Staying Put or Going with the Flow

The atmospheric temperature at a fixed spot may change in two ways. First, heat sources or sinks may increase or decrease the thermal energy; for example, sunshine may warm the air or radiation at night may cool it. Second, warmer or cooler air may be transported to the spot by the air flow in a process called advection. Normally, the two mechanisms act together, sometimes negating and sometimes reinforcing each other. What is true for temperature is also true for other quantities: pressure, density, humidity and even the flow velocity itself. This last effect may be described by saying that “the wind blows the wind” [TM132 or search for “thatsmaths” at].


Hurricane Ophelia approaching Ireland, 16 October 2017, 1200Z. Image from

There are two complementary ways to model fluid flow. We can focus on a fixed spot and study how variables like pressure and temperature there change with time. Alternatively, we can “go with the flow”, following a parcel of air as it is carried along by the wind and study how its properties change over time. These two points of view lead to two distinct formulations of the mathematical equations governing the atmosphere. The fixed-point view is modelled by the Eulerian equations, named after the Swiss mathematician Leonhard Euler. The perspective following the flow leads to the Lagrangian equations, named for the French-Italian mathematician Joseph-Louis Lagrange.

EulLagNumerical Algorithms

To forecast the weather, we must compute the solution of the equations using a numerical algorithm – a step-by-step recipe yielding the flow variables at successive moments. The forecast progresses incrementally through small steps forward in time. Traditionally, algorithms were based on the Eulerian equations. This approach yields an accurate solution, but at a high price: the Eulerian algorithms are subject to a severe restriction on the time interval over which the solution is computed. Typically, this “stability criterion” limits the permissible time step to a few minutes, whereas the time scale of the flow is more like an hour or more. Thus, a large number of steps are required to complete the forecast. The result is a heavy computational burden, delaying the delivery of the prediction. 


Andre Robert

Around 1980, the Canadian meteorologist André Robert showed that, by using a numerical algorithm based on Lagrange’s form of the equations, we could liberate ourselves from the shackles of the stability criterion. A larger time step could be used so that fewer steps were needed to reach a given forecast range, substantially reducing the overall computational cost and making the forecast available sooner.

Into Operations

The first weather centre to exploit this idea was the Irish Meteorological Service – now Met Éireann – in a project led by Professor Ray Bates. Ireland was the first of many countries to use a so-called semi-Lagrangian advection scheme in an operational setting. This method is now used internationally in many numerical weather prediction centres. Since timeliness is vital for operational forecasting, the Lagrangian approach to solving the equations was a significant scientific advance.


As a historical footnote, the Eulerian description of fluid flow was first introduced by Daniel Bernoulli, and the Lagrangian description was first used by Euler. We have here an example of Stigler’s law of eponymy: no scientific discovery is named after its original discoverer. Of course, Stigler’s Law did not originate with Stephen Stigler himself!


Bates, J R and  A McDonald, 1982: Multiply-Upstream, Semi-Lagrangian Advective Schemes: Analysis and Application to a Multi-Level Primitive Equation Model. Monthly Weather Review, 110, 1831-1842.

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