Eulerian and Lagrangian averaging

Eulerian averaging

The Eulerian weighted temporal mean of some scalar $f$ is defined as

\[\begin{equation} \label{fbarEdef} \bar{f}^{\mathrm{E}}(\vb*{x},t) = \int_{-\infty}^{\infty} G(t-s)f(\vb*{x},s) \, \mathrm{d} s\,, \end{equation}\]

where $G(t)$ is some weight function, also referred to as a filter kernel or impulse response. The Fourier transform of $G(t)$ is denoted $\hat{G}(\omega)$, and is the frequency response of the weight function. Here, we will generally want $G(t)$ to represent a low-pass filter, which retains the low frequencies and removes high frequencies. For example, two possible choices of weight function are

1. Top-hat impulse response: $G(t) = \begin{cases} 1/T \,, \hspace{1cm} -T/2 < t < T/2 \\ 0\,, \hspace{1cm} \mathrm{otherwise}\end{cases}\,,$
2. Top-hat frequency response: $\hat{G}(\omega) = \begin{cases} 1 \,, \hspace{1cm} -\omega_c < \omega < \omega_c \\ 0\,, \hspace{1cm} \mathrm{otherwise}\end{cases}\,.$

The Eulerian mean $\bar{f}^{\mathrm{E}}(\vb*{x},t)$ is the field found by taking an average in time at a fixed spatial location $\vb*{x}$.

Note that OceananigansLagrangianFilter can also be used to find the Eulerian mean. Two different ways to do this are described in helpful tips.

Lagrangian averaging

In contrast, the Lagrangian mean finds the temporal average whilst moving with the flow on an (imaginary) fluid particle. We define the Lagrangian weighted temporal mean as

\[\begin{equation} \label{fbardef} \bar{f}^{\mathrm{L}}(\bar{\vb*{\varphi}}(\vb*{a},t),t) = \int_{-\infty}^{\infty} G(t-s)f(\vb*{\varphi}(\vb*{a},s),s) \, \mathrm{d} s\,, \end{equation}\]

where the flow map $\vb*{\varphi}(\vb*{a},t)$ is the position at time $t$ of a particle with label $\vb*{a}$ (which could be the initial position of the particle such that $\vb*{\varphi}(\vb*{a},0) = \vb*{a}$). The mean flow map $\bar{\vb*{\varphi}}$ is defined by

\[\begin{equation}\label{phibardef} \bar{\vb*{\varphi}}(\vb*{a} ,t) = \int_{-\infty}^{\infty} G(t-s)\vb*{\varphi}(\vb*{a},s)\,\mathrm{d} s\,. \end{equation}\]

The definition \eqref{fbardef} ensures that $\bar{f}^{\mathrm{L}}$ is the true generalised Lagrangian mean, in that (for strict band-pass filters) applying the same averaging procedure to the mean flow itself leaves it unchanged (Baker et al., 2025). However, we also define an alternative Lagrangian mean, which is a rearrangement of $\bar{f}^{\mathrm{L}}$:

\[\begin{equation}\label{fstardef} f^*(\vb*{\varphi}(\vb*{a},t),t) = \int_{-\infty}^{\infty} G(t-s)f(\vb*{\varphi}(\vb*{a},s),s) \, \mathrm{d} s\,. \end{equation}\]

While $\bar{f}^{\mathrm{L}}(\vb*{x},t)$ describes the average along a particle trajectory whose mean position is $\vb*{x}$, $f^*(\vb*{x},t)$ defines the average along a particle trajectory whose position is $\vb*{x}$ at time $t$. It is often more desirable, or more convenient, to find $f^*$ instead. If $\bar{f}^{\mathrm{L}}$ is also needed, it can be found by a rearrangement of $f^*$ using a map

\[\begin{equation} \vb*{\Xi}(\vb*{\varphi}(\vb*{a},t),t) = \bar{\vb*{\varphi}}(\vb*{a},t)\,. \end{equation}\]