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Can tomorrow's weather be calculated, the way an astronomer calculates an eclipse? In 1904 Vilhelm Bjerknes argued that it could, in principle: treat the atmosphere as a physical system and calculate its future from the laws of mechanics and heat. This post covers who he was, what his 1904 paper actually says, a worked example of "stepping forward" in time, and what you can take from his approach.
Who was Vilhelm Bjerknes?
The Royal Society's obituary notice describes him as the "doyen of meteorologists". It records that he died in Oslo on 9 April 1951, at the age of eighty-nine, and that his father, C. A. Bjerknes, showed marked mathematical ability and taught at the university. A short Physics Today obituary calls him a leading Norwegian physicist and the founder of the Bergen School of Meteorology.
What did his 1904 paper say?
The paper appeared in German in Meteorologische Zeitschrift. An English translation by Esther Volken and Stefan Bronnimann was published in 2009, and it is short enough to read in one sitting. Bjerknes opens with two conditions for a "rational" forecast. First, you must know the state of the atmosphere at a given time accurately enough. Second, you must know the laws by which one state develops from another.
He then admits the first condition was not yet met. Weather stations were on land only, although the sea covers most of the surface, and observations were made at ground level only. He suggested filling the gaps with reports from steamships by radiotelegraphy and with observations of the upper air.
Why seven variables and seven equations?
Bjerknes says the state of the atmosphere at any point is known when you have its velocity, density, pressure, temperature and humidity. Velocity has three components, so there are seven unknowns. For these he lists seven equations:
three equations of motion (Newton's second law for air),
the continuity equation (conservation of mass),
the equation of state, linking density, pressure, temperature and humidity,
two equations from the first and second laws of thermodynamics.
Only the equation of state is a simple algebraic relation. The other six are partial differential equations. Bjerknes says there is no hope of solving them exactly, pointing out that even three bodies attracting each other by Newton's law already exceed the mathematics of his day. Instead he proposed working from maps of the initial state and drawing new maps "from hour to hour", splitting each step into a mechanical part and a thermal part.
Worked example: two pieces of Bjerknes's method
Part 1: the equation of state. For dry air, p = ρRT, with R = 287 J kg^-1 K^-1. Take p = 101,325 Pa and T = 288 K. ρ = p ÷ (RT) = 101,325 ÷ (287 × 288) = 101,325 ÷ 82,656 ≈ 1.23 kg/m³. NASA's Earth fact sheet lists a surface density of 1.217 kg/m³ and an average temperature of 288 K, so the estimate is close.
Part 2: one step forward in time. The numbers here are sample values to show the idea. A town has a temperature of 20 °C. Fifty kilometres upwind, it is 14 °C. A steady wind of 10 m/s blows from the cold side towards the town.
Temperature gradient: (20 − 14) ÷ 50 km = 0.12 °C per km.
Distance the air travels in one hour: 10 m/s × 3,600 s = 36 km.
If nothing heats or cools the air on the way, the air reaching the town in an hour started about 36 km upwind, where it was 20 − 0.12 × 36 ≈ 15.7 °C.
So the forecast for one hour ahead is a drop of about 4.3 °C. A real model then corrects this "mechanical" step with a "thermal" step for sunlight, clouds and evaporation, exactly the split Bjerknes described. Our guide to differential equations for JEE covers the maths behind rates of change.
Where do weather fronts come in?
In 1919 the same issue of Monthly Weather Review carried Vilhelm's article Weather forecasting47%3C90b:WF%3E2.0.CO;2) and a paper by J. Bjerknes, On the structure of moving cyclones47%3C95:OTSOMC%3E2.0.CO;2). Fronts are now part of your geography syllabus: NCERT's Class 11 chapter on atmospheric circulation and weather systems defines a front as the boundary zone where two different air masses meet, and names four types: cold, warm, stationary and occluded.
What mistakes do students make about forecasting?
Thinking forecasts are guesses. They are calculations from physical laws, started from measurements.
Thinking better computers alone fix everything. Bjerknes's first condition, an accurate initial state, still limits forecasts. Our post on Edward Lorenz and the butterfly effect shows why small errors grow.
Forgetting units in p = ρRT. Pressure must be in pascals and temperature in kelvin.
What can a student learn from Bjerknes?
He began, as his paper says, "with the simplest formulation of the problem", counted unknowns against equations, and was honest about what was missing. That is a habit you can use on any long numerical: list the unknowns, list the equations, and only then start solving.
If you struggle to set up multi-step problems like this, the Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can compare your set-up with a full solution.
Try it now: redo Part 2 with a wind of 5 m/s and a two-hour step.
