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, { \frac{\partial C}{\partial t}} . (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:

    \[ { \frac{\partial C}{\partial t}} = -\underline{\textbf{u}} \cdot \nabla C  + K \nabla^2 C + Source(x,y,z,t) \]


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 fluidThis 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, -\underline{\textbf{u}} \cdot \nabla C, from both sides of the Eulerian advection diffusion equation

Key Takeaways

The Lagrangian advection-diffusion equation is:

    \[{ \frac{D C}{Dt}}={ \frac{\partial C}{\partial t}}+\underline{\textbf{u}} \cdot \nabla C  = K \nabla^2 C + Source(x,y,z,t) \]

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