3.1. Single ice floe dynamics

The motion of individual ice floes with different sizes was analysed in a TG vortex field and compared with the passive tracer case (figure 1). Three floe–eddy size ratios were considered, $R_f/R_e=0.1, 0.5$ and $1.0$ . In all cases, the centre of mass of the ice floes was initially positioned at $r_f = 0.5R_e$ , where $r_f$ denotes the radial position of the floe centre of mass, to capture the influence of ocean fields both inside and outside the vortex cell.

The trajectories of free-drifting ice floes are shaped by oceanic and atmospheric forcing, the effects of floe inertia, the Coriolis force and the pressure gradient force due to the sea surface tilt. Under low to negligible wind speeds and low sea ice concentration, ice floe motion is expected to be predominantly driven by oceanic forcing. However, it has been hypothesised that ice floe inertia plays an important role in setting the direction in which floes translate by delaying their response to changes in the underlying ocean flow field. From the cases tested, the ice floe with a size ratio of $R_f/R_e=0.5$ exhibited the best performance at resembling a passive tracer and forming a closed-loop trajectory over the TG vortex cell (figure 1 $a$ ). While inertia does result in an outward tilt of the ice floe velocity $\mathbf{u}_f(t)$ relative to the ocean velocity averaged over the floe area at $t$ , $\overline {\mathbf{u}}_o(t)$ (a subset of figure 1 $a$ ), the resulting force from combining the Coriolis force and the pressure gradient force due to the sea surface tilt, $\hat {k}\times (\overline {\mathbf{u}}_{o} - {\mathbf{u}}_{f})$ , reduces this spiralling effect, effectively driving the ice floe to form a closed loop (figure 1 $a$ ). Finally, ice floes were observed to undergo alternating periods of acceleration and deceleration relative to the equilibrium state in which all forces balance instantaneously to zero. As a result, the radial component of floe position oscillates (figure 1 $b$ ).

Changes in floe size with respect to vortex size result in a deviation of floe motion with respect to the eddying motion underneath. For example, smaller ice floes, for which $R_f/R_e=0.1$ , followed an outwardly spiral trajectory. In this case, the ice floe velocity and the averaged ocean velocity vectors were better aligned, reducing the inward effect from the Coriolis and pressure gradient forces compared with floes with larger size ratios. This is similar to the behaviour of millimetre-sized particles in a Taylor vortex (Wereley & Lueptow Reference Wereley and Lueptow1999; Deng et al. Reference Deng, Wang and Smith2006; Qiao et al. Reference Qiao, Deng and Wang2015) and a Rankine vortex (Varaksin & Ryzhkov Reference Varaksin and Ryzhkov2022), in which the role of inertia is linked to their resulting spiral trajectories relative to the flow direction. As the floe–eddy size ratio increases to $R_f/R_e=1.0$ , discrepancies between ice floe and averaged ocean velocities become more pronounced due to filtration of measured ocean velocities over larger floe areas. This results in larger forces directed toward the vortex centre compared with smaller floes, leading to an inward spiralling motion.

Figure 2. Motions of ice floes with different sizes in a TG vortex field. ( $a,b$ ) Normalised rotation rates, $\Omega _f$ , ( $c,d$ ) speeds, $V_f$ and ( $e,f$ ) angle deviations of ice floes for floe–eddy size ratios of $R_f/R_e=$ 0.1 (blue dashed lines), 0.5 (red dot-dashed lines) and 1.0 (green double dot-dashed lines) are compared with the passive tracer case (black solid lines). Floe rotation rates are normalised by the ( $a$ ) ocean vorticity averaged over the floe area, $\overline {\omega }_o$ , and ( $b$ ) ocean vorticity at the floe centre of mass, $\omega _{o,C}$ . Floe speeds are normalised by ( $c$ ) ocean speed averaged over the floe area, $\overline {V}_o$ , and ( $d$ ) ocean speed at the floe centre of mass, $V_{o,C}$ . Angle deviations are calculated by subtracting the angle of the ( $e$ ) averaged ocean velocity, $\Delta \theta _f$ , and the ( $f$ ) centre-of-mass ocean velocity, $\Delta \theta _{f,C}$ , from that of the ice floe velocity. The ice floes are initially released at $r_f/R_e = 0.5$ .

The kinematics of ice floes with different sizes were examined along their trajectories (figure 2). We present results using two normalisations: (i) considering ocean quantities averaged over the floe area, including averaged ocean vorticity, $\overline {\omega }_o$ , and averaged ocean speed, $\overline {V}_o$ and (ii) ocean quantities at the floe centre of mass, such as ocean vorticity, $\omega _{o,C}$ , and ocean speed, $V_{o,C}$ . Note that the floe rotation rates are normalised by half of the ocean vorticity. Angle deviations are calculated by subtracting the angle of the ocean velocity from that of the ice floe velocity, $\Delta \theta _f = \theta _f - \overline {\theta }_o$ and $\Delta \theta _{f,C} = \theta _f - \theta _{o,C}$ , where $\theta _f$ denotes the angle of the ice floe velocity, $\overline {\theta }_o$ denotes the angle of the ocean velocity averaged over the ice floe area and $\theta _{o,C}$ denotes the angle of the ocean velocity at the centre of mass of the ice floe. All angles are calculated using the direction of each velocity vector with respect to the positive $x$ -axis, measured in the counter-clockwise direction.

Normalised ice floe rotation rates, velocities and orientations oscillated over time, periodically crossing the equilibrium values. Small ice floes with $R_f/R_e = 0.1$ show higher sensitivity to local ocean quantities, leading to normalised rotation rates and speeds close to unity and minimal angle deviations (figure 2). Ocean quantities averaged over the floe area and at the floe centre of mass exhibit similar values, resulting in comparable normalised rotation rates (figures 2 $a$ and 2 $b$ ) and speeds (figures 2 $c$ and 2 $d$ ). As ice floes move farther from the vortex centre along their outwardly spiral trajectories, the normalised rotation rates begin to decrease at later times. The angle deviations have negative values for most of the evolution time due to the outwardly spiral shape of the trajectory (figures 2 $e$ and 2 $f$ ). Contrasting the ice floe case, the passive tracer perfectly mirrors the vorticity and velocity of the underlying flow field with zero angle deviations.

As the floe–eddy size ratios increase to $R_f/R_e = 1.0$ , the normalised ice floe rotation rates and speeds exhibit larger deviations from unity (figure 2 $a$ – $d$ ). With larger floe–eddy size ratios, the floe rotation rates normalised by the averaged ocean vorticity increase because the floe samples low-vorticity regions near cell boundaries, reducing the averaged ocean vorticity (figure 2 $a$ ). Conversely, the floe speeds normalised by the averaged ocean speed decrease due to high-velocity regions near the cell boundaries covered by the floe area, resulting in higher averaged ocean speed (figure 2 $c$ ). In the context of ocean quantities at the floe centre of mass, the normalised rotation rates and speeds of ice floes decrease significantly for larger floe–eddy size ratios. This occurs because the ocean vorticity and velocity at the floe centre of mass are considerably higher compared with the values averaged over the floe area (figures 2 $b$ and 2 $d$ ). For both types of ocean quantities, the angle deviation becomes more pronounced as the floe–eddy size ratios increase (figures 2 $e$ and 2 $f$ ). With larger ratios, the resultant forces from a combination of the Coriolis force and the pressure gradient force due to the sea surface tilt become larger, resulting in a shift of floe trajectories from an outwardly spiral shape to an inwardly spiral shape as the ratios increase. As a result, the angle deviations at $R_f/R_e=0.5$ are smaller than in the other two cases since the floe trajectories form nearly closed loops. Floes with $R_f/R_e=1.0$ show positive angle deviations since they have inwardly spiralling trajectories.

The motion of a single floe provides insight into the instantaneous response of ice floes to an underlying ocean eddy flow field. Our results demonstrate the importance of floe–eddy size ratios modulating the distribution of forces due to drag, Coriolis and the pressure gradient on ice floes as they drift over ocean eddy fields.

3.2. Statistics of ice floe kinematics in an idealised vortex

Based on the statistics of ice floes in a TG vortex field, we derived analytical expressions describing the kinematic link between ice floe rotation and the ocean vorticity underneath. Appendix A outlines the criteria for identifying trapped ice floes. Given the time duration of the simulation and the requirements for floe selection, ice floes with floe–eddy size ratios ranging from 0.05 to 1.4 were considered for further analysis.

Figure 3. Rotational motion of centred ice floes in a TG vortex field. Ice floe rotation rates, $\Omega _f$ , normalised by the ( $a$ ) ocean vorticity averaged over the floe area, $\overline {\omega }_o$ , and the ( $b$ ) the ocean vorticity at the floe centre of mass, $\omega _{o,C}$ , for different floe–eddy size ratios, $R_f/R_e$ . The simulation results are compared with analytical relations using square-shape approximation (dashed lines) and Taylor series expansion (dashed-dotted lines), and the passive tracer case (solid lines).

We examined ice floes positioned at the centre of a TG vortex representing an idealised scenario (figure 3). In this case, the ice floes are initially released at the centre of the vortex, exhibiting a distinct trend in their rotation driven solely by spatial variations in ocean vorticity within the vortex core. The ice floe rotation rate normalised by the ocean vorticity averaged over the floe area shows a monotonic growth with increasing floe–eddy size ratios (figure 3 $a$ ). This observed rise in rotation rates can be attributed to the gradual decline in local ocean vorticity from the centre of a vortex cell toward its periphery. As floe–eddy size ratios increase, the averaged ocean vorticity decreases, increasing normalised rotation rates. In contrast, the rotation rate normalised by the ocean vorticity at the floe centre of mass decreases for larger floe–eddy size ratios (figure 3 $b$ ). This is due to the fact that large ice floes rotate at slower rates while the ocean vorticity is maximised at the centre of mass of the rotating ice floes.

Building upon the physical interpretation of ice floe rotation, we established analytical relations for the normalised rotation rates by balancing the torques acting on the ice floes. In the present analysis, the quadratic drag terms in the angular momentum equation (2.10) are substituted with linear drag, thereby enabling the derivation of explicit relations describing the dependence of the normalised rotation rates on floe–eddy size ratios. We investigated the effect of this substituted parameterisation on the rotational relationship using the Rankine vortex, a simplified case for which solutions for both drag parameterisations can be derived. The analytical relation for the normalised rotation rate of ice floes shows minimal sensitivity to the choice of drag parametrisation, as detailed in Appendix B, such that a linear drag law was used in deriving an analytical relation.

By introducing a so-called size parameter, $\gamma = R_f/R_e$ , we derived an analytical relationship for the normalised rotation rate of an ice floe located at the centre of the TG vortex. Near the centre of the TG vortex ( $2\pi x/L_{TG}$ , $2\pi y/L_{TG} \ll 1$ ), the ocean surface velocity can be approximated using a Taylor series expansion. In the absence of wind, ice floe rotation is dominated by ice–ocean stress for which the induced equilibrium of torques leads to the relation

(3.1) \begin{equation} \iint _{A^*} \mathbf{r}_{f}'^* \times ({\mathbf{u}}_{f}^* - {\mathbf{u}}_{o}^*) {\textrm {d}} A^* = \textrm 0. \end{equation}

Substituting the second-order Taylor series expansion into (3.1) yields the following expression for the ice floe rotation rate normalised by the averaged ocean vorticity:

(3.2) \begin{equation} \frac {\Omega _f}{\overline {\omega }_o/2} = \frac {\displaystyle \left [1- \frac {\pi ^2}{8} \left (\frac {\gamma ^2}{3}\right ) + \frac {\pi ^4}{64} \left ( \frac {\gamma ^4}{48} \right ) \right ]} {\displaystyle \left [1- \frac {\pi ^2}{8} \left (\frac {\gamma ^2}{2}\right ) + \frac {\pi ^4}{64} \left ( \frac {\gamma ^4}{24}\right ) \right ]}. \end{equation}

Given that the ocean vorticity at the centre of mass of the centred floe in the TG vortex is $\omega _{o,C} = 2A\pi ^2/L_{\textit{TG}}^2$ , the ice floe rotation rate normalised by the ocean vorticity at the floe centre of mass can be expressed as

(3.3) \begin{equation} \frac {\Omega _f}{\omega _{o,C}/2} = \left [1- \frac {\pi ^2}{8} \left (\frac {\gamma ^2}{3}\right ) + \frac {\pi ^4}{64} \left ( \frac {\gamma ^4}{48}\right ) \right ]. \end{equation}

Equation (3.2) is in good agreement with the simulation results for the rotation rates normalised by the averaged ocean vorticity provided $R_f/R_e \leqslant 0.7$ (figure 3 $a$ ). In this range, the Taylor series approximation effectively restores the original functions. However, we observed noticeable deviations in the normalised rotation rate when the ice floe size exceeded the eddy size due to limitations of the Taylor series expansion near the edge of the vortex cell. Equation (3.3) aligns well with simulation results for the rotation rate normalised by the ocean vorticity at the floe centre of mass for different floe–eddy size ratios because the ocean vorticity remains constant (figure 3 $b$ ).

We derived another analytical relation for the normalised rotation rate of the centred ice floe by approximating the floe shape as a square with the same characteristic length scale. With the equilibrium condition for ice–ocean stress torques, the analytical relation can be expressed as

(3.4) \begin{equation} \frac {\Omega _f}{\overline {\omega }_o/2} = \frac {12}{\pi ^2\gamma ^2} \left [1 - \frac {\pi \gamma }{2} {\textrm {cot}} \left (\frac {\pi \gamma }{\textrm {2}} \right )\right ]. \end{equation}

We positioned square-shaped floes centred with respect to the origin. This simplification allows for the direct use of trigonometric functions. As a result, the ice floe rotation rate normalised by the ocean vorticity at the floe centre of mass can be written as

(3.5) \begin{equation} \frac {\Omega _f}{\omega _{o,C}/2} = \frac {\displaystyle 12}{\displaystyle \pi ^2\gamma ^2} \left [\frac {\displaystyle {\textrm {sin}}\left ({\frac {\pi \gamma }{2}}\right )}{\displaystyle \frac {\pi \gamma }{2}} \right ]^2 \left [1 - \frac {\pi \gamma }{2} {\textrm {cot}} \left (\frac {\pi \gamma }{\textrm {2}}\right )\right ]. \end{equation}

Equation (3.4) closely matches the simulation results for the rotation rates normalised by the averaged ocean vorticity because it incorporates trigonometric functions fully in the solution (figure 3 $a$ ). However, (3.5) deviates from the simulation results for the rotation rates normalised by the centre-of-mass ocean vorticity due to the square-shape approximation (figure 3 $b$ ). It is worth noting that, for the centred ice floes, the derived analytical relations (3.2)–(3.5) are functions solely of the floe–eddy size ratios.

We also examined a more realistic scenario of an ice floe being positioned off centre with respect to the TG vortex cell. We conducted simulations with over 2000 randomly distributed ice floes for each floe–eddy size ratio. While centred ice floes remain at the vortex centre, off-centred ice floes have the potential to drift away, depending on their radial positions. For the present analysis, only trapped ice floes were considered.

The normalised rotation rates of off-centred floes are dependent on the position of the floes relative to the vortex core. Therefore, to analyse this case, we computed probability density functions (PDFs) over all floe positions for each bin of floe–eddy size ratios (figure 4). As discussed in the single floe analyses (§ 3.2), the observed variability arises from floe inertia, causing a delay in ice floe response to changes in the underlying ocean eddy field. This delay results in a discrepancy between the rotation of the ice floes and the underlying ocean eddies. Considering ice floe rotation rates normalised by the averaged ocean vorticity, ice floes behave as passive tracers provided $R_f/R_e \leqslant 0.7$ (figure 4 $a$ ). However, as the floe–eddy size ratio increases and approaches 1.4, the peaks in the normalised rotation rates shift to greater values, reaching approximately 1.8, similar to the case of centred ice floes (figure 3 $a$ ). Around $R_f/R_e \approx 0.5$ , the distribution narrows and exhibits higher peaks, suggesting that a solid body rotation approximation accurately captures floe motion within this range of floe–eddy size ratios. This behaviour can be attributed to the ice floe area filtering out ice floe–ocean stress, thereby reducing ice floe responsiveness to any variability within the vortex cell. As floe–eddy size ratios increase, excessive filtering of ocean information over the ice floe diminishes sensitivity to underlying ocean characteristics, resulting in a broadening of the PDF. For ice floe rotation rates normalised by the ocean vorticity at the floe centre of mass, ice floes behave as passive tracers when $R_f/R_e \leqslant 0.25$ . The peaks of the PDF shift to smaller values as the size ratios increase, reaching approximately 0.25 at $R_f/R_e=1.4$ (figure 4 $b$ ). Similar to the behaviour observed for averaged ocean vorticity, the distributions are more dispersed for small floes, while clear peaks are evident for $R_f/R_e \geqslant 0.25$ , as the centre-of-mass ocean vorticity for larger floes tends to exceed the area-averaged vorticity.

Figure 4. Rotational motion of off-centred ice floes in a TG vortex field. The PDFs of ice floe rotation rates, $\Omega _f$ , normalised by the ( $a$ ) ocean vorticity averaged over the floe area, $\overline {\omega }_o$ , and the ( $b$ ) ocean vorticity at the floe centre of mass, $\omega _{o,C}$ , for different floe–eddy size ratios. The simulation results are compared with analytical relations using square-shape approximation (dashed lines) and Taylor series expansion (dashed-dotted lines), and the passive tracer case (solid lines).

Following the same approach as for centred ice floes, we derived analytical relations for the off-centred floes. By employing the Taylor series expansion, the analytical relations for the ice floe rotation rate normalised by the averaged ocean vorticity can be derived as

(3.6) \begin{equation} \frac {\Omega _f}{\overline {\omega }_o/2} = \frac {\displaystyle \left \{1- \frac {\pi ^2}{8} \left [\frac {\gamma ^2}{3} + \left (\frac {r_f}{R_e}\right )^2 \right ] + \frac {\pi ^4}{64} \left [ \frac {\gamma ^4}{48} + \frac {\gamma ^2}{6}\left (\frac {r_f}{R_e}\right )^2 + \left (\frac {x_f}{R_e}\right )^2\left (\frac {y_f}{R_e}\right )^2\right ] \right \}} {\displaystyle \left \{1- \frac {\pi ^2}{8} \left [\frac {\gamma ^2}{2} + \left (\frac {r_f}{R_e}\right )^2 \right ] + \frac {\pi ^4}{64} \left [ \frac {\gamma ^4}{24} + \frac {\gamma ^2}{4}\left (\frac {r_f}{R_e}\right )^2 + \left (\frac {x_f}{R_e}\right )^2\left (\frac {y_f}{R_e}\right )^2\right ] \right \}},\end{equation}

where $x_f$ and $y_f$ are the horizontal Cartesian coordinates of the floe centre of mass, respectively, and $r_f = \sqrt {x_f^2 + y_f^2}$ is its radial position. Similarly, the ice floe rotation rate normalised by the centre-of-mass ocean vorticity can be then expressed as

(3.7) \begin{equation} \frac {\Omega _f}{\omega _{o,c}/2} = \frac {\displaystyle \left \{1- \frac {\pi ^2}{8} \left [\frac {\gamma ^2}{3} + \left (\frac {r_f}{R_e}\right )^2 \right ] + \frac {\pi ^4}{64} \left [ \frac {\gamma ^4}{48} + \frac {\gamma ^2}{6}\left (\frac {r_f}{R_e}\right )^2 + \left (\frac {x_f}{R_e}\right )^2\left (\frac {y_f}{R_e}\right )^2\right ] \right \}} {\displaystyle \left [1- \frac {\pi ^2}{8} \left (\frac {r_f}{R_e}\right )^2 + \frac {\pi ^4}{64} \left (\frac {x_f}{R_e}\right )^2\left (\frac {y_f}{R_e}\right )^2 \right ]}.\end{equation}

In contrast to the centred ice floe cases, relations (3.6) and (3.7) not only depend on the floe–eddy size ratio, but also on the radial positions of the ice floes. Therefore, the normalised rotation rates were averaged over all $r_f$ values for each of the floe–eddy size ratio bins. As $r_f \rightarrow 0$ , the analytical relations for the off-centred floes (3.6) and (3.7) converge to the relations for the centred floes (3.2) and (3.3). Equation (3.6) is in good agreement with the PDF peaks for small floe–eddy size ratios (figure 4 $a$ ). Their differences become evident around $R_f/R_e \approx 0.5$ and become more pronounced for larger size ratios due to the issues of the Taylor series expansion near the cell edges. Conversely, (3.7) closely aligns with the PDF peaks of ice floe rotation rates normalised by the centre-of-mass vorticity for most floe–eddy size ratios (figure 4 $b$ ).

The square-shape approximation for off-centred floes yields the same analytical relations as for centred floes. Thus, (3.4) and (3.5) are also used to describe the rotation rates of off-centred floes. These equations match the PDF peaks over a wide range of floe–eddy size ratios and only show minor deviations at ratios around $R_f/R_e=1.4$ (figures 4 $a$ and 4 $b$ ). Overall, the square-shape approximation provides accurate estimates of the PDF peak compared with the Taylor series approximation.

Figure 5. Translational motion of off-centred ice floes in a TG vortex field. The PDFs of ice floe speed, $V_f$ , normalised by the ( $a$ ) ocean speed averaged over the floe area, $\overline {V}_o$ , and the ( $b$ ) ocean speed at the floe centre of mass, $V_{o,C}$ , and ( $c$ and $d$ ) angle deviations. For these last quantities, ice and ocean velocity angles are calculated using the direction of each velocity vector with respect to the positive $x$ -axis, measured in the counter-clockwise direction. Then, deviations are computed by subtracting the angle of ( $c$ ) the averaged ocean velocity, $\Delta \theta _f$ , and the ( $d$ ) centre-of-mass ocean velocity, $\Delta \theta _{f,C}$ , from that of the ice floe velocity, for different floe–eddy size ratios. The simulation results are compared with analytical relations using square-shape approximation (dashed lines) and Taylor series expansion (dashed-dotted lines), and the passive tracer case (solid lines).

The PDFs of normalised ice floe speeds and floe–ocean angle deviations depend on the floe–eddy size ratios (figure 5), corresponding to the results of single floe analyses depicted in figure 2. For the ice floe speed normalised by the ocean speed averaged over the floe area, the PDF peaks at unity ( $V_f/\overline {V}_o = 1$ ), resembling the behaviour of a passive tracer (figure 5 $a$ ). As the floe–eddy size ratio increases and approaches 1.4, the normalised speed at the PDF peaks gradually decreases to $V_f/\overline {V}_o = 0.75$ . In contrast to the ice floe rotation rate results, the PDFs of normalised floe speeds for small-sized floes exhibit narrow distributions. This trend can be attributed to the fact that the Coriolis force and the pressure gradient force due to the sea surface tilt depend on the velocity difference, $\Delta {\mathbf{u}}_{f} = \overline {\mathbf{u}}_{o}-{\mathbf{u}}_{f}$ , while the torque generated by these forces depends solely on ocean velocity, $\overline {\mathbf{u}}_{o}$ , with no contribution from the Coriolis force. As a result, floe speed is less responsive to changes in ocean speed. This is reflected in the third term of (2.9), which is derived using the geostrophic balance between the pressure gradient force due to sea surface tilt and the Coriolis force.

Overall, the ocean speed at the floe centre of mass is greater than the ocean speed averaged over the floe area. The peaks of the PDF for the floe speed normalised by the centre-of-mass ocean speed decrease to 0.25 as the floe–eddy size ratio increases to 1.4 (figure 5 $b$ ). It is useful to note that the velocity difference between the ice floe and the underlying ocean remains relatively constant across different floe–eddy size ratios, such that $\lvert \Delta {\mathbf{u}}_{f} \rvert \approx C_{\textrm {0}}$ , where $C_0$ denotes a constant. This value can be derived by equating the magnitude of the ice floe–ocean stress (the first term on the right-hand side of (2.9)) to a combination of the Coriolis force on the ice floe and the pressure gradient force due to the sea surface tilt (the third term on the right-hand side of (2.9)) under torque equilibrium, as follows: $\lvert \Delta {\mathbf{u}}_{f}^{*} \rvert ^2 /H_{f,o}^* \sim \lvert {\Delta } {\mathbf{u}}_{f}^{*} \rvert /Ro$ . Then, the floe speed normalised by the averaged ocean velocity can be expressed as

(3.8) \begin{equation} \frac {V_f}{\overline {V}_{o}} \approx 1 - \frac {C_0}{\overline {V}_{o}}, \end{equation}

implying that, for lower averaged ocean speeds as in the case of larger floes, there is a reduction in normalised floe speeds.

Similar to (3.6)–(3.7), the analytical relation for ice floe speed normalised by the ocean speed averaged over the floe area and by the ocean speed at the floe centre of mass can be derived using the Taylor series expansion and the square-shape approximation. In relation (3.8), the averaged ocean speed can be approximated using a Taylor series expansion as follows:

(3.9) \begin{equation} \overline {V}_o = \left (\frac {\pi }{2}\right )^2 \left (\frac {A_{\textit{TG}}}{R_e}\right ) \left (\frac {r_f}{R_e}\right ) M_o, \end{equation}

where $M_o=[(x_f/r_f)^2M_{o,1}^2 + (y_f/r_f)^2M_{o,2}^2]^{1/2}$ . The parameters $M_{o,1}$ and $M_{o,2}$ are given by

(3.10a) \begin{align} M_{o,1} & = 1 - \frac {1}{12}\left (\frac {\pi }{2}\right )^2 \left \{3\gamma ^2 + 2\left [\left (\frac {x_f}{R_e}\right )^2 + 3\left (\frac {y_f}{R_e}\right )^2\right ]\right \} \nonumber\\ &\quad + \frac {1}{96} \left (\frac {\pi }{2}\right )^4 \left \{\gamma ^4 + 2\gamma ^2 \left [\left (\frac {x_f}{R_e}\right )^2 + 3\left (\frac {y_f}{R_e}\right )^2\right ] + 8\left (\frac {x_f}{R_e}\right )^2\left (\frac {y_f}{R_e}\right )^2 \right \}, \end{align}

(3.10b) \begin{align} M_{o,2} & = 1 - \frac {1}{12}\left (\frac {\pi }{2}\right )^2 \left \{3\gamma ^2 + 2\left [3\left (\frac {x_f}{R_e}\right )^2 + \left (\frac {y_f}{R_e}\right )^2\right ]\right \} \nonumber\\ &\quad + \frac {1}{96} \left (\frac {\pi }{2}\right )^4 \left \{\gamma ^4 + 2\gamma ^2 \left [3\left (\frac {x_f}{R_e}\right )^2 + \left (\frac {y_f}{R_e}\right )^2\right ] + 8\left (\frac {x_f}{R_e}\right )^2\left (\frac {y_f}{R_e}\right )^2 \right \}. \end{align}

Similarly, the ice floe speed normalised by the centre-of-mass ocean speed can be expressed using the averaged ocean speed as follows:

(3.11) \begin{equation} \frac {V_f}{V_{o,C}} = \left (1 - \frac {C_0}{\overline {V}_{o}}\right )\left ( \frac {\overline {V}_{o}}{V_{o,C}} \right ). \end{equation}

The centre-of-mass ocean speed can also be approximated using the Taylor series expansion

(3.12) \begin{equation} V_{o,C} = \left (\frac {\pi }{2}\right )^2 \left (\frac {A_{\textit{TG}}}{R_e}\right ) \left (\frac {r_f}{R_e}\right ) M_{o,C}, \end{equation}

where $M_{o,C}$ is given by

(3.13) \begin{align} \begin{split} M_{o,C}^2 & = 1 - \frac {1}{3}\left (\frac {\pi }{2}\right )^2\left (\frac {r_f}{R_e}\right )^2 \left [1 + 4\left (\frac {x_f}{r_f}\right )^2\left (\frac {y_f}{r_f}\right )^2 \right ] \left [1 + \frac {1}{12}\left (\frac {\pi }{2}\right )^4\left (\frac {x_f}{R_e}\right )^2\left (\frac {y_f}{R_e}\right )^2\right ] \\ & + \frac {1}{36}\left (\frac {\pi }{2}\right )^4\left (\frac {r_f}{R_e}\right )^4 \left [1 + 18\left (\frac {x_f}{r_f}\right )^2\left (\frac {y_f}{r_f}\right )^2 + \frac {1}{4}\left (\frac {\pi }{2}\right )^4\left (\frac {r_f}{R_e}\right )^4\left (\frac {x_f}{r_f}\right )^4\left (\frac {y_f}{r_f}\right )^4 \right ]. \\[-12pt] & \end{split} \end{align}

Note that, for comparison with the simulation results, the normalised floe speeds in the analytical relations were averaged over all $r_f$ values for each bin of floe–eddy size ratios. Similarly, using the square-shape approximation, the averaged ocean speeds can be approximated as the following:

(3.14) \begin{align} \begin{split} \overline {V}_o &= \left (\frac {\pi }{2}\right ) \left (\frac {A_{\textit{TG}}}{R_e}\right ) \left [\mathrm{sin}\left (\frac {\pi \gamma }{2}\right ) \bigg / \left (\frac {\pi \gamma }{2}\right ) \right ]^2 \\ &\quad \times \left [ \mathrm{cos}^2 \left (\frac {\pi }{2} \frac {x_f}{R_e}\right ) \mathrm{sin}^2\left (\frac {\pi }{2} \frac {y_f}{R_e}\right ) + \mathrm{sin}^2 \left (\frac {\pi }{2} \frac {x_f}{R_e}\right ) \mathrm{cos}^2 \left (\frac {\pi }{2} \frac {y_f}{R_e}\right ) \right ]^{1/2}. \end{split} \end{align}

The centre-of-mass ocean speed can also be obtained as

(3.15) \begin{equation} V_{o,C} = \left (\frac {\pi }{2}\right ) \left (\frac {A_{\textit{TG}}}{R_e}\right ) \left [\mathrm{cos}^2 \left (\frac {\pi }{2} \frac {x_f}{R_e}\right ) \mathrm{sin}^2 \left (\frac {\pi }{2} \frac {y_f}{R_e}\right ) + \mathrm{sin}^2 \left (\frac {\pi }{2} \frac {x_f}{R_e}\right ) \mathrm{cos}^2 \left (\frac {\pi }{2} \frac {y_f}{R_e}\right ) \right ]^{1/2}. \end{equation}

The analytical relation using the square-shape approximation ((3.14) and (3.15)) shows good agreement with the peaks of the normalised speeds, while the analytical relation using the Taylor series expansion ((3.9) and (3.12)) exhibits deviations for $R_f/R_e \geqslant 1$ (figures 5 $a$ and 5 $b$ ). The constant $C_0=0.15$ was chosen, producing the best-fit to the PDF peaks. Note that as $r_f \rightarrow 0$ , the analytical relations for the normalised floe speed ((3.8) and (3.11)) converge to the centred floe case, as expected.

The PDFs of the angle deviation between ice floe velocity and ocean velocity are also contingent on the floe–eddy size ratio (figures 5 $c$ and 5 $d$ ). As discussed in the single floe analysis (figure 2), the velocity of larger floes exhibits greater deviations from ocean velocity, including the averaged ocean velocity and the centre-of-mass velocity, thus resulting in more dispersed distributions for larger floe–eddy size ratios. While the vectors of the averaged ocean velocity and the centre-of-mass ocean velocity differ slightly in magnitude, their orientations are the same. This observation can be readily confirmed using the square-shape approximation, as follows:

(3.16) \begin{equation} \mathrm{tan}\overline {\theta }_o = - \mathrm{tan}\left (\frac {\pi }{2}\frac {x_f}{R_e}\right ) \bigg/ \mathrm{tan}\left (\frac {\pi }{2}\frac {y_f}{R_e}\right ) = \mathrm{tan}\theta _{o,C}, \end{equation}

indicating that the angle of the averaged ocean velocity is equal to the angle of the centre-of-mass ocean velocity.

In addition to calculating ocean velocity and vorticity beneath single ice floes, we can also compute local ocean vorticity from Green’s theorem by using ice floe trajectory information. Passive tracers trapped in an eddy exhibit closed-loop trajectories. In this case, we can estimate the averaged ocean vorticity of the region enclosed by the trajectory as $\tilde {\omega }_o = \int \mathbf{u}_{tr} \cdot \mathrm{d}\mathbf{S}_{tr}$ , where $\mathbf{u}_{tr}$ is the tracer velocity, $\mathbf{S}_{tr}$ is the tracer trajectory and $\sim$ denotes the averaged quantity over the closed region. Here, since $\mathbf{u}_{tr} = \mathbf{u}_{o}$ , the true ocean vorticity, $\tilde {\omega }_o$ , can be obtained. Similarly, trapped ice floes have been observed to form nearly closed-loop patterns, with the endpoints slightly offset from the starting points. However, as previously discussed, the velocity of ice floes corresponds to the filtered ocean velocity over the floe area but is not exactly the same. As a result, any averaged ocean vorticity estimate within a trajectory-enclosed region, $\tilde {\omega }_f$ , does not match the true averaged ocean vorticity of the same region, $\tilde {\omega }_f \neq \tilde {\omega }_o$ . Nonetheless, their ratios can be approximated using the ice floe and underlying ocean velocities as

(3.17) \begin{equation} \frac {\tilde {\omega }_f}{\tilde {\omega }_o} = \frac {\oint \mathbf{u}_{f} \cdot \mathrm{d}\mathbf{S}_{f}} {\oint \mathbf{u}_{o,C} \cdot \mathrm{d}\mathbf{S}_{f}} \approx \frac {V_f}{V_{o,C}}, \end{equation}

where $\mathbf{u}_{o,C}$ is the ocean velocity at the floe centre of mass and $\mathbf{S}_f$ is the ice floe trajectory.

Floe trajectories are not perfectly closed, with just a short segment between the start and endpoints. Hence, the line integration in (3.17) is conducted from the first to the last points. The true averaged ocean vorticity can be calculated by integrating the centre-of-mass ocean velocity along the floe trajectory. The ratio of the ocean vorticity estimate based on ice floe velocity to the true ocean vorticity, $\tilde {\omega }_f / \tilde {\omega }_o$ , can be approximated by the normalised ice floe speed, $V_f/V_{o,C}$ (3.17).

The PDFs of the ratio between the ocean vorticity estimate and the true ocean vorticity depend on floe–eddy size ratios (figure 6). The floe speed normalised by the centre-of-mass ocean speed significantly decreases for larger floe–eddy size ratios (figure 5 $b$ ), resulting in smaller vorticity ratios. Furthermore, the peaks of the PDF for most floe–eddy size ratios align with the analytical relation for normalised floe speed using the square-shape approximation ((3.11), (3.14) and (3.15)). The relation using the Taylor series expansion ((3.11)–(3.13)) is in good agreement with the PDF peaks for $R_f/R_e \leqslant 0.75$ , showing a discrepancy for larger ratios due to the coverage limit of the Taylor series expansion near the edge of the vortex cell.

Figure 6. Averaged ocean vorticity estimates from ice floe trajectories in a TG vortex field. The PDFs of ratios between the averaged ocean vorticity of the region enclosed by ice floe trajectories, $\tilde {\omega }_f$ , and true averaged ocean vorticity of the region, $\tilde {\omega }_o$ , for different floe–eddy size ratios. The simulation results are compared with analytical relations using the square-shape approximation (dashed line) and Taylor series expansion (dashed-dotted line), and the passive tracer case (solid line).

3.3. Statistics of ice floe kinematics in a QG flow field

We performed simulations of ice floes in a QG flow field to apply the derived analytical relations to a more realistic ocean eddy field (figure 7). Passive tracers were released into the flow field, and ocean vorticity was interpolated to the Lagrangian tracer positions. Then, a LAVD-based eddy detection method (Haller et al. Reference Haller, Hadjighasem, Farazmand and Huhn2016) was used to identify the boundaries of eddies by searching for the outermost closed contour of the LAVD, which indicates local rotation relative to mean rotation. The LAVD was computed by averaging the vorticity deviation along the Lagrangian tracer trajectory as follows:

(3.18) \begin{equation} \mathrm{LAVD} \left (x_0,y_0\right ) = \omega _o' \left [X\left (x_0,y_0\right ),Y\left (x_0,y_0\right )\right ], \end{equation}

where $(X,Y)$ are the coordinates of tracers with an initial position of $(x_0,y_0)$ , and $\omega _o'$ is the vorticity deviation from the spatial average over the whole domain. Given that the ocean field is time-independent in the present study, the time component is not considered in (3.18). A total of 70 eddies were identified in the QG flow field (figure 7 $a$ ); their morphologies become more apparent in the LAVD field (figure 7 $b$ ). Note that the LAVD has been widely used to capture vorticity-dominated structures in atmospheric and oceanic flows and has also been used as a baseline for evaluating trajectory-based diagnostics to quantify the kinematics of an underlying fluid (Aksamit et al. Reference Aksamit, Encinas-Bartos, Haller and Rival2024). By leveraging the rotational relationship between floes and underlying eddies, the LAVD has the potential to be applied to the trajectories and rotation rates of ice floes for eddy detection purposes.

Figure 7. Motion of ice floes in a QG flow field. Boundaries (red lines) and centres (red dots) of the detected eddies are shown in the field of ( $a$ ) ocean vorticity and ( $b$ ) LAVD normalised by the Coriolis parameter $f$ . ( $c$ – $f$ ) Representative cases of trajectories (black dashed lines) of trapped ice floes within local eddies marked as $A$ to $D$ with white dashed boxes in the QG and LAVD flow fields, respectively. The colour and arrows in the figure correspond to the magnitude of the normalised vorticity and LAVD and the magnitude and direction of the velocity at a given location, respectively. The floes (white circle) are positioned at the endpoint of their trajectories.

We selected four varying-size representative eddy cases to examine the kinematic link between drifting ice floes and the underlying flow field (figure 7 $c$ – $f$ ). To this end, we randomly released over 2,000 ice floes near each eddy, identifying floes trapped within vortex cores (Appendix A). Ice floes released near the eddy boundary exhibited nearly closed-loop trajectories resembling the shapes of the eddies (figure 7 $c$ – $f$ ). Hence, the trajectories of individual trapped ice floes provide a direct estimate of local eddy length scales, whereby the size of the largest enclosed region by ice floe trajectories, $R_{traj}=\sqrt {A_{traj}/\pi }$ , was chosen as a trajectory-derived length scale for each local eddy. Here, $A_{traj}$ is the area of the largest enclosed region by ice floe trajectories. This length scale closely matches the eddy size in the TG vortex.

Figure 8. Rotational motion of trapped ice floes in local QG eddies. ( $a$ – $d$ ) The PDFs of ice floe rotation rates, $\Omega _f$ , normalised by the averaged ocean vorticity over the floe area, $\overline {\omega }_o$ , for different ratios between floe size and trajectory-derived length scale, $R_f/R_{traj}$ , in local eddies marked as $A$ to $D$ with white dashed boxes in a QG flow field, respectively. The simulation results are compared with analytical solutions using square-shape approximation (dashed line), Taylor series expansion (dashed-dotted line) and the passive tracer case (solid line).

In QG eddies, ice floes simultaneously covering both the inside and outside regions of an eddy often become entrapped by the eddy, resulting in minor discrepancies between eddy size and trajectory-derived length scales. These differences arise due to variations in vorticity and velocity distributions among different QG eddies, leading to diverse trajectory shapes and trajectory-derived length scales. In the selected QG eddies, $R_f/R_{traj}$ range from 0.05 to 1.25. Beyond these ranges, ice floes generally remain translating near the eddies or drift away from them over the simulation period.

We investigated the rotational relationship between trapped ice floes and the underlying local eddies for the four eddies marked in figure 7. Despite variations in eddy size and shape, the normalised rotation rates of ice floes showed similar qualitative and quantitative trends to those in the idealised vortex cases (figure 8). Notably, the PDFs of ice floe rotation rates, normalised by the ocean vorticity averaged over the floe area, peak at unity when $R_f/R_{traj} \leqslant 0.5$ for all four cases (figure 8). Note that the distributions of the normalised floe rotation rates in the QG eddy cases are more dispersed than in the TG vortex cases. We attribute this discrepancy to the highly nonlinear, deformed eddies dominant in the QG flow field. The PDF peaks become more pronounced in the QG cases as the floe size approaches half of the trajectory-derived length scale ( $R_f/R_{traj} = 0.5$ ). The peaks shift toward 1.5 as $R_f/R_{traj}$ hits 1.25. Analytical relations derived for the TG vortex effectively characterise this behaviour. The relation from the Taylor series expansion (3.6) aligns with PDF peaks for smaller $R_f/R_{traj}$ , while the relation from the square-shape approximation (3.4) agrees well with the peaks across most size ratios.

The PDFs of ice floe rotation rates normalised by the ocean vorticity at the floe centre of mass exhibit similar qualitative behaviours compared with the idealised vortex case (figure 9). As $R_f/R_{traj}$ increases to 1.25, the PDF peaks decrease to 0.25, consistent with the TG vortex cases (figure 4 $b$ ). However, analytical relations (3.5) and (3.7), derived for the TG vortex, exhibit discrepancies with the PDF peaks due to differences in the eddy structures of the flow fields. Vorticities near the centre of QG eddies exhibit smaller spatial gradients compared with the TG vortex, leading to smaller normalised rotation rates in the QG eddy cases. In addition, the centre-of-mass ocean vorticity changes abruptly across regions, contrasting with the averaged ocean vorticity, such that the rotation rates normalised by the centre-of-mass ocean vorticity show more discrepancies with the TG vortex cases and analytical relations derived for them.

Figure 9. Rotational motion of trapped ice floes in local QG eddies. ( $a$ – $d$ ) The PDFs of ice floe rotation rates, $\Omega _f$ , normalised by the ocean vorticity at the floe centre of mass, $\omega _{o,C}$ , for different ratios between floe size and trajectory-derived length scale in local eddies marked as $A$ to $D$ with white dashed boxes in a QG flow field, respectively. The simulation results are compared with analytical solutions using square-shape approximation (dashed line), Taylor series expansion (dashed-dotted line) and the passive tracer case (solid line).

The ocean vorticity estimate of the region enclosed by ice floe trajectories in QG eddies exhibits similar trends to the estimate in the TG vortex (figure 10). The PDFs of the ratio between the ocean vorticity estimate and true ocean vorticity reach their peaks at unity. However, these peaks decrease to 0.25 for larger $R_f/R_{traj}$ , reaching up to 1.25. This trend is likely due to a greater discrepancy between the averaged ice floe velocity and the ocean velocity at the floe centre of mass in QG eddies. Overall, analytical relation (3.17) captures the qualitative behaviour of the PDF peaks, albeit with minor quantitative discrepancies observed for certain $R_f/R_{traj}$ .

Figure 10. Averaged ocean vorticity estimate using ice floe trajectories in local QG eddies. ( $a$ – $d$ ) The PDFs of ratios between averaged ocean vorticity of the floe trajectory-enclosed region, $\tilde {\omega }_f$ , and true averaged ocean vorticity of the same region, $\tilde {\omega }_o$ , for different ratios between floe size and trajectory-derived length scale in local eddies marked as $A$ to $D$ with white dashed boxes in a QG flow field, respectively. The simulation results are compared with analytical solutions using square-shape approximation (dashed line), Taylor series expansion (dashed-dotted line) and the passive tracer case (solid line).

The observed ice floe motions and their analytical relations to the ocean, derived from the idealised eddies, offer insights into the rotational relationship between ice floes and local eddies in the QG flow field. These findings demonstrate the potential applicability of these derived relations to analyse ocean eddies and estimate their vorticity using ice floes. The complexity in the QG case stems from how each eddy exhibits a unique structure, velocity and vorticity distribution, as illustrated in figure 7. These fundamental differences between the TG vortex and QG eddies limit the direct application of these relationships to QG fields. Nonetheless, further investigation, including analysing the effects of deformed eddy configurations and different vortex types, holds promise for developing a comprehensive framework to characterise the ocean eddy field from ice floe satellite remote sensing observations.