The Q vector and its application: Frontogenesis / frontolysis and related
circulations
The conventional method of quasi-geostrophic diagnosis has the disadvantage that we have to
look at two main forcing functions, namely vorticity and temperature advection, which, of
course, can have opposite signs and therefore tend to cancel each other out. There is an
approximation with only one forcing term, the advection of vorticity by the thermal wind, but
by this method the effects of the deformation with regard to the release of vertical motions are
neglected. Since these effects are important for frontogenesis and frontolysis, it is better to use
the alternative form of diagnosis which contains also only one forcing, but includes the effects
just mentioned. This form uses the so-called Q vector introduced by Hoskins et al. (1978).
According to its definition, the Q vector describes temporal changes of the temperature
distribution at a pressure surface resulting from horizontal variations of the geostrophic wind.
The other factor for changing this distribution is by horizontally differing vertical motions.
For a parcel within a geostrophic flow omega vanishes and the change of the temperature
gradient is totally determined by Q. The parcel can experience the temporal change either
locally or along its path through the temperature pattern.
Fig.1: On the manual estimation of the Q vector
The Q vector can easily be computed, but also manually estimated. For that the change of the
geostrophic wind along an isotherm has to be considered (Fig. 1). The Q vector points
perpendicular to the right of the vector change. Its amount is proportional to the magnitude of
the rate of vector change, and to the magnitude of the temperature gradient (Sanders and
Hoskins, 1990).
Using the Q vector, the modified quasi-geostrophic omega equation has a form containing
only one forcing function on the right hand side, namely the convergence or divergence of
the Q vectors. A forcing of ascent results in areas with convergence, a forcing of descent in
areas with divergence of Q. This term replaces the two adiabatic forcing functions of the
conventional form of the omega equation and allows a direct computation of the total forcing
without any simplifications.
But the use of the Q vector has also other advantages for diagnostic purposes. The
temperature gradient can change either its direction or amount or both. In order to estimate
these changes, the Q vector should be split into its components along and perpendicular to the
isotherms (Fig. 2). It can be shown that the component of Q along the isotherms (Qs) reflect
changes of the direction and the component perpendicular to the isotherms (Qn) changes of
the amount of the temperature gradient.
Fig.2: The components of the Q vector
For changes of the direction of the temperature gradient, i.e. a rotation of the isotherms, the
component of the geostrophic wind transverse to the isotherms (Vgn) must vary along the
isotherms. This is the case in fields with vorticity and – with corresponding orientation of the
isotherms – deformation. A cyclonic rotation is indicated by Qs pointing into the direction of
the thermal wind, an anticyclonic rotation by Qs pointing in the opposite direction.
Qn reflects changes of the amount of the temperature gradient and therefore frontogenetic and
frontolytic effects in the geostrophic wind field. An increase of the amount, i.e. frontogenesis
results there where the component of the geostrophic wind transverse to the isotherms shows
a convergence across the isotherms. It is indicated by Qn pointing towards the warmer air. A
decrease, i.e. frontolysis takes place with a divergence of this wind component. In this case Qn
points towards the colder air. The variations of Vgn across the isotherms decisive in this case
are, in the geostrophic wind field, solely caused by deformation.
According to this, also the total omega forcing (FQ) can be split into contributions from a part
FQs resulting from Q vector vergences along the isotherms and a part FQn which becomes
effective in the case of Q vector vergences across the isotherms.
Fig.3: Distribution of Qs and FQs in an unstable baroclinic wave
FQs comes into play when vorticity and/or deformation vary along the isotherms. It is
therefore mainly determined by the above-mentioned forcing term, namely the advection of
vorticity by the thermal wind. Accordingly, vergences of Qs are found, e.g., in the short waves
of the upper current, whereby generally a forcing of ascent results between the trough and the
ridge downstream and a forcing of descent between the trough and the ridge upstream. It is
typical that the areas with ascent and descent forced by FQs alternate with each other along
the frontal zone. FQs is normally the trigger for stronger developments. As an example the
situation outlined in Fig. 3 shows a growing baroclinic wave. Within the trough Qs points into
the direction of the thermal wind indicating the cyclonic rotation of the isotherms, whereas Qs
has an opposite direction in the ridges due to the anticyclonic rotation working there. As a
result, there is a convergence of Qs between trough and ridge downstream leading to the
ascent of the relatively warmair, and a divergence of Qs between trough and ridge upstream
causing the descent of the relatively cold air air.
Fig.4: (a) Distribution of Qn and FQn in a frontogenetic pattern.
(b) Scheme of the transverse circulation in a frontogenetic pattern
Vergences of Qn are linked to the processes of frontogenesis and frontolysis in the geostrophic
wind field. The resulting vertical motions are oriented belt-like parallel to the frontal zone and
produce, together with the ageostrophic transverse motions below and above the level of
strongest vertical motion, closed circulations across the frontal zone – solenoidally direct with
ascent of the warmer and descent of the colder air in the case of frontogenesis, solenoidally
indirect with descent of the warmer and ascent of the colder air in the case of frontolysis. As
an example Fig. 4 shows the frontal zone of a cold front embedded in a diffluent flow. The
frontal zone undergoes a frontogenetic effect indicated by Qn pointing from the colder to the
warmer air. Since the magnitude of Qn is maximised there where the temperature gradient is
strongest, divergence of Qn results at the cold side and convergence of Qn at the warm side of
the frontal zone indicating the descending and ascending motions, resp., working there. Qn
itself reflects the direction of the lower part of the circulatory motion.
Fig.5, l.h.s.: Scheme of an upper-level jet streak with isohypses at 500 hPa (solid lines) and
isopleths of relative vorticity (dashed lines); r.h.s.: Distribution of Q vectors and total omega
forcing (dashed lines) superimposed on 500 hPa isohypses being identical to isotherms of
the mean temperature (Taken from Pedder, 1997)
As another example, Fig.5 shows a schematic representation of an upper-level jet streak with
confluent entrance and diffluent exit regions. Due to the combination of shear and curvature,
there are vorticity extremes at the flanks of the jet maximum, and a square-like distribution of
vorticity advection results. Assuming isotherms running parallel to the isohypses, there is no
temperature advection effective, and the vorticity advection points to a forcing of ascent at the
anticyclonic part of the entrance and the cyclonic part of the exit region, and a forcing of
descent at the other parts.
That is confirmed by the Q vector divergence shown at the r.h.s. of Fig.5. The orientation of
the Q vectors, however, clearly reveals frontogenetic and frontolytic wind field effects as
main cause of the square-like distribution of the omega forcing. The Q vectors have a great
component across the isotherms (Qn), pointing to the warmer air and indicating frontogenesis
in the entrance area, but pointing to the colder air and indicating frontolysis in the exit region.
The resulting vertical motions therefore mainly belong to the ageostrophic transverse
circulations released by these effects. The component of Q along the isotherms (Qs) is
releatively small and reflects the cyclonic and anticyclonic curvature of the isotherms at both
flanks of the jet maximum. Its divergence (FQs), however, has the same sign as the circulatory
forcing (FQn) and enhances it further.
The next example (Fig.6, taken from Kurz(1998)) should demonstrate the advantage of the Q
vector diagnostics in cases when the forcing functions of the conventional omega equation
tend to cancel each other out. It is a case with a big trough just approaching western Europe
with strong PVA, but also CA ahead of it. The CA works behind a surface cold front running
from Norway over the Low Countries and France towards the Iberian Peninsula.
A qualitative reasoning with regard to vertical motions and the related cyclogenetic or
anticyclogenetic effects using the advections alone is impossible in this case, and it would be
necessary to compute both terms directly in order to know the sign of the resultant total
forcing. However, that can be more easily done by computing the Q vector divergence which
clearly indicates a forcing of ascent ahead of the trough and behind the surface front. And the
comparison between FQs and FQn shows that the ascending motion is mainly driven by a
direct circulation released by a strong frontogenesis in the wind field ahead of the trough and
covering the frontal zone of the cold front. In this case, there was indeed a broad band of deep
cloud masses mainly to the rear of the surface front.
Fig.6a: Weather situation from 23-02-89, 00 UTC. Top: Absolute topography 500 hPa
with isohypses (solid lines), isotherms (broken lines) and surface and surface front.
Middle row: Vorticity advection 500 hPa. Bottom: Temperature advection 700 hPa
Fig.6b: Frontogenesis parameter (-Qn) at 500 hPa
Fig.6c: FQ, FQn and FQs at 500 hPa (from left to right)
As final example, Fig.7 contains the complete Q vector diagnosis for the mature stage of a
cyclonic development. The conventional analyses show a warm sector cyclone near surface
between a short-wave trough and the ridge downstream at 500 hPa. That is the typical
position for an intensification of the cyclone.
The frontal zone at 500 hPa runs north of the surface low with a warm bulge just above it. The
Q vectors are greatest in the frontal zone and their component parallel to the isotherms is
dominating. However, components across the isotherms are also significant in some places.
From the change of the direction, Q vector convergence north-east of the surface low is
immediately visible, indicating a forcing of ascent there as depicted in the distribution of FQ.
The ascent also covers the area above the centre of the low, whereas west of it a strong
forcing of descent results. According to the convergence being effective below the area of
ascent and the divergence below the area of descent, a further intensification of the surface
cyclone, but also a quick movement in easterly direction can be expected.
In the framework of the expanded Q vector diagnostics we see that with regard to the forcing
of ascent the contribution by FQs is the dominant one. That is due to the fact that the vorticity
maximum at 500 hPa lies west of the surface low so that there is positive advection of
vorticity by the thermal wind or – considered relative to the movement – warm advection by
the relative winds just above the low and its immediate fore part. The circulatory part FQn is
generally weaker, but significantly influences the total forcing in some places. That is
especially true for the descent in the cold frontal zone behind the low which results from an
Fig.7a: Weather situation from 20-10-1986, 12 UTC. L.h.s.: Surface map with isobars
(solid Lines), relative isohypses 500/1000 hPa (broken lines) and surface fronts.
R.h.s.: Absolute topographie 500 hPa with isohypses (solid lines), isotherms (broken
lines) and distribution of high cloud according to the IR-satellite image
Fig.7b: Q vector diagnostics 500 hPa with geostrophic wind and isotherms,
Q vectors and isotherms, frontogenesis parameter (-Qn), omega forcing
contributions by FQs and FQn and total omega forcing FQ.
indirect circulation(with ascent of the colder and descent of the warmer air). This circulation
is due to frontolysis in this part of the frontal zone as indicated by the cross-components of Q
pointing towards the colder air and by the distribution of the frontogenesis parameter (-Qn).
The latter also shows a frontogenetic effect in the warm frontal zone ahead of the low causing
a direct circulation. The ascending branch of this circulation supports the lifting of the air in
the warm bulge of the isotherms due to FQs whereas the forcing for descent is overcompensated by the other effect. Frontolysis and frontogenesis result from the fact that the horizontal
winds are strongest in the warm air and the component of the winds transverse to the
isotherms therefore increase across the cold part of the frontal zone and decrease across the
warm part – seen in the direction of the movement of the frontal zone. This constellation is
rather typical for the mature stage of a cyclogenesis.
As a whole the omega forcing has a comma-like shape with the strongest ascent north-east of
the low extending far backwards to the north-west and south-west, and a tongue of descent
immediately upstream of the centre of the cyclone. Remembering the fact that the air in these
upper levels undergoes a relative motion from the rear to the fore part of the cyclone, the
distribution is in accordance with the IR-satellite image of that date which is characterised by
the conspicuous tongue of dry air aloft reaching from the rear up to the position of the surface
centre.
References:
Hoskins,B.J., Draghici,I., Davies,H.C.(1978): A new look at the ω equation. Quart. J. Roy.
Meteor. Soc. 104, 31-38
Sanders,F. and Hoskins,B.J.(1990): An Easy Method for Estimation of Q-Vectors from
Weather Maps. Weather and Forecasting 5, 346-353
Pedder,M.A.(1997): The omega-equation: Q-G interpretations of simple circulation features.
Meteorol.Appl. 4, 335-344
Kurz,M.(1998): Synoptic Meteorology. Training Guidelines of the German Meteorological
Service Nr.8. Deutscher Wetterdienst Offenbach a.M.