Quasi-geostrophic diagnosis of baroclinic waves (I)
1. Basic statements
Starting with the quasi-geostrophic vorticity equation, we obtain a correlation between local
vorticity changes through advection and/or divergences and the distribution of local
geopotential changes. A relative maximum of geopotential fall (rise) is correlated either with
positive (negative) vorticity advection and/or convergence (divergence). We can then define
cyclogenetic and anticyclogenetic effects, resp., as local increase (decrease) of vorticity
correlated with relatively strong geopotential fall (rise).
For the new origin of cyclonic vorticity or the intensification of a maximum of cyclonic
vorticity, convergence in the wind field is necessary; on the other hand, divergence is a
prerequisite for the production or intensification of anticyclonic vorticity.
Once built up, the vorticity extremes are advected by the horizontal wind. But their movement
can also be effected by means of vorticity production through suitable divergence effects.
Maxima of cyclonic vorticity in lows or troughs move in the direction of positive advection or
to where cyclonic vorticity is produced by convergence, maxima of anticyclonic vorticity in
highs or ridges, on the other hand, in the direction of negative advection or to where
anticyclonic vorticity is produced by divergence.
On the closed circulation systems of the lower troposphere the advection is often weak, so
that the vorticity extremes in the lows or highs are mainly moved by convergence and
divergence, resp., on their flanks. In contrast to that, the movement of the troughs and ridges
in the mid troposphere takes place mainly through advection, since divergence is generally
weak in these levels. Higher up in the short waves of the jet stream level the advection of
vorticity is strongest, but is often more or less compensated in its effect on the movement
through counter working divergence.
Divergences are connected with vertical motions. Information about the distribution of macroscale vertical motions is provided by the omega equation. Using its conventional adiabatic
form, we find as forcing functions for the release of vertical motions the vorticity advection
with its vertical variation and the temperature advection with its horizontal distribution. A
forcing for ascending motion results in a region with positive vorticity advection (PVA)
increasing with height and/or in a region with maximised warm air advection (WA). A forcing
for descent results in a region with vertically increasing negative vorticity advection (NVA)
and/or maximised cold air advection (CA). With the same forcing, the smaller the stability,
the greater are the vertical motions. If, in addition to both adiabatic forcing terms, diabatic
effects are at work, a forcing for ascent (descent) follows in the region of biggest diabatic
heating (cooling). In the macro-scale the release of latent heat by condensation is of special
importance, since through that the primary working ascending motion can be considerably
strengthened.
Using the frame of a two-layer-model and combining the statements of the omega equation
with those of the vorticity equation and taking into account that the vorticity change through
advection is generally weak in the lower troposphere, we can formulate the following rules for
lower and upper troposphere, resp., shown in Table 1.
Table 1
In order to apply this knowledge, we need analyses (or forecasts) of the vorticity advection
aloft (in 500 or 300 hPa) and of the temperature or thickness advection in the mid
troposphere.
By neglecting the effects of deformation, the two adiabatic forcing terms of the omega
equation can be combined into one, the advection of vorticity by the thermal wind. A forcing
of ascent results in an area of positive advection with the thermal wind, i.e. along the
isotherms. It is connected with a cylogenetic effect in the lower troposphere and an
anticyclogenetic one aloft. Descent, on the other hand, connected with an anticyclogenetic
effect in the lower layers and a cyclogenetic one aloft, has to be expected with negative
advection by the thermal wind. This correlation has a great advantage since the problem of
possible cancellation between the two original terms does not appear and sign and relative
strength can be easily estimated by a comparison of the fields of vorticity and temperature at a
given pressure level.
When considering a development in a co-ordinate system moving with a baroclinic wave, it
results that the relative motion in the mid troposphere where the zonal wind corresponds to
the phase velocity of the wave, is mainly determined by the vorticity of the wind field.
Therefore the relative streamlines are almost identical to the isopleths of vorticity at this level.
Accordingly, the just described simplified forcing of vertical motions can be altered into a
form in which the temperature advection with the relative winds is decisive. A forcing of
ascent (descent) has to be expected in areas with WA (CA) by the relative winds.
The list given above can be expanded by the effects caused by orography and friction. The
forced ascending motion on the windward side of a mountain ridge leads to horizontal
divergence and subsequent decrease of vorticity above. That is an anticyclogenetic effect. On
the lee side of the mountains we find the opposite conditions: descent, convergence and
increase of vorticity in the layers above which is a cyclogenetic effect.
Surface friction reduces not only the wind speed but also the vorticity in the boundary layer
and therefore works cyclolytically and anticyclolytically, resp. But the friction also influence
the layers above. It causes ascending (descending) motion at the top of the boundary layer in
areas with cyclonic (anticyclonic) vorticity. That leads to divergence (convergence) in the
layers above and therefore reduces the vorticity also there. Contrary to this, friction may also
lead to production of vorticity when it shows a significant horizontal gradient, e.g. across a
coast line or at the flanks of a mountain ridge.
2. Dynamics of baroclinic waves
With the aid of the correlations just derived, we can easily explain and understand the basic
dynamics of the baroclinic waves, the typical macro-scale feature of the westerlies. In the
following, we consider only waves in which the advection of relative vorticity dominates in
the mid troposphere and which therefore progressively follow the basic westerly current. One
can then distinguish between stable wave types, the amplitude or vorticity of which remains
constant during their movement, and unstable waves, the amplitude or vorticity of which is
subjected to bigger changes in time – either growing or damped.
Fig. 1 shows schematically the structure of the different wave types. Stable waves show
upright axes of troughs and ridges. The fields of vertically averaged temperature and
geopotential have to be then either in phase or must have a phase difference of 180o. In the
first case the coldest air – compared in zonal direction - lies in the troughs, the warmest air in
the ridges. Vorticity and zonal wind increase with height and so does the vorticity advection.
As a consequence, a forcing of ascending motion results between trough and ridge
downstream, a forcing for descending motion between trough and ridge upstream. The
temperature advection is often weak within such waves, but with CA ahead and WA at the
rear of the troughs it can have a compensating effect with regard to the vertical motions.
Connected with the vertical motions, convergence (divergence) works ahead (at the rear) of
the trough in lower levels, enhancing the effect of the weak advection for the movement of the
vorticity maximum in the trough. The opposite distribution of divergence and convergence,
resp., results in compensating the effect of strong vorticity advection in upper levels. As a
consequence, the phase speed of the wave remains the same in all levels and is proportional to
the vorticity advection in middle levels. Since no divergences are at work at the axes of the
wave, no development can occur.
Waves with the warmest air in the troughs and coldest air in the ridges show vertically
decreasing vorticity, so that with vertically increasing wind only small differences of the
advection result in most cases. Accordingly, the dynamics of these waves is mainly
determined by the temperature advection: The WA ahead of the troughs causes ascending
motion, the CA at the rear descending motion connected with the relevant convergence and
divergence, resp., in lower and upper levels.
Both stable wave types are in some cases detectable through the whole troposphere, but they
are frequently confined to single layers. Waves with cold troughs appear preferably in the mid
and upper troposphere, whereas the waves with the warm troughs in many cases are confined
to the lower troposhere. In the latter, the baroclinicity near the surface is often concentrated in
defined frontal zones. A superposition of both wave types in the vertical is also observed.
In baroclinic unstable waves the fields of mean temperature and geopotential show phase
differences of +/- 90o, so that the vertical axes of the troughs and ridges are tilted either
backward or forward. The wave with backward tilted axes is the one in which cyclogenesis
and anticyclogenesis occur. Through the combined effects of vorticity and temperature
advection the warm air between the trough in the mid troposphere and the ridge downstream
is lifted, whereas the cold air between trough and ridge upstream descends, Since the axes are
tilted backwards, the ascent due to PVA aloft works just above the surface trough or low
leading to convergence and intensification of the vorticity there. The same is valid for the
surface ridge or high through the divergence resulting from the descent due to NVA in upper
levels. On the other hand the WA is concentrated below the upper ridge, the CA below the
upper trough, so that through the relevant divergence/convergence in these levels the vorticity
of both can also grow. The movement of the surface wave is at first mainly determined by the
effects of the temperature advection, whereas the upper wave follows the vorticity advection.
It can be easily shown that in reality the intensification of the wave not only takes place in
lower and upper levels, but in all levels, due to convergence along the tilted trough axis and
divergence along the tilted ridge axis.
With a forward tilt of the axes the conditions are just inverse, so that in all levels the vorticity
is diminished and the wave is damped.
The schemes of Fig. 1 can describe, of course, only in a first raw approximation the real
structure of all baroclinic waves appearing day by day in the synoptic analyses. In reality
there are many modifications especially through asymmetries in the distribution of wind and
temperature. As regards the wind field, the modifications result mainly from the fact that pure
sine-shaped waves as in the schemes are seldom encountered in nature. Often a superposition
with confluence and diffluence patterns is found. In addition, baroclinicity is normally not
uniformly distributed, but concentrated in frontal zones from which a jet-stream character of
the upper current results. Through that the shear comes into play in addition to the curvature
and the distribution of vorticity and vorticity advection is accordingly modified.
Fig. 2 contains some typical combinations. In troughs and ridges with confluent entrance and
diffluent exit and wind maximum in the axis area, the areas with strongest vorticity advection
and subsequently also strongest omega forcing have a position near to the axes, whereby the
extremes lie at the cyclonic or anticyclonic flank of the jet stream, resp., due to the effect of
the shear. On the other hand, the vorticity advection is mainly weak in troughs and ridges with
minima of the wind speed. The pattern around a straight jet maximum can be considered as
combination of a trough and ridge with confluent entrance and diffluent exit so that a squarelike distribution of the vorticity advection and omega forcing results.
In cases with asymmetries in the wind field across the troughs and ridges the vorticity
extremes possibly lie outside their axes, so that changes of their intensity can occur not only
through divergence, but also through advection of higher or lower vorticity into the trough or
ridge area.
Additional effects are at work if the temperature advection around a low or high is very
different. If the WA dominates at the flanks of a pressure centre, a contribution for descending
Fig.2: Distribution of the vorticity advection in waves with superimposed
confluence and diffluence (jet axis thick marked)
motion results just over the centre from which we should expect a weakening if it is a low, but
an intensification if it is a high. In cases in which CA dominates, the opposite is true.
3. Change of wave type; release of baroclinic instability
For weather forecasting the question of a release of the baroclinic instability is essential, i.e.
the question as to whether and under what conditions stable waves become unstable and how
this process occurs.
To begin with, we consider a stable trough with upright axis covering the whole troposphere
and travelling with uniform phase speed eastwards. It is embedded in a baroclinic zonal flow
with vertically increasing wind speed. According to the advection of vorticity along the
zonally oriented isotherms, ascending motion takes place east of the trough and descending
motion west of it.. With the aid of Fig.3a we see, that through the convergence and divergence
connected with the vertical motions and the resulting vorticity changes in the lower and upper
troposphere, the trough would aquire a backward tilted axis. With that an unstable wave with
growing amplitudes would develop.
But this is not the only effect. We have also to consider the simultaneous vorticity advection
at work, whereby the advection relative to the moving trough is important. Since it moves
with a phase velocity comparable to the wind speed in the mid troposphere, we find a relative
current from the fore part to the rear, i.e. from east to west, in the lower troposphere and in the
opposite direction, from west to east, in the upper troposphere. As shown by Fig.3b, waveshaped streamlines result from the combination of the relative translation with the cyclonic
vorticity of the trough. Due to the advection within these relative currents, a tendency results
contrary to the effect of the divergences, namely to a forward tilting of the trough axis. If this
Fig.3a: Diagram of the time-related change of vorticity starting from an isolated vorticity
maximum by divergence (top) and during a downstream development (bottom).
After Simmons and Hoskins (1979)
Fig.3b: Relative motions in upper, mid and lower troposphere (top) and time-related
change of vorticity due to relative advection (bottom)
effect would dominate, the trough would be damped in the future. If both effects are just
about equal, then the trough keeps its upright axis and remains stable.
In order to maintain a hydrostatic balance, the described changes of the tilt must be connected
with changes of the temperature distribution. For a backward tilted axis of the trough there
must be a warming ahead and a cooling behind of it. Starting with zonally oriented isotherms,
this can be managed by the WA ahead and the CA behind of the trough being effective in the
absolute system as well as relative to the moving trough, provided that this effect is stronger
than the compensating effect of the vertical motions contributing to the opposite temperature
changes. Altogether the lower trough is displaced by the convergence to the developing warm
bulge and the upper trough to the developing cold bulge of the isotherms of the mean
temperature (see Fig.4).
For a forward tilt, on the other hand, there must be a cooling ahead and a warming behind of
the trough which requires that the effect of the vertical motions is stronger than the horizontal
advection.
According to Simmons and Hoskins (1979), the ratio of the effects of divergence and relative
advection for the vorticity changes at the flanks of the trough depends strongly on the length
scale of the systems under consideration. Due to the influence of static stability, a defined
threshold cannot be given, but the shorter the wave length, the greater the possibility for the
advection to make the larger effect, whereas in longer waves the divergence dominates the
Fig. 4: Distribution of mean temperature in a wave with upright position, backward
tilt and forward tilt of the trough axis (from left to right)
vorticity changes. Concerning the temperature changes, the vertical motions must then be
more effective in shorter waves, whereas the advection prevails in longer ones.
With a limited vorticity maximum as in Fig. 3, the change to a backward tilt also leads to the
origin of new vorticity extremes, namely an upper vorticity minimum above the area with
ascending motion ahead of the trough and a lower vorticity minimum below the area with
descent behind of the trough. The effect of divergence normally also dominates at the outer
fringe of the disturbance, so that in the future not only the primary extremes are intensified,
but also new regions of positive and negative vorticity develop downstream and upstream of
it. The development of one trough or one wave in a certain region then causes either a new
formation of waves or an intensification of already existing waves in the neighbourhood. This
possibly very effective process is called “downstream resp. upstream development”.
It is clear, that the question whether a stable baroclinic wave seen in the operational analysis
becomes unstable or not, cannot be decided by the qualitative diagnosis discussed here, but
need numerical simulation. There is, however, another frequently observed possible origin of
an unstable wave and onset of cyclogenesis to which Petterssen (1956) has already pointed
and which can easily monitored by the use of diagnostic methods. It follows from the
coupling of two stable waves which are at first independent of each other, namely a cold
trough in the upper troposphere and a frontal trough or a wave with warm air in the lower
troposphere (cf. Fig.5).
A precondition for the coupling is a faster movement of the upper trough and with that an
approach to the lower system. Since the phase velocity depends on the wave length and is
proportional to the velocity of the basic current in the relevant level, a higher phase speed of
the upper wave can be expected due to the vertical increase of velocity of the baroclinic basic
current if the wave lengths of the two systems are comparable. However, in many cases both
waves are laterally displaced with regard to the basic flow – the upper one at its cold side, the
lower one at its warm side. Therefore a relative movement across the current is often
necessary in order that the vorticity maximum of the upper trough comes to the rear of the
lower wave and does not laterally pass by. Decisive for that are outbreaks of deep cold air
which can cause a corresponding component of the movement of the upper maximum due to
descending motion and upper convergence. The effect of WA in the lower levels is similarly
favourable for a lateral turning off of the lower wave.
First PVA ahead of the upper trough and CA at the rear of the lower wave become superimposed in the joining up process. In accordance with Table 1 this means compensation
Fig.5: Relative motion and coupling of baroclinic waves. Top: Map illustration with fronts,
surface isobars (solid lines), relative isohypses (broken lines) and position of upper
level trough (Tr); bottom: Schematic cross-section with vorticity extremes, vertical
motions and relative motion of extremes (Taken from Kurz (1998)).
with regard to the vorticity tendency near the ground, i.e. still no development. Due to
opposite vertical motions – ascending motion above, descending motion below – convergence
nevertheless already occurs in the intermediate layers so that cyclonic vorticity is produced
there and a continuous trough axis with backward slope can originate. Ahead of this trough
ascent is now released also in the mid troposphere. It works directly above the lower trough
and triggers its intensification. This coupling process often leads to a very intense cyclone
within short time.
4. Further development of the unstable wave
Once in motion, the processes of cyclogenesis and anticyclogenesis run accelerated. With
increase of vorticity in the upper trough and decrease in the upper ridge the vorticity
difference between the two increases. The same is true for the vorticity advection which leads
to increased vertical motions above surface low and surface high and causes their further
intensification. The strengthening of the winds associated with this leads to a stronger
temperature advection as a result of which the upper current is again effected.
The described processes continue as long as the phase difference between temperature and
geopotential waves is kept and the tilts of trough and ridge maintain their backward position.
For that it is necessary that the temperature advection relative to the moving wave remains
stronger or at least equal compared with the opposite effect of the vertical motions on the
temperature change.
In this respect the release of greater amounts of latent heat in the ascending warm air is very
important. Due to this effect, the adiabatic cooling of the warm air is reduced to a greater or
lesser extent. It allows therefore a considerably stronger ascent without the temperature
distribution being changed more rapidly as a result. Through the increased convergence under
the area of ascent the production of cyclonic vorticity near surface is further enhanced. The
considerable intensities which cyclones in the westerlies may reach can to a large extent be
traced back to this effect. In contrast to this, the adiabatic warming during the subsiding
motion in the area of a developing anticyclone cannot be reduced by a similar effect.
There is, however, another and even more important difference between the development of
the cyclonic and anticyclonic systems. As can be read from the complete vorticity equation,
the production of cyclonic vorticity is accelerated with increasing vorticity, while conversely
the production of anticyclonic vorticity is slowed down more and more. As a result of this,
surface cyclone and upper trough can reach considerably greater intensity, compared by their
relative vorticity, as surface high and upper ridge. It has to be noted that the quasi-geostrophic
approximation of the vorticity equation does not include this effect.
Because of this asymmetry, the extremes of temperature advection move nearer and nearer to
the core of the cyclone during the intensification phase. As a result, the structure of the upper
wave is changed by a spreading out of upper trough and ridge in the direction of the low. The
distance between them is therefore shortened just over the depression giving rise to a further
increase of PVA there. Near the centre of the anticyclone, however, temperature advection
remains relatively weak.
According to these facts, the future development of cyclones and anticyclones shows clear
differences. For strongly developing cyclones it is typical that the increasing temperature
advection dominates the temperature change within the moving cyclone against the opposite
effect of the vertical motions for a longer time. It is, however, not the advection in the
absolute system, but the advection by the relative winds which is decisive. Since – as already
mentioned – the vorticity mainly determines the relative motion in the mid troposphere, a
cyclonic rotation of the air masses around the vorticity maximum is released. Within this
rotation WA by the relative winds spreads out from the fore part to the movement-related left
fringe of the depression and finally to its rear, while CA relative to the moving system
becomes effective at the opposite side. Starting from small bulges of the isotherms,
pronounced cyclonically bent tongues of warmer and colder air may develop during this
process as shown in Fig.6.
Fig.6, l.h.s.: Relations between geostrophic streamlines (thin broken lines), translation
field (thick broken lines) and relative streamlines (solid lines) in the mid troposphere in
the area of a strongly developed cyclone; r.h.s.: Time-related change of an isotherm in
the relative flow field (Taken from Kurz (1998)).
It should be noted that the temperature advection by the relative winds may significantly
differ from the advection seen in the absolute system. Since according to Fig.6 the vorticity
maximum and with that the center of the relative vortex is shifted to the flank of the low with
the strongest winds, WA by the relative winds also takes place in the center of the low, where
zero advection is indicated in the absolute field!
As already pointed out, the temperature advection by the relative winds also determines the
forcing of vertical motions in a first approximation: The relative WA is connected with an
ascending motion, the relative CA with descending motion. It can be shown, however, that
during the described process also deformation becomes effective leading to typical
frontogenetic and frontolytic effects at the frontal zone and causing meso-scale circulations
across it. This, of course, influences the temperature distribution as well as the regime of
vertical motions. Concerning the latter, the total omega forcing might be somewhat modified,
but not totally changed by this contribution in most cases. That means, that together with the
cyclonic rotation of the air masses areas of ascent with cloud and precipitation and, on the
other hand, areas of descent with few cloud only move around the vorticity maximum in the
mid troposphere.
Since the temperature changes due to the vertical motion reduce the effect of relative
advection, the movement of the isotherms proceeds more slowly than by advection alone. In
addition the temperature contrast between the tongues of warmer and colder air is more and
more diminished. Since the cooling of the warmer air by lifting can be minimised through
release of latent heat, the warming of the descending colder air dominates in many cases. That
contributes to the fact that the core of many well developed cyclones maintains a relatively
high temperature until the end.
Also in lower levels the relative streamlines around the deepening low lose their initial wave
shape and assume circular or oval forms. Therefore also the surface frontal system undergoes
a cyclonic rotation whereby the warm sector becomes even narrower due to horizontal
convergence until occlusion sets in. Conversely, the descending colder air diverges near the
ground and fills the core area of the surface depression more and more. In upper levels at the
same time a corresponding contraction of the cold air and an expansion of the lifted warm air
takes place. Since there, however, the relative streamlines normally maintain their wave form,
the warmer air moves away towards the ridge downstream, whereas the cold air is advected
into the upper trough.
Fig.7: Movement of the isotherms of mean temperature and of the vorticity extremes
in the upper and lower troposphere relative to the vortex in the mid troposphere
The movement of surface and upper level systems is, of course, decisively affected by the
cyclonic rotation of the air masses. As schematically shown by Fig.7, the surface low and its
vorticity maximum follow the tongue of warmer air to the left and backwards, whereas the
upper trough shears out to the right and moves forward following the tongue of colder air. In
doing so the vorticity extremes approach more and more the centre of the relative vortex in
the mid troposphere. Altogether the vertical axis of vorticity makes a cyclonic rotation around
the vortex in the middle levels before it straightens up fully. This motion takes place relatively
quickly since here the effect of the vergences point in the same direction as the advection with
the relative winds.
Contrary to the described process of cyclone development, anticyclogenesis proceeds less
dramatic. Since the production of anticyclonic vorticity by divergence is limited by the
threshold (-f), the rotation of the air masses around the centre of anticyclonic vorticity in the
mid troposphere remains rather weak. The temperature changes due to relative advection are
often fully compensated or even over-compensated by the effect of vertical motions. That is
due to the fact, that with the same forcing the resulting vertical velocities are biggest in air
with anticyclonic vorticity.
Concerning the relative motions in lower and upper troposphere, the streamlines maintain
their wave shape there. As a result, there is advection of colder air into the surface high and
advection of warmer air into the upper ridge.
Concerning the vorticity changes, the relative advection in lower and upper levels soon
predominates the decreasing effect of divergence, so that the vertical axis rises and becomes
upright after some time. The cold surface high below the warm upper ridge and the mid
troposphere, warmed due to the previous descending motion, are normally found as final
condition of anticyclogenesis.
After the axis has straighten up, the behaviour of the pressure systems is determined by
vorticity and temperature advection on their fringes. If closed centres have also formed in
upper levels which have approximately circular-symmetrical velocity distribution, advection
is slight at all levels and the pressure systems become stationary. They are subject to the
influence of surface friction which leads to a more or less rapid weakening and dissipation of
the system near the ground while the vortices aloft can continue for a longer time. If the
vorticity advection remains strong in upper levels, a de-coupling of the systems in lower and
upper troposphere is possible, i.e. the upper wave moves away and can eventually give rise to
a new interaction with another baroclinic system further downstream.
References:
Petterssen, S. (1956): Weather Analysis and Forecasting Vol.I. Mc Graw-Hill Book Company
Simmons, A.J. and Hoskins, B.J. (1979): The downstream and upstream development of
unstable baroclinic waves. J.Atm.Sci 36, 1239-1254
Kurz, M. (1998): Synoptic Meteorology. Training Guidelines of the German Meteorological
Service Nr.8. Deutscher Wetterdienst Offenbach a.M.