouling and allow long term prediction of forces and their distributions in marine structures, combining hydrodynamic, biological and ecological processes. We address the challenges of such processes happening in significantly different timescales and the stochasticity associated with them. The proposed model can be used for simple or complex biological growth scenarios, as well as for measured or estimated data. This model can be used to provide a more accurate estimate of extreme forces than currently used methods.

An ensemble of these randomly generated growth models can be used to provide estimates of the possible force distributions in time when combined with wave forecasting. Site specific inspection can also be used to reduce the uncertainty associated with marine growth and therefore force. Future improvements to this method would involve a more complex model for the calculation of the Morison coefficient. As can be seen from Figs. 2 and 3 we are still reliant on falling into a range of values for roughness. Further wave basin roughness tests and machine learning based approach to prediction are possible solutions in reducing this uncertainty. Ideally a large database of Morison coefficient values would be created for various values of Reynolds number, Keulegan-Carpenter number and relative roughness providing more points and increased accuracy for Morison coefficient estimates but
until that is done there is a certain amount of uncertainty associated with each Morison equation estimate.

The current biological model accounts for the treatment of hard bodied growth and future work should be done to account for the effects of deformable soft bodied growth which come with their own challenges in fluid structure interaction. Future models can also attempt to account for species-species interaction as well as the possibility of detaching events in extreme conditions.

### CRediT authorship contribution statement

**Cian Warby:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Conceptualization. **Fred-eric Dias:** Writing – review & editing, Supervision, Software, Resources, Methodology, Investigation, Funding acquisition, Conceptualization. **Franck Schoefs:** Writing – review & editing, Supervision, Resources, Methodology, Investigation, Funding acquisition, Conceptualization. **Vikram Pakrashi:** Writing – review & editing, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors confirm that there exists no conflict of interest.

### Data availability

Data will be made available on request.

### Acknowledgements

This work was funded by Science Foundation Ireland (SFI) under Marine Renewable Energy Ireland (MaREI), the SFI Research Centre for Energy, Climate and Marine (grant 12/RC/2302-2), Capacités SAS and project SIMAR/JUNIC co-founded by Région Pays de la Loire, la CARENE, WEAMEC, Les Ports de Loire Atlantique and CAP 44. Vikram Pakrashi would also like to acknowledge Sustainable Energy Authority of Ireland TwinFarm RDD/604 as well as SFI NexSys, Ireland 21/SPP/3756 and Interreg SISdATA, Ireland EAPA-0040/2022.

### References

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