Anyone who lives by the sea is familiar with the regular ebb and flow of the tides. But we all live at the bottom of an ocean of air. The atmosphere, like the ocean, is a fluid envelop surrounding the Earth, and is subject to the influence of the Sun and Moon. While sea tides have been known for more than two thousand years, the discovery of tides in the atmosphere had to await the invention of the barometer [TM202 or search for “thatsmaths” at irishtimes.com].
Posts Tagged 'Geophysics'
We are living at the bottom of an ocean
Published January 7, 2021 Irish Times Leave a CommentTags: Geophysics
Ireland’s Mapping Grid in Harmony with GPS
Published November 19, 2020 Irish Times Leave a CommentTags: Geophysics, Maps, Spherical Trigonometry
The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential [TM199 or search for “thatsmaths” at irishtimes.com].
Transverse Mercator projection with central meridian at Greenwich.
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Weather Forecasts get Better and Better
Published November 5, 2020 Irish Times Leave a CommentTags: Geophysics, Numerical Weather Prediction
Weather forecasts are getting better. Fifty years ago, predictions beyond one day ahead were of dubious utility. Now, forecasts out to a week ahead are generally reliable [TM198 or search for “thatsmaths” at irishtimes.com].
Anomaly correlation of ECMWF 500 hPa height forecasts over three decades [Image from ECMWF].
A Ring of Water Shows the Earth’s Spin
Published February 13, 2020 Occasional Leave a CommentTags: Geophysics, Mechanics
Around 1913, while still an undergraduate, American physicist Arthur Compton described an experiment to demonstrate the rotation of the Earth using a simple laboratory apparatus.
Spin-off Effects of the Turning Earth
Published June 6, 2019 Irish Times Leave a CommentTags: Fluid Dynamics, Geophysics, Numerical Weather Prediction
Gaspard-Gustave de Coriolis (1792-1843).
On the rotating Earth, a moving object deviates from a straight line, being deflected to the right in the northern hemisphere and to the left in the southern hemisphere. The deflecting force is named after a nineteenth century French engineer, Gaspard-Gustave de Coriolis [TM164 or search for “thatsmaths” at irishtimes.com].
Coriolis was interested in the dynamics of machines, such as water mills, with rotating elements. He was not concerned with the turning Earth or the oceans and atmosphere surrounding it. But it is these fluid envelopes of the planet that are most profoundly affected by the Coriolis force.
Joseph Fourier and the Greenhouse Effect
Published March 21, 2019 Irish Times Leave a CommentTags: Applied Maths, Fourier analysis, Geophysics
Jean-Baptiste Joseph Fourier, French mathematician and physicist, was born in Auxerre 251 years ago today. He is best known for the mathematical techniques that he developed in his analytical theory of heat transfer. Over the past two centuries, his methods have evolved into a major subject, harmonic analysis, with widespread applications in number theory, signal processing, quantum mechanics, weather prediction and a broad range of other fields [TM159 or search for “thatsmaths” at irishtimes.com].
Greenhouse Effect [Image Wikimedia Commons]
From a Wide Wake to the Width of the World
Published January 24, 2019 Occasional Leave a CommentTags: Geophysics, Navigation
The finite angular width of a ship’s turbulent wake at the horizon enables the Earth’s radius to be estimated.
By ignoring evidence, Flat-Earthers remain secure in their delusions. The rest of us benefit greatly from accurate geodesy. Satellite communications, GPS navigation, large-scale surveying and cartography all require precise knowledge of the shape and form of the Earth and a precise value of its radius.
A Zero-Order Front
Published August 30, 2018 Occasional Leave a CommentTags: Fluid Dynamics, Geophysics, modelling
Sharp gradients known as fronts form in the atmosphere when variations in the wind field bring warm and cold air into close proximity. Much of our interesting weather is associated with the fronts that form in extratropical depressions.
Below, we describe a simple mechanistic model of frontogenesis, the process by which fronts are formed.
The Miraculous Spiral on Booterstown Strand
Published August 16, 2018 Irish Times Leave a CommentTags: biology, Geometry, Geophysics
We all know what a spiral looks like. Or do we? Ask your friends to describe one and they will probably trace out the form of a winding staircase. But that is actually a helix, a curve in three-dimensional space. A spiral is confined to a plane – it is a flat curve. In general terms, a spiral is formed by a point moving around a fixed centre while its distance increases or decreases as it revolves [see TM145, or search for “thatsmaths” at irishtimes.com].
The spiral sandbank on Booterstown strand (satellite image digitally enhanced by Andrew Lynch).
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Tides: a Tug-of-War between Earth, Moon and Sun
Published August 2, 2018 Irish Times Leave a CommentTags: Fluid Dynamics, Geophysics
All who set a sail, cast a hook or take a dip have a keen interest in the water level, and the regular ebb and flow of the tides. At most places the tidal variations are semi-diurnal, with high and low water twice each day [see TM144, or search for “thatsmaths” at irishtimes.com].
Animation of tide prediction machine, showing outputs for New York (semi-diurnal tides) and Kuril Islands (diurnal tides) [Source: American Mathematical Society (see below)].
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Trigonometric Comfort Blankets on Hilltops
Published July 5, 2018 Irish Times Leave a CommentTags: Geophysics, Maps, Spherical Trigonometry
On a glorious sunny June day we reached the summit of Céidín, south of the Glen of Imall, to find a triangulation station or trig pillar. These concrete pillars are found on many prominent peaks throughout Ireland, and were erected to aid in surveying the country [see TM142, or search for “thatsmaths” at irishtimes.com].
Trig pillar on summit of Croaghan Moira, Wicklow [Image from https://mountainviews.ie/%5D.
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Waves Packed in Envelopes
Published April 26, 2018 Occasional Leave a CommentTags: Fluid Dynamics, Geophysics, Wave Motion
In this article we take a look at group velocity and at the extraction of the envelope of a wave packet using the ideas of the Hilbert transform.
Geodesics on the Spheroidal Earth-II
Published April 19, 2018 Irish Times Leave a CommentTags: Geometry, Geophysics, Navigation, Pythagoras
Geodesy is the study of the shape and size of the Earth, and of variations in its gravitational field. The Earth was originally believed to be flat, but many clues, such as the manner in which ships appear and disappear at the horizon, and the changed perspective from an elevated vantage point, as well as astronomical phenomena, convinced savants of its spherical shape. In the third century BC, Eratosthenes accurately estimated the circumference of the Earth [TM137 or search for “thatsmaths” at irishtimes.com].
Geodesic at bearing of 60 degrees from Singapore. Passes close to Quito, Ecuador. Note that it is not a closed curve: it does not return to Singapore.
Geodesics on the Spheroidal Earth – I
Published April 12, 2018 Occasional Leave a CommentTags: Geometry, Geophysics, Navigation, Pythagoras
Both Quito in Ecuador and Singapore are on the Equator. One can fly due eastward from Singapore and reach Quito in due course. However, this is not the shortest route. The equatorial trans-Pacific route from Singapore to Quito is not a geodesic on Earth! Why not?
A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.
Staying Put or Going with the Flow
Published February 1, 2018 Irish Times Leave a CommentTags: Algorithms, Geophysics, Numerical Analysis
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 irishtimes.com].
Hurricane Ophelia approaching Ireland, 16 October 2017, 1200Z. Image from https://earth.nullschool.net/
Energy Cascades in Van Gogh’s Starry Night
Published January 4, 2018 Irish Times Leave a CommentTags: Fluid Dynamics, Geophysics
“Big whirls have little whirls that feed on their velocity,
And little whirls have lesser whirls, and so on to viscosity.“
We are all familiar with the measurement of speed, the distance travelled in a given time. Allowing for the direction as well as the magnitude of movement, we get velocity, a vector quantity. In the flow of a viscous fluid, such as treacle pouring off a spoon, the velocity is smooth and steady. Such flow is called laminar, and variations of velocity from place to place are small. By contrast, the motion of the atmosphere, a fluid with low viscosity, can be irregular and rapidly fluctuating. We experience this when out and about on a gusty day. Such chaotic fluid flow is called turbulence, and this topic continues to challenge the most brilliant scientists [TM130 or search for “thatsmaths” at irishtimes.com].
Vincent Van Gogh’s Starry Night.
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Inertial Oscillations and Phugoid Flight
Published July 13, 2017 Occasional Leave a CommentTags: Applied Maths, Fluid Dynamics, Geophysics
The English aviation pioneer Frederick Lanchester (1868–1946) introduced many important contributions to aerodynamics. He analysed the motion of an aircraft under various consitions of lift and drag. He introduced the term “phugoid” to describe aircraft motion in which the aircraft alternately climbs and descends, varying about straight and level flight. This is one of the basic modes of aircraft dynamics, and is clearly illustrated by the flight of gliders.
Glider in phugoid loop [photograph by Dave Jones on website of Dave Harrison].
The Water is Rising Fast
Published April 20, 2017 Irish Times Leave a CommentTags: Fluid Dynamics, Geophysics
Seventy percent of the Earth is covered by water and three quarters of the world’s great cities are on the coast. Ever-rising sea levels pose a real threat to more than a billion people living beside the sea. As the climate warms, this is becoming a greater threat every year [TM113 or search for “thatsmaths” at irishtimes.com].
Mean Sea level in Seattle from 1900 to 2013
Mercator’s Marvellous Map
Published May 21, 2015 Irish Times Leave a CommentTags: Geometry, Geophysics, Spherical Trigonometry
Try to wrap a football in aluminium foil and you will discover that you have to crumple up the foil to make it fit snugly to the ball. In the same way, it is impossible to represent the curved surface of the Earth on a flat plane without some distortion. [See this week’s That’s Maths column (TM068): search for “thatsmaths” at irishtimes.com].
![Mercator projection of the Earth, truncated at 75 degrees North and South [Wikimedia Commons, author: Strebe].](https://thatsmaths.files.wordpress.com/2015/05/mercator-projection-75deg.jpg)
Mercator projection of the Earth, truncated at 75 degrees North and South [Wikimedia Commons, author: Strebe].
The Hodograph
Published April 9, 2015 Occasional Leave a CommentTags: Astronomy, Fluid Dynamics, Geophysics
The Hodograph is a vector diagram showing how velocity changes with position or time. It was made popular by William Rowan Hamilton who, in 1847, gave an account of it in the Proceedings of the Royal Irish Academy. Hodographs are valuable in fluid dynamics, astronomy and meteorology.
Hodograph plot of wind vectors at five heights in the troposphere. This indicates vertical wind shear and also horizontal temperature gradients. Since the wind veers with height between V2 and V3, it is blowing warmer air north-eastwards to a colder region (image source: NOAA).
Mode-S: Aircraft Data improves Weather Forecasts
Published April 2, 2015 Irish Times Leave a CommentTags: Algebra, Geometry, Geophysics
A simple application of vectors yields valuable new wind observations for weather forecasting [see this week’s That’s Maths column (TM065) or search for “thatsmaths” at irishtimes.com].
The air speed is A (blue), the wind speed is W (black) and the ground speed is G (red). Since the ground speed is the resultant (vector sum) of air speed and wind speed, a simple vector subtraction gives the wind speed: W= G – A.
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Falling Bodies [1]: Sky-diving
Published November 13, 2014 Occasional Leave a CommentTags: Fluid Dynamics, Geophysics, Mechanics
Aristotle was clear: heavy bodies fall faster than light ones. He arrived at this conclusion by pure reasoning, without experiment. Today we insist on a physical demonstration before such a conclusion is accepted. Galileo tested Aristotle’s theory: he dropped bodies of different weights simultaneously from the Leaning Tower of Pisa and found that, to a good approximation, they hit the ground at the same time.
Aristotle and Galileo.
El Niño likely this Winter
Published November 6, 2014 Irish Times Leave a CommentTags: Fluid Dynamics, Geophysics, modelling, Physics
This week’s That’s Maths column in The Irish Times (TM056 or search for “thatsmaths” at irishtimes.com) is about El Niño and the ENSO phenomenon.
In 1997-98, abnormally high ocean temperatures off South America caused a collapse of the anchovy fisheries. Anchovies are a vital link in the food-chain and shortages can bring great hardship. Weather extremes associated with the event caused 2000 deaths and 33 million dollars in damage to property. One commentator wrote that the warming event had “more energy than a million Hiroshima bombs”.
Patterns of Pacific Ocean sea surface temperature during El Niño and La Niña episodes. Image courtesy of Climate.gov.
Gauss’s Great Triangle and the Shape of Space
Published July 10, 2014 Occasional Leave a CommentTags: Gauss, Geophysics, Maps
In the 1820s Carl Friedrich Gauss carried out a surveying experiment to measure the sum of the three angles of a large triangle. Euclidean geometry tells us that this sum is always 180º or two right angles. But Gauss himself had discovered other geometries, which he called non-Euclidean. In these, the three angles of a triangle may add up to more than two right angles, or to less.
10 Deutschmark currency note (front)
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Breaking Weather Records
Published April 24, 2014 Occasional Leave a CommentTags: Geophysics, Number Theory, Probability
In arithmetic series, like 1 + 2 + 3 + 4 + 5 + … , each term differs from the previous one by a fixed amount. There is a formula for calculating the sum of the first N terms. For geometric series, like 3 + 6 + 12 + 24 + … , each term is a fixed multiple of the previous one. Again, there is a formula for the sum of the first N terms of such a series. Continue reading ‘Breaking Weather Records’
Simulating the Future Climate
Published March 6, 2014 Irish Times Leave a CommentTags: Computer Science, Geophysics, Numerical Analysis
The Earth’s climate is changing, and the consequences may be very grave. This week, That’s Maths in The Irish Times ( TM040 ) is about computer models for simulating and predicting the future climate.
Interesting Bores
Published January 2, 2014 Irish Times Leave a CommentTags: Applied Maths, Fluid Dynamics, Geophysics, Mechanics, Wave Motion
This week’s That’s Maths column in the Irish Times ( TM036 ) is about bores. But don’t be put off: they are very interesting.
A Hole through the Earth
Published August 29, 2013 Occasional Leave a CommentTags: Geophysics, Mechanics
“I wonder if I shall fall right through the earth”, thought Alice as she fell down the rabbit hole, “and come out in the antipathies”. In addition to the author of the “Alice” books, Lewis Carroll – in real life the mathematician Charles L. Dodgson – many famous thinkers have asked what would happen if one fell down a hole right through the earth’s centre.
Galileo gave the answer to this question: an object dropped down a hole piercing the earth diametrically would fall with increasing speed until the centre, where it would be moving at about 8 km per second, after which it would slow down until reaching the other end, where it would fall back again, oscillating repeatedly between the two ends.
Continue reading ‘A Hole through the Earth’
