# 16 Eulerian and Lagrangian rates of change

The terms **Eulerian** and **Lagrangian** in this context refer to whether one is considering the time rate of change is within a control volume that is stationary, or within one that is moving with the fluid.

The **Eulerian** rate of change is observed by many common instruments. It is geo-referenced to a single point in space and evolving in time. While nothing is perfectly fixed in space, some approximate Eulerian observers are:

- current meters (anchored to the seafloor)
- satellites that repeatedly sample the same points in space
- CTDs that profile vertically, with the horizontal position held fixed (as closely as possible).
- Fixed organisms like barnacles

Key Takeaways

The Eulerian rate of change corresponds mathematically to the partial derivative with respect to time, . (Recall that partial derivatives assume all the other variables, in this case the spatial coordinates, are held constant.)

The advection-diffusion equation for the Eulerian time rate of change was derived in the previous section:

The **Lagrangian** rate of change is observed by instruments or organisms that drift with the current. It is the rate of change experienced by a water parcel. While nothing perfectly drifts with the current, some examples of approximate Lagrangian observers are:

- Satellite-tracked surface drifters
- Argo floats, when floating at their parking depth
- Plankton (assuming no biological control over their movement)

The control volume corresponding to a water parcel has no advective flux across its surface, because it is moving *at the same velocity *as the surrounding fluid*. *This means there is no advective flux divergence either. We can thus find the Lagrangian rate of change by subtracting that part due to advective flux divergence, , from both sides of the Eulerian advection diffusion equation

Key Takeaways

The Lagrangian advection-diffusion equation is:

Lagrangian observers only record changes due to diffusion and source/sink terms.

We use the capital “*D*” here to indicate the Lagrangian rate of change. It is equivalent to the *total derivative* in multi-variate calculus. It is also called the *material derivative.*

Want more? See this 5-minute YouTube video with more visually-oriented explanations of the difference between Eulerian and Lagrangian fields