Lagrangian filtering for wave-mean flow decomposition

TL;DR

Geophysical flows exhibit fast waves embedded in slower mean motions, complicating wave–mean separation in simulations. The paper generalizes Lagrangian filtering by incorporating arbitrary convolutional weights to define a weighted Lagrangian mean, enabling on-the-fly wave–mean decomposition without particle tracking. It provides three PDE-based strategies to compute the generalized Lagrangian mean alongside the flow, and demonstrates their use in a rotating shallow-water model with geostrophic turbulence and Poincaré waves, including multiple wave–mean decompositions. The results show that Lagrangian filtering can recover a clean wave field and robustly separate wave and mean components, while highlighting trade-offs in accuracy, boundary handling, and computational cost across strategies.

Abstract

Geophysical flows are typically composed of wave and mean motions with a wide range of overlapping temporal scales, making separation between the two types of motion in wave-resolving numerical simulations challenging. Lagrangian filtering - whereby a temporal filter is applied in the frame of the flow - is an effective way to overcome this challenge, allowing clean separation of waves from mean flow based on frequency separation in a Lagrangian frame. Previous implementations of Lagrangian filtering have used particle tracking approaches, which are subject to large memory requirements or difficulties with particle clustering. Kafiabad and Vanneste (2023, KV23) recently proposed a novel method for finding Lagrangian means without particle tracking by solving a set of partial differential equations alongside the governing equations of the flow. In this work, we adapt the approach of KV23 to develop a flexible, on-the-fly, PDE-based method for Lagrangian filtering using arbitrary convolutional filters. We present several different wave-mean decompositions, demonstrating that our Lagrangian methods are capable of recovering a clean wave-field from a nonlinear simulation of geostrophic turbulence interacting with Poincaré waves.

Figures (8)

• Figure 1: Schematic of a particle trajectory (black) with label $\bm{a}$ in the interval $[t^*-T,t^* + T]$, with positions labelled by the flow map $\bm{\varphi}(\bm{a},t)$. The mean particle trajectory on the same interval is shown in blue, with positions labelled by the mean flow map $\bar{\bar{\bm{\varphi}}}(\bm{a},t^*)$. Red arrows indicate the maps $\bm{\Xi}^{i \mapsto j}$ from position $i$ to position $j$, where position 1 is the trajectory endpoint $\bm{\varphi}(\bm{a},t^* +T)$, position 2 is the trajectory mean $\bar{\bar{\bm{\varphi}}}(\bm{a},t^*)$, and position 3 is the trajectory midpoint $\bm{\varphi}(\bm{a},t^*)$.
• Figure 2: Shallow water relative vorticity for a simulation over 40 time units ($T=20$). The mode-1 wave frequency is $\omega = 4.17$, and the low-pass filters use a cut-off frequency of $\omega_c = 2$. a) Instantaneous vorticity at the interval midpoint $t^* = 20$. b) Lagrangian and c) Eulerian low-pass at $t^* = 20$. e) Lagrangian and f) Eulerian top-hat mean at $t^* = 20$, computed over the interval $[18,22]$, i.e. $T = 2$. d) $G(t)$ for the low-pass and top-hat means, showing that $T = 2$ is an appropriate averaging interval for the top-hat to compare it to the low-pass. The directory including the Jupyter notebook that generated this figure can be accessed at https://cocalc.com/share/public_paths/bdc0d1617e113644a25e3ba4c0b91b8fad20701f/Figure-2.
• Figure 3: As in figure \ref{['fig:mean_comparison']}, showing the time ($t^*$) evolution of each field at $y = 2.8$. The directory including the Jupyter notebook that generated this figure can be accessed at https://cocalc.com/share/public_paths/bdc0d1617e113644a25e3ba4c0b91b8fad20701f/Figure-3.
• Figure 4: Comparison of calculation of $\overline{\zeta}^\mathrm{L}$ using strategies 1 and 3 with $T=20$. a) $\tilde{\zeta}$, found using strategy 1, b) $\overline{\zeta}^\mathrm{L}$, found by remapping $\tilde{\zeta}$ using $\bm{\Xi}^{1 \mapsto 2}$, and c) the $y$ component of $\bm{\Xi}^{1 \mapsto 2}$. d) $\zeta^*$, found using strategy 3, e) $\overline{\zeta}^\mathrm{L}$, found by remapping $\zeta^*$ using $\bm{\Xi}^{1 \mapsto 3}$, and f) the $y$ component of $\bm{\Xi}^{1 \mapsto 3}$. $x$ and $y$ axes correspond to $x$ and $y$ coordinates of the full domain. The directory including the Jupyter notebook that generated this figure can be accessed at https://cocalc.com/share/public_paths/bdc0d1617e113644a25e3ba4c0b91b8fad20701f/Figure-4.
• Figure 5: The four different wave decompositions: (top) Eulerian, (second row) semi-Eulerian, (third row) Lagrangian first definition, and (bottom) Lagrangian second definition. For each row, the middle 'mean' field is subtracted from the left 'instantaneous' field to give the right 'wave' field. The flow parameters are as for figure \ref{['fig:mean_comparison']}, and strategy 3 is used. The directory including the Jupyter notebook that generated this figure can be accessed at https://cocalc.com/share/public_paths/bdc0d1617e113644a25e3ba4c0b91b8fad20701f/Figure-5.