You turn on the television, and often the first channel that pops up is the weather. It’s going 24/7 with predictions that go from weekly all the way down to hourly, with conditions that go from humidity to temperature. But what goes on behind the scenes is heavily entrenched in mathematics– meteorology’s backbone is a technique called Numerical Weather Prediction (NWP). This process uses the power of computers and complex algorithms to model the future state of the atmosphere and make a forecast. It uses variables such as temperature, pressure, wind, and rainfall collected from various instruments such as satellites, buoys, radars, and ground-based observing stations all to make those television forecasts with which we are so familiar.
The roots of Numerical Weather Prediction can be traced back to the work of physicist Vilhelm Bjerknes. Born in Norway, he was a professor of applied mechanics and mathematical physics at the University of Stockholm and is considered to be the father of modern meteorology. His most notable achievement other than the fundamental relationship between fluid dynamics and thermodynamics was his 1904 paper on weather forecast. He theorized that a system of nonlinear partial differential equations could be used for long-term weather prediction, and idea previously unheard of.
Bjerknes’ theories were largely left untouched until 1922, when Lewis Fry Richardson spent three years practicing and researching Bjerknes’ techniques. While he was a member of an ambulance unit during World War I, he did a six-week computation that would predict the change in pressure at a single point over six hours. It ended up being very inaccurate, but it was the first application of Bjerknes techniques. Richardson foresaw a fantastical “forecast factory” that would use 64,000 humans in a great hall as computers to forecast the weather all over the globe. The invention of the computer made this prediction seem silly, but at the time it was revolutionary.
Small gains were made in the 20th century, such as the weather balloon and the confirmed existence of the jetstream by American WWII pilots, but it wasn’t until the invention of the first computer that NWP really became what it is today. John von Neumann, the developer of the ENIAC computer, recognized that his invention was ideal for the computations necessary for NWP that had been developed by Bjerknes and Richardson so long ago. He assembled a group of theoretical meteorologists and made the first day nonlinear weather prediction in April of 1950. Weather centers spread across the globe during the 20th century after this discovery, making leaps and bounds in accuracy with weather satellites and increasing amount of computer power and data.
The mathematics behind NWP lies in the idea of the nonlinear partial differential equation. It is a vast branch of mathematics used in many fields such as neurology, gas dynamics, fluid mechanics, relativity, ecology, and thermodynamics. But its benefit for NWP is that it allow one to look at how a single variable changes in a single direction or when other variables are constant. It also allows one to determine the gradient of a field, or essential the direction one needs to move in to see the greatest temperature increase, for example. Essentially, taking a partial derivative is only possible in multivariable equations because one needs to differentiate with respect to only one variable and treat the rest as constants. Partial derivatives are denoted by ∂, called “del” instead of the “d” we usually see in dy/dx. To visualize, just imagine a curve in 3D space cut by a plane of single value of y. The resulting curve’s slope would be the partial derivative of x.
The nonlinear aspect of the partial differential equations essentially represents their increased complexity and chaotic, unpredictable nature. Often, nonlinear equations in physics and branches of mathematics cannot be solved, or only sometimes can be solved. The mathematical definition of nonlinear is that the change in output of a function is not proportional to the change in input, and often is found in equations that use chaos theory–numerical weather prediction algorithms are one of these. Chaos theory has allowed for a greatly increased accuracy in forecast models as well as ensemble forecasts that allow for small statistical/probability changes.
Nonlinear partial differential equations come most into play in the Primitive Equations of meteorology. The Primitive Equations are the set of nonlinear differential equations used to approximate atmospheric flow, and contain three main sets: continuity equations, conservation of momentum/Navier-Stokes equations, and thermal energy equations. Each represents conservation of mass, hydrodynamic flow on the surface of a sphere, and thermal energy, respectively. Each of these uses partial derivatives of several variables, such as: T- temperature; P- pressure, in millibars; θ – potential temperature; U and V– for horizontal velocities (vector quantities), can also represent vector components of a single velocity when combined with W (vertical velocity) to give a three dimensional wind field; ρ (the Greek letter rho)- to mean density (a function of temperature and composition); and RH- relative humidity.
The primitive equations are used to make vast computational algorithms to produce forecast models. The simplified process of how forecasts are made and were made by Richardson and Neumann starts with observation. Given field values of the necessary values, one can solve the nonlinear partial differential equations and calculate the atmospheric tendencies. Newer and newer data sets are used to recompute the tendencies and set trends in the data. Extrapolation of these tendencies allows for short-term prediction, but use of complex algorithms with Chaos theory and six-layer atmospheric models today allows for accurate weather prediction months into the future.
The NOAA (National Oceanic and Atmospheric Association) heads most NWP today. They have over 210 million field observations processed by supercomputers and plugged into algorithms each day, harkening back to the 64,000-person forecast theater Richardson foresaw. Current operational computers run with a computational speed of 14 trillion calculations per second and 14.8 million model fields per day. This incredible speed allows for the malleable forecasts we see on our phones and televisions– ones that are weekly, daily, and even hourly. The future of Numerical Weather Prediction is vast. The frontier of new research is environmental modeling. Forecast predictions will son be able to include ecosystem forecasts rather than just climate, weather, and water, and the climate change problem so presently at the forefront of our modern world may be able to be predicted– and therefore avoided. Not much farther off is the idea of space weather predictions, allowing for space travel to become a more possible venture in the next few years.
The next time you check your phone for the high temperature tomorrow, know that behind that number are thousands of data inputs, calculations, and some partial derivatives. Behind every forecast is layers of mathematics we don’t even realize is there.