2.1.1. Buoyancy and added mass

We decompose total pressure $p$ into an undisturbed component $p_{undisturbed}$ and a disturbed component $p_{disturbed}$ owing to the presence of the object. Assuming an object that is small relative to the wavelength ($D/\lambda _0\ll 1$), the undisturbed pressure varies as $p_{undisturbed}=\rho _f g(\eta (x,t)-z)$ on the scale of the object with $\rho _f$ the density of the fluid, so that the dynamic free surface boundary condition $p_{undisturbed}(z=\eta )=0$ is satisfied, the variation with depth is hydrostatic, and any depth-dependent variation owing the waves (cf. $\exp (k_0 z)$ with $k_0$ the wavenumber) is ignored.

The undisturbed pressure integrated around the wetted surface results in a buoyancy force acting in the normal direction to the free surface,

(2.11)\begin{equation} B_n(t)=\frac{g m }{\beta} \frac{V_{{s}}}{V}\varXi^{{-}1}_p = \frac{g m }{\beta}\left(3\left(\frac{s(t)}{D}\right)^2-2\left(\frac{s(t)}{D}\right)^3\right)\varXi^{{-}1}_p , \end{equation}

where $g$ is the gravitational constant, $V_s$ is the submerged, $V$ the total volume of the sphere and $\beta \equiv \rho _o/ \rho _f$ is the ratio of object to fluid density. By including $\rho _f g\eta (x,t)$ in the undisturbed pressure, we have included the Froude–Krylov force resulting from the waves.

The disturbed component of pressure leads to added-mass terms, as derived by Maxey & Riley (Reference Maxey and Riley1983)

(2.12a,b)\begin{equation} M_\tau = \frac{C_{m,\tau}(s) m}{\beta} (\dot{u}_\tau(\boldsymbol{\tilde{x}}_p,t)-\dot{v}_\tau)\quad \text{and}\quad M_n = \frac{C_{m,n}(s)m}{\beta} (\dot{ u}_n(\boldsymbol{\tilde{x}}_p,t)-\dot{v}_n), \end{equation}

where $\boldsymbol {C}_m=(C_{m,\tau },C_{m,n})$ is the added-mass coefficient, which is deliberately left as an unspecified function of submergence $s(t)$ at this stage of the derivation.

The small-diameter assumption leaves the vertical location, where we should evaluate the velocity of the surrounding fluid in (2.12), unspecified. We set this location to be at the free surface, $\tilde {\boldsymbol {x}}_p=(x_p,\eta _p)$.

2.1.2. Gravity forces

The gravity force acts in the vertical direction, and has the following components in the moving coordinate system:

(2.13a,b)\begin{equation} G_\tau(t)={-}m g\partial_x \eta|_{x_p} \varXi_p(t) \quad \text{and}\quad G_n(t)={-}m g\varXi_p(t). \end{equation}

2.1.3. Resistance forces

The resistance terms are caused by drag on the object when it has a velocity relative to that of the surrounding liquid. To begin, we assume viscous drag. We assume this drag depends on the submergence of the object and, specifically, we assume the drag is proportional to the submerged projected area of the sphere in the tangential and normal directions (see figure 2). Other drag formulations are discussed and examined in § 4. The resistance force in the tangential direction is,

(2.14)\begin{equation} R_{\tau}= 3{\rm \pi} \rho_f \nu D \hat{A}_{{s},\tau}(u^*_{\tau}-v^*_{\tau}), \end{equation}

where $u^*_{\tau }$ and $v^*_{\tau }$ are the velocity components in the $\tau$-direction of the surrounding fluid and the object velocity respectively (in the moving reference frame). The normalised area in the tangential direction $\hat {A}_{{s},\tau }$ is the projected area of the submerged sphere,

(2.15)\begin{equation} A_{{s},\tau}=\frac{D^2}{8}\left(\zeta-\sin(\zeta)\right) \quad \text{with}\ \zeta\equiv 2\cos^{{-}1}\left(1-2s/D\right), \end{equation}

normalised by the maximum projected area $A= {\rm \pi} D^2/4$, so that $\hat {A}_{{s}, \tau } = A_{{s},\tau }/A$. Assuming the drag is proportional to the submerged projected area following Beron-Vera et al. (Reference Beron-Vera, Olascoaga and Lumpkin2016), which has been validated for steady flows (Miron et al. Reference Miron, Medina, Olascoaga and Beron-Vera2020; Olascoaga et al. Reference Olascoaga, Beron-Vera, Miron, Triñanes, Putman, Lumpkin and Goni2020), we evaluate the fluid velocity $u^*_{\tau }$ at the free surface, $\tilde {\boldsymbol {x}}_p=(x_p,\eta _p)$.

Figure 2.

Diagrams of (a) the submerged volume $V_{{s}}$ as a function of the variable submergence $s(t)$; (b) the projected area of a submerged sphere moving in the normal direction ($\boldsymbol {e}_n$); and (c) the projected area of a submerged sphere moving in the tangential direction ($\boldsymbol {e}_\tau$). All diagrams are shown in the ($\tau$, $n$) coordinate system.

Similar to the $\tau$-direction, we have for the $n$-direction,

(2.16)\begin{equation} R_{n}=3{\rm \pi} \rho_f \nu D \hat{A}_{{s},n}(u^*_{n}-v^*_{n}), \end{equation}

where we have evaluated the velocity of the surrounding fluid at the same location $\tilde {\boldsymbol {x}}_p$ as for the tangential resistance force. The submerged projected area of a sphere in the normal direction is given by (see figure 2)

(2.17)\begin{equation} A_{{s},n}={\rm \pi} s(t)\left(D-s(t)\right), \end{equation}

which again, is normalised by the maximum projected area of a sphere $A= {\rm \pi} D^2/4$, so that $\hat {A}_{{s},n}= A_{{s},n}/A$. Later, in § 4, other drag formulations are considered to examine the robustness of the model's predictions.

3.2.1. The tangential direction

To first order of approximation, the velocity and acceleration in the horizontal coordinate $x$ and the tangential coordinate $\tau$ are equal, i.e. $\dot {x}_p^{(1)}=v_x^{(1)}=v_{\tau }^{(1)}$ and $\ddot {x}_p^{(1)}=\dot {v}_x^{(1)}=\dot {v}_{\tau }^{(1)}$. The only forces that play a role are the tangential components of the added mass, gravity and the resistance force. The first-order added-mass terms in the tangential direction are

(3.4)\begin{equation} M_\tau^{(1)}=\frac{C_m m}{\beta}( \dot{u}_x^{(1)} -\ddot{x}_p^{(1)}), \end{equation}

where we now assume for simplicity that the added-mass coefficient $C_m$ is a constant and independent of direction. Other added-mass formulations are discussed and examined in § 4.

In a potential flow, a fully submerged sphere has an added-mass coefficient of $1/2$. Instead of deriving the complicated dependence of $C_m$ on the object's density, we interpolate linearly between the values for a sphere that is fully submerged ($\beta =1$, $C_m=1/2$) and a sphere that is entirely out of the water ($\beta =0$, $C_m=0$) and set $C_m= \beta /2$. The robustness of this assumption is investigated numerically in § 4.

The resistance force (2.14) can be approximated as

(3.5)\begin{equation} R_\tau^{(1)}=\varGamma_R m\omega_0 \hat{A}_{{s},\tau}^{(0)}(u_x^{(1)}|_{\tilde{\boldsymbol{x}}_{p}^{(0)}}-\dot{x}_p^{(1)}) \quad \text{with}\ \varGamma_R\equiv \frac{3 {\rm \pi} \nu D}{\beta V \omega_0},\end{equation}

where the non-dimensional coefficient $\varGamma _R$ measures the importance of the resistance force.

From the object's equation of motion (2.1) we thus obtain

(3.6)\begin{equation} \left(1+\frac{C_m}{\beta}\right) \ddot{x}_p^{(1)}=\frac{C_m}{\beta} \dot{u}_x^{(1)}|_{\tilde{x}_p^{(0)}} - g \partial_x \eta^{(1)}|_{x_{p}^{(0)}}+\varGamma_R \hat{A}^{(0)}_{{s},\tau}\omega_0 (u_x^{(1)}|_{\tilde{\boldsymbol{x}}_{p}^{(0)}}-\dot{x}_p^{(1)}). \end{equation}

We seek a solution to the forced second-order ordinary differential equation (3.6) of the form $x_p^{(1)}=\mathcal {R}(i {X}^{(1)}a_0 \exp (\textrm {i} \varphi _{p}^{(0)}))$ with $\varphi _{p}^{(0)}=k_0x^{(0)}_p-\omega _0 t+\varphi _0$ and $\varphi _0=\arg (A_0)$, ignoring initial transients. The complex coefficient $X^{(1)}$ represents the amplitude and phase change of the horizontal motion of the object relative to those of an idealised Lagrangian object under the influence of waves at the same order, $x_{L}^{(1)}=\mathcal {R}(i a_0 \exp (\textrm {i} \varphi _{p}^{(0)}))$. We obtain $X^{(1)}=1$, i.e. there is no horizontal motion amplification compared to that of a Lagrangian particle.

3.2.2. The normal direction

Expressing the submergence depth $s$ in terms of the vertical coordinate $z_p$, we have, without approximation, that $s=D/2-(z_p-\eta _p)\varXi _p$. Therefore, the velocity and acceleration in the vertical coordinate $z$ and the normal coordinate $n$ are related to first order by

(3.7a,b)\begin{equation} \dot{z}_p^{(1)}=v_z^{(1)}={-}\dot{s}^{(1)}+\dot{\eta}_p^{(1)} \quad \text{and}\quad \ddot{z}_p^{(1)}=\dot{v}_z^{(1)}={-}\ddot{s}^{(1)}+\ddot{\eta}_p^{(1)}. \end{equation}

We first approximate the buoyancy force (2.11) by

(3.8)\begin{equation} B_n^{(1)}= \varGamma_B m\omega_0^2 s^{(1)} \quad \text{with} \ \varGamma_B\equiv \frac{6}{\beta k_0 D}\left(\frac{s^{(0)}}{D}-\left(\frac{s^{(0)}}{D}\right)^2\right), \end{equation}

the added-mass terms by

(3.9)\begin{equation} M_n^{(1)}=\frac{C_m m}{\beta} \ddot{s}^{(1)}, \end{equation}

and the resistance force (2.16) by

(3.10)\begin{equation} R_n^{(1)}=\varGamma_R m\omega_0 \hat{A}_{{s},n}^{(0)}\dot{s}^{(1)}, \end{equation}

where we have used $u_z^{(1)}(z=0)=\dot {\eta }_p^{(1)}$ from the linearised kinematic free surface boundary condition and $v_n^{(1)}=\dot {z}_p^{(1)}$. The new non-dimensional coefficient $\varGamma _B$ measures the strength of dynamic buoyancy, and $\varGamma _R$ measures the strength of the resistance force, as for the tangential resistance force in (3.5). From the object's equation of motion (2.1) we thus obtain

(3.11)\begin{equation} \left(1+\frac{C_m}{\beta}\right)(\ddot{\eta}^{(1)}_p-\ddot{s}^{(1)}) =\frac{C_m}{\beta}\dot{u}_z^{(1)}|_{\tilde{x}_p^{(0)}} +\varGamma_B \omega_0^2s^{(1)} +\varGamma_R \hat{A}^{(0)}_{{s},n}\omega_0 \dot{s}^{(1)}, \end{equation}

where we note gravity only enters at zeroth order. As for the tangential direction, we seek a solution to the forced second-order ordinary differential equation (3.11) of the form $s^{(1)}=\mathcal {R}(\mathcal {S}^{(1)}a_0 \exp (\textrm {i} \varphi _{p}^{(0)}))$ with $\varphi _{p}^{(0)}=k_0x^{(0)}_p-\omega _0 t+\varphi _0$ and $\varphi _0=\arg (A_0)$, ignoring initial transients. We find for the non-dimensional submergence at first order $\mathcal {S}^{(1)}$

(3.12)\begin{equation} \mathcal{S}^{(1)}=\frac{1+\displaystyle \frac{C_m}{\beta}-\varGamma_B-\textrm{i}\varGamma_R \hat{A}^{(0)}_{{s},n}}{\left(1+\displaystyle \frac{C_m}{\beta}-\varGamma_B\right)^2+(\varGamma_R \hat{A}^{(0)}_{{s},n})^2}. \end{equation}

Figures 3 and 4 respectively show the magnitudes and arguments of the first-order solutions for the horizontal motion amplification $X^{(1)}$ and the variable submergence $\mathcal {S}^{(1)}$. In these figures, the purely Lagrangian limit, in which the object is simply transported with the Stokes drift and floats on the moving surface, corresponds to $X^{(1)}=1$, $\mathcal {S}^{(1)}=0$. This limit is obtained as the object size tends to zero. Note that the phase of variable submergence in this limit is non-zero, $\text {arg}(\mathcal {S}^{(1)})\rightarrow {\rm \pi} /2$. This is because both imaginary and real parts of the variable submergence tend to zero, with the imaginary part approaching zero at a faster rate. As our model is only valid for objects that are small relative to the wavelength, we truncate the $x$-axis at $D/\lambda _0=6\,\%$. Diffraction of the wave field typically only becomes important for $D/\lambda _0>20\,\%$.

Figure 3.

For viscous drag, magnitudes of the first-order horizontal motion amplification $X^{(1)}$ (a) and the variable submergence $\mathcal {S}^{(1)}$ (b) as functions of dimensionless object size $D/\lambda _0$ for different density ratios $\beta =\rho _o/\rho _f$, where the density ratio for each colour is shown in the legend. We have set $C_m= \beta /2$. Numerical and analytical solutions from perturbation theory are denoted by crosses and solid lines, respectively.

Figure 4.

For viscous drag, arguments of the first-order horizontal motion amplification $X^{(1)}$ (a) and the variable submergence $\mathcal {S}^{(1)}$ (b) as functions of dimensionless object size $D/\lambda _0$ for viscous drag and for different density ratios $\beta =\rho _o/\rho _f$, as shown in the legend. We have set $C_m= \beta /2$. Numerical and analytical solutions from perturbation theory are denoted by crosses and solid lines, respectively.

As confirmed in figure 3(a), the magnitude of the horizontal motion $|X^{(1)}|$ is equivalent to that of a purely Lagrangian tracer. Turning to figure 4(a), the argument of the horizontal motion $\arg (X^{(1)})$ is evidently also zero. As shown in figure 3(b), the magnitude of the variable submergence $|\mathcal {S}^{(1)}|$ increases monotonically with object size and does so at a larger rate for density ratios closer to unity. Variable submergence is driven by the free surface elevation and governed by drag, dynamic buoyancy and (added) mass, which are respectively the resistance, spring and inertia terms of a forced spring–mass–damper system (cf. (3.11)). The larger the object, the more dominant is the acceleration of the free surface, which acts as an apparent force in the moving reference frame in which the variable submergence is defined, thus increasing the ‘bobbing’ of the object. The lower the density ratio, the stronger the buoyancy force and the stiffer the ‘spring’. The response in variable submergence for a stiffer ‘spring’ is smaller. The argument of variable submergence $\arg (\mathcal {S}^{(1)})$ decreases monotonically with object size and growing importance of inertia but is dependent on the density ratio, as shown in figure 4(b).

At first order in steepness the tangential and normal directions are independent, and so it is possible for there to be a significant change in first-order variable submergence whilst the first-order horizontal motion remains unchanged. As can be seen in the next section, a change in first-order variable submergence results in a change in horizontal motion at second order.

3.3.1. Tangential and normal directions

In the tangential direction, the added-mass terms at second order can be obtained from the combination of an expansion in the horizontal and vertical displacements of the object, a coordinate transformation and evaluation of the advective derivative, respectively

(3.14)\begin{align} M_\tau^{(2)} &= \frac{C_m m}{\beta} (\dot{u}_x^{(2)} +x_p^{(1)}\partial_x \dot{u}_x^{(1)}|_{\tilde{x}_p^{(0)}} +\eta_p^{(1)} \partial_z \dot{u}_x^{(1)}|_{\tilde{x}_p^{(0)}}\nonumber\\ &\quad +\dot{u}_z^{(1)}|_{\tilde{x}_p^{(0)}} \partial_x \eta^{(1)}|_{x_p^{(0)}} +\dot{x}_p^{(1)} \partial_x u_x^{(1)}|_{\tilde{x}_p^{(0)}} +\dot{\eta}_p^{(1)} \partial_z u_x^{(1)}|_{\tilde{x}_p^{(0)}} -\dot{v}_{\tau}^{(2)}). \end{align}

In addition to the added-mass terms, the tangential force consists of a correction to the tangential component of gravity due to the object's horizontal displacement,

(3.15)\begin{equation} G_{\tau}^{(2)}=\left.-mg\partial_{xx}\eta^{(1)}\right|_{x_p^{(0)}}x_{p}^{(1)}, \end{equation}

and a tangential resistance force,

(3.16)\begin{equation} R_{\tau}^{(2)}=3{\rm \pi}\rho_f\nu D(\hat{A}_{{s},\tau}^{(1)}(u_{\tau,p}^{(1)}-v_{\tau}^{(1)}) +\hat{A}_{{s},\tau}^{(0)}(u_{\tau,p}^{(2)}-v_{\tau}^{(2)})). \end{equation}

For the first-order velocity components, we have $u_{\tau ,p}^{(1)}=u_x^{(1)}|_{\tilde {\boldsymbol {x}}_p^{(0)}}$ and $v_{\tau }^{(1)}=\dot {x}_{p}^{(1)}$. Noting from the coordinate transformation that $u_\tau =u_x+\partial _x\eta |_{x_p}u_z+{O}(\alpha ^3)$, we obtain for the second-order accurate horizontal fluid velocity component at the object position

(3.17)\begin{equation} u_{\tau,p}^{(2)}= u_{x}^{(2)}|_{\tilde{\boldsymbol{x}}_p^{(0)}} +\partial_x u_{x}^{(1)}|_{\tilde{\boldsymbol{x}}_p^{(0)}}x_p^{(1)} +\partial_z u_{x}^{(1)}|_{\tilde{\boldsymbol{x}}_p^{(0)}}\tilde{z}_p^{(1)} +\partial_x\eta^{(1)}|_{x_p^{(0)}}u_z^{(1)}|_{\tilde{\boldsymbol{x}}_p^{(0)}}. \end{equation}

We set the second-order Eulerian wave-induced velocity $u_{x}^{(2)}$ to zero for the regular waves considered here. The object's horizontal velocity component at second order is

(3.18)\begin{equation} v_{\tau}^{(2)}=\dot{x}_{p}^{(2)}+\partial_x \eta^{(1)}|_{x_p^{(0)}}\dot{z}_{p}^{(1)}, \end{equation}

where $\dot {x}_{p}^{(2)}$ is the quantity that is ultimately of interest. Combining (3.17) and (3.18) and substituting into (3.16) gives

(3.19)\begin{align} R_{\tau}^{(2)}&=3{\rm \pi}\rho_f\nu D( \hat{A}_{{s},\tau}^{(1)}(u_x^{(1)}|_{\tilde{\boldsymbol{x}}_p^{(0)}}-\dot{x}_{p}^{(1)})\nonumber\\ &\quad +\hat{A}_{{s},\tau}^{(0)}( \partial_x u_{x}^{(1)}|_{\tilde{\boldsymbol{x}}_p^{(0)}}x_p^{(1)} +\partial_z u_{x}^{(1)}|_{\tilde{\boldsymbol{x}}_p^{(0)}}\eta_p^{(1)} -\dot{x}_{p}^{(2)}+\partial_x \eta^{(1)}|_{x_p^{(0)}}\dot{s}^{(1)})),\end{align}

where we have substituted $u_x^{(2)}=0$, $\dot {z}_{p}^{(1)}=\dot {\eta }_p^{(1)}-\dot {s}^{(1)}$ and $u_z^{(1)}|_{\tilde {\boldsymbol {x}}_p^{(0)}}=\dot {\eta }_p^{(1)}$ from the linearised kinematic free surface boundary condition. We use the notation $\hat {A}^{(1)}_{{s},\tau }=\hat {A}^{\prime (0)}_{{s},\tau }(s^{(1)}/D)$ with $\hat {A}^{\prime (0)}_{{s},\tau }\equiv \partial _{\hat {s}} \hat {A}_{{s},\tau }(\hat {s})|_{\hat {s}^{(0)}}$ and $\hat {s}\equiv s/D$ according to (3.3).

In the normal direction, the total force at first order consists of a buoyancy force, an added mass and a resistance force already evaluated in (3.8), (3.9) and (3.10), respectively.

3.3.2. The wave-induced drift

Substituting the first-order solutions for $x_p^{(1)}$ (i.e. $X^{(1)}=1$) and for $s^{(1)}$ from (3.12) and for the wave quantities from table 1 and averaging over the waves, we obtain the following expression from (3.13) for the wave-induced drift of the object $\bar {v}_x={\bar {\dot {x}_p^{(2)}}}$:

(3.20)\begin{align} \bar{v}_x=\frac{u_{S}}{2}\left[\overbrace{2-\underbrace{\mathcal{R}(\mathcal{S}^{(1)})}_{\substack{\text{Increases}\\\text{drift}}}}^{\text{Adjusted Stokes drift}} +\frac{1}{\hat{A}^{(0)}_{{s},\tau}\varGamma_{R}}\left(\overbrace{\underbrace{-\varGamma_B \mathcal{I}(\mathcal{S}^{(1)})}_{\text{Increases drift}}}^{\substack{\text{Buoyancy} \\ \text{resolved into} \\ \text{the } x\text{-direction}}} + \overbrace{\underbrace{\frac{C_m \mathcal{I}(\mathcal{S}^{(1)})}{\beta }}_{\text{Negligible effect}}}^{\text{Added mass}}\right) +\overbrace{\underbrace{\frac{\hat{A}^{(0)}_{{s},n}}{\hat{A}^{(0)}_{{s},\tau}}\mathcal{R}(\mathcal{S}^{(1)})}_{\text{Reduces drift}}}^{\text{Normal drag}}\right], \end{align}

where $u_S=k_0\omega _0a_0^2$ is the Stokes drift. We define the drift amplification factor $X^{(2)}\equiv {\bar {v_x}}/u_S$, so that $X^{(2)}$ corresponds to the terms inside the square brackets in (3.20) divided by $2$. Equation (3.20) is the main result of this paper, and we will interpret it below. The text above the terms explains their physical origins, and the text below their effect on the wave-induced drift of the object compared to the Stokes drift.

We begin by examining the wave-induced drift amplification factor $X^{(2)}$ as a function of object size and for different density ratios in figure 5. It is evident that the drift is enhanced and increasingly so for larger and heavier objects. Figure 6 examines the contributions to $X^{(2)}$ of the four components in (3.20): the adjusted Stokes drift, buoyancy resolved in the $x$-direction, normal drag and added mass, which we will discuss in turn. In (3.20) and figure 6, $X^{(2)}=1$ corresponds to objects that do not experience an increase in drift and are simply transported with the Stokes drift (i.e. $\bar {v}_x=u_{S}$).

Figure 5.

For viscous drag, wave-induced drift amplification $X^{(2)}$ as a function of dimensionless object size $D/\lambda _0$ for different density ratios $\beta =\rho _o/\rho _f$ (see legend). We have set $C_m= \beta /2$. Numerical and analytical solutions from perturbation theory are denoted by crosses and solid lines, respectively.

Figure 6.

For viscous drag, contributions to the wave-induced drift amplification $X^{(2)}$ from the five components in (3.20) as a function of non-dimensional object size $D/\lambda _0$ for density ratio $\beta =0.8$ and $C_m= \beta /2$.

3.3.3. Adjusted Stokes drift

The adjusted Stokes drift terms in (3.20) reflect change in linear object trajectory. For unmodified horizontal motion ($X^{(1)}=1$) and zero variable submergence ($\mathcal {S}^{(1)}=0$), we obtain $X^{(2)}=1$ from the adjusted Stokes drift terms alone. For larger objects, the increase in the vertical motion due to ‘bobbing’ of the object effectively enhances the Stokes drift, as shown in figure 6. This mechanism occurs because the linear variable submergence changes the object's orbit and hence its velocity and time spent under trough and crest. Integration of the linear velocity component along the linear orbit results in Stokes drift. Hence, changes to velocity and orbit result in an adjusted Stokes drift.

3.3.4. Buoyancy resolved in the $x$-direction

The mechanism through which buoyancy, when resolved in the $x$-direction and averaged over the wave cycle, can increase the drift of an object is illustrated in figure 7. Without variable submergence (left column), the dynamic buoyancy force is simply zero. With variable submergence but without drag in the normal direction (middle column), the first-order buoyancy force resolved in the $x$-direction does not result in a net force on the object, as the first-order buoyancy force and the first-order slope required to resolve this force into the $x$-direction are out of phase. It is only in the presence of a drag component in the normal direction (right column) that a phase lag in the submergence depth arises and a net force results. As shown in figure 6, the buoyancy force thus makes a relatively large contribution to the object's drift.

Figure 7.

Schematics of the object trajectory (red) and free surface (blue) for three cases: no variable submergence, variable submergence with no normal drag and variable submergence with normal drag. The schematics illustrate the physical mechanism for increased drift arising from variable submergence $s^{(1)}$, where variable submergence and drag are in the $n$-direction, and a mean motion in the $x$-direction is created due to the slope of the free surface $\partial _x\eta ^{(1)}$. For this illustration, we have chosen a density ratio $\beta =1/2$.

3.3.5. Normal drag

Although normal drag is required to create the phase difference that leads to the net buoyancy force resolved in the $x$-direction, normal drag also acts to reduce the magnitude of the ‘bobbing’ mechanism and thus reduces the drift motion, as shown in figure 3. The horizontal direction component of normal drag opposes the horizontal direction component of buoyancy force, with the balance resulting in a drift that is greater than the adjusted Stokes drift discussed above. Tangential drag, through the inverse dependence of $X^{(2)}$ on the projected area $\hat {A}_{s,\tau }^{(0)}$ and the effective drag coefficient $\varGamma _R$ in (3.20), acts to reduce the increase in object drift, by effectively ‘anchoring’ the object to the fluid and its Stokes drift.

3.3.6. Added mass

At first order, the object accelerates in the normal direction, experiencing an inertia force in addition to the buoyancy force and the normal drag discussed above, and so an added-mass term has to be take into account. As shown in figure 3, the contribution by added mass is relatively small and acts to reduce drift.

4.2.1. Laboratory-scale results

At laboratory scale, we set $f_0=1.25\ \textrm {Hz}$, corresponding to $\lambda _0=1.0\ \textrm {m}$ and $\alpha =0.1$. With object diameters up to $D=60\ \textrm {mm}$, we obtain $D/\lambda _0=6\,\%$, where the limit of validity for viscous drag is $D/\lambda _0=0.2\,\%$ (see § 3.4). At laboratory scale, figure 8 compares the analytically predicted linear motion using viscous drag with the corresponding numerical results using non-viscous drag. The response in the normal direction is unchanged because the forcing is inertial with little effect from drag. As the object size increases, inertia increasingly dominates over drag. A small decrease in horizontal linear motion is evident reaching a few per cent for larger objects. The results for small objects are the same because the non-viscous drag recovers viscous drag in the small-object limit.

Figure 8.

Laboratory-scale numerical simulation results using non-viscous drag for magnitudes of the first-order horizontal motion amplification $X^{(1)}$ (a) and the variable submergence $\mathcal {S}^{(1)}$ (b) as functions of dimensionless object size $D/\lambda _0$ for different density ratios $\beta =\rho _o/\rho _f$, where the density ratio corresponding to each colour is listed in the legend. Here, $C_m= \beta /2$. Numerical and analytical solutions from perturbation theory are denoted by crosses and solid lines, respectively.

The drift amplification increases slightly when using non-viscous drag for larger objects, as seen in figure 9. This is because the (tangential) drag force for larger objects is lower for non-viscous drag than for viscous drag, resulting in reduced resistance to increased drift compared to the Stokes drift. The maximum Reynolds number reached in the numerical solutions at laboratory scale was $Re_{{max}}=3.1 \times 10^4$.

Figure 9.

Laboratory-scale numerical simulation results using non-viscous drag for wave-induced drift amplification $X^{(2)}$ as a function of dimensionless object size $D/\lambda _0$ and for different density ratios $\beta =\rho _o/\rho _f$ (see legend). Here, $C_m=\beta /2$. Analytical solutions using viscous drag from perturbation theory are denoted by solid lines.

4.2.2. Field-scale results

We set a wave frequency of $f_0=0.2\ \textrm {Hz}$ and a steepness of $\alpha =0.05$ to represent a typical wind wave at field scale. The frequency of $0.2\ \textrm {Hz}$ corresponds to the peak in the spectrum with $\alpha =0.05$ at the upper end of the steepness range for wind waves in the ocean (Toffoli & Bitner-Gregersen Reference Toffoli and Bitner-Gregersen2017). This steepness corresponds to a dimensional wave amplitude of $a_0=0.3\ \textrm {m}$. The difference between viscous and non-viscous drag results will be larger at field scale owing to the higher value of Reynolds numbers, which reached a maximum of $Re_{{max}}= 7.3\times 10^5$ in the numerical simulations.

Figure 10(a) shows the linear horizontal motion, which is mostly unchanged from the perturbation theory result. The magnitude of variable submergence is inertia driven and thus very similar to the viscous analytical result shown in figure 10(b).

Figure 10.

Field-scale numerical simulation results using non-viscous drag for magnitudes of the first-order horizontal motion amplification $X^{(1)}$ (a) and variable submergence $\mathcal {S}^{(1)}$ (b) as functions of dimensionless object size $D/\lambda _0$ for different density ratios $\beta =\rho _o/\rho _f$, where the density ratio corresponding to each colour is shown in the legend. Field scale here denotes a $0.2\ \textrm {Hz}$ wave with a steepness of $\alpha =0.05$. Here, $C_m= \beta /2$. Numerical and analytical solutions from perturbation theory are denoted by crosses and solid lines, respectively.

The drift amplification for field-scale simulations using non-viscous drag shown in figure 11 is greater than the perturbation theory result based on viscous drag, and even more so than at laboratory scale. This is because the non-viscous drag force is now considerably smaller than its viscous equivalent (taken outside the range of Reynolds numbers for which it is valid). The (tangential) drag force obtained for larger objects is lower for non-viscous drag than for a viscous drag formulation, resulting in reduced resistance to increased drift compared to the Stokes drift.

Figure 11.

Field-scale numerical simulation results using non-viscous drag for the wave-induced drift amplification $X^{(2)}$ as a function of dimensionless object size $D/\lambda _0$ for different density ratios $\beta =\rho _o/\rho _f$ (see legend). Field scale is modelled by a $0.2\ \textrm {Hz}$ wave with a steepness of $\alpha =0.05$. Here, $C_m=\beta /2$. Analytical solutions using viscous drag from perturbation theory are denoted by solid lines.

Using the results from field-scale numerical simulations for non-viscous drag, a $1\ \textrm {m}$ diameter object of density $\rho _p = 0.9\ \text {g}\,\text {cm}^{-3}$ leads to a $50\,\%$ increase in drift ($X^{(2)}=1.5$). This is a significant increase compared to the Stokes drift infinitesimal objects would experience. By comparison, a $0.1\ \textrm {m}$ diameter object in the same wave field does not experience any drift amplification ($X^{(2)}=1$) and behaves as a perfectly Lagrangian tracer.