GG3069: Climate and Weather Systems

Lecture 4: Lapse Rates, Stability and Instability


A large amount of cloud formation and much of the dynamic behaviour of the atmosphere depends upon vertical movements of air. The tendency of air masses to move up or down is termed its stability. Unstable air masses are prone to vertical movements, while stable air resists vertical motion. The stability of air is a function of its buoyancy with respect to the surrounding air, which is in turn dependent on their relative densities. In Lecture 2 we saw that density is related to both pressure and temperature: for constant temperature, density increases along with pressure; and for constant pressure density decreases with increasing temperature. In other words, warm air will tend to expand and become less dense, making it more buoyant than cooler, denser air, and causing it to rise. Conversely, cool air will tend to contract and become more dense, decreasing its buoyancy and causing it to sink. A huge amount of atmospheric behaviour follows from these simple relations.

Lapse rates

To understand more about the stability of air masses, we need to examine the temperature changes undergone by an air mass as it rises or sinks, which in turn relate to pressure changes. The simplest case concerns unsaturated air, i.e. air which carries all the available moisture in gas form (Lecture 3). Relationships between pressure and temperature lead to a simple linear relation between temperature and altitude for rising or sinking air. This is known as the Dry Adiabatic Lapse Rate (DALR), and is equal to 9.8°K km-1. Air lifted up will cool at this rate due to reduction in pressure, air sinking will warm at this rate due to pressure increases. This was demonstrated in the pressure vessel (Lecture 2). The word 'adiabatic' is derived from the Greek word for 'impassable', and it refers to a system which does lose or gain energy. Thus, rising air is said to cool or warm adiabatically when its temperature changes are due entirely to pressure changes. In reality, some degree of energy exchange will always take place, but these are generally small on short timescales.

When condensation or evaporation occur in the air, however, lapse rates of rising or falling air differ from this value. As we have seen in Lecture 1, latent heat is released by condensation and consumed by evaporation (2,500 J g-1). This alters the adiabatic lapse rate: because energy is released by condensation, rising air will cool more slowly if condensation is occurring. Thus, there is a smaller change in temperature with height than would be the case for unsaturated air. The modified lapse rate is termed the Saturated Adiabatic Lapse Rate (SALR). The varying amounts of water vapour that can be held in air at different temperatures means that the SALR is non linear. The SALR is lowest at high temperatures, because of much higher saturation mixing ratios: i.e: greater amounts of energy are released at the vapour/droplet transition, therefore temperature changes with altitude are reduced. At low temperatures, the SALR is more similar to DALR: smaller amounts of moisture are available for condensation, so the modification of the lapse rate is less.

Adiabatic Lapse Rates are commonly different to the real vertical change in temperature, known as the Environmental Lapse Rate (ELR). The ELR is influenced by patterns of heating, cooling and mixing, and the past history of an air mass. Actual vertical temperature gradients in the atmosphere are thus highly variable, and can even show an increase in temperature with height, a situation known as a temperature inversion.

Conditions Determining Air Stability

The stability of air masses depends on the relative values of the ELR and the appropriate Adiabatic Lapse Rate.

Air is stable if the ELR less than the ALR. If, for any reason, a parcel of air is uplifted, it will cool to lower temperatures than its new surroundings along the ALR. Hence the air parcel will be denser than its surroundings and will tend to fall back to its original level. This situation is encouraged by a small ELR or a temperature inversion.

There are two cases:

Absolute instability
In this case, the ELR is greater than both DALR and SALR. Uplifted air cools relatively slowly, and will thus be warmer and less dense than its new surroundings. It will therefore tend to continue to rise.

Conditional instability
In this case, the ELR is less than the DALR but greater than the SALR. Air will be stable unless forced to rise to altitude where condensation occurs, whereupon spontaneous uplift will occur.

Air Stability and Potential Temperature

In Lecture 2, we introduced the idea of potential temperature. Potential temperature is denoted q (Greek theta), and can be defined as:

the temperature that an air mass would have if it were moved dry adiabatically to a level at which pressure is 1000 hPa.

The equivalent concept for saturated air is the wet-bulb potential temperature (qw), which is the temperature that air would have if it moved to a level where the pressure is 1000 hPa, along the Saturated Adiabatic Lapse Rate.

Potential temperature is an extremely important concept, because it allows us to directly compare air masses regardless of their altitude or pressure, and thus allows us to predict how air masses will interact. To illustrate the concept, let us re-examine the conditions for air stability, from the point of view of potential temperature. For simplicity, we will consider only the case of dry air.

A stable atmosphere is one in which potential temperature increases with altitude. That is, if the environmental lapse rate is such that potential temperature increases with altitude, then the atmosphere will be stable. This is the same as saying that the ELR is less than the ALR. Examples of this situation are when the lower levels of the air are cooled by a cold ground surface, or if warm air is advected over cool air. It is also the case in the stratosphere, where the air is heated from above by UV bombardment: this is why the stratosphere is so stable.

An unstable atmosphere is one in which potential temperature decreases with altitude. In this case, the lowest levels have the highest potential temperature: this upsets the hydrostatic equilibrium (Lecture 2), and the lower air will thus tend to rise. This is the situation in which air is heated from below by longwave emission from the ground surface. It is equivalent to the case where the ELR is greater than the ALR.

A situation we have not yet considered is a neutral atmosphere: in this case, potential temperature is constant with altitude. This is equivalent to saying that the ELR = ALR. This situation is quite common in windy, well mixed conditions in the lower troposphere. Air heated at the ground is rapidly mixed upwards by convection and turbulent winds, thus equalising potential temperature.

Representing Air Stability

One of the most versatile and useful ways of representing air stability is based on plots of actual temperature against potential temperature for vertical transects through the atmosphere. Such temperature-potential temperature diagrams are known as T f diagrams or Tephigrams. (This is because potential temperature can be regarded as equivalent to entropy, which is denoted by the Greek letter f (Phi)). On the basic Tephigram, temperature is plotted on the vertical axis, and potential temperature on the horizontal axis. Air pressure then plots as a series of gently curving diagonal lines slanting up from bottom right to top left. Because air pressure decreases with altitude, it is useful to rotate such diagrams until the pressure isolines are approximately horizontal, with the highest pressures at the bottom (1050 hPa) and the lowest at the top (usually 200 hPa) so that the diagram then appears as a vertical slice through the atmosphere. The shape of temperature profiles then shows at a glance whether the air is stable or not. Furthermore, these diagrams also permit the exact calculation of the behaviour of air masses.

Above: Tephigram showing lines of equal temperature (rising from left to right), potential temperature (rising from right to left), pressure (sub-horizontal curved lines), and saturated idiabats (steep dashed curved lines). Also shown are temperature curves derived from soundings over Northern Ireland (red) and the sahara (blue). The Irish curve closely follows a saturated adiabat through most of the atmosphere, characteristic of a well-mixed, cloudy atmosphere. The abrupt change in direction just below 300 mbar is the tropopause: the abrupt change in thermal characteristics of the atmosphere between the troposphere nad stratosphere. The Sahara line (blue) is parallel to a dry adiabat (line of equal potential temperature: this is characteristic of a dry atmopshere well mixed by convection.

Important concepts to note in connection with Tephigrams:


Consequences of Instability: Convection and formation of cumulus clouds.

Most of the heating of the atmosphere is accomplished by longwave radiation from ground or water surfaces. This means that, in the troposphere, maximum energy receipts are commonly at the lowest levels. This will raise temperatures (and potential temperatures) there, upsetting the hydrostatic balance and creating instability. This one fact accounts for a huge amount of atmospheric behaviour. It explains why vertical motions are so prevailent in the troposphere: the atmosphere is constantly mixed to evacuate energy from lower levels to the upper troposphere, where it can lose energy by longwave radiation into space. Thermally-driven vertical mortions are known as convection.

Convection is initiated by heating at a ground or water surface. The vertical dimension of convective cells is determined by the temperature profile of the atmosphere, and the moisture content of the air. The temperature profile (or, as we have seen, the vertical potential temperature gradient) is the ultimate determinant of stability. Limited convection can occur in a generally stable atmosphere if excess heating occurs near the ground. In this case, energy can be gently lofted upwards in dry thermal cells. Such thermals can be hundreds or thousands of metres high in some warmed dry atmospheres, providing ideal conditions for parapenting and other aerial pursuits.

The most vigorous convective cells involve the formation of cumulus clouds. This is because cloud formation involves the release of latent heat which, as we have seen, provides an extra source of energy during condensation. Indeed, latent heat release provides the bulk of the energy involved in large cumulus systems.

Small cumulus are the visible portion of small convective cells, which can form in the lower part of the troposphere due to heating from below. Small cumulus are preferentially developed over land, due to greater heating compared with water surfaces where latent heat is consumed during the evalopration of water. Cumulus have a cellular form, either with the cloud in the centre of the cell (closed cells) or around cell boundaries (open cells). In the latter case, clear air sinks in the centre of the cell, and rises between cells. The type of cell pattern relates to air properties and rates of energy exhange.

Open convection cells seen from space

Cloud streets are elongated convection cells which form when there is horizontal transport of a convecting air mass. Cloud streets are most common where a cool airstream blows over a warm surface; e.g. northerly winds in the n. hemisphere mid latitudes.

We have seen that for stable air (where potential temperature increases with height), air that is forced to rise will return to its original altitude. One of the most beautiful consequences of this behaviour is lee waves or mountain waves downwind of large obstacles to the flow. As air is blown against a mountain, it is forced to rise. If it is stable, then on the lee side of the mountain it will fall again. However, its momentum is such that it will shoot past its original altitude and go lower than it was before. It then is forced to rise again by its disequilibrium with hydrostatic conditions. It will again overshoot, this time going too high, and so on. If the resulting wave intersects the condensation level of the air, clouds will form at the crests of the waves. These clouds, among the most beautiful in nature, are commonly seen downwind of mountain ranges in stable, windy conditions. They resemble great plates or lenses. Look out for them in the wide skies of St Andrews Bay: they are quite common. If the moisture content of the air varies with altitude, such lee wave clouds can form vertical stacks of lenses, called piles des assiettes, or 'piles of plates'.

Solitary wave cloud, Aberdeenshire

Lee waves over Corsica from space

The basic mathematical description of these waves is very elegant. Consider first the vertical movements of the wave. The air oscillates up and down at a characteristic frequency, which is controlled by the temperature profile of the air. In particular, it depends on the vertical gradient in potential temperature. If potential temperature increases upwards (stability), disturbed air will oscillate up and down at the Brunt - Vaisala Frequency N / 2p, where

N = ((g/q) x (Dq/Dz))1/2

g is gravity
q is the potential temperature, and
Dq/Dz is the vertical gradient of potential temperature;
the exponent 1/2 means 'the square root'.

The units of N / 2p is sec-1, or cycles per second. The equation shows that the oscillation frequency is greatest for larger gradients in potential temperature. This clearly makes sense: the steeper the potential temperature gradient, the greater the disequilibrium experienced by displaced air, therefore it will tend to rise or fall with higher velocity.

Now consider the horizontal motion. The air moves horizontally with a wind speed V. For a wave, the wavelength l is equal to the velocity divided by the frequency:

l = V / f

Since the wave frequency is N / 2p,

l =  2p V / N

This shows that the wavelength (spacing) of lee wave clouds will be greatest for high wind speeds and slightly stable air (small increase of potential temperature with altitude). Closely spaced waves will result from low wind speeds and high potential temperature gradients. However, the waves will not propagate forever. Eventually, friction between the oscillating layer and the surrounding air will damp the oscillation, and the lee wave train will eventually die out.

Even if you don't completely follow the maths, you have to admit lee waves are pretty cool clouds.

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