Atmospheric Pressure Simulator

What Is the Atmospheric Pressure Simulator?

How to use the atmospheric pressure simulator

1. Climb. Drag Altitude h anywhere from 0 to 20,000 m; it moves in single metres, so real summits such as 8,849 m land exactly. The figure beside the slider echoes the height, the cream marker line in the picture rises to it, and the gold curve shows how much of the sea-level pressure survives at every height on the way.
2. Read the pressure. Pressure at altitude is the main readout, in kilopascals to four significant figures, with the same value in hectopascals, atmospheres and psi on the line beneath it. Fraction of sea-level pressure gives the ratio as a percentage.
3. Watch the barometer. The tube on the right is a mercury barometer scaled to 800 mm, and Mercury barometer column prints its height in mm Hg with inches of mercury underneath; Water column equivalent says how tall a water barometer would have to be. Both columns take about two seconds to settle after a change, while the numbers update at once.
4. Read the air itself. ISA air temperature shows the model's temperature at that height in degrees Celsius and kelvin, and Air density the mass of a cubic metre of air there, from the ideal gas law.
5. Change the weather. Drag Sea-level pressure P0 between 950 and 1050 hPa in quarter-hectopascal steps. Every pressure, both columns and the density scale with it; the fraction and the temperature do not, which is the point the whole lab is built to make.
6. Start again. Reset returns the sliders to 0 m and 1013.25 hPa and replays the settle from the ground. The five preset places above are the quicker route to a specific real location.

Worked example: change one thing at a time

Begin from the Reset state and move one slider per step. Every figure in the table is the string the lab itself displays for those slider positions; if the table and the lab ever disagree, the lab is right.

The first four rows are one climb with the weather held standard. Row 1 to row 2: the first 2,000 m take the share of sea level from 100.0 % to 78.5 % and knock the mercury from 760.0 mm down to 596.3 mm, while the temperature drops from 15.0 °C to 2.0 °C, thirteen degrees for two kilometres, which is the 6.5 °C per kilometre lapse rate at work.

Row 2 to row 3: the next 6,849 m up to the summit of Everest cost another 47.5 points, leaving 31.0 %, a 235.8 mm column and air at -42.5 °C. Row 3 to row 4: the last 2,151 m to the tropopause remove only 8.7 more points, because by now there is so little air left overhead to lose. Push the slider on past 11,000 m and the temperature readout sticks at -56.5 °C while the pressure keeps sliding: 19.33 kPa at 12,000 m, 12.04 kPa at 15,000 m and 5.475 kPa at the top of the slider.

Rows 5 and 6 put the altitude back to zero and move only the weather. At sea level the pressure readout is simply whatever the sea-level slider says, 95.00 kPa in the deep low and 105.0 kPa in the strong high, and the barometer swings from 712.6 mm to 787.6 mm between them. The share of sea level reads 100.0 % in both rows and the temperature stays at 15.0 °C, because neither quantity has anything to do with the weather in this model. The two rows also bracket what a real barometer does over a year: a swing of about a tenth of the pressure, and seventy-odd millimetres of mercury.

One more slider position is worth finding. The atmosphere is often said to be half gone by about five and a half kilometres, and the lab lets you check: at 5,500 m it reads 50.51 kPa and 49.8 %, so the halfway point sits just below that height on a standard day. Kilimanjaro, at 5,895 m, is already on the thinner side of it at 47.2 %.

Formula and symbol reference

Everything the lab prints follows from three relationships: the standard-atmosphere pressure profile P = P0·(1 - L·h/T0)^(g·M/(R·L)) below 11 km, with an exponential continuation above it; the ideal gas law rho = P·M/(R·T) for the density; and the hydrostatic balance P = rho·g·h, run backwards, for the height of each liquid column. The ranges marked “in this lab” are the simulator's own displayed values at the ends of its slider travel.

The physics: why pressure falls with height

Take any thin horizontal slice of the atmosphere. It is not accelerating, so the air below must push up on it slightly harder than the air above pushes down, and the difference is the slice's own weight per unit area. Written for an infinitesimal slice that is the hydrostatic balance dP/dh = -rho·g: pressure falls with height at a rate set by the local density.

In a lake the density is the same at every depth and the balance integrates to the straight line P = rho·g·h, which is why the lab's water column rises in exact proportion to the pressure. Air will not sit still for that, because its density is not a constant but is itself decided by the pressure it is under.

The ideal gas law closes the loop: rho = P·M/(R·T), density proportional to pressure at a given temperature. Substituting it turns the balance into dP/P = -(g·M/(R·T))·dh, which says the atmosphere loses a fixed fraction of its pressure per metre rather than a fixed amount.

That is the whole reason the gold curve in the picture bends the way it does. Near the ground the air is dense and every metre of climb sheds a lot of weight; high up the air is thin and the same metre sheds very little. You can read it off the table above: the first 2,000 m take 21.5 percentage points off the sea-level pressure, and the 9,000 m from the tropopause to the top of the slider take only 16.9.

If the temperature were the same all the way up, the fraction lost per metre would be constant and the profile would be a perfect exponential. The troposphere is not like that; it cools with height at roughly 6.5 °C per kilometre, because sunlight warms the ground and the air above it expands and cools as it rises.

Feeding T = T0 - L·h into the balance and integrating gives the power law P = P0·(1 - L·h/T0)^(g·M/(R·L)), whose exponent works out at 5.256 for dry air, and that is the formula the lab evaluates below 11 km. Slide the altitude upwards and the temperature readout falls in step, 15.0 °C at the ground, 2.0 °C at 2,000 m, -42.5 °C on Everest.

At 11,000 m the cooling stops. The tropopause is where the standard model switches to a constant -56.5 °C, and with T fixed the balance integrates to a plain exponential, P = P(11 km)·exp(-g·M·(h - 11000)/(R·T11)). Pressure is continuous across the join, so nothing jumps in the readouts as you cross it; only the temperature freezes, and the slope of the curve changes by an amount too small to see at this scale. The lab stops the slider at 20,000 m because that is where the isothermal layer ends and the standard atmosphere starts to warm again.

The barometer is the same hydrostatic balance run in reverse. A closed tube of mercury standing in an open dish has vacuum above the column and the atmosphere pressing on the dish, so the column settles where rho(Hg)·g·h equals the air pressure; with mercury at 13,595 kg/m³ that is 760.0 mm at 101.3 kPa, and it is why the lab's column-top label and its pressure readout always agree. Mercury earns its place by being dense: the water column readout shows that the same pressure would need 10.33 m of water, taller than a two-storey house, which was the original puzzle that led Torricelli to the barometer in the first place.

Where the standard atmosphere breaks down

The lab shows an idealised average column: dry air, a fixed temperature profile and constant gravity. Every real day departs from it, and each departure leaves a recognisable fingerprint on the numbers.

Real days are not standard days

The ISA air temperature readout is what the model assumes, not a forecast. A warm air column is less dense, so the pressure falls more slowly with height than the lab shows and a given pressure is met at a greater altitude; a cold column does the reverse. Aviation carries a rule of thumb of about 4 % of the height above the pressure datum for every 10 °C the air differs from ISA, which at Everest is several hundred metres.

Symptom: a barometer at a known height reads a few per cent off the lab even with the correct sea-level pressure dialled in.

Weather moves the whole curve

The model contains no weather; you supply it through the sea-level slider. Its 950 to 1050 hPa span is roughly five per cent either side of standard and covers a deep depression to a strong anticyclone, which is why the lab lets you set it. Mean sea-level pressure also drifts by a few hectopascals through a single day, and near the ground each hectopascal is worth roughly eight to nine metres of altitude: slide from 0 to 500 m and the readout falls from 101.3 kPa to 95.46 kPa.

Humidity

The molar mass the lab uses, 0.0289644 kg/mol, is for dry air. Water vapour, at 0.018 kg/mol, is lighter than the nitrogen and oxygen it displaces, so humid air is a little less dense and its pressure falls a little more slowly with height. In hot, humid air the effect can pass one per cent of the density; the lab ignores it.

Above 20 km there are more layers

The isothermal layer the lab uses between 11 km and 20 km is the second of seven in the 1976 standard. Above 20 km the stratosphere warms, first slowly and then quickly, because ozone absorbs sunlight there, and each layer has its own lapse rate and its own exponent. The slider stops at 20,000 m rather than extrapolating an exponential that would soon be wrong.

Geopotential versus geometric altitude

The standard atmosphere is tabulated against geopotential height, which treats g as 9.80665 m/s² all the way up. Real gravity weakens with distance from the Earth's centre, so the geometric height above sea level is slightly larger than the geopotential value: about 60 m larger at the top of the slider, and a negligible amount at the heights people actually stand on.

The lab's own limits

The altitude slider moves in whole metres and the sea-level slider in quarter hectopascals, and every readout is rounded for display, four significant figures on the pressure, one decimal place on the temperature and the mercury column, two on the water column. The mercury density is fixed at its 0 °C value; a warm barometer's mercury is slightly less dense and stands a fraction of a per cent taller. The drawn columns take about two seconds to settle while the printed numbers are instant, so during the animation the label on the column top is the answer and the column beneath it is still catching up.

Where atmospheric pressure is actually used

Pressure altimeters and QNH

An aircraft altimeter is a barometer with its dial marked in feet or metres according to this very model, and the small window on its face, the subscale, is the lab's sea-level slider. Set the subscale to the reported sea-level pressure, called the QNH, and the instrument reads altitude above the sea; set it to 1013.25 hPa, as every aircraft does above the transition altitude, and it reads pressure altitude, the basis of flight levels. In the lab, dialling 950 hPa with the altitude at zero drops the reading to 95.00 kPa, which an altimeter still set to 1013.25 would interpret as being several hundred metres up.

Weather reports reduced to sea level

A station's barometer measures the pressure at the station's own height, and forecasters convert it to an equivalent sea-level value so that stations at different heights can be drawn on one map. The lab runs that conversion the other way round. Load the Denver preset: with the map showing a standard 1013.25 hPa, a barometer in the city itself reads 83.42 kPa, or 834.2 hPa, and the fraction readout says why, 82.3 %.

Cabin altitude

An airliner at cruise does not hold its cabin at sea-level pressure; airworthiness rules require only that the cabin altitude stays below 8,000 ft, about 2,400 m, in normal operation. Set the slider to 2,400 m and the lab shows what passengers breathe: 75.63 kPa, 74.6 % of sea level. Now load the Airliner cruise preset for the air outside the window at 11,000 m, 22.63 kPa, and the difference between the two is the load the fuselage is built to carry.

Cooking and the boiling point

Water boils when its vapour pressure reaches the pressure of the air above it, so the boiling point tracks the lab's main readout. At the 79.50 kPa the lab gives for 2,000 m water boils close to 93 °C, and at the 31.44 kPa of the Everest summit at about 70 °C, too cool to cook many foods properly; the exact pairing comes from steam tables, not from this lab. A pressure cooker is the same physics run upwards.

Breathing at altitude

Oxygen is a fixed fifth of dry air by volume at every altitude in this model, so the Fraction of sea-level pressure readout is also the fraction of sea-level oxygen pressure that each breath delivers. Many people notice the thinner air above roughly 2,500 m, where the lab reads a little under three-quarters of sea level, and above about 8,000 m, where it reads about a third, unacclimatised humans cannot survive for long without supplementary oxygen. The precise thresholds vary from person to person; the physics of the pressure does not.

Where to go next

Put numbers to a specific height with the atmospheric pressure calculator, which runs the same model with the working written out and will also solve the altimeter problem, altitude from a measured pressure, directly. For the concept from the ground up, including how a barometer is built and why you do not feel ten tonnes of air, read Atmospheric Pressure Explained, and for the definition it all rests on, force spread over area, the guide to pressure in physics. The pressure simulator shows the same hydrostatic balance building with depth in water instead of thinning with height in air, and everything else we have built is in the library of physics simulations.