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Chapter 1
Meteorological Background Information
1.1
Introduction
This chapter contains information that may be useful when carrying out the Goatfell activity, the meteorological activity and later in your studies. This section provides material which may be helpful in analysing
and interpreting atmospheric profiles.
1.2
Potential Temperature
Air pressure decreases with height and generally, so does temperature. Even if the air flow is very turbulent
and thoroughly mixed vertically, this is still true. This is because if a parcel of air is moved upwards it
expands, and expansion without addition of extra heat results in a lowering of the temperature according
to the ideal gas law.
Potential temperature (denoted by θ) is a quantity which does not change for an air parcel unless we add
or remove heat (e.g. by solar heating). It is related to the pressure, the absolute temperature (in Kelvins)
and the surface pressure by:
p
ps
R/Cp
p
ps
0.286
θ=T
=T
(1.1)
Physically, θ is the temperature an air parcel would have if it were brought down to a level where the
pressure is 1000 hPa (in practice, close to the surface) and hence compressed without adding heat.
Because potential temperature doesn’t change for an air parcel unless we add heat, we call it a conserved
quantity.
Close to the surface (the lowest km or so, in the boundary layer), lots of vertical mixing results in approximately constant potential temperature with height. Above the boundary layer, potential temperature
generally increases with height. This is a stable situation. If potential temperature decreases with height
we have convective instability. This is discussed in more detail in the section on skew-T plots (Subsection 1.3.8 p15).
By examining the ‘environmental’ vertical profile of θ (such as that measured by a radiosonde) we can
determine the static stability of the atmosphere. When the environmental θ (or θenv ) increases with height,
air parcels that are vertically displaced within the layer will tend to return to their initial position, which
is a statically stable situation (see left-hand panel of Figure 1.1). Consider a small volume, or ‘parcel’
of air within a stable layer that is displaced upwards. The potential temperature of the parcel (θparcel ) is
2
Chapter 1. Meteorological Background Information
initially equal to that of its environment, and remains unchanged as the parcel is displaced because it is
conserved. Because θenv increases with height, the lifted parcel becomes colder than its environment
(θparcel < θenv ). As a consequence of the ideal gas law, this relatively cold parcel is also denser and
heavier than its environment and hence negatively buoyant. As shown by the blue arrow, this negative
buoyancy returns the parcel back towards its original position. On the other hand, if the parcel is displaced
downwards from its initial position, it becomes positively buoyant and is drawn upwards back to its original
position.
Figure 1.1: Vertical profiles of potential temperature (θ) and an illustration of parcel stability
By contrast (middle panel of Figure 1.1), when θenv decreases with height, an air parcel that is displaced
upwards (downwards) will become positively (negatively) buoyant and will continue rising (sinking), which
is a statically unstable situation because the parcels accelerate in the direction of their displacement.
When statically unstable layers develop in the atmosphere, such as when the ground is heated by the
sun during the daytime, convective eddies develop that rapidly mix θ within the layer to neutralise this
instability. A statically neutral layer is one where θenv remains constant with height. Air parcels that are
displaced vertically through a neutral layer will remain at their new level without undergoing any buoyancydriven acceleration because their θ remains identical to their environment. Due to turbulent air motions
that vertically mix potential temperature (and other properties), the boundary layer is often characterised
by neutral stability.
Note that the above analysis of parcel stability only holds when the parcel is unsaturated. If the parcel
saturates due to lifting (say, as it ascends a mountain), latent heat is released within it due to condensation
and/or deposition that causes its θ to increase. Likewise, a parcel cools down if it experiences evaporation
and/or sublimation of liquid or frozen water particles.
1.2.1
The Hydrostatic Equation
To a good approximation, the atmospheric pressure varies with height according to the hydrostatic equation. This essentially states that the pressure at any height balances the weight of the overlying atmosphere
and so neglects the pressure fluctuations associated with vertical air motion.
The upward pressure on the bottom of a thin layer of air is slightly greater than the downward pressure on
the top. This creates an excess upward pressure which is balanced by the weight of the layer.
In Figure 1.2, consider a column of the atmosphere of cross-sectional area 1 m2 and consider in particular
the slice through this column between the levels z and z + δz . The weight of this slice is:
δW = ρgδz
(1.2)
1.2 Potential Temperature
3
Figure 1.2: Force balance on a thin layer of atmosphere
This must balance the upward pressure force on the slice which is:
p − (p + δp) = −δp
(1.3)
In these equations, p is the pressure, ρ is the density, g is the acceleration due to gravity (9.81 ms−2 ) and
z is height. For a balance to exist:
pdδz = −δp
(1.4)
Dividing through by δz and taking the limit of δz → 0 (very small) we get the hydrostatic equation:
dp
dz
= −ρg
(1.5)
The ideal gas law for air states that:
p = ρRT
(1.6)
where R is the gas constant for air (287 J kg−1 K−1 ) and T is the absolute temperature. We can use
Equation 1.6 to eliminate ρ from Equation 1.5:
pg
dp
=−
dz
P RT
(1.7)
If we measure pressure, p1 at height, z1 and pressure p2 at height z2 , then we can integrate Equation 1.7
between the limits:
Z
p2
p1
dp
=
p
Z
z2
z1
g
RT
(1.8)
On the right-hand side, T is really dependent on height z, but the range of absolute temperature variation
is small in relative terms (e.g. just a few degrees compared with absolute values of 280-290 K). Hence we
can replace the variable T by a constant average value, for example the temperature half way up the hill.
4
Chapter 1. Meteorological Background Information
This allows us to take the temperature outside the integral on the right hand side. Evaluating the integrals
gives:
h
ln p
ip2
p1
=
g h iz2
z
RT
z1
(1.9)
Hence:
ln(p2 ) − ln(p1 ) =
g
(z1 − z2 )
RT
(1.10)
This can be rearranged to give:
RT
(z1 − z2 ) =
ln
g
1.2.2
p2
p1
(1.11)
Hydrostatic Equation for Moist Air
In the above analysis R depends on the chemical composition of the air. As the molecular weight of the
gas increases, R decreases. This is meteorologically important because moist air contains water vapour
(H2 O), which has a lower molecular weight than the primary molecules in air (N2 and O2 ) and hence
a larger R. At a given p and T, Equation 1.6 implies that moist air tends to be lighter than dry air. For
completely dry air R is given by Rd = 287 J kg−1 K−1 , whilst for pure water vapour Rv = 461.5 J kg−1 K−1 .
When applying Equation 1.6 to moist air the convention is to use the gas constant for dry air Rd and to
replace T with a fictitious temperature called the virtual temperature, Tv . This represents the temperature
to which dry air would have to be heated to have the same density as the moist air being considered.
Consider an air parcel containing a mixture of dry air, water vapour, and condensed water. The full density
of the air mixture may be written as
ρ=
md + mv + ml
Va + Vl
(1.12)
where, md , mv and ml are the respective masses of the dry air, water vapour, and condensed water,Va
and Vl are the respective volumes of the unsaturated air and the liquid water. Dividing through by md and
neglecting the small ratio of liquid volume to air volume gives
mv
ml
ρ = ρd 1 +
+
md md
pd
=
(1 + rv + rl )
Rd T
(1.13)
where ρd is the dry-air density, pd and e are the partial pressures of dry air and water vapour, and the
mass ratios rv and rl are known as the water-vapour mixing ratio and the liquid-water mixing ratio. rl is
usually much less than rv so we can neglect the final term in Equation 1.13. Applying this approximation
and performing some algebraic manipulation, the ideal gas law for a moist atmosphere becomes:
p = ρRd Tv
(1.14)
where
1 + rv /
1 + rv
≈ T (1 + 0.61rv )
Tv = T
(1.15)
where = Rd /Rv . Note that because mass is a conserved quantity, and because md and mv are the
masses of dry air and water vapour in Equation 1.13, rv is also conserved for unsaturated air motions.
This does not hold for saturated air motions because water vapour may be gained or lost as water changes
phase between gas, liquid, and solid form. We can now continue to use Equations 1.7 to 1.11 as if the air
1.2 Potential Temperature
5
Figure 1.3: Schematic plan-form view of flow acceleration through a gap
Figure 1.4: Schematic illustration of flow deflection around a mountain peak and the Bernoulli effect. When
winds are forced to detour around a mountain peak, they slow down just upwind of the peak then speed
up around its edges
were dry, taking account of the moisture via Tv . Calculation of Tv requires the mixing ratio rv which can
be calculated from the relative humidity q as follows:
q = 100x
rv
rs
(1.16)
where rs is the saturation mixing ratio.
1.2.3
Gap Flow and Bernoulli’s Equation
Air is commonly observed to travel faster and maintain lower pressure within gaps and valleys in a mountain
range than over surrounding areas. These increased wind speeds follow straightforwardly from conservation of mass and momentum. Mass conservation requires that the flux of air entering a control volume
must equal the flux of air exiting it. Consider Figure 1.3, which shows air funnelling through a gap such as
that between two adjacent mountain ridges.
On the upstream side, air with a density of ρ1 and a wind speed of U1 is confined between two rigid walls
separated by a distance L1 . This air must squeeze together as it passes through the narrowing channel
and exits the other end of width L2 with a density of ρ2 and a speed of U2 . If all of the air displacements are
confined to the horizontal plane in the figure, conservation of mass requires that the flux of mass entering
the control volume must equal that exiting it, or
6
Chapter 1. Meteorological Background Information
ρ1 U1 L1 = ρ2 U2 L2
(1.17)
If we further assume that the flow is incompressible, or that the density of an air parcel does not change
along its path, then ρ1 = ρ2 , which gives the following simple relation:
U2 = U1
L1
L2
(1.18)
Thus the flow accelerates within the gap by an amount that is determined by the width of the gap. If we
further assume the flow to be frictionless and steady in time, momentum conservation allows Bernoulli’s
equation,
1
p + ρU 2 + ρgz = constant along a streamline
2
(1.19)
to hold for each streamline of air that passes through the gap. Because the flow is horizontal, the third
term on the left side is constant, which gives:
1
p + ρU 2 = constant along a streamline
2
(1.20)
1
1
p1 + ρ1 U12 = p2 + ρ2 U22
2
2
(1.21)
The above may be rewritten as:
For incompressible flow Equation 1.21 may be rearranged to give:
p2 = p1 +
1
U12 − U22
2
(1.22)
which may be further simplified by substituting Equation 1.18:
1
p2 = p1 + ρ1 U12
2
1−
L1
L2
2 !
(1.23)
Because L1 > L2 the pressure at the gap exit must be lower than that of the impinging flow. Note that this
effect (often called the Bernoulli effect) explains the changes in wind velocities when air detours around
the sides of a mountain peak ( Figure 1.4). The incoming air, which is unable to pass over the mountain,
is thus forced to converge around the sides of the mountain where it passes through a smaller effective
volume and accelerates.
1.2.4
Orographic Flow Regimes
Air that impinges on a mountain can either rise over it (‘unblocked’ flow) and or detour around it (‘blocked’
flow). The flow response depends largely on the Froude number (Fr):
Fr =
U
NH
(1.24)
where U is a characteristic upstream wind speed, H is the mountain height, and N is the Brunt-Väisälä , or
‘buoyancy’ frequency, which is given by:
1.2 Potential Temperature
7
g dρ
ρ0 dz
g dθ
≈
θ0 dz
N2 ≈
(1.25)
where ρ = ρ(z) and θ = θ(z) are the vertical density and potential temperature profiles of the undisturbed
flow and ρ0 and θ0 are reference density and potential temperature values (for example, that of air at the
surface). Note that N 2 > 0 when the atmosphere is statically stable (because θ increases with height),
N 2 < 0 when the atmosphere is statically unstable, and N 2 = 0 when the atmosphere is statically neutral.
The simplest interpretation of Fr is that it represents the ratio of the kinetic energy of the impinging flow
to the potential energy required to ascend the mountain. Because density tends to decrease with height,
N 2 is usually positive and the atmosphere is stable; that is, when air parcels are displaced vertically, they
accelerate back towards their original position. When stable air is forced to ascend a mountain, it becomes
cold and dense relative to its surroundings (or ‘negatively buoyant’), and is pulled downwards.
How does the above expression for Fr come about? The kinetic energy of the incident flow is proportional
to ρ0 U 2 and the potential energy increase in lifting surface air to the mountain top is ∆ρgH , where ∆ρ is
the density difference between the lifted surface air and the undisturbed surrounding air at the hill top. The
density perturbation caused by lifting the air upwards by δz may be written as the difference between the
original air density and ρ(z) at the new level:
∆ρ = ρ0 − ρ
dρ
= ρ0 − ρ0 + δz
dz
(1.26)
If the vertical gradient of ρ is constant with height, then the perturbation density at the mountain top
becomes:
dρ
H
dz
1 dρ
=−
δ0 H
ρ0 dz
∆ρ = −
(1.27)
Thus the change in PE becomes:
∆(P E) = −∆ρgH
= −ρ0 N 2 H 2
(1.28)
The ratio of kinetic to potential energy then becomes:
KE
ρo U 2
=
PE
ρ0 N 2 H 2
= F R2
1.2.5
(1.29)
Unblocked Flow
When Fr> 1 the impinging flow possesses enough momentum to overcome the negative buoyancy that it
acquires over the upwind slope, which allows it to complete its ascent over the mountain. In this ‘unblocked’
8
Chapter 1. Meteorological Background Information
flow regime, the wind may either accelerate or decelerate on the upwind side of the mountain depending
on another type of Froude number based on the width L of the mountain:
FrH =
U
NL
(1.30)
Figure 1.5: Comparison of the vertical structures of evanescent (left) and mountain-wave (right) flows
The vertical flow structure is illustrated in Figure 1.5. When the impinging winds are strong, with weak
stability and a narrow mountain (left panel), the flow passes over the mountain too rapidly to be affected by
buoyant forces. Instead it squeezes together in the vertical as it passes over the mountain, which causes
it to accelerate by the Bernoulli effect discussed earlier. This response is often referred to as ‘evanescent’
because the mountain disturbance decays rapidly above the ridge. By contrast, when the flow is weaker,
the stability is stronger, and the mountain is wider, the flow spends enough time over the mountain to feel
the buoyant effects. Ascent over the upwind side of the mountain is resisted by the negative buoyancy
that is created by the lifting in a stable atmosphere (N 2 > 0), which causes the flow to decelerate. Over
the lee slope the flow accelerates because the negatively buoyant air is free to descend. The ‘mountain
waves’ created in this situation are not confined to the area directly over the mountain, but can propagate
both upwards and downwind into the atmosphere. Mountain waves are associated with many important
atmospheric phenomena including downslope windstorms, trapped waves, internal wave breaking, and a
drag on the atmospheric circulation.
1.2.6
Blocked Flow
When the flow is too weak to overcome the retarding effects of negative buoyancy it detours around the
mountain. In this ‘blocked’ flow regime, the low-level winds tend to lie parallel to contours of terrain height
rather than perpendicular to them. For this reason, wind speeds are generally low on the upwind side of
the mountain but may increase around the mountain edges due to the Bernoulli effect (see Figure 1.6).
The winds may also strengthen over the crest due to the ambient increase in wind speed with height and/or
wave activity aloft.
1.2.7
Lower Boundary Condition on the Flow at the Hill Surface
Close to the hill surface we need to be concerned with exactly how the flow behaves because this affects
issues like where the streamlines have come from upwind. At the finest scale, the approach to take is
that described in the section on ‘Wind and Energy Profiling’ in ?? ??. The flow can be examined in detail
1.2 Potential Temperature
9
Figure 1.6: Comparison of low-level responses to an isolated mountain in the unblocked (left) and blocked
(right) regimes
when analysing the data from the 10 m mast at the field centre. However, on a slightly larger scale (say
the scale of Goat Fell) we need to know how the hill affects the flow without requiring the details of the
turbulence within the surface layer. On this scale, the flow is often assumed to approximate a ‘no-slip
boundary condition’.
1.2.8
The No-Slip Boundary Condition
Actually on the surface, viscous forces mean that the flow is stationary. Within a few metres of the surface
the effect is communicated to the flow by turbulent eddies. Just above this however, the dominant effect
of a surface is to prevent flow passing through the surface. In other words, the air flow slips past the
surface, parallel to the orientation of the surface. The condition that this should happen is called the
no-slip boundary condition.
How can we quantify this condition? The simple way is to recognise that under the no-slip condition, for an
air parcel close to the surface, it moves such that its altitude is the altitude of the hill surface at its current
horizontal position and that rate of change of altitude is the parcel’s vertical velocity. In mathematical terms,
suppose that the hill surface is defined by:
z = h(x, y)
(1.31)
Dh
=w
Dt
(1.32)
The no-slip condition says:
Recall that the ‘material derivative’, D/Dt is the rate of change following an air parcel and in terms of the
air velocity with components (u, v, w) it is given by:
D
δ
δ
δ
δ
=
+u +v +w
Dt
δt
δx
δy
δz
(1.33)
Applying this to Equation 1.32 and noting that the hill is not moving (does not depend on time) we get as
the no-slip lower boundary condition:
u
δh
δh
+v
=w
δx
δy
on z = h(x, y)
(1.34)
10
Chapter 1. Meteorological Background Information
The significance of this condition for the current exercise is that if we have a topographic dataset giving
the altitude of the hill surface as a function of horizontal position, and we measure the horizontal wind (u
and v) we can deduce the vertical velocity, w. This is important because direct measurement is difficult,
involving precision instruments such as the 3-D sonic anemometer used in the 10 m tower exercise.
1
1 dC
=
HCO
C dz
1
=√
τk
1.3
1.3.1
(1.35)
Weather Forecasting
Introduction
This section provides material which will be helpful to you in preparing weather forecasts. Please note that
this information is for forecasting in the northern hemisphere, specifically for the UK.
1.3.2
Features of Synoptic-Scale Structures in Mid-Latitudes
There are several features of synoptic scale flow in the mid-latitudes, which will be key to understanding
the large-scale conditions which influence the day-to-day weather in Arran. These will be most obvious
on the Met Office analyses and forecast charts, but will have signatures in other diagnostics too. A range
of these features is shown in Figure 1.7. Some basic principles follow, which will provide useful synoptic
information and which will help you with your forecasts.
1.3.3
Geostrophic Flow
In normal fluid flow air moves from areas of high pressure towards areas of low pressure due to the
pressure gradient force. As the air begins to move it is deflected by the Coriolis force resulting from the
rotation of the Earth. When the pressure gradient and Coriolis forces are in balance the air no longer
moves from high to low pressure areas; instead it moves along an isobar and is said to be in geostrophic
balance. This geostrophic flow neglects the effects of friction, making it a good approximation for synoptic
scale (horizontal scales of greater than 1000 km) mid-latitude atmospheric flow. Furthermore, the closer
the isobars are to each other, the faster the flow. Hence, by looking at the isobars on a surface analysis
chart we can immediately gain an impression of the wind speed and direction. The scale in the top lefthand corner of each chart enables an estimation of the wind speed.
1.3.4
Thickness
Air pressure decreases with height so it is possible to measure the height difference between two levels
above a point on the surface. Thickness, in synoptic meteorology, refers to the height difference between
two standard pressure levels, 1000 hPa (near the surface) and 500 hPa (usually around 5-6 km). Thickness
is usually measured in units of tens of metres or decametres (dam). The significance of the thickness
measurement is that the colder the atmosphere the more dense it becomes, and therefore the shallower
the layer between 1000 and 500 hPa is. Thickness is a useful measure of the large-scale temperature
through the lower part of the atmosphere. A useful rule-of-thumb in the UK is that if the thickness is less
than 528 dam, any precipitation is likely to fall as snow, rather than rain. The angle between the thickness
and surface pressure contours can also give an indication of the sign and amount of thermal advection.
1.3 Weather Forecasting
11
Figure 1.7: Typical synoptic features in the mid-latitudes. Taken from Met Office surface analysis for
1800GMT on 27 October 2002
1.3.5
Cyclones and Anticyclones
Regions of low surface pressure are called cyclones or depressions, or sometimes simply ‘lows’, while
regions of high surface pressure are called anticyclones or ‘highs’.
Cyclones are regions of large-scale ascent of air. As the air rises it expands and its temperature falls. The
lower the temperature, the less capacity the air has to hold water vapour and so condensation occurs,
forming clouds. Hence cyclones are regions of cloud formation and rainfall. Conversely, anticyclones
are regions of large-scale descent of air. Descending air is compressed and increases in temperature,
increasing its capacity for retaining water vapour. Hence anticyclones are often (but not always) regions of
clear skies and little rain. In winter there are certain conditions (particularly in which the anticyclone builds
from the south-west) in which a great deal of shallow cloud or ‘anticyclonic gloom’ is associated with an
anticyclone.
In the northern hemisphere, the direction of air flow in a cyclone is anti-clockwise, and clockwise in an anti
cyclone.
You will be able to see the cyclones and anticyclones on the surface pressure analysis and forecasts, and
also in the satellite pictures which show the associated cloud patterns. In a chart such as the one shown
in Figure 1.7 they are marked by H and L symbols, usually accompanied by a central pressure value.
12
Chapter 1. Meteorological Background Information
Figure 1.8: Schematic cross-sections through warm and cold fronts
Figure 1.9: Cross-sections through warm and cold occluded fronts
1.3.6
Fronts and Frontal Progression
These are surfaces, often only a few kilometres thick, between air masses of different properties. The term
is particularly used to describe interfaces between air masses of different temperatures. Frontal surfaces
are generally inclined at a small angle to the horizontal, with colder air under-cutting warmer air. Their
surface positions will be plotted on surface analysis charts, but you will also see the transition between
warm and cold air masses through analysis of the radiosonde ascent data from the field site.
The direction of motion of the front determines whether, as it passes a fixed point, it marks a change to
warmer or cooler conditions. There are three different types of frontal structure. A warm front represents
a transition from a cold to a warm air mass at the surface, while a cold front represents a transition from a
warm to a cold air mass at the surface. The third type, the occluded front, represents a more complicated
structure where two or more frontal features have combined. Occluded fronts often have little surface
structure, but are still important because they are correlated with rainfall.
Fronts are generated when horizontal convergence of the air flow brings surfaces of different air properties
close together. Figure 1.8 indicates air motion associated with fronts. For both warm and cold fronts, warm
air is rising above cold air causing the warm air to cool as it expands, and the formation of clouds; thus
fronts are potential regions of rainfall. Clouds tend to form along the frontal surface, with the cloud base
close to the location of the front. Hence a typical frontal progression at a fixed site can often be observed
as a gradual lowering or rising of the cloud base as the front passes by. Typical slopes are 1:50 for the
cold front and 1:200 for the warm front. Note that these diagrams of fronts (Figure 1.8) are idealised, and
that real warm and cold fronts can have a much more complicated vertical structure. Warm and cold fronts
can be seen in Figure 1.7.
More complicated vertical structures also occur in occluded fronts which are common over the UK (Figure 1.9). Determining the temperature structure of the air mass, and hence the precipitation and cloud
structures, requires examination and comparison of radiosonde ascents, surface analyses and satellite
images.
1.3 Weather Forecasting
13
Typical stages in the lifecycle of a mid-latitude cyclone are shown in Figure 1.10. Again these stages are
highly idealised and each individual cyclone will be subtly different to the others. Mid-latitude cyclones form
on the temperature gradient between the equator and pole. In an idealised sense, this gradient can be
thought of as one long frontal feature between warm air to the south and cold air to the north (Figure 1.10a).
Cyclones often form as ‘so-called’ waves which initially look like small kinks in the front but which grow into
angular points marking where the cold and warm fronts meet (Figure 1.10b). As the system begins to
mature, the central pressure at the point where the fronts meet decreases, forming a ‘warm-sector’ with
generally overcast cloud conditions but little precipitation (Figure 1.10c). During the final stage of cyclone
development the cold front begins to catch the warm front at the surface and form an occluded front, close
to the centre of the cyclone (Figure 1.10d).
So-called secondary cyclones can develop in a similar way on pre-existing trailing fronts of mother cyclones. Their evolution is usually much faster and they are smaller in scale, which makes them a difficult
feature to forecast.
Figure 1.10: Stages in the development of a mid-latitude cyclone. (a) Pre-existing front. (b) Wave forms
on front. (c) Fully-developed cyclone with ‘warm sector’. (d) Occluded stage.
1.3.7
Local Influences on Forecasting
The features described above act on the synoptic scale. However, there are other effects that act on a
more local scale, and are related to orography.
1.3.7.1
Orographic Wind Effects
Steep orography can have significant impacts on the local flow, either through a direct dynamical impact on
the flow, or through a thermodynamic generation of flow in mountain valleys as a result of variable heating
and cooling in different parts of the valley. You should be aware of these potential local impacts on your
14
Chapter 1. Meteorological Background Information
forecast given the location of our measuring site. In particular channelling of the wind down the valley may
have important impacts on your wind direction forecasts. See p6 for more information.
1.3.7.2
Orographic Enhancement of Precipitation
It is often observed that hills as low as 50 m can produce significant enhancement of precipitation on
their downstream side. This can be in the order of 10-40% greater than the precipitation occurring on the
upstream side.
Although the interaction of orography and the synoptic and mesoscale flow is complex; the simplest explanation for this is the seeder-feeder mechanism. When air is forced to ascend as it meets the hill a
small non-precipitating cloud, called a feeder cloud, is formed. When a significant precipitating cloud, like
that seen in a frontal feature, passes overhead falling precipitation from the seeder cloud can collect more
moisture from the feeder cloud before it hits the ground. This results in higher rainfall totals for the same
storm in hilly regions. Various mechanisms for orographic precipitation, including the seeder-feeder effect,
are shown in Figure 1.11.
Figure 1.11: Mechanisms of orographic precipitation: (a) seeder-feeder mechanism; (b) upslope condensation; (c) upslope triggering of convection; (d) upstream triggering of convection; (e) thermal triggering
of convection; (f) leeside triggering of convection; (g) leeside enhancement of convection. Slanted lines
below cloud base indicate precipitation. [From Fig. 12.24 [?] ].
1.3 Weather Forecasting
1.3.8
15
Skew-T/log(P) Diagrams
Thermodynamic diagrams are ways of representing the vertical structure of the atmosphere in a way which
allows different variables to be related in a quick graphical format and which aid calculations of properties
such as stability. There are at least five variations of these diagrams; perhaps the most common are the
tephigram (used by the UK Met Office and some other meteorological services) on which the axes are
temperature and potential temperature and the skew-T/log(p) (often simply called skew-T and used quite
widely internationally) on which the axes are temperature and log(pressure). These diagrams have the
property that equal areas on the diagram represent equal amounts of energy; this enables the calculation
of a wide range of atmospheric processes to be carried out graphically. Here we will concentrate on the
skew-T diagram.
Figure 1.12 is an example of a blank skew-T diagram. The axes are temperature (horizontal) and pressure
on a logarithmic scale (vertical). Five other sets of lines are shown:
1. Isobars - lines of constant pressure. These are horizontal.
2. Isotherms - lines of constant temperature. These are straight and slope from lower left to upper right.
3. Dry adiabats - lines of constant potential temperature (see Equation 1.36).
p
p0
−R/Cp
p
p0
−288
θ=T
≈T
(1.36)
These are curved lines sloping from the lower right to upper left. They indicate the rate of temperature
change in a parcel of dry air rising or descending adiabatically, i.e. with no loss or gain of heat.
4. Saturated adiabats - lines of constant equivalent potential temperature. They are significantly curved.
They represent the rate of temperature change in a parcel of saturated air, assuming that all the
condensed water vapour is liquid and is removed as precipitation, and no other heat exchange takes
place. The equivalent potential temperature is related to the potential temperature by:
θe = θ exp
lrc
Cp T
(1.37)
5. Saturation mixing ratio lines - lines representing constant values of the maximum capacity of an air
parcel to hold water vapour. The lines slope upwards from lower left to upper right and are slightly
curved. Mixing ratio is usually measured in g/kg. The mixing ratio is the mass of water vapour to the
mass of dry air, i.e.:
r=
mv
md
(1.38)
The skew-T diagram is used to represent the thermodynamic state of a profile through the atmosphere in
the form of plots of temperature and dew point. These are represented as red (temperature) and green
(dew point) lines. If the temperature of a parcel of air is reduced, its ability to hold water vapour is reduced.
The dew point temperature is the temperature to which a parcel of air at a constant pressure would have
to be reduced in order to become saturated, assuming that no water is added or removed. In the example
shown in Figure 1.15, the temperature and dew point are nearly equal up to about 500 hPa, so the air is
virtually saturated - in practice this means that there is cloud from the surface up to 500 hPa.
At a given temperature and pressure, suppose that the maximum mass of water vapour which an air parcel
can hold is mvs. Then the saturation mixing ratio, rs , is given by:
rs =
mv s
md
(1.39)
16
Chapter 1. Meteorological Background Information
Figure 1.12: Blank skew T diagram
The relative humidity is expressed as a percentage and is given by
q = 100x
1.3.9
r
rs
(1.40)
Lifting Condensation Level
Consider what would happen if a parcel of unsaturated air close to the surface were forced to rise whilst
not exchanging any heat or moisture. This could happen, for example, during flow over a mountain. The
pressure of the parcel would decrease and its volume would increase, but because of the absence of heat
exchange, it would follow a dry adiabat on the skew-T diagram (dark red line in Figure 1.14).
Another quantity which would not change is the mixing ratio, since no moisture is exchanged. The value
of the mixing ratio can be read off the skew-T diagram from the dew point curve at the surface. Suppose
1.3 Weather Forecasting
17
we follow a saturated mixing ratio line on the skew-T diagram upwards from this point at the surface (black
line in Figure 1.16). Where the parcel’s dry adiabat meets this line, the air parcel will be saturated. This is
the level where cloud would be expected to condense out - it is called the Lifting Condensation Level.
1.3.10
Ascent above the Lifting Condensation Level
Suppose that an air parcel near the surface is again forced to ascend without exchange of heat and
initially without exchange of moisture. It will follow a dry adiabat until it reaches the lifting condensation
level (shown as the straight brown line in Figure 1.17). Any further ascent, with its associated cooling, will
result incondensation of water vapour to form cloud droplets (which may produce rain). The phase change
from vapour to liquid is associated with the release of latent heat, causing the cooling to be reduced. The
air parcel will now be following the saturated adiabat (the curved brown line in Figure 1.14).
Note that for the example shown in Figure 1.17, the temperature of our lifted air parcel will always be lower
than that of the surrounding air (we usually call this the environment temperature). Since it is cooler than
its surrounding; the air parcel will be denser and hence will have a tendency to sink back towards the
ground. This indicates that the air is stable. For the case in Figure 1.17, this is true no matter how far the
parcel is forced to ascend; we call this state unconditional stability.
Consider now the slightly different case in Figure 1.18, where the environmental temperature decreases
with pressure more quickly than the dry adiabat starting from the surface. Here, the temperature of the
ascending air parcel will, even if it is forced to rise only a small distance, be higher than the environmental
temperature and so the parcel will be less dense and as a result, positively buoyant. It will continue to rise
without any additional forcing. In this state we call the atmosphere unconditionally unstable - the smallest
vertical perturbation of an air parcel will cause it to continue to rise away from its original level.
1.3.11
Conditional Instability
Let us now consider a case superficially rather similar to the stable one in Figure 1.17, but where the
atmosphere is slightly moister. This is shown in Figure 1.19.
An air parcel from near the surface is forced to rise initially along the dry adiabat until it reaches the lifting
condensation level. Above that it follows the saturated adiabat. In this case however, it reaches a level
where the parcel temperature becomes greater than the environment temperature and therefore the parcel
becomes less dense than its surroundings. Above this level therefore, the parcel will continue to ascend
freely by buoyancy. The level where the parcel temperature first becomes greater than the environment
temperature is called the level of free convection. This case is another one which is unstable but unlike
the previous case, the instability can only start after the parcel has been forced through a stable layer to
the level of free convection. This type of instability is called conditional instability. Usually, as the air parcel
continues to rise by free convection, it will eventually reach another level where its temperature once again
becomes lower than that of the environment. At this level, positive buoyancy will cease; the level is called
the equilibrium level.
1.3.12
Convective Available Potential Energy (CAPE) and Convective Inhibition (CIN)
Let us continue to study the example of conditional instability. In order to force our air parcel from the
surface up to the level of free convection (LFC), it would be necessary to do work against the tendency
for the parcel, which is denser than its environment, to sink under gravity. We can calculate the amount
of work needed, and it turns out to be proportional to the area on the skew-T diagram between the parcel
temperature and the environment temperature. This is shown in light blue in Figure 1.20. Specifically, the
work required is called the Convective Inhibition (CIN), and is given by:
18
Chapter 1. Meteorological Background Information
Z
pL F C
(Tenv − Tparcel ) d ln(p)
CIN = R
(1.41)
p0
or equivalently:
Z
CIN = g
0
LF C
Tenv − Tparcel
dz
Tenv
(1.42)
CIN is measured in J kg−1 . Its magnitude measures how difficult it is for the parcel to reach the level of
free convection, or in other words, how strongly free convection are suppressed.
The opposite happens after the parcel reaches the level of free convection. The parcel can extract energy
from its surroundings by buoyancy all the way from the LCL to the equilibrium level, EL. The energy
which can be extracted is again the area on the skew-T diagram between the parcel temperature and the
environment temperature and is called the Convective Available Potential Energy, CAPE. It is shown in
pink in Figure 1.20, and is given by:
Z
CAPE =
pE L
(Tenv − Tparcel ) d ln(p)
R
(1.43)
pL F C
or equivalently:
Z
CAPE =
EL
g
LF C
Tenv − Tparcel
dz
Tenv
(1.44)
A value of CAPE above 1000J kg−1 indicates significant instability and above 2500J kg−1 shows that the
atmosphere is very unstable. Under these conditions, deep convective clouds and thunderstorms are likely
to develop.
1.3.13
Representation of the Wind on Skew-T Diagrams
The wind speed and direction are often displayed alongside a skew-T diagram in the form of wind vectors
consisting of lines and barbs. The line indicates the direction from which the wind is coming. The barbs
on the end of the line indicate the speed. Each full barb indicates 10 ms−1 and each half barb indicates 5
ms−1 . A triangle indicates 50 ms−1 .
1.3 Weather Forecasting
Figure 1.13: The five principal sets of lines of the skew-T diagram
19
20
Chapter 1. Meteorological Background Information
Figure 1.14: Skew-T diagram with example lines labelled.
Figure 1.15: Example skew-T diagram from a sounding made at Lochranza on 13 September 2008.
1.3 Weather Forecasting
21
Figure 1.16: Example skew-T diagram (from Achern, Germany on the 14 June 2007) showing lifting condensation level construction.
Figure 1.17: The brown line shows the path taken by an air parcel forced to ascend from the surface. In
this case the atmosphere is unconditionally stable.
22
Chapter 1. Meteorological Background Information
Figure 1.18: Note that in the hatched layer, the environment temperature is lower than the temperature of
the air parcel forced to ascend from the surface following the dry adiabat.
1.3 Weather Forecasting
23
Figure 1.19: The air parcel from near the surface is forced to rise initially along the dry adiabat until it
reaches the lifting condensation level. Above that it follows the saturated adiabat.
24
Chapter 1. Meteorological Background Information
Figure 1.20: CAPE (pink) and CIN (light blue)
1.3 Weather Forecasting
25
Figure 1.21: Example wind profile and symbols.