Wingtip vortices

The cores of vortices spin at very high speed and are regions of very low pressure. To first approximation, these low-pressure regions form with little exchange of heat with the neighboring regions (i.e., adiabatically), so the local temperature in the low-pressure regions drops, too.[6] If it drops below the local dew point, there results a condensation of water vapor present in the cores of wingtip vortices, making them visible.[6] The temperature may even drop below the local freezing point, in which case ice crystals will form inside the cores.[6]

The phase of water (i.e., whether it assumes the form of a solid, liquid, or gas) is determined by its temperature and pressure. For example, in the case of liquid-gas transition, at each pressure there is a special "transition temperature" ${\displaystyle T_{c}}$ such that if the sample temperature is even a little above ${\displaystyle T_{c}}$, the sample will be a gas, but, if the sample temperature is even a little below ${\displaystyle T_{c}}$, the sample will be a liquid; see phase transition. For example, at the standard atmospheric pressure, ${\displaystyle T_{c}}$ is 100 °C = 212 °F. The transition temperature ${\displaystyle T_{c}}$ decreases with decreasing pressure (which explains why water boils at lower temperatures at higher altitudes and at higher temperatures in a pressure cooker; see here for more information). In the case of water vapor in air, the ${\displaystyle T_{c}}$ corresponding to the partial pressure of water vapor is called the dew point. (The solid–liquid transition also happens around a specific transition temperature called the melting point. For most substances, the melting point also decreases with decreasing pressure, although water ice in particular - in its I_h form, which is the most familiar one - is a prominent exception to this rule.)

Vortex cores are regions of low pressure. As a vortex core begins to form, the water in the air (in the region that is about to become the core) is in vapor phase, which means that the local temperature is above the local dew point. After the vortex core forms, the pressure inside it has decreased from the ambient value, and so the local dew point (${\displaystyle T_{c}}$) has dropped from the ambient value. Thus, in and of itself, a drop in pressure would tend to keep water in vapor form: The initial dew point was already below the ambient air temperature, and the formation of the vortex has made the local dew point even lower. However, as the vortex core forms, its pressure (and so its dew point) is not the only property that is dropping: The vortex-core temperature is dropping also, and in fact it can drop by much more than the dew point does.

To first approximation, the formation of vortex cores is thermodynamically an adiabatic process, i.e., one with no exchange of heat. In such a process, the drop in pressure is accompanied by a drop in temperature, according to the equation

${\displaystyle {\frac {T_{\text{f}}}{T_{\text{i}}}}=\left({\frac {p_{\text{f}}}{p_{\text{i}}}}\right)^{\frac {\gamma -1}{\gamma }}.}$[6]

Here ${\displaystyle T_{\text{i}}}$ and ${\displaystyle p_{\text{i}}}$ are the absolute temperature and pressure at the beginning of the process (here equal to the ambient air temperature and pressure), ${\displaystyle T_{\text{f}}}$ and ${\displaystyle p_{\text{f}}}$ are the absolute temperature and pressure in the vortex core (which is the end result of the process), and the constant ${\displaystyle \gamma }$ is about 7/5 = 1.4 for air (see here).

Thus, even though the local dew point inside the vortex cores is even lower than in the ambient air, the water vapor may nevertheless condense — if the formation of the vortex brings the local temperature below the new local dew point.[6]

For a typical transport aircraft landing at an airport, these conditions are as follows: ${\displaystyle T_{\text{i}}}$ and ${\displaystyle p_{\text{i}}}$ have values corresponding to the so-called standard conditions, i.e., ${\displaystyle p_{\text{i}}}$ = 1 atm = 1013.25 mb = 101${\displaystyle \,}$325 Pa and ${\displaystyle T_{\text{i}}}$ = 293.15 K (which is 20 °C = 68 °F). The relative humidity is a comfortable 35% (dew point of 4.1 °C = 39.4 °F). This corresponds to a partial pressure of water vapor of 820 Pa = 8.2 mb. In a vortex core, the pressure (${\displaystyle p_{\text{f}}}$) drops to about 80% of the ambient pressure, i.e., to about 80 000 Pa.[6]

The temperature in the vortex core is given by the equation above as ${\displaystyle T_{\text{f}}=\left({\frac {\scriptstyle 80\,000}{\scriptstyle 101\,325}}\right)^{\scriptscriptstyle 0.4/1.4}\,T_{\text{i}}=0.935\,\times \,293.15=274\;{\text{K}},}$ or 0.86 °C = 33.5 °F.

Next, the partial pressure of water in the vortex core drops in proportion to the drop in the total pressure (i.e., by the same percentage), to about 650 Pa = 6.5 mb. According to a dew point calculator, that partial pressure results in the local dew point of about 0.86 °C; in other words, the new local dew point is about equal to the new local temperature.

Therefore, this is a marginal case; if the relative humidity of the ambient air were even a bit higher (with the total pressure and temperature remaining as above), then the local dew point inside the vortices would rise, while the local temperature would remain the same. Thus, the local temperature would now be lower than the local dew point, and so the water vapor inside the vortices would indeed condense. Under the right conditions, the local temperature in vortex cores may drop below the local freezing point, in which case ice particles will form inside the vortex cores.

The water-vapor condensation mechanism in wingtip vortices is thus driven by local changes in air pressure and temperature. This is to be contrasted to what happens in another well-known case of water condensation related to airplanes: the contrails from airplane engine exhausts. In the case of contrails, the local air pressure and temperature do not change significantly; what matters instead is that the exhaust contains both water vapor (which increases the local water-vapor concentration and so its partial pressure, resulting in elevated dew point and freezing point) as well as aerosols (which provide nucleation centers for the condensation and freezing).[7]