Julius Busse — Research
Eigenbasis Picard iteration for nonlinear particle dynamics (work in progress, June 2026)
In the infinite-depth limit, through a sequence of transformations we reduce the particle system and arrive at \[ \begin{aligned} \frac{\mathrm{d}\sigma}{\mathrm{d}t} &= -i\sigma\omega - ib\varepsilon\,\sigma\bar{\sigma}, \\[4pt] \frac{\mathrm{d}\omega}{\mathrm{d}t} &= \frac{1}{\mathrm{St}}\!\left(\varepsilon(1-b)\bar{\sigma} - \omega\right) - \left(\frac{1}{\mathrm{St}} + i(1-b)\right). \end{aligned} \] We can restate this problem, using an ansatz that decomposes \(\sigma,\omega\) along the eigenvectors, arriving at \[ PD\dot{A} = N(\Xi), \] where \(N(\Xi) = B(\Xi,\Xi)\) with \(B\) a bilinear form. We are investigating the asymptotic and convergence properties of Picard iteration applied to this system, and its relation to classical multiple-scales methods.
Wave-averaged transport of inertial particles in finite-depth water waves (April 2026)
We derive wave-averaged asymptotics for inertial particle motion under linear finite-depth water waves. The dimensionless flow field is \[ \vec{u} = \frac{1}{\sinh h} \begin{pmatrix} \cosh(z+h)\cos(x-t)\\ \sinh(z+h)\sin(x-t) \end{pmatrix}, \] and particle motion is governed by the simplified Maxey–Riley equations, \[ \dot{\vec{x}} = \vec{v}, \qquad \dot{\vec{v}} = \frac{1}{\mathrm{St}}\!\left(\varepsilon\vec{u} - \vec{v}\right) - \frac{1-b}{\tanh h}\,\vec{e}_z + b\,\frac{\mathrm{d}(\varepsilon\vec{u})}{\mathrm{d}t}, \] where \(\varepsilon\) is the wave steepness, \(\mathrm{St}\) the Stokes number, and \(b \in [0,1]\) controls added mass and settling. We assume \(\epsilon, \mathrm{St}, (1-b)\ll 1\). We expand \(\vec{v}\) in \(\mathrm{St}\), evaluate at the zeroth-order particle path, Taylor-expand in \(\varepsilon\) and wave-average to obtain \[ \dot{x} = \frac{\varepsilon^2}{2\sinh^2\!h}\cosh(2z^*) \left[1 + b'\mathrm{St}^2 \frac{-b' - \tanh^2\!h + 2b'\tanh h\tanh(2z^*)}{\tanh^2\!h} \right] + O(\varepsilon^3 + \mathrm{St}^3 + \varepsilon\,\mathrm{St}\,b'), \] \[ \dot{z} = -\frac{\mathrm{St}\,b'}{\tanh h} \left(1 + \frac{\varepsilon^2}{2\sinh^2\!h} \bigl(\cosh(2z^*) + \tanh h\sinh(2z^*)\bigr) \right) + O(\varepsilon^3 + \mathrm{St}^3 + \varepsilon\,\mathrm{St}\,b'), \] where \(z^* = h + z_0 - b'\mathrm{St}\,t/\tanh h\) and \(b' = 1-b\). The \(O(\varepsilon^2\mathrm{St}^2)\) correction to \(\dot{x}\) reduces the Stokes drift for inertial particles; the \(O(\varepsilon^2\mathrm{St})\) term in \(\dot{z}\) describes enhanced settling near the free surface. The asymptotics can be integrated explicitly and the final advection distance determined via numerical root finding. [preprint]
Numerical averaging methods for oscillatory particle trajectories (April 2026)
To verify the asymptotics numerically, one must extract the slow drift from a trajectory dominated by fast wave-induced oscillations. We compare three procedures: single Lagrangian averaging, double averaging and Gaussian averaging. Single Lagrangian averaging is sensitive to the choice of phase at which the Lagrangian period is evaluated; we show that no single phase reproduces both components of the asymptotics simultaneously. Double and Gaussian averaging do not suffer from this issue and agree well with the derived asymptotics across a wide range of \(\mathrm{St}\) and \(\varepsilon\).
Explosion of stochastic delay differential equations (June 2026)
Joint work with Cónall Kelly (University College Cork). We study scalar SDDEs with a single discrete delay, \[ \mathrm{d}x(t) = b(x(t),\,x(t-\tau))\,\mathrm{d}t + \sigma(x(t),\,x(t-\tau))\,\mathrm{d}W(t), \] and the question of whether solutions explode in finite time with positive probability. Under a regularity condition on \(b\) and a Yamada–Watanabe-type condition on \(\sigma\), we show that this question reduces to the same question for the ordinary SDE obtained by freezing the delayed argument at a constant. The argument combines the Ikeda–Watanabe comparison theorem for SDEs with the blow-up analysis of Ishiwata and Nakata for delay differential equations.