Optics 012 · A ship and the setting Sun through layered air

Mirage & Green-Flash Observatory

How can layers of warm and cool air make distant objects sink, rise or appear several times? Temperature changes the air's density and with it the refractive index, and light bends toward the cooler, denser air. Over hot ground the lowest air is hottest, so rays curve upward: distant objects sink and the sky seems to lie on the road, an inferior mirage. When the lowest air is colder, rays curve down and distant objects loom into view. Several temperature layers stack several images.

Look at a ship 20 km out through cold sea air: its light bends back down, so the ship looms up, stacks and flips. The same air stretches the green flash.

Images of the ship (simulated) $\text{—}$

Ship’s apparent height: this air / standard air $\text{—}$

Strongest bending of a level ray $\text{—}$

Height of the green flash as the red edge sets $\text{—}$

Height of that strongest bending $\text{—}$

How much lower the Sun sinks before its green edge sets than before its red edge sets $\text{—}$

Apparent altitude of the vanishing line, 580 nm $\text{—}$

Steepest temperature gradient below 150 m $\text{—}$

Gradient that bends a level ray along the sea, at the camera $\text{—}$

Largest relative drift of Bouguer’s invariant along the traced rays $\text{—}$

Rays traced for this record $\text{—}$

Optics 012

How can layers of warm and cool air make distant objects sink, rise or appear several times?

Temperature step across the layer

\Delta T

8\,\mathrm{K}

Height of the layer above the sea

h_{\mathrm{c}}

25\,\mathrm{m}

Spread of the layer

a

3\,\mathrm{m}

Camera height above the sea

h_{\mathrm{eye}}

10\,\mathrm{m}

Distance to the ship

D

20\,\mathrm{km}

Air temperature at sea level without the layer

T_0

0\,{}^{\circ}\mathrm{C}

A positive step puts warmer air above cold air at the sea; a negative step puts hot air at the surface.

Model of the mirage and green-flash calculations of van der Werf, Können and Lehn (Applied Optics 42, 367, 2003) and the transfer characteristic of Lehn (JOSA 73, 1622, 1983). The air follows the US Standard Atmosphere 1976 started from the chosen sea-level temperature, plus one layer whose temperature changes by the chosen step as a smooth Fermi step around its central height. Pressure follows from hydrostatic balance with inverse-square gravity, and the refractive index from pressure and temperature with the constants of van der Werf et al. for 650, 580 and 520 nm; humidity is left out. Rays are traced in the vertical plane over a spherical Earth of radius 6371 km by a fourth-order Runge–Kutta method; Bouguer’s invariant stays constant along every ray to better than one part in a hundred million. The layer is horizontally uniform for 50 km and then ends, where the paper’s layer starts to weaken; light still low there continues through the same air without the layer. If the layer went on for ever, a level ray from 15 m under a layer 40 m up would be trapped from a 3.64 K step, where the paper finds the threshold between 3.7 and 4.0 K. Extinction is left out, although along the long low paths of a trapping layer real air lets mostly red light through. The ship is a 60 m freighter 21 m tall; the camera image uses 241 rays in 580 nm light and ignores turbulence, diffraction and the camera lens. In the scene, heights are drawn about 30 to 270 times taller than distances, and the sea’s curve uses the same height scale.

Physics tutorial

See a ship rise, stack and flip over the sea

BackgroundLight travels a little more slowly in dense air than in thin air. Cold air is denser than warm air, and over a calm sea the temperature can change by several degrees within a few metres of height.

Why it mattersThrough such air a ray from a distant ship does not run straight: it curves towards the denser air. Sailors have long reported ships hanging upside down in the sky, coasts rising from beyond the horizon and a green flash at sunset. In 2003 van der Werf, Können and Lehn traced rays through layers of this kind to explain both the mirages and the flash.

Start with the essentials

Focus question: A camera on the shore looks at a ship 20 km away. How can a layer of air a few metres thick show the ship several times, some of the images upside down?
One-sentence intuition: Trace each ray from the camera to the plane of the ship and plot the height $z$ where it lands against the elevation $\beta$ at which it left the camera: this is the transfer characteristic (Lehn 1983). In plain air $z$ rises steadily with $\beta$ and the ship looks normal. A layer that bends rays more than the sea curves folds the curve. Each stretch where $z$ rises is an upright image, each stretch where it falls an inverted one.

Core mathematical model

The refractive index of air

n-1=A(\lambda)\,\frac{P}{T}

Pressure $P$ is in hectopascals and temperature $T$ in kelvin. Van der Werf et al. (2003) give $A=7.87\times10^{-5}\,\mathrm{K/hPa}$ at 580 nm, 0.4 % more at 520 nm and 0.3 % less at 650 nm. At sea level $n-1$ is about $3\times10^{-4}$ , larger in cold air than in warm air.

One layer on the standard atmosphere

T(h)=T_{\mathrm{US}}(h)-\Delta T+\frac{\Delta T}{1+e^{-(h-h_{\mathrm{c}})/a}}

The US Standard Atmosphere 1976 cools by 6.5 K per kilometre. The layer adds a step $\Delta T$ centred at the height $h_{\mathrm{c}}$ , with 90 % of it inside a band $6a$ thick (van der Werf et al. 2003, Eq. 7). A positive step leaves cold air under warmer air; a negative step leaves hot air at the surface.

Following a ray around the Earth

\begin{aligned}\frac{\mathrm{d}h}{\mathrm{d}\psi}&=(R+h)\tan\beta\\\frac{\mathrm{d}\beta}{\mathrm{d}\psi}&=1+\frac{R+h}{n}\,\frac{\mathrm{d}n}{\mathrm{d}h}\end{aligned}

The ray climbs at the elevation $\beta$ above the local horizontal while it advances through the angle $\psi$ about the Earth’s centre, with $R=6371\,\mathrm{km}$ (van der Werf et al. 2003, Eqs. 4 and 5). Together the two equations keep Bouguer’s invariant $n\,(R+h)\cos\beta$ constant; the traced rays hold it to better than $10^{-8}$ .

When a layer traps light

\begin{gathered}k=-\frac{R}{n}\,\frac{\mathrm{d}n}{\mathrm{d}h}\\k>1\iff\frac{\mathrm{d}T}{\mathrm{d}h}>\frac{T^2}{A\,P\,R}-B\end{gathered}

The bending $k$ compares the curvature of a level ray with that of the Earth; standard air at 15 °C gives $k\approx0.17$ . Where $k>1$ , a level ray curves faster than the sea falls away and is trapped. That takes warming of about 0.11 K per metre of height (van der Werf et al. 2003, Eq. 12). Where the temperature falls faster than $B=0.034\,\mathrm{K/m}$ , $k<0$ and rays bend upwards, as over a hot road.

Height of the green flash

\Delta\beta_{\mathrm{g}}=\int\mathbf{1}\!\left[\alpha_{\mathrm{g}}(\beta)\le\min\alpha_{\mathrm{r}}\right]\mathrm{d}\beta

Each apparent altitude $\beta$ maps to the true altitude $\alpha$ in which its ray leaves the atmosphere. Once the Sun’s top edge sinks below the lowest true altitude red light can come from, red is gone; green light, bent slightly more, still arrives over the range $\Delta\beta_{\mathrm{g}}$ . In the standard-air preset that range is 14 arcseconds; next to a layer that almost traps rays, the nearly flat transformation curve stretches it more than tenfold.

Common difficulties

A mirage is not a reflection

Typical misconceptionThe inverted image is the ship reflected in the sea or in a layer of air.

Better mental modelNo surface reflects the light. In the default preset the inverted image hangs above the ship, made by rays that bend smoothly through the layer and cross on the way; the transfer characteristic shows where they cross.

Warm air does not always lift things

Typical misconceptionWarm air makes distant things look higher.

Better mental modelLight bends towards the colder, denser air. Hot air at the surface bends rays upwards, so things sink and an inverted image appears below; cold air over the sea bends rays down and lifts distant things into view.

The green flash is not in the eye

Typical misconceptionThe green flash is an afterimage of the red sunset.

Better mental modelAir bends green light slightly more than red, so the Sun’s green image sits a little higher and sets last. This simulation has no eye at all and the green still outlasts the red; a layer close to trapping stretches it into a flash tall enough to see.

Run the experiment

01
Watch the ship stack and flip
Keep the default preset, Cold sea air: the ship stacks and flips, and look at the camera image under the scene.
What to observe: The camera sees three images of the freighter, the middle one upside down, together 16.1 arcminutes tall. Without the layer the ship would be 2.85 arcminutes tall.
02
Read the folded curve
Look at the plot of where each ray meets the ship, then drag the camera up and down.
What to observe: The curve rises, falls and rises again; the falling stretch, in magenta, is the inverted image. At the layer, 25 m up, a level ray bends 4.9 times as sharply as the Earth curves. Moving the camera changes which rays cross, so the images grow, shrink and merge.
03
Heat the surface
Load Hot surface: an inverted image below.
What to observe: Hot air in the lowest metre bends rays upwards. The bottom 5.8 m of the ship disappear, and an inverted copy of the rest hangs below it, like the sky seen on a hot road.
04
Let the ship loom
Load Gentle warm layer: the ship looms up.
What to observe: The step is spread over many metres, so the curve never folds and there is one upright image. It is stretched to 5.24 arcminutes instead of 2.95.
05
Bring back a hidden ship
Load A ship beyond the horizon reappears.
What to observe: At 30 km the ship lies behind the curve of the sea in standard air. A sharp step 14 m up traps rays beneath it, and the camera sees three thin images of the ship between 3.6 and 11.2 m above the waterline, two of them inverted.
06
Stretch the green flash
Load Near-trapping layer: a tall green flash, then Standard air: no mirage, and compare the sunset frames.
What to observe: Just below the 3.64 K step that would trap level rays, the green edge of the Sun still spans 162 arcseconds of sky at the moment the red edge sets. In standard air the green rim is 14 arcseconds high.

Mirage & Green-Flash Observatory

See a ship rise, stack and flip over the sea

Core mathematical model

The refractive index of air

One layer on the standard atmosphere

Following a ray around the Earth

When a layer traps light

Height of the green flash

Common difficulties

A mirage is not a reflection

Warm air does not always lift things

The green flash is not in the eye

Run the experiment

Watch the ship stack and flip

Read the folded curve

Heat the surface

Let the ship loom

Bring back a hidden ship

Stretch the green flash

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