Numerical integrators for Lagrangian oceanography

Abstract

Abstract. A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field precalculated with an ocean model. Mathematically, this is simply numerical integration of an ordinary differential equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field in both space and time, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km, and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge–Kutta integrators of orders 1–4, as well as three variable-step Runge–Kutta methods (Bogacki–Shampine 3(2), Dormand–Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, which are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand–Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. Equivalently, the special-purpose integrators can provide the same accuracy as standard methods at a reduced computational cost. The best results are seen for coarser resolutions (4 and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations on data from atmospheric models.

Exploring the origin and fate of surface and sub-surface marine microplastics in the Canary Islands region

“…In this case, 60501 particles were released at 1062 m depth in the vicinity of Cape Sant Vincent and followed for 4 years (2017-2020). In both cases, surface and intermediate levels, the Lagrangian trajectories are defined by the ODE ∂x ∂ t = ṽ (x, y, t), with the initial condition x(t 0 )=x 0 (van Sebille et al, 2018;Nordam and Duran, 2020). An explicit Runge-Kutta scheme of order 4-5 in both space and time was then used to solve the ODE (Tyranowski and Desbrun, 2019;Komen et al, 2020).…”

On the Advection of Upwelled Water on the Western Yucatan Shelf

“…Trajectories were integrated back in time for 40 days, from July 19 to June 10, in one experiment using the 2018 velocities, and in another experiment using the 2003-2014 HyCOM climatological velocity described in Duran et al (2018). The almost 7,000 trajectories in each experiment were computed using a fourth-order Runge-Kutta with cubic interpolation, a combination that is efficient and accurate when integrating discrete data (Nordam and Duran, 2020). The 2018 velocity was a two-dimensional field at 5 m depth, while the climatological velocity was a surface field.…”

Speeding up Python-based Lagrangian Fluid-Flow Particle Simulations via Dynamic Collection Data Structures

“…The tracing is done by Lagrangian advection, where particles follow a given background (Eulerian) advective flow field [9] and whose trajectories are numerically integrated over time as given in eq. (2.1), implemented in e.g., an Euler-forward-or Runge-Kutta scheme [19].…”