DriftCaster Finite Inertia Mathematical Basis

Physics based finite-inertia buoy trajectory framework

1. Basic idea

A common, simple way to predict a drifting buoy trajectory is to assume that its velocity is directly prescribed by the environment:

\mathbf V_b
=
\mathbf U_c
+
\mathbf U_S
+
\alpha\mathbf U_{10}

Here, the buoy is effectively treated as a massless tracer. If the current changes direction instantaneously, the buoy changes direction instantaneously.

A real buoy has mass and therefore momentum. Wind, currents, and waves exert forces on it, causing it to accelerate rather than instantaneously adopt a new velocity.

The proposed framework therefore predicts two things simultaneously:

$\frac{d{\mathbf{x}}_b}{dt}={\mathbf{V}}_b$

and

M_{\rm eff}\frac{d\mathbf V_b}{dt}
=
\text{sum of forces acting on the buoy}

where:

• ${\mathbf{x}}_b$ = buoy position
• ${\mathbf{V}}_b$ = buoy velocity
• $M_{\mathrm{eff}}$ = effective mass of the buoy and surrounding water that must be accelerated

This is a simplified version of the Maxey-Riley / Beron-Vera-Olascoaga-Miron (BOM) approach for finite-inertia particles at the ocean surface.

The important principal is that there is no fitted memory or relaxation timescale. The buoy’s response time emerges naturally from its mass, size, drag, immersion, and environmental conditions.

2. The environmental velocities

At every timestep, and at the buoy’s evolving position, we obtain three environmental velocity fields.

Ocean current

${\mathbf{U}}_c({\mathbf{x}}_b,t)$

This is the Eulerian ocean current supplied by the ocean model. It represents the motion of the water itself.

The buoy is strongly coupled to this flow through its submerged portion.

Stokes drift

${\mathbf{U}}_S({\mathbf{x}}_b,t,z)$

Stokes drift represents the net Lagrangian transport associated with surface waves.

It is calculated from the directional wave spectrum rather than parameterized from wind speed.

Because Stokes drift decreases rapidly with depth, the quantity relevant to the buoy should ideally be averaged over its submerged portion:

\overline{\mathbf U_S}
=
\int_{-h}^{0}W(z)\mathbf U_S(z)\,dz


where:

• $h$ = buoy draft
• $z$ = depth
• $W(z)$ = weighting describing how strongly different portions of the submerged buoy interact with the surrounding water

A simple first approximation could use a uniform average over the buoy draft.

Wind

${\mathbf{U}}_a={\mathbf{U}}_{10}({\mathbf{x}}_b,t)$

where ${\mathbf{U}}_{10}$ is the atmospheric wind vector at 10m.

Importantly, wind is not converted directly into buoy velocity. Instead, the wind exerts aerodynamic drag on the portion of the buoy above the water. This eliminates the need to prescribe something such as:

${\mathbf{U}}_{\mathrm{windage}}=0.012{\mathbf{U}}_{10}$

The effective windage instead emerges from the the buoy’s physical properties.

3. The water velocity experienced by the buoy

For the first version of the framework, we define an effective Lagrangian water velocity:

\mathbf U_L
=
\mathbf U_c+\overline{\mathbf U_S}

This combines:

• Eulerian transport by the current
• wave-induced Stokes transport

The relative velocity between the water and the buoy is therefore:

\mathbf U_{\rm rel,w}
=
\mathbf U_L-\mathbf V_b

If ${\mathbf{V}}_b={\mathbf{U}}_L$, there is little relative water motion and therefore little hydrodynamic drag.

If the current suddenly accelerates while the buoy retains its previous velocity, $|{\mathbf{U}}_L-{\mathbf{V}}_b|$ becomes large, producing a force that accelerates the buoy toward the new flow.

This is how the buoy naturally retains momentum while still eventually following the surrounding water.

4. Hydrodynamic drag from the water

The submerged portion of the buoy experiences drag:

\mathbf F_{D,w}
=
\frac12
\rho_w
C_{D,w}(Re_w)
A_w
|\mathbf U_L-\mathbf V_b|
(\mathbf U_L-\mathbf V_b) ${\rho }_w$ : water density

• ${\rho }_w\sim 1025\mathrm{kg}{\mathrm{m}}^{-3}$ for sea water

$A_w$: submerged reference area

• the effective frontal area of the portion of the spotter below the water.
• This comes from the physical geometry of the buoy.
• A larger submerged area means stronger coupling to the water.

$C_{D,w}$: hydrodynamic drag coefficient

• this describes how efficiently the submerged buoy converts relative water motion into drag.
• We do not necessarily make $C_D$ constant.
• Instead, $C_{D,w}=C_D(R{e}_w)$ where Re_w is the particle Reynolds number.

$R{e}_w$: water-side Reynolds number

• Re_w=
\frac{L_w|\mathbf U_L-\mathbf V_b|}{\nu_w}
• where:
  • $L_w$ = characteristic submerged dimension of the buoy
  • ${\nu }_w$ = kinematic viscosity of seawater
  • $|{\mathbf{U}}_L-{\mathbf{V}}_b|$ = water velocity relative to the buoy
• This tells us what hydrodynamic regime the buoy is experiencing.
• At low Reynolds number, $C_D\to \frac{24}{Re}$, which causes the drag law to approach linear stokes drag, consistent with the classical Maxey-Riley theory.
• At high Reynolds number, the drag becomes increasingly nonlinear and approximately quadratic.
• Therefore, the same equation works from weak conditions into TC conditions.

5. Aerodynamic drag from the wind

The portion of the buoy above the water experiences a separate aerodynamic force:

\mathbf F_{D,a}
=
\frac12
\rho_a
C_{D,a}(Re_a)
A_a
|\mathbf U_{10}-\mathbf V_b|
(\mathbf U_{10}-\mathbf V_b)

The structure is similar to water drag, but the physical quantities are different.

${\rho }_a$ : Atmospheric density, approximately $1.2\mathrm{kg}{\mathrm{m}}^{-3}$

• This is nearly 1,000 times smaller than seawater density.

$A_a$ : the exposed aerodynamic reference area of the Spotter

• This depends on its freeboard and geometry.

$C_{D,a}$ :Aerodynamic drag coefficient of the exposed buoy

• This is not the same quantity as the air–sea drag coefficient used for calculating ocean wind stress.
• It describes air flowing around the physical Spotter.

$R{e}_a$

• Re_a=
\frac{L_a|\mathbf U_{10}-\mathbf V_b|}{\nu_a}
• This allows aerodynamic drag to change appropriately between weak and extreme winds.

6. Windage now emerges naturally

Eventually, $\frac{d{\mathbf{V}}_b}{dt}\approx 0$. The air may be pushing the bnuoy downwind while water drag resists that motion.

The buoy settles into a velocity where these forces approximately balance: ${\mathbf{F}}_{D,w}+{\mathbf{F}}_{D,a}\approx 0.$

Therefore, an effective relationship resembling \mathbf V_b
\approx
\mathbf U_c+\mathbf U_S+\alpha\mathbf U_{10} can naturally emerge.

But now $\alpha$ isn’t prescribed. It emerges from quantities such as ${\rho }_a,{\rho }_w,A_a,A_w,C_{D,a},C_{D,w}$.

Consequently, the effective windage can change with environmental conditions while remaining physically determined.

7. Buoy mass

The actual Spotter has mass $m_b$.

Newton’s second law gives $F=m_{b}a$.

A more massive buoy requires more force to produce the same acceleration.

This is the fundamental source of the buoy’s mechanical momentum.

But $m_b$ isn’t quite the entire inertial mass.

8. Added mass

When the buoy accelerates through water, it must also accelerate some surrounding water.

This produces added mass:

$m_A=C_A{\rho }_w{V}_{\mathrm{sub}}$

where:

• $C_A$ = added-mass coefficient determined primarily by geometry
• $V_{\mathrm{sub}}$ = submerged buoy volume
• ${\rho }_w$ = water density

For an ideal sphere, $C_A\approx 0.5$.

A Spotter is not a sphere, so a geometry-appropriate value would eventually be preferable.

The effective inertial mass is then approximately

$M_{\mathrm{eff}}=m_b+m_A$

This means the environmental forces must accelerate both the buoy and some surrounding water. Again, this creates inertia without introducing an arbitrary relaxation parameter.

9. Acceleration of the surrounding water

Maxey–Riley dynamics also recognize that the water itself may be accelerating.

The material acceleration of the water is

\frac{D\mathbf U_L}{Dt}
=
\frac{\partial\mathbf U_L}{\partial t}
+
(\mathbf U_L\cdot\nabla)\mathbf U_L

The first term, $\frac{\partial {\mathbf{U}}_L}{\partial t},$ represents the current changing with time at a fixed location.

The second, $({\mathbf{U}}_L\cdot \nabla ){\mathbf{U}}_L$, represents acceleration associated with moving through spatial gradients in the flow.

This term may become especially important in rapidly evolving TC conditions.

A complete reduced Maxey–Riley formulation therefore contains a force associated with the acceleration of the surrounding fluid in addition to ordinary relative-velocity drag.

The exact coefficient should be retained from the chosen reduced Maxey–Riley/BOM derivation rather than fitted.

10. Coriolis acceleration

Because trajectories occur on a rotating Earth, the horizontal momentum equation also contains Coriolis acceleration:

$f\hat{\mathbf{z}}\times {\mathbf{V}}_b$

where

$f=2\Omega \sin \varphi .$

Here:

• $\Omega$ = Earth’s rotation rate
• $\varphi$ = latitude
• $\hat{\mathbf{z}}$ = vertical unit vector

Coriolis causes moving objects to experience an apparent acceleration perpendicular to their motion.

It becomes increasingly relevant as the timescale of the trajectory increases.

Importantly, this is particle Coriolis dynamics, not the same thing as adding an empirical inertial oscillation to the trajectory.

The ocean model may already contain near-inertial oscillations in ${\mathbf{U}}_c$, while the buoy’s own momentum equation still needs to be formulated consistently in the rotating reference frame.

11. The resulting framework

The buoy state is $({\mathbf{x}}_b,{\mathbf{V}}_b)$.

At each timestep we know its current position and velocity.

At that location and time, we sample: \mathbf U_c,\qquad
\mathbf U_{10},\qquad
E(f,\theta).

The wave spectrum provides $\overline{{\mathbf{U}}_S}.$

We then construct \mathbf U_L
=
\mathbf U_c+\overline{\mathbf U_S}.

From the relative velocities, ${\mathbf{U}}_L-{\mathbf{V}}_b$ and ${\mathbf{U}}_{10}-{\mathbf{V}}_b,$ we calculate the appropriate Reynolds numbers and drag forces.

The general momentum equation then has the structure

M_{\rm eff}
\left[
\frac{d\mathbf V_b}{dt}
+
f\hat{\mathbf z}\times\mathbf V_b
\right]
=
\mathbf F_{\rm fluid\,acceleration}
+
\mathbf F_{D,w}
+
\mathbf F_{D,a}

The resulting acceleration updates the buoy velocity: \mathbf V_b(t)
\rightarrow
\mathbf V_b(t+\Delta t),

and that velocity updates its position: \mathbf x_b(t)
\rightarrow
\mathbf x_b(t+\Delta t).

At the new position, the environmental fields are sampled again and the process repeats.

Goal

The central hypothesis is:

A freely drifting surface buoy is not a perfectly passive Lagrangian tracer. Its finite mass, buoyancy, immersion, aerodynamic exposure, hydrodynamic drag, added mass, wave forcing, and Earth’s rotation cause its velocity to respond dynamically to the evolving environment.

The framework therefore replaces

\text{environmental velocity}
\rightarrow
\text{buoy velocity}

with

\text{environment}
\rightarrow
\text{forces}
\rightarrow
\text{buoy acceleration}
\rightarrow
\text{buoy velocity}
\rightarrow
\text{buoy position}

That is the key conceptual difference.

It also gives the model a useful physical constraint: every mechanism added to the trajectory must correspond to an identifiable force or environmental transport process, rather than being introduced solely because it improves trajectory agreement.