Beta-plane Hydraulics of the Northern Hemisphere Jet

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Beta-plane Hydraulics of the Northern Hemisphere Jet
Beta-plane Hydraulics of the
Northern Hemisphere Jet Stream
A diploma thesis submitted to the
Department of Meteorology and Geophysics,
University of Innsbruck
for the degree of
Magister der Naturwißenschaften
(Master of Science)
presented by
Michael Riffler
March 2005
Abstract
Blocking is the breakdown of a strong zonal atmospheric flow into a quasi-stationary
meridional current. The primary synoptic feature is a stable and strong anticyclone,
mainly found in two distinct regions: over the East Pacific and the Euro-Atlantic.
It was discovered nearly one century ago due to its impact on the climate in the
affected areas. Despite numerous efforts to identify the mechanisms causing this
stable flow configuration, no generally accepted explanation has been found yet.
Rossby (1950) treated blocking analogous to a hydraulic jump in open-channel
flow. Armi (1989) extended this theory for the beta-plane. The hydraulic analysis is
an integral approach, which retains the essential nonlinear aspects of the flow. The
aim of this study is the verification of this theory with the help of European Centre
for Medium-Range Weather Forecasts (ECMWF) T511 analysis and ERA-40 reanalysis data. For simplicity data at the 300 hPa level are used. Correctly, isentropic
levels need to be used, since this ensures the conservation of energy, momentum
and mass.
In the first part, a new dynamical index is presented that describes the behavior
of the Pacific jet stream throughout the winter. Compared to other measures, like
geopotential height indices using nearly fixed mean latitudes to build up differences,
this index encompasses the seasonal variability in the latitudinal and longitudinal
position of the jet stream.
The second part deals with the verification of the hydraulic jump theory. February/March 1995 and 2003 with a similarly strength of the Pacific jet stream but
different blocking frequency over North America are compared to each other. The
distinguishing factor is the state of the flow in the section downstream of the block
(over the Atlantic). Once this part exhibits stable supercritical flow conditions, the
upstream block can form. This finding is in agreement with open-channel hydraulic
theory which states that the conditions for a jump are determined by a control
acting further downstream. Nonetheless, the data used for the study is not fully
consistent with the theory.
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Zusammenfassung
Blocking ist durch den Übergang von einer zonalen in eine quasi-stationäre meridionale Strömung gekennzeichnet. Primäres Merkmal ist eine außergewöhnlich stabile
Anticyclone, die bevorzugt in zwei Regionen auftritt: im Ostpazifik und im EuroAtlantischen Raum. Aufgrund seiner Auswirkungen auf das Wetter und Klima
wurde dieses Phänomen erstmals vor rund 100 Jahren bemerkt. Seit Beginn wurden
die unterschiedlichsten Hypothesen zur Entstehung von Blockinglagen aufgestellt,
doch bis heute gibt es noch keine allgemein anerkannte Theorie.
Rossby (1950) versuchte mit seiner Theorie Blocking analog zum hydraulischen
Sprung in einem Kanal zu erklären. Armi (1989) entwickelte dieses Prinzip auf
einer β-Ebene weiter. Die hydraulische Betrachtung der Strömung ist ein integraler
Ansatz, d.h. nach Integration über den Jetstream bleiben wichtige nicht-lineare
Aspekte der Strömung erhalten. Das Ziel dieser Studie ist die Verifikation der Theorie anhand von European Centre for Medium-Range Weather Forecasts (ECMWF)
T511 Analyse- und ERA-40 Reanalysedaten. Der Einfachheit halber werden Daten
auf der 300 hPa Fläche verwendet. Um Energie-, Massen- und Impulserhaltung zu
berücksichtigen, müssten jedoch isentrope Daten verwendet werden.
Der erste Teil behandelt einen neu entwickelten dynamischen Index, der
unabhängig von der geographischen Lage des Jetstreams Auskunft über den
Strömungszustand gibt. Dies ist eine Verbesserung gegenüber sogenannten ”height
indices”, welche nur durch die Differenzenbildung des Geopotentials zwischen zwei
bestimmten, meist fixierten, geographischen Breiten gebildet werden.
Im zweiten Teil wird die hydraulische Theorie verifiziert. Hierfür werden
Februar/März 1995 und 2003 verglichen, da diese Zeiträume einen ähnlich starken
Pazifikjet, jedoch unterschiedliche Häufigkeiten von Blocking aufweisen. Der bestimmende Faktor ist der Strömungszustand stromabwärts des Blocks (über dem
Atlantik). Sobald dieser eine stabile überkritische Strömung aufweist, kann sich
stromaufwärts ein Block bilden. Dies stimmt mit der hydraulischen Theorie in
einem Kanal überein, welche besagt, dass die Bedingungen für einen hydraulischen
Sprung durch einen stromabwärts gelegenen Einflussfaktor bestimmt werden. Dennoch muss einschränkend hinzugefügt werden, dass die verwendeten Daten nicht zur
Gänze mit der Theorie konvergieren.
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Contents
Abstract
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Zusammenfassung
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Contents
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1 Purpose and Motivation
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2 Literature review and hydraulic jump theory
2.1 Literature in general . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Identification of a block - Climatologies . . . . . . . .
2.1.2 Theories about the development of blocking . . . . .
2.1.3 Predictability of blocking . . . . . . . . . . . . . . . .
2.2 Hydraulic theory . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Hydraulic jumps in open channel flow . . . . . . . . .
2.2.2 The hydraulic jump theory for atmospheric blocking .
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3 Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream”
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The hydraulic theory for the atmospheric jet stream and blocking . .
3.2.1 Hydraulic jumps . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Data and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Data used . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Hydraulic analysis . . . . . . . . . . . . . . . . . . . . . . . .
3.4 A detailed analysis of the winter 1994-95 with a strong Pacific jet . .
3.5 The Atlantic jet and the European block . . . . . . . . . . . . . . . .
3.6 Evolution of a block and the role of jumps . . . . . . . . . . . . . . .
3.6.1 Evolution of a block . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Jump results . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . .
3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
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References
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Appendix
A
Impact of the different datasets (T319/T511 vs. ERA-40) . . . . . .
B
Additional analyzed winter cases - Comment on the automatical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C
Mathematical Derivations . . . . . . . . . . . . . . . . . . . . . . .
C.1
Momentum equation . . . . . . . . . . . . . . . . . . . . . .
C.2
Froude/Rossby number Roβ . . . . . . . . . . . . . . . . . .
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Acknowledgement/Danksagung
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Curriculum Vitae
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Chapter 1
Purpose and Motivation
Blocking is the breakdown of a strong zonal atmospheric flow into a quasi-stationary
meridional current (Berggren et al., 1949). The primary synoptic feature is a stable
and strong anticyclone which extends throughout the troposphere (Schneider, 1996).
This flow pattern can persist for several days or even weeks leading to anomalous
weather conditions in the affected region (Rex, 1950a,b). The main interest in
blocking started in the beginning of the last century and was first noted by Garriott
(1904). In the Northern Hemisphere it is mainly found in two distinct regions: over
the North Pacific and the Euro-Atlantic. Therefore, numerous studies focused on
this topic and the mechanism which causes the stable flow configuration but non of
them is generally accepted yet. A good theory has to describe all the features of
such a block: the splitting of the jet stream, the persistence of the pattern and the
preferred geographical location.
The purpose of this study is the verification of a theory which describes blocking
analogous to a hydraulic jump in open-channel flow. This idea was first presented
by Rossby in 1950. Four decades later, Armi (1989) took Rossby’s initial thought
further and developed the beta-plane equivalent of the open-channel hydraulics.
The laboratory verification succeeded but at that time not enough upper air data
were available to test the theory on earth’s atmosphere. Testing the theory is now
possible using the extensive reanalysis and analysis sets of various global models, e.g.
European Centre for Medium-Range Weather Forecasts (ECMWF). A subset of the
ERA-40 reanalysis and T511 deterministic forecast analysis fields is used to develop
a new dynamical index describing the state of the jet stream. The hope is that
this index could support the prediction of blocked flow. Furthermore, longer term
forecasts might be eventually possible if integral aspects of the flow are reasonably
stable over time scales longer than the usual limit of about one week for computation
of the complete details of the flow.
A comprehensive survey of the literature about atmospheric blocking is found in
chapter 2. It describes the mechanisms to identify blocks used so far, climatologies
Purpose and Motivation
2
about the frequency and impact of blocking, theoretical studies trying to explain
the mechanism of blocking and the skill of general circulation models (GCM) in
predicting the stable flow pattern. A general introduction to the theory of hydraulic
jumps in open-channel flow and in the atmosphere is included as well.
Chapter 3 is the main part of this work, the draft of an article to be submitted to
the Quarterly Journal of the Royal Meteorological Society. This article presents the
preliminary results of the hydraulic analysis for a number of Northern Hemisphere
winters. A short summary of the blocking literature can be found as well as a short
description of the hydraulic theory. It also includes a brief description of the data
used for this study.
Some extra details about the impact of the different datasets used for the study
can be found in the appendix. It further contains figures of all the other winter
periods which are not described in the article. At the end, detailed derivations of
two main equations in the hydraulic theory can be found.
Chapter 2
Literature review and hydraulic
jump theory
The draft version of the article in chapter 3 contains a short review of the literature
about atmospheric blocking as well as a summary of the hydraulic background.
Nevertheless, the brevity required of an article necessitated to skip some details.
Therefore, a full literature review will be found in section 2.1. A comprehensive
overview of the hydraulic treatment of flow in an open channel and on a beta plane
is presented in section 2.2.
2.1
Literature in general
Berggren et al. (1949) describe blocking as the breakdown of a strong zonal atmospheric flow into a quasi-stationary meridional current. In the AMS Glossary
(Geer, 1996) one can find the following description for blocking: ”The obstructing,
on a large scale, of the normal west-to-east progress of migratory cyclones and anticyclones. A blocking situation is attended by pronounced meridional flow in the
upper levels, often comprising one or more closed anticyclonic circulations at high
latitudes and cyclonic circulations at low latitudes (cut off highs and cut off lows).
This anomalous circulation pattern (the block) typically remains nearly stationary
or moves slowly westward, and persists for a week or more.” And finally in the Encyclopedia of Climate and Weather (Schneider, 1996) the phenomenon of blocking is
explained as ”a persistent weakening or even reversal of the mid-latitude westerlies
over a broad region along with a strong meridional (north-south) flow around the
same region.” The primary synoptic feature of a block is a strong quasi-stationary
warm anticyclone which extends throughout the troposphere. This blocking high
changes the wind fields in the upper troposphere significantly shifting the main jet
stream axis 20 to 30 degree poleward (Schneider, 1996). In many cases a second jet
Literature review and hydraulic jump theory
4
around the subtropical flank of the blocked region can be found.
2.1.1
Identification of a block - Climatologies
Historically, there have been two main approaches to the identification of a block.
One is based on the subjective blocking criteria established by Rex (1950a), which
are described in the next paragraph. The second approach uses dynamical concepts
to describe the state of the flow configuration.
Height Indices
The main interest in blocking started in the late 1940s, when better aerological data
became available and gave rise to more detailed synoptical studies. During this
time, Rex (1950a,b; 1951) compiled a unique and much quoted climatology of 112
blocking cases for the Atlantic and the Pacific. In these studies, he established a
much used definition of blocking. A block has to fulfill the following characteristics:
a.) westerly current must split into two branches
b.) each branch must transport an appreciable mass
c.) the double-jet system must extend over at least 45◦ of longitude
d.) sharp transition from zonal type flow upstream to meridional type flow downstream
e.) the pattern must persist with recognizable continuity for at least ten days
Especially the criteria mentioned in c.) and d.) are somewhat arbitrary values. In
this climatology Rex discusses major climatological characteristics during blocked
flow over Europe. The main feature is, that the storm tracks over Europe are
distorted to the North and South. Therefore, widespread areas over whole Europe
are drier than usual and only few spots over Scandinavia show higher than average
precipitation amounts. He also describes the prevailing type of precipitation due
to blocking, which is mainly of orographic or convective manner. Another feature
is that warm air advection takes place in the northwest and north of Europe with
surface temperature anomalies up to +12 K over East-Greenland. The remaining
parts of Europe are mostly under the influence of cold air advection and show
negative temperature anomalies, which is caused by the advection of continental
or polar air masses with north-easterly or easterly winds. Furthermore, Rex found
the maximum blocking activity between 30◦ W to 0◦ E for the Atlantic (peaking at
10◦ W) and between 170◦ to 130◦ W over the Pacific Ocean (peaking at 150◦ W).
Literature review and hydraulic jump theory
5
Based on the Rex-criteria, Lenjenäs and Økland (1983) performed a 30 years
climatological study where they use a so-called height index to identify blocking situations. A height index (or zonal index) is simply the geopotential height difference
at two certain latitudes, in this case the difference in the 500 hPa geopotential height
between 60◦ and 40◦ N. A blocked region is specified (after Lenjenäs and Økland)
with a negative index for at least 30 degrees of longitude. Furthermore, they do not
constrain the number of days a block has to exist (cf. last Rex-criterion). However,
it is mentioned that the longer the duration of the blocking, the more it is concentrated around certain longitudes. In contrast to Rex’s study, this concentration is
shifted to the west with the peak of maximum activity between 160◦ E to 160◦ W.
The Euro-Atlantic region is similar to the region found by Rex with the highest
blocking frequency between 20◦ W to 20◦ E. Additional results are presented for the
prevailing months of blocking, which are found from February through March over
Europe and in January over the Pacific. Tibaldi and Molteni (1990) and Tibaldi et
al. (1994) apply the same criteria as Lenjenäs and Økland (1983) but allow slight
variations in the latitude for the computation of the height index which can be
written as follows:
Z(φ0 ) − Z(φS )
GHGS =
(2.1)
φ0 − φN
Z(φN ) − Z(φ0 )
GHGS =
(2.2)
φN − φ0
where φN = 80◦ N + ∆, φ0 = 60◦ N + ∆, φS = 40◦ N + ∆ and ∆ = −4◦ , 0◦ or 4◦ .
A longitude is then defined as blocked at a specific instant of time if the following
conditions are satisfied for at least one value of ∆:
(1) GHGS > 0,
(2) GHGN < −10m/deg lat.
Chen and Yoon (2002) used NCEP-NCAR reanalysis data for the period from
1957 to 1997. Their intention was to show that the North Pacific blocking activity
had undergone a noticeable interdecadal change over the past four decades. They
recognized an increase in blocking days and a shift to the east of blocking highs (6.5
days and 8.7◦ longitude within 40 years). In order to identify blocked flow, they
used a similar method as Lenjenäs and Økland (1983). It is further mentioned that
this eastward shift in the blocking patterns could have implications to the climate
in the affected areas. Since the major precipitation in the Northwest Pacific occurs
during the winter season, the large scale subsidence due to the blocking high may
explain the decreasing precipitation trend and warmer than usual temperatures over
the past four decades in this region.
One major problem with the definition of a height index is the (nearly) fixed
latitude which is used to build the geopotential height difference. As it is shown
Literature review and hydraulic jump theory
6
Figure 2.1: Schematic illustration of the potential vorticity (PV) index. The solid line
displays a contour line of constant potential temperature (Θ) on a surface of constant PV
= 2 PVU. φ0 is the central blocking latitude, Θnorth the mean potential temperature for
the northern part, Θsouth for the southern part (from Pelly and Hoskins 2003a).
in other studies (e.g. Pelly and Hoskins, 2003a) and in the following chapters of
this work, the jet stream and the associated eddy kinetic energy (EKE), which
represents the synoptic activity of the storm tracks, varies with latitude and season
(cf. Fig. 5 in section 3.4). Thus, a fixed latitude is not an appropriate method for
the computation of a zonal index and dynamical indices are more accurate for the
identification of blocked flow.
Dynamical Indices
Rossby (1950) came up with the idea to treat atmospheric blocking analogous to a
hydraulic jump in open channel flow. Armi (1989) took these thoughts further and
derived a Froude/Rossby number (similar to the Froude number in open-channel
hydraulics) dividing the flow in a super and subcritical part. Blocking is a typical
subcritical flow, whereas the strong westerly jet stream is supercritical (more about
this theory in section 2.2 and chapter 3).
Another dynamical way of describing blocking is presented by Pelly and Hoskins
(2003a). They use differences of potential temperature along a potential vorticity
(PV) surface. As it is illustrated in Fig. 2.1, their PV theory poses blocking as a
wave breaking phenomenon where subtropical air with high potential temperature
is shifted poleward with the blocking anticyclone. Further downstream, polar air
with low potential temperature is brought to the south and a dipole-like flow pattern
builds up. By measuring the differences in the potential temperature blocking can be
identified as a positive potential temperature anomaly. An advantage of this study
is also the use of a variable blocking latitude as described above. After evaluating a
Literature review and hydraulic jump theory
7
five year period it turns out that the blocking frequency is highest during summer
and autumn in the Euro-Atlantic region, the Pacific shows a distinct maximum
during summer. They also found seasonal variations in the longitudinal position of
the highest blocking frequency, especially for the Pacific. While during summer the
peak is found around the date line (180◦ W), all other seasons exhibit a maximum at
125◦ W, which is consistent with older synoptical studies form the 1950s. The EuroAtlantic region shows only slight seasonal variations. However, this is in contrast to
the results found by Lenjenäs and Økland (1983) and Tibaldi and Molteni (1990).
These height index studies show a pronounced maximum during winter and the
Pacific maximum farther west in the vicinity of the date line or even west of it,
which is the sector of the strongest mean westerly tropospheric jet stream. Pelly and
Hoskins (2003a) demonstrate that this behavior is primarily due the fixed blocking
latitude. Not only the geographical location is different from the results gained with
the dynamical index, but also the absolute frequency differs from the zonal index.
The PV-index identified more blocking situations than the height index. This can
be explained with the occurrence of Omega-blocks which are not really captured by
the height indices (compared to the high over low blocks with the dipole structure).
Moreover, the height indices also capture features which cannot be explained as
blocking situations, for example low pressure centers on the northern flank of the jet
(Pelly and Hoskins, 2003a). A comparison between the Atlantic and the Pacific ends
up with a higher blocking frequency in the former region, as it has been described
in older investigations.
2.1.2
Theories about the development of blocking
For more than half a century the development and maintenance of blocked flow have
been debated but a general accepted theory is still lacking. A good and satisfactory
theory has to describe all the features of such a block: the splitting of the jet stream,
the persistence of the pattern and the preferred geographical location.
The first theory to be mentioned was formulated by Berggren et al. (1949)
and Rossby (1950), which is also the theory investigated in this work. It treats
atmospheric blocking analogously to a hydraulic jump in open-channel flow. A
detailed discussion is deferred to section 2.2.
Egger (1978) uses quasi-geostrophic inviscid channel flow on a β-plane to investigate the dynamics of blocking highs. The model has low spectral resolution in
the zonal direction and includes orographic forcing as well. Both, barotropic and a
two-level baroclinic flow are evaluated. In his experiments he shows that blocking
can be explained as non-linear interactions between waves of low phase speeds and
orographically forced standing waves. Therefore, geographically fixed forcing seems
Literature review and hydraulic jump theory
8
to be a necessary condition for the formation of a block. This is in agreement with
the preferred geographical locations of atmospheric blocking and the model seems
to reproduce the main features of a block. Nevertheless, the model atmosphere and
topographic distribution used by Egger is extremely simple. Other physical process
like diabatic heating and frictional dissipation, which are important factors in the
decay of blocked flow, are neglected as well.
Charney and DeVore (1979) apply a barotropic channel model with a homogeneous β-plane atmosphere and a free surface. They argue that all the phenomena to be considered have their counterparts in a simpler barotropic atmosphere.
Therefore, questions involving topographic forcing, resonance and non-linear interaction via advective terms in the equations of motions may be studied barotropically.
Since the motions under investigation are of large scale, the flow is considered quasigeostrophic and governed by conservation of potential vorticity. With the help of
this simple barotropic model they show that more than one stable, steady solution
is possible for a given external forcing. The forcing is due to zonal flow forced over
topography or by thermal asymmetries of the lower troposphere. The types of flow
to be observed are a high index flow with strong zonal winds and small amplitudes
in the eddies, as well as a low index flow with large amplitudes and weaker zonal
component. Between these two stable states there has to be a transition from one to
the other, as it is observed in the real atmosphere. They hypothesize that the two
flow configurations are not entirely stable, indeed, they are baroclinically unstable
to smaller scale perturbations. They further hypothesize that these instabilities are
causing an additional forcing which drives the flow system from the vicinity of an
attractor basin of one equilibrium state into that of another. Two years later, Charney et al. (1981) tried to verify the barotropic blocking theory described above with
the help of atmospheric observation. Some minor modifications are made to the
model used in the former study. In order to identify blocked flow in the atmosphere
and model they use geopotential height anomalies which is different to the studies
mentioned before. Nonetheless, this is still a semi-objective criterion, since it can
not a priori be assumed that the height anomaly is due to blocked flow or weaker
(stronger) low (high) pressure centers (e.g. Aleutian Low, Island Low, Northwest
Pacific High). With their model, they reproduce 19 of the investigated 34 blocking
cases. Despite this effort, they are not able to explain the regional character of
blocked flow, which is a main characteristic of blocking. It is suggested that the
inclusion of longitudinal variations of external forcing and dissipation may account
for this important property. The second phenomenon which can not be explained is
the mechanism of transition from normal to blocked situations and vice versa.
The hypothesis put forth in Charney and DeVore (1979) that blocking may be
unstable to travelling waves was tested in Charney and Straus (1980). According to
Literature review and hydraulic jump theory
9
this work, blocked flow can be produced by a thermal driving in a two layer baroclinic
model. The resulting stable flow pattern is unstable to propagating waves with the
same zonal wave number.
Tung and Lindzen (1979) try to explain blocking as a linear, resonant amplification of planetary-scale waves. The forcing in their theory is due to topography
and surface heating and the waves in the atmospheric flow can interact resonantly
with the stationary forcing. They further suggest that during years with normal
circulation (little blocking) there exists a significant destructive interference in the
planetary waves between the forcing due to topography and due to land-sea differential heating. Hence, when the usual sea surface temperature distribution is altered,
the two forcing mechanisms may act in harmony to reinforce instead of cancel each
other. The advantage of this mechanism, if proven correct, could have a predictive
value in the sense that if an abnormal ocean surface temperature distribution is
observed during fall, one can reasonably state that the likelihood for the occurrence
of major large-scale blocking is higher than during normal years. This is in contrast
to the results found by Mullen (1989). He uses the National Center for Atmospheric
Research (NCAR) Community Climate Model (CCM) to study Pacific sea surface
temperatures (SST) anomalies and their impact on blocking in the Pacific region.
According to his paper, these anomalies do not affect significantly the total number
of blocking days over the North Pacific (in the model) but apparently can affect
the preferred region where blocking ridges are likely to occur. A warm tropical
SST anomaly enhances the blocking activity over the NE-Pacific and suppresses it
over the central N-Pacific, whereas a cold anomaly tends to suppress blocking over
the central Pacific. The combination of both anomalies is more effective in modifying blocking action over the mid-latitudes of the North Pacific than either SST
anomaly alone.
There are more recent studies (e.g. Hoskins, 1997) which describe blocking as
a breaking wave of potential temperature. As described in the previous section, the
ingredient for a block after this theory is a substantial mass of tropical air (with its
low PV on a given Θ-surface) which is advected poleward ahead of a large amplitude
slowly moving cyclone. Similar to a PV anomaly, this air develops its own anticyclonic circulation building a cut off high. Consequently, this anticyclone influences
the upstream weather systems to elongate meridionally and deposit more low PV
air on the poleward side and high PV air on the equatorward side, thereby acting to
reinforce the block. An important factor for the decay of such stationary and stable
anticyclones are diabatic heating effects. For example, above a region of latent heat
release the PV is decreased and below it the PV is increased. Thus, the latent heating in a mid-latitude low pressure center can lead to significant enhancement of the
low level cyclonic circulation and of the upper ridge in the air moving ahead of it.
Literature review and hydraulic jump theory
2.1.3
10
Predictability of blocking
Since blocking situations often last a week or more and bring persistent weather
in most of the affected areas, there is a substantial interest in the skill of Global
Circulation Models (GCM) in predicting atmospheric blocking. One main problem is that since no generally accepted theory exists explaining the developing and
maintaining mechanisms of blocking, their correct implementation in the models
cannot be tested. This uncertainty carries over to the skill of forecast models to
predict the onset and decay of blocking, which was studied by Tibaldi and Molteni
(1990), Tibaldi et al. (1994); D’Andrea et al. (1998), Pelly and Hoskins (2003b)
and Mauritsen and Källén (2004).
Tibaldi and Molteni (1990; Tibaldi et al., 1994) as well as Pelly and Hoskins
(2003b) investigated the ability of the ECMWF operational forecast model. The
former study discusses the deterministic forecasts and the latter compares the Ensemble Prediction System (EPS; Molteni et al., 1996) with the deterministic model.
Both studies mention that the GCMs tend to a climate of their own (e.g. ECMWF
with a westerly bias) with increasing integration time and clearly underestimate the
frequency of blocking . In 1990 a skillful deterministic forecast for blocked flow was
possible for three to four days in advance, in 2003 for about five days. A skillful
forecast means a forecast which is better than predicting the climatological possibility of blocking for a certain day. The above mentioned skills are valid for cases
where blocked flow is already apparent in the initial conditions. The prediction of
the onset and decay is less successful, whereas the latter can be predicted one day
earlier than the former. This reflects the problem of the model with the transition
from zonal flow to a stationary blocked pattern. In such a case the transition is often
too slow. An additional problem is the duration of blocking which is also underestimated. Pelly and Hoskins (2003b) found out that with the use of probabilistic
forecasts, namely the EPS, one is able to make skillful forecasts of blocked flow up
to ten days. Again, the prediction of the onset and decay is less reliable with six
and seven days respectively. This is a major improvement, but one problem with
the study is that they use only one year of available data and during this period two
major changes in the EPS took place. Mauritsen and Källén (2004) also focused on
the EPS forecast skill and used a similar index as Charney and DeVore (1979) in
order to identify blocked flow. They show that during the period from 1 December
2000 to 28 February 2001 the EPS reveals 30 percent fewer blockings than in the
analysis. This is, however, a major improvement compared to the 50 percent under
prediction found by Tibaldi and Molteni (1990) in earlier versions of the ECMWF
model.
D’Andrea et al. (1998) present a general study about 15 GCMs and their
behavior in the prediction of blocking. Nine years of forecasts are verified against
Literature review and hydraulic jump theory
11
Figure 2.2: Comparison of longitudinal blocking frequency during December, January
and February for 15 GCMs. The dashed line (a) is the observed blocking frequency used
for comparison, obtained from ECMWF analyses. The different forecast models are displayed form b to r, with following abbreviations: b) CCC (Canadian Climate Centre),
c) COLA (Center for Ocean-Land-Atmosphere Studies), d) CSIRO (Commonwealth Scientific and Industrial Research Organization), e) CSU (Colorado State University), f)
DERF/GFDL (DERF model of Geophysical Fluid Dynamics Laboratory), g) ECHAM
(Max-Plank-Institut für Meteorologie), h) ECMWF (European Centre for Medium-Range
Weather Forecasts), i) JMA (Japan Meteorological Agency), l) METEOFR (Centre National de Recherches Météorologiques), m) MRI (Meteorological Research Institute), n)
NCAR (National Center for Atmospheric Research), o) NMC (National Meteorological
Center), p) RPN (Recherche en Prévision Numérique), q) UGAMP (UK Universities
Global Atmospheric Modelling Programme), r) UKMO (Hadley Centre for Climate Prediction, UK Meteorological Office). (from D’Andrea et al., 1998)
ECMWF analyses. The models use a wide spectrum of horizontal and vertical
resolutions, numerical techniques and physical parameterizations. With a modified
version of the Tibaldi and Molteni (1990) index, they assess a climatology and
verify the skill of the 15 different models. The results are displayed in Fig. 2.2.
Despite the large variety of the models, a few common features are found such as
the tendency to underestimate both the total frequency and average duration of
blocked flow. A dependency of this problem on the models horizontal resolution is
not evident. However, if the attention is limited to grid-point models only (UKMO,
MRI, CSU), the UKMO model performs better than the other two models of lower
Literature review and hydraulic jump theory
12
resolution. There are also some cases of model overestimation, especially in the West
Atlantic region. The CSIRO shows an excessive peak around 35◦ W and only few
blocking in usual Euro-Atlantic region. This peak could therefore represent a strong
westward shift of Euro-Atlantic blocking and is connected with a large amplitude of
the Greenland ridge at the eastern edge of the North American continent in certain
models. In general, many models tend to shift the Euro-Atlantic blocking to the
east of the observed position, whereas no such general tendency is found for the
Pacific region. Among the better models with respect to reproduce the annual cycle
of blocking frequency are the UKMO, ECHAM and NCAR; the ECMWF is found
in the middle.
One main problem with the studies about the predictability of blocked flow
with the help of GCMs is that they all use different indices for the identification of
blocking situation. In this context, especially the works using hight indices should be
regarded carefully. As explained before, this type of index is somewhat problematic
for the determination of blocking.
2.2
Hydraulic theory
Hydraulic jumps are a well known phenomenon for hydraulic engineers. They are
used for example to dissipate energy in water flowing over dams, weirs and other
hydraulic structures (Chow, 1959). The application of hydraulic jump theory to
the atmosphere is also well established for flow over obstacles, like hills or even
mountain ranges. The violent turbulent motion in the lee of a mountain ridge is
described as a hydraulic jump in the atmosphere (Prandtl, 1944). Before considering
hydraulic jumps in the atmosphere one should first think about the phenomenon and
its energetics in an open channel.
2.2.1
Hydraulic jumps in open channel flow
A stationary hydraulic jump in a rectangular open channel can be seen in Fig. 2.3
showing the cross section through an open channel. Across the jump, certain criteria
have to be fulfilled (Acheson, 1992; Henderson, 1966):
1. conservation of mass
h1 U1 = h2 U2 = q
h...depth of the fluid
U ...fluid velocity
q...discharge per unit width of the channel
(2.3)
13
Literature review and hydraulic jump theory
Figure 2.3: (a) Schematic illustration of a hydraulic jump with h1 the depth and U1 the
velocity of the fluid at section A, h2 and U2 at section B (U1 > U2 ). (b) demonstrates
the course of momentum (conserved) and specific energy (loss) through the jump (see Eq.
2.4 and 2.5).
2. conservation of momentum (more detailed derivation in Appendix C)
M=
h21 h1 U12
h2 h2 U22
+
= 2+
2
g
2
g
(2.4)
g...gravitational acceleration
3. loss of energy owing to the turbulence within the jump.
The equation for the energy is the specific energy of an open channel flow and can
be written as follows:
U2
(2.5)
E = gh +
2
or with the use of q from Eq. 2.3
E = gh +
q2
.
2h2
(2.6)
At this point the Froude number is defined as
U
Fr = √ ,
gh
(2.7)
the ratio of the stream velocity to the velocity of gravity waves along the top of
the fluid (or inertia force to gravity force; Kundu and Cohen, 2002; Sabersky et
al., 1999). A flow is called subcritical with F r < 1 and supercritical with F r > 1.
14
Literature review and hydraulic jump theory
5
M′
E′
4.5
dim.less specific energy and momentum
4
3.5
3
2.5
∆E
2
A
B
1.5
1
0.5
0
0
0.5
1
1.5
2
Fr
2.5
3
3.5
4
Figure 2.4: Course of specific energy and momentum across a hydraulic jump. Displayed are the non-dimensional specific energy (E 0 ) and momentum (M 0 ) as a function
of the Froude number. Point A is in supercritical flow upstream of the jump, point B
in subcritical flow downstream of the jump (cf. Fig. 2.3). ∆E is the energy loss due to
turbulence in the hydraulic jump.
If the flow is subcritical, disturbances or gravity waves are able to propagate upstream, whereas in a supercritical flow they cannot. Consequently, subcritical flow
is subject to up and downstream control and supercritical flow to upstream control
only. Figure 2.4 helps to understand the energy loss through the hydraulic jump.
Displayed are the curves of the dimensionless specific energy E 0 and momentum M 0 .
The dimensionless form of Eq. 2.6 can be obtained by dividing through ghc , the
critical height (F r = 1), and using Eq. 2.3 and the Froude number (Eq. 2.7)
E
h
q2
=
+
ghc
hc 2gh2 hc
U2
q2
hc
F r2 =
= 3 = ( )3
gh
gh
h
1
E 0 = F r−2/3 + F r4/3
2
(2.8)
(2.9)
(2.10)
with E 0 = E/ghc .
The dimensionless form of the momentum equation can be derived by dividing Eq.2.4
through h2c and the use of F r from Eq. 2.9
1
M 0 = F r−4/3 + F r2/3
2
(2.11)
with M 0 = M/h2c . Figure 2.4 illustrates that the jump is only possible in one direction, from a high to a low Froude number. In order to explain this behavior, we
should remember the criteria to be fulfilled in such a jump. First, the conservation
15
Literature review and hydraulic jump theory
of mass. The second criterion can be seen in Fig. 2.4, if one starts at point A with
a high F r (supercritical flow) and regard the conservation of momentum, demonstrated with the horizontal line to point B (low F r and subcritical flow). If now the
energy is compared at point A and B, one can see a loss of energy ∆E. On the other
hand, if one would start with a subcritical F r, the conservation of momentum has
to lead to an increase in the specific energy, which would require external energy.
Types (strength) of hydraulic jumps
Since momentum is conserved across the jump, one can derive an equation for the
hydraulic jump with Eq. 2.4:
M1 = M2
h1 U12
h2 h2 U22
+
= 2+
2
g
2
g
h21
(2.12)
(2.13)
or in terms of discharge q = hU
h21
q2
h22
q2
+
=
+
2
gh1
2
gh2
¶
2 µ
³
´
1
q
1
1 2
−
=
h2 − h21
g h1 h2
2
2
q
1
=
(h2 + h1 )
gh1 h2
2
(2.14)
(2.15)
(2.16)
Now, the substitution q = U1 h1 results in
U12 h1
1
=
(h2 + h1 )
gh2
2
!
Ã
U12
1 h2 h2
=
+1
gh1
2 h1 h1
F r12 =
Ã !

2
1  h2
h2
+ 
2
h1
h1
(2.17)
(2.18)
(2.19)
Equation 2.19 is quadratic in h2 /h1 , the solution is given by
µ
¶
1 q
h2
=
1 + 8F r12 − 1 .
h1
2
(2.20)
Equation 2.16 can also be solved for q = U2 h2 with the resulting solution:
µ
¶
1 q
h1
2
=
1 + 8F r2 − 1 .
h2
2
(2.21)
Subscript 1 stands for the upstream and subscript 2 for the downstream conditions
of the hydraulic jump (see Fig. 2.3). Both, the depth ratio h2 /h1 and F r12 can be
seen as a measure of the strength of the jump (Henderson, 1966). In fact, hydraulic
Literature review and hydraulic jump theory
16
Figure 2.5: Various types of hydraulic jumps: a) Undular jump, b) weak jump, c)
oscillating jump, d) steady jump and e) strong jump.
jumps are classified into different categories with the help of the upstream Froude
number F r1 (Henderson, 1966; Chow, 1959). Figure 2.5 illustrates several distinct
types of jumps:
1. F r1 = 1.0, the flow is critical, hence no jump can form.
2. F r1 = 1.0 to 1.7, the jump shows undulations (unbroken waves) and is therefore called undular jump.
3. F r1 = 1.7 to 2.5, is the range of surface turbulence only, the wave is broken
and the energy loss is still low, thus the jump is called weak jump.
4. F r1 = 2.5 to 4.5 is called oscillating jump due to the upstream flow penetrating
the turbulent front as an oscillating jet and producing large irregular waves at
the surface.
5. F r1 = 4.5 to 9.0 is the region of a steady jump with large energy dissipation.
This sort of jump is well balanced and the position very stable.
6. F r1 > 9.0, a strong jump which produces rough surface waves and a very
effective mechanism for the dissipation of energy.
17
Literature review and hydraulic jump theory
Finally, it should be mentioned that equations 2.20 and 2.21 contain three independent variables. Therefore, it is necessary to know two of them in advance to compute
the third one. If for example the upstream conditions are known, the downstream
depth may be calculated. Nevertheless, the downstream depth is not caused by the
upstream conditions but rather from a control acting further downstream. If this
control produces the required depth h2 , a jump will form. This is a very important
fact to realize (Henderson, 1966).
2.2.2
The hydraulic jump theory for atmospheric blocking
One of the first applications of open channel hydraulics to the atmosphere was by
Berggren et al. (1949). They tried to explain atmospheric blocking as a hydraulic
jump in the atmosphere. In a more detailed study, Rossby (1950) picked up this
idea again and investigated similarities between aspects of the jet stream and open
channel flow hydraulics. He noticed that sometimes the well-defined zonal jet in the
mid-latitudes all of a sudden breaks down in the form of a blocking wave. If the wave
is large enough, a retrograde motion takes place destroying the zonal jet to the west
and forming a series of cyclones and anticyclones to the east. He further describes
this behavior as a hydraulic jump in the atmosphere. In his study, the vertical
component of open channel flow is replaced by the horizontal component, and gravity
waves are replaced by Rossby waves. In order to explain the mechanism slowing
down the jet in the blocking wave, Rossby starts with the pressure distribution in a
non-divergent geostrophic jet in an incompressible fluid
∂p
= −ρf0 u,
∂y
(2.22)
with a constant density ρ, which means that the fluid is barotropic, hence no temperature gradient does exist. u is the zonal wind speed and f0 the constant Coriolis
parameter. The jet is assumed to be symmetric with the origin of the coordinate
system in the center of the jet with half width a (see Fig. 2.6). At the outer boundaries of the jet (−a and +a) the zonal speed is zero. The pressure difference across
the jet is given by the integration from the southern (−a) to the northern (+a)
boundary
p−a − pa = ρf0 V,
(2.23)
where V is the volume transport per unit mass, which can be written as
V =
Z +a
−a
udy.
(2.24)
This holds for any shape of u(y). In Eq. 2.23 no existing mechanism is obvious,
being able to slow down the fluid. If the Coriolis parameter is expanded such that
18
Literature review and hydraulic jump theory
y
North
+a
East
0
u(y)
−a
Figure 2.6: Idealized cross section through a zonal jet stream.
the current is now considered on a Rossby β-plane (f = f0 + βy), the pressure
distribution can be written as follows:
p(y, a) = p(−a) − ρf0
Z y
−a
{z
|
0
0
u(y )dy −ρβ
Z y
}|
−a
u(y 0 )y 0 dy 0 .
{z
(2.25)
}
p0
p
Equation 2.25 consists of two parts, one part which is independent of the jet width
(p) and the second one, p0 , the pressure excess, which is not. For westerlies, p0 is a
maximum in the center of the jet and vanishes at the outer borders. Moreover, the
pressure excess increases with increasing jet width. Due to p0 , a negative pressure
gradient is exerted on a particle travelling along a streamline from west to east and
thereby will be decelerated.
Now, we want to consider the energy flux in the jet stream, which will be treated
similar to open channel flows (cf. Henderson, 1966; Armi, 1989):
G=
Z +a ³ 2
u
−a
u
2
|{z}
+
p ´
dy.
ρ
(2.26)
|{z}
KE
PE
The total energy G is given by the integration of the kinetic energy KE and potential
energy P E across the current from −a to +a (see Fig. 2.6). If p in P E is replaced
by Eq. 2.25, the total energy flux becomes:
G=
Z +a
1
|
−a
2
3
u(y) dy − β
Z +a
−a
{z
u(y)
Z y
−a
u(y 0 )y 0 dy 0 dy +
}
1
p−a V
− f0 V 2 ,
ρ
2
(2.27)
G0
with the specific energy flux G0 . For further computations a Froude/Rossby number
Roβ similar to the Froude number F r in open channel hydraulics is used, which
relates inertia forces to pressure forces arising from width variations of the zonal jet
stream (a more detailed derivation can be found in Armi, 1989):
R
+a 1 3
2 −a
u dy
2KE
=
Roβ =
R +a
R y2
PE
−β −a u(y) −a u(y 0 )y 0 dy 0 dy
(2.28)
19
Literature review and hydraulic jump theory
6
5
total
4
C
A
G′0
B
D
kinetic
3
2
potential
1
0
0
1
2
3
4
5
Roβ
Figure 2.7: Non-dimensional specific energy flux, G00 (solid line), for eastward flowing
zonal current. Contributions due to kinetic energy (dash-dotted) and potential energy
(dashed) are shown separately. The dotted vertical line separates subcritical flow (Roβ <
1) from supercritical flow (Roβ > 1). Loss of energy through (weak) dissipation brings
both supercritical flow (A → B) and subcritical flow (C → D) towards the critical state
(further explanation see text).
The non-dimensional specific energy flux for a zonal current in terms of Roβ is
(detailed derivation in Appendix C)
2/3
−1/3
G00 = Roβ + 2Roβ
,
(2.29)
and is displayed in Fig. 2.7. For an easier understanding of this figure it shall be
mentioned that for a symmetric velocity profile, as in Fig. 2.6, the Froude/Rossby
number can also be written as
Roβ =
u
α0 βa2
,
(2.30)
where u is the mean zonal wind speed and α0 a factor taking the shape of the velocity
distribution into account (Armi, 1989). Since Fig. 2.7 is valid for a fixed volume
flux, one can see that a flow with Roβ < 1 is broad and slow, whereas with Roβ
greater than unity it is a fast and narrow zonal jet.
Furthermore, the downstream effect of weak dissipation can immediately be seen
from this figure. With the conservation of the volume flux, the specific energy flux
(total) can only be reduced by frictional dissipation. Hence, an initial supercritical
flow will decrease Roβ from point A to B. Thus, the potential energy has to rise and
the kinetic energy is reduced, which is accompanied by a broadening of the jet and a
reduction in the zonal wind speed. In contrast, for an initially subcritical flow, Roβ
will increase from point C to D with an associated acceleration and narrowing of
Literature review and hydraulic jump theory
20
the jet (increase of kinetic energy and reduction of potential energy). In both case
Roβ tends toward unity, where the flow is critical.
Chapter 3
Draft of the article ”Beta-plane
Hydraulics of the Northern
Hemisphere Jet Stream”
3.1
Introduction
For more than half a century, the similarities between aspects of the jet stream and
open channel flow hydraulics have been noted. The first use of this analogy is due
to Rossby (1950) who suggested the hydraulic jump analogue to treat the blocking
phenomenon. This treatment was reviewed by Rex (1950a,b) in his celebrated paper
on blocking, in which a first climatology of these events can be found. The primary
advantage and disadvantage of the hydraulic treatment to be used here is that it is
an integral approach. The properties of the flow to be considered will be the total
transport, center latitude, transport of both the kinetic and potential energy of
the flow and associated width and representative speed. The advantage is that after
integration across the jet stream, essential nonlinear aspects of the flow are retained,
in particular conditions of the flow and whether the flow is sub or supercritical with
respect to a dimensionless Froude/Rossby number. Small scale structure of the
jet stream is however ignored in this approach. A full theoretical description with
laboratory experiments can be found in Armi (1989).
The hope is that with the integral hydraulic like approach longer term forecasts
might be eventually possible if integral aspects of the flow are reasonably stable over
time scales longer than the usual limit of about one week for computation of the
complete details of the flow. Testing the theory is now possible using the extensive
reanalysis data available from the European Center for Medium-Range Weather
Forecasts (ECMWF). The preliminary results of these analyses for a number of
Northern Hemisphere winters are the subject of this paper.
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 22
The integral concepts were developed many years ago; first for compressible flow
through a nozzle (Hugoniot, 1886, and Reynolds, 1886) and later for flows in an open
channel. These concepts have been reviewed in many texts either on gas dynamics
(cf. Liepmann and Roshko, 1957) or open channel flows (c.f. Henderson, 1966). For
zonal flow hydraulics, the width of the jet will take on the role of the depth of the
analogous open channel flow. Many dynamical concepts carry over, in particular
the importance of a dimensionless Froude/Rossby number relating a representative
zonal velocity of the jet and its width, analogous to the Froude number for open
channel flow. However, for zonal flows the flow direction is crucial since long Rossby
waves, by which information concerning critical controls and other aspects of the
flow is transported, can only propagate to the west. For eastward flowing jets such
as the westerlies, both subcritical and supercritical flows are possible in complete
analogy to open channel flows. The analogy does not carry over completely to
westward flows, since they are flowing in the same direction as all the long Rossby
waves and are in this sense supercritical (Armi, 1989).
The main aim of this study is to describe the state of the atmospheric jet stream
in a dynamical way. We want to present a more accurate method to determine the
flow characteristics and behavior than it is done by other zonal indices. Most of the
other indices (e.g. Lejenäs and Økland, 1983; Tibaldi and Molteni, 1990; Tibaldi et
al., 1994) are based on geopotential height differences which are fixed in latitude and
do not take variations in the latitudinal position of the jet stream into account. The
exemplary individual cases and summaries of variations during a whole winter period
will demonstrate that the position of the jet has substantial variations throughout
the year. A further goal will be the application to the blocking phenomenon. The
anomalous flow pattern can have a major impact on climate in the blocked regions as
it is described by Rex (1950a,b) in the first climatology about atmospheric blocking.
A dynamical description of blocking different from ours has been presented in
a recent study by Pelly and Hoskins (2003a). They use differences in the potential
temperature on a potential vorticity (PV) surface. The PV theory states blocking as
a wave breaking phenomenon, where air with high potential temperature is shifted
poleward with the blocking anticyclone. Further downstream, air with low potential
temperature is brought to the south and a dipole-like flow pattern builds up. By
measuring the differences in the potential temperature, blocking can be identified
as a negative potential temperature anomaly. An advantage of this study is also
the use of a variable blocking latitude as described above. Egger (1978) proposed
that blocking is a phenomenon due to non-linear interaction of forced waves and
slowly moving free waves. He uses quasi-geostrophic inviscid channel flow on a βplane to investigate the dynamics of blocking highs. Charney and DeVore (1979)
and Wiin-Nielsens (1979) show that a low-order, non-linear barotropic system, with
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 23
external and/or mountain forcing and dissipation may posses multiple stable states.
Topographical forcing can exhibit two totally different flow situations, namely a
high index flow with strong zonal winds and small amplitudes in the eddies, as
well as a low index flow with large amplitudes and weaker zonal component. In a
following study Charney et al. (1981) tried to verify their barotropic blocking theory
with atmospheric observations. However, one main feature of blocking, the regional
character, cannot be explained with their model. Tung and Lindzen (1979) try to
explain blocking as a linear, resonant amplification of planetary-scale waves. The
forcing in their theory is due to topography and surface heating and the waves in
the atmospheric flow can interact resonantly with the stationary forcing.
Since blocking situations often last a week or more and bring persistent weather
in most of the affected areas, there is a substantial interest in the skill of Global
Circulation Models (GCM) in predicting atmospheric blocking, as can be seen in the
studies of Tibaldi and Molteni (1990), Tibaldi et al. (1994) and recently by Pelly
and Hoskins (2003b), all of which investigated the skill of the ECMWF operational
forecast model. Pelly and Hoskins (2003b) found that with the use of probabilistic
forecasts one is able to make skillful forecasts of blocked flow up to ten days. The
beginning can be predicted six days in advance and the decay more than seven days.
3.2
The hydraulic theory for the atmospheric jet
stream and blocking
Aspects of the analogy between the jet stream system and open channel hydraulic
flow were first pointed out by Rossby (1950) and reviewed by Rex (1950) in consideration of blocking waves (cf. Berggren et al., 1949). Rossby describes this behavior
as a hydraulic jump in the atmosphere for which two dynamically possible states
may exist which are compatible with continuity and momentum conservation as
in an open channel hydraulic jump. The depth of the flow and associated long
gravity waves are replaced by the width of the jetstream and the associated waves
are now known as long Rossby waves. In order to explain the mechanism slowing
down the jet in the blocking wave, Rossby starts with the pressure distribution in a
non-divergent geostrophic jet in an incompressible fluid
∂p
= −ρf u,
∂y
(3.1)
with a constant density ρ, the constant zonal wind speed u and the Coriolis parameter f . The jet is assumed to be symmetric with the origin of the coordinate system
in the center of the jet and a half width a (see Fig. 3.1).The pressure difference
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 24
y
North
+a
East
0
u(y)
−a
Figure 3.1: Idealized cross section through a zonal jet stream.
across the jet is given by
ps − pn = ρf V,
(3.2)
where ps and pn are the pressure at the southern and northern boundary of the jet
and V is the volume transport per unit mass, which can be written as
V =
Z +a
−a
udy.
(3.3)
If the Coriolis parameter is expanded such that the current is now considered on a
Rossby beta-plane (f = f0 + βy) the pressure distribution can be written as follows:
p(y, a) = p(−a) − ρf0
|
Z y
{z
0
−a
0
u(y )dy −ρβ
Z y
−a
}|
u(y 0 )y 0 dy 0 .
{z
(3.4)
}
p0
p
p is independent of the jet width, whereas p0 , the pressure excess, is not. For
westerlies, p0 is a maximum in the center of the jet and vanishes at the outer borders.
Moreover, the pressure excess increases with increasing jet width.
Now, we want to consider the energy flux in the jet stream, which will be treated
as for open channel flows (cf. Henderson, 1966):
G=
Z +a ³ 2
u
−a
u
2
|{z}
KE
+
p ´
dy.
ρ
(3.5)
|{z}
PE
The total energy G is given by the integration of the kinetic energy KE and potential
energy P E across the current from −a to +a (see Fig. 3.1). If p in P E is replaced
by Eq. 3.4, the total energy flux becomes:
G=
Z +a
1
|
−a
2
3
u(y) dy − β
Z +a
−a
{z
u(y)
Z y
−a
u(y 0 )y 0 dy 0 dy +
}
1
p−a V
− f0 V 2 ,
ρ
2
(3.6)
G0
with the specific energy flux G0 . For further computations a Froude/Rossby number
Roβ similar to the Froude number F r in open channel hydraulics is used. It relates
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 25
6
5
4
G′0
total
3
kinetic
2
potential
1
0
0
1
2
3
4
5
Roβ
Figure 3.2: Non-dimensional specific energy flux, G00 (solid line), for eastward flowing
zonal current. Contributions due to kinetic energy (dashed) and potential energy (dashdotted) are shown separately. The dotted vertical line indicates the separation between
subcritical flow Roβ and supercritical flow Roβ .
inertia forces to pressure forces arising from width variations of the zonal jet stream
(a more detailed derivation can be found in Armi, 1989):
R
+a 1 3
2 −a
u dy
2KE
Roβ =
=
R +a
R y2
PE
−β −a u(y) −a u(y 0 )y 0 dy 0 dy
(3.7)
The non-dimensional specific energy flux for a zonal current in terms of Roβ is
2/3
−1/3
G00 = Roβ + 2Roβ
,
(3.8)
and is displayed in Fig. 3.2. For an easier understanding of this figure it shall be
mentioned that for a symmetric velocity profile, as in Fig. 3.1, the Froude/Rossby
number can also be written as
Roβ =
u
,
α0 βa2
(3.9)
where u is the mean zonal wind speed, and α0 a factor taking the shape of the
velocity distribution into account (Armi, 1989). Since Fig. 3.2 is valid for a fixed
volume flux, one can see that a flow with Roβ < 1 is broad and slow, whereas with
Roβ greater than unity it is a fast and narrow zonal jet.
Furthermore, the downstream effect of weak dissipation can immediately be
seen from this figure. With the conservation of the volume flux, the specific energy
flux can only be reduced by frictional dissipation. Hence, an initial supercritical
flow will decrease Roβ , accompanied by a broadening of the jet and a reduction in
the zonal wind speed. In contrast, for an initially subcritical flow, Roβ will increase
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 26
with an associated acceleration and narrowing of the jet. In both cases Roβ tends
toward unity, where the flow is critical.
3.2.1
Hydraulic jumps
Hydraulic jumps are a well known phenomenon for hydraulic engineers. They are
used for example to dissipate energy in water flowing over dams, weirs and other
hydraulic structures. We review some of the fundamental aspects of hydraulic jumps
because many will carry over to the analogue of the atmospheric block. Across a
hydraulic jump certain criteria have to be fulfilled: (1) the conservation of mass, (2)
the conservation of momentum and (3) the loss of energy owing to the turbulence
within the jump. With the help of the momentum function and the knowledge of
the conservation of momentum, one can derive an equation for the hydraulic jump
(Chow, 1959):
µ
¶
h2
1 q
2
=
1 + 8F r1 − 1 ,
(3.10)
h1
2
where h is the depth of the fluid and F r the corresponding Froude number. Subscript
1 (2) stands for the upstream (downstream) conditions. Both, the depth ratio h2 /h1
and F r12 can be seen as a measure of the strength of the jump (Henderson, 1966).
In fact, hydraulic jumps are classified into different categories with the help of F r1
(Chow, 1959; Henderson, 1966) as listed below:
1. F r1 = 1.0, the flow is critical, hence no jump can form.
2. F r1 = 1.0 to 1.7, the water shows undulations (unbroken waves) and is therefore called undular jump.
3. F r1 = 1.7 to 2.5 is the range of surface turbulence only, the wave is broken
and the energy loss is still low, thus the jump is called weak jump.
4. F r1 = 2.5 to 4.5 is called oscillating jump, due to the upstream flow penetrating the turbulent front as an oscillating jet and producing large irregular
waves at the surface.
5. F r1 = 4.5 to 9.0 is the region of a steady jump with strong energy dissipation.
This sort of jump is well balanced and the position very stable.
6. F r1 > 9.0 is a strong jump, which produces rough surface waves and a very
effective mechanism for the dissipation of energy.
Finally, it should be mentioned that Eq. 3.10 contains three independent variables. Therefore, it is necessary to know two of them in advance to compute the
third one. If for example the upstream conditions are known, the downstream depth
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 27
may be calculated. Nevertheless, the downstream depth is not caused by the upstream conditions but rather from a control acting further downstream. If this
control produces the required depth h2 , a jump will form (Henderson, 1966).
3.3
3.3.1
Data and analysis
Data used
The data used in this study are the geopotential and the wind field at 300 hPa from
1992 to 2003 for the Northern Hemisphere. Since we want to focus on the winter
periods of 1994-95 and 2002-03, which both exhibit a similar behavior in the Pacific
jet stream but are different in the occurrence of blocking, we have to deal with
two different datasets. This is owing to the ECMWF ERA-40 reanalysis dataset
(Simmons and Gibson, 2000) which is only available until August 2002. Therefore,
we use ERA-40 from September 1992 to May 1998 and the analysis fields of the
deterministic forecast model (T319/T511) from September 1998 to May 2003. The
impact of the different resolution has been evaluated and except for a few outliers the
differences for two overlapping years are only minor. The analysis was performed for
daily 00 UTC from the beginning of September to the end of May (termed ”winter”),
since this period captures the development of a strong jet during fall and the decay
of it during spring.
3.3.2
Hydraulic analysis
One aim of the study is to develop an analysis which is easy to apply to atmospheric
weather data. The geopotential and wind field on certain pressure levels are standard numerical weather prediction (NWP) products in weather services. The use of
constant pressure levels is a convenient but limited approximation in atmospheric
sciences. The isobaric analysis is not fully consistent with hydraulic theory, the use
of an isentropic surface would be more accurate since the vorticity and Bernoulli
equation are conserved on such a surface. The hydraulic analysis is primarily applied over the Pacific Ocean due to the occurrence of a single jet at this particular
pressure level. In order to determine the state of the flow, the following properties
are computed for a cross section through the jet stream:
1. Froude/Rossby number (Roβ ): The Froude/Rossby number, the ratio of the
kinetic energy transport KE to potential energy transport P E, integrated
across the current given by Eq. 3.5.
2. Volume transport per unit mass V : The volume transport (simply derived
from Eq. 3.3) has to be conserved. It can be used to check whether the
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 28
downstream cross section is chosen with the right width and how accurate the
use of an isobaric surface is.
3. Center latitude φc of the volume transport: This is the latitude of the median
of the volume transport V . It is used to set the origin of the coordinate system
for the computation of Roβ . Furthermore, as it will be shown in section 3.4 it
is a measure for the strength of winter.
4. Longitudinal position and extent of the (zonal) jet stream over the Pacific.
One complication of the analysis used for this study is the complicated jet structure
over the Atlantic region at this pressure level (cf. figures in section 3.4). Hence,
the hydraulic analysis is performed in two different ways. First, individual cases are
used to study both the Atlantic and the Pacific jet stream, where the jet stream
details are identified manually. In a second step, the Pacific current is analyzed
automatically for the period 1992 to 2003. A short comparison between the analysis
for the Pacific and the Atlantic is shown in section 3.5.
Case studies
The case studies are characterized by a subjective determination of the jet borders
and are used to verify and fully understand the behavior of Roβ as a dynamical
index. For this reason, exemplary stages of the evolution of the jet stream are
picked out and described in section 3.5. Special attention was turned on cases with
blocked flow over the Eastern Pacific. Moreover, the mechanism of the hydraulic
jump phenomenon is investigated with this type of analysis in section 3.6.
Automatic jet analysis
With the automatic analysis, the subjective determined jet borders are replaced by
an objective, reproducible and less extensive method. One problem is that the flow
pattern in subcritical regions (e.g. blocking) and over the Atlantic is too complex
in order to be analyzed automatically with our procedure and no accurate method
has been found yet. Therefore, the auto jet analysis was only successful for the
supercritical part of the Pacific jet stream where the zonal current has clear boundaries to the North and South. This part is found in the region of 120◦ to 180◦ East
which is also characterized by little blocking frequency during the months under
investigation (Pelly and Hoskins, 2003a).
In order to compute the jet properties of the supercritical part of the Pacific
jet stream, it is desirable to find a cross section through the jet which represents
the main part of the mass transport. Consequently, we divide the region from
120◦ to 180◦ E in two parts and search for the latitude with the highest mean
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 29
1.8
1.7
1.6
1.5
1.3
β
Ro′ , V
′
1.4
1.2
1.1
1
0.9
0.8
0
5
10
15
cut off in percent
20
25
30
Figure 3.3: Test of different cut off values: Shown are relative values of the
Froude/Rossby number Ro0β = Roβ /Roβ0 and the volume transport V 0 = V /V0 . Roβ0
and V0 are the values for the full cross section with zero percent cut off, Roβ and V for
10, 20, 25 and 30 percent cut off.
zonal wind speed. With the resulting latitude, the cross section through the jet is
now determined along constant longitude with the northern and southern boundary
limited by a drop of the wind speed below 20 percent of the maximum value gained
before. The cut off at 20 percent is due to the flattening of the velocity distribution
in the outer regions of the jet which we are not interested in. This procedure shall
ensure that only the main part of the jet is picked out. The resulting error due
to this cut off is shown in Fig. 3.3. This figure was compiled with four real cross
sections through the Pacific jet stream and demonstrates that the error due to the
cut off in Roβ is substantial higher than for the volume flux. For example, the cut
off we use for the automatical jet analysis results in an error of similar magnitude
for Roβ of about 20 to 30 percent, while the error in the volume flux is only at five
percent. The figure further demonstrates that Roβ is very sensitive to variations
in the width of the jet. In contrast, the volume flux is a stable measure, since the
main mass transport is due to the interior regions of the jet. With the help of this
test, the values for Roβ and V gained with the automatical jet analysis might be
corrected.
The next step is to set the origin of the coordinate system, a critical important
point in the computation of the specific energy flux. This point is determined by
the latitude of the median of the volume transport. The impact off the cut off was
also tested for this point (not shown), but the latitude remains very stable. Again,
this is due to the low contribution of the outer regions to the total transport.
One additional feature is computed in the automatical analysis, the zonal extent
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 30
of the jet stream. It shall help to identify the strength of the zonal current and is
defined as the zone with a zonal wind speed higher than 50 ms−1 . The threshold
is arbitrary, but allows to compare different occurrences of the jet (for additional
details see Appendix B).
3.4
A detailed analysis of the winter 1994-95 with
a strong Pacific jet
In this section we demonstrate the behavior of the hydraulic parameters described
in section 3.3. We use the winter period 1994-95, which is an example with a strong
Pacific jet stream but low blocking frequency. Moreover, this winter will then be
compared with the winter period 2002-03, when the jet was similarly strong but
differed in the occurrence of blocking. We start with the description of individual
cases during this particular winter, where Fig. 4a to 4f are exemplary stages in the
development of the jet stream. A summary of the whole winter is given in Fig. 5.
Figure 4a is a typical example at the beginning of winter. Over the Pacific
Ocean, the Froude/Rossby number (Roβ ) is highly supercritical at 10.2 . The cross
section perpendicular to the jet shows the triangular shape at the velocity distribution typical for a supercritical jet. Maximum wind speed (u-component) is nearly
88 ms−1 . Further downstream, just west of North America, the jet broadens and
the corresponding Roβ reaches a subcritical value of 0.73 over North America at
the marked location. The shape of the cross-sectional velocity distribution at this
location is very irregular as is typical for the near critical or subcritical part of
the jet. Moving further downstream the flow reaccelerates and Roβ reaches 10.0.
Downstream of this section the flow is decelerated by a high-over-low situation over
Europe. Over Asia the jet reaches a critical state with Roβ = 1.03. The accelerations are due to the land-sea-contrast (cf. Mullen, 1989), whereas the decelerations
are caused by the widening of the jet or a blocking situation. The transport, V , as
well as the latitude of the median of the volume flux, φc , should remain constant for
the jet stream system. Deviations in this case are due to the fact that we use a fixed
pressure level instead of isentropic surfaces. Furthermore, the atmosphere in the
hydraulic theory is considered to be barotropically compared to the real baroclinic
atmosphere. An underestimation of the volume flux like in this example leads to
higher Froude/Rossby numbers compared to a case where the volume flux in the
upstream and downstream section are identical.
Figure 5 summarizes the whole winter period 1994-95. This figure shows
the three properties described above at the longitude of maximum velocity: the
Froude/Rossby number (Fig. 5a), the volume transport (Fig. 5b) and the center
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 31
Figure 4a: Circumpolar chart from the Northern Hemisphere on 22 October 1994, displayed are the 300 hPa Geopotential [100m2 s−2 ] and the u-component [ms−1 ] of the
wind field. The lines in the chart indicate the cross sections with R the corresponding Froude/Rossby number (Roβ ). The plots below are the cross sections through the jet
with the first plot starting over the Pacific Ocean (the following are in order of the flow
direction). The abscissa shows the wind speed in [ms−1 ], the bar in the first graph is the
scale for the isotachs in the wind field on the circumpolar chart above. The solid line
in this cross section indicates the position of the center (φc ) of the volume flux V of the
current at this location in units of 108 m2 s−1 .
latitude of the volume transport (Fig. 5c). In addition, the zonal extent of the
jet stream (Fig. 5d) has been computed, where the dots indicate the longitudinal position of the jet maximum. The horizontal lines in these figures shall help
to distinguish certain periods characterized by similar flow properties. In contrast
to our individual manually analyzed cases (Fig. 4a to Fig. 4f), Fig. 5 shows a
slightly higher Roβ (lower V ) in this automatically generated analysis. This is due
to the cut off at 20 percent of the maximum wind speed, as it is explained in section
3.3.2. Another limitation of this figure is that only the Pacific jet stream has been
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 32
Figure 4b: Same as for Fig. 4a on 12 November 1994.
analyzed. The structure of the Atlantic jet on a single pressure level is often too
complicated to be analyzed with our automated procedure. Therefore, the behavior
of the downstream section of the Pacific jet and the Atlantic will only be studied
for individual cases as part of section 3.6.
Fig. 4a belongs to the beginning of this period. During early fall, Roβ exhibits
some scatter with variations from 4 up to values as high as 25. The high values
are associated with a narrow jet and low mass transport. Furthermore, the center
latitude of the jet, φc , is between 40 and 50 degree north. The zonal extent is short
and highly variable.
We call the next period from the end of October until November the doldrums.
It is often observed in other winter periods. Figure 4b is an example out of this
period. Compared to Fig. 4a one can see that the jet over the Pacific (R=4.4) and
over the Atlantic (R=5.0) is weaker in this particular case. The center latitude φc
is further to the north.
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 33
Figure 4c: Same as for Fig. 4a on 13 December 1994.
A remarkable change takes place at the end of November. The jet moves to
the south, reaching a latitude φc of 30 to 35 degree north; the zonal extent is
also longer. At the same time Roβ fluctuates less and the volume transport has
increased to values with less scatter above 108 m2 s−1 . Figure 4c at the end of this
period shows that the jet over the Pacific is much stronger than in the former cases
with a maximum wind speed close to 80 ms−1 and with V = 1.49·108 m2 s−1 . The
subcritical portion further downstream over the Eastern Pacific exhibits a series of
travelling troughs and ridges during this time, none of them constrained to a stable
position yet.
A few days later, again a noticeable change takes place. The zonal extent of
the jet has increased appreciably (Fig. 4d). A strong Pacific jet extends around
approximately one third of the Northern Hemisphere. The whole jet moved further
downstream. Over the West Coast of North America, more stationary cyclones and
anticyclones form in the wake of the jet. Moreover, the Atlantic jet is still weak. In
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 34
Figure 4d: Same as for Fig. 4a on 22 December 1994.
Fig. 5a we can see that Roβ shows fewer variability during this time with values
between six and twelve.
The next change is obvious at the end of January. Most of the fluctuations in
Roβ are in a small band between eight and ten, a little lower than in the period
before. The flow configuration in the section downstream of the jet exit region
is even more stable. During February most of the time a high-over-low situation
alternating with a stable ridge is found in this region. Figure 4e exemplifies this
period. The core of the jet is nearly 30 degree north, the volume transport is high
with V above 1.5·108 m2 s−1 and the maximum wind speed is 89 ms−1 . Additionally,
the jet splits into two branches with most of the volume flux in the northern branch.
Going further downstream, a new acceleration to a supercritical jet takes place over
the West Atlantic and again the jet splits west of Europe (accompanied by a block
over Southwest Europe). Compared to Fig. 4d, where only a Pacific jet exists, the
Atlantic jet is now well developed.
The last period starts at mid-March and can bee seen as the breakdown of
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 35
Figure 4e: Same as for Fig. 4a on 4 February 1995.
winter. Similar to the beginning of the winter period, Roβ in Fig. 5a scatters widely,
this time mostly in the lower range. The volume flux converges to 1.0·108 m2 s−1 ,
the latitudinal position returns north and the zonal extent decreases. Figure 4f
demonstrates the flow behavior during this period. Most of the cross sections show
Froude/Rossby numbers close to one, except over the Pacific where the current is
slightly supercritical (Roβ =2.5). As before at the beginning of this winter period φc
is at a very stable latitudinal position all around the hemisphere.
3.5
The Atlantic jet and the European block
A high blocking frequency over the Atlantic characterizes the winter of 2002-03. It
starts similar to the winter of 1994-95 (c.f. section 3.4). Values of Roβ are high
due to the narrow jet with a low mass transport, mostly between 0.5·108 m2 s−1 to
1.0·108 m2 s−1 (Fig. 6). The central jet latitude φc (Fig. 6c) is far north. The zonal
extent is short at this time. The second period in this winter is characterized by
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 36
Figure 4f: Same as for Fig. 4a on 14 April 1995.
lower fluctuations of the Froude/Rossby numbers and a longer zonal extent than
during fall 1994. A look at Fig. 7a shows the typical flow configuration during
this time. A comparison with the same period during winter 1994-95 (Fig. 4b)
demonstrates that the jet stream over the Pacific is much stronger in winter 200203. The so called doldrums are absent. The strength of the current continues to
increase from the beginning of this particular winter. The jet reaches a southerly
position early in the winter, which seems to be an important factor in blocking.
Compared to the winter period 1994-95 the first half of this winter is characterized
by an approximately 10 percent higher mass transport, a period of relatively little
variability of the Froude/Rossby numbers (from the end of October to the beginning
of December) and a southerly position.
A comparison between the individual cases displayed in Fig. 4b and 4d for
winter 1994-95 and Fig. 7a and 7b for winter 2002-03 demonstrates that the Atlantic
jet is more developed in the latter winter period. Additional examples for the second
half are shown in section 3.6.
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 37
(a)
(c)
(b)
(d)
05/01/95
←04/14
04/01/95
03/01/95
←02/04
02/01/95
01/01/95
←12/22
←12/13
12/01/94
←11/12
11/01/94
←10/22
10/01/94
09/01/94
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
2.5 20
Volume flux per m (108 m2s−1)
30
40
50
latitude/deg.
−120
−180
120
longitude/deg.
60
Figure 5: The four properties of the jet stream shown for the winter periods 1994-95.(a)
Froude/Rossby number (Roβ ), (b) the volume flux per unit mass, (c) the center latitude
of the volume flux (φc )and (d) the longitudinal extent of the jet stream and position of the
maximum. The positions and dates marked on the r.h.s are the individual cases described
in section 3.4.
(a)
(c)
(b)
(d)
05/01/03
04/01/03
03/01/03
02/01/03
01/01/03
←12/28
12/01/02
←11/17
11/01/02
10/01/02
09/01/02
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
Volume flux per m (108 m2s−1)
2.5 20
30
40
50
latitude/deg.
−120
−180
120
longitude/deg.
Figure 6: Same as Fig. 5 for the winter periods 2002-03.
60
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 38
Figure 7a: Same as Fig. 4a on 17-Nov-2002.
The following period (January) is very similar to January 1995: Roβ has fewer
fluctuations in 2003 and φc is closer to 30 degree north. A comparison of the mass
transport shows approximately similar values for both winters. However, Roβ suddenly drops in the next period (February, March). This drop is studied in more
detail in the following section.
The winter 2003 ends approximately two weeks later than in the year 1995,
indicated by the return to the north of the jet center and the increased scatter in
Roβ .
If one compares the whole period for both winters, it is obvious that the Atlantic
jet in the second case is more developed than in the first winter. To demonstrate the
difference of the flow characteristic over the Atlantic (30◦ to 90◦ W) we want to refer
to Fig. 8. This figure exhibits the center latitude (median) of the mass transport
(φc ) for both winter cases. The other properties of the jet stream are not shown
since the method of the automatic jet analysis does not work well for them in this
region. In winter 1994-95, φc varies strongly at the beginning of winter, whereas in
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 39
Figure 7b: Same as Fig. 4a on 28-Dec-2002.
winter 2002-03 the scatter is less and on average the position further south. The
differences persist for the second half of the winter. While the main current in 2003
remains mostly in a southern position with view outliers, in 1995 the center latitude
of the jet varies a lot.
3.6
3.6.1
Evolution of a block and the role of jumps
Evolution of a block
In this section we want to demonstrate the development of a typical blocking situation over the East Pacific. As described in the previous section, winter 2002-03
had an exceptional high blocking frequency over the Pacific as well as over the Atlantic sector. We confine ourselves to the Pacific cases as they are easier to capture
at the level of 300 hPa. Figures 9a to 9f show certain states in the evolution of
blocked flow over the East Pacific/western part of North America in the period of
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 40
(b)
(a)
05/01/95
05/01/03
←04/14
04/01/95
04/01/03
03/01/95
03/01/03
←02/04
02/01/95
01/01/95
02/01/03
01/01/03
←12/28
←12/22
←12/13
12/01/94
12/01/02
←11/17
←11/12
11/01/94
11/01/02
←10/22
10/01/94
10/01/02
09/01/94
09/01/02
20
30
40 50 60
latitude/deg.
70
20
30
40 50 60
latitude/deg.
70
Figure 8: Center latitude of the volume transport shown for the Atlantic (90◦ W to 30◦ W)
for the winter period 1994/95 (a) and 2002/03 (b).
February and March 2003. Figure 10 is a summary of the whole period with the
three Froude/Rossby numbers Pacific upstream (PU), Pacific downstream (PD) and
Atlantic (A) for every other day. The notation Pacific downstream will refer to the
subcritical part, usually between 90◦ to 150◦ West. The notation Atlantic will refer
to the region between 30◦ to 90◦ West.
Figure 9a shows the beginning of the blocking period. At 1 February 2003
the Pacific jet is well developed and strong with a Roβ of 8.0 and a high V of
1.46·108 m2 s−1 . Around 150◦ W the jet stream broadens, leaving an extended subcritical zone in its wake with Roβ of 0.49 at PD. Even the Atlantic is nearly critical
with Roβ 1.3. During the next four days an Omega block forms in the PD section.
Figure 9b shows the fully developed block on 5 February. In this case, the upstream
Roβ (PU) has slightly decreased to a value of 7.6, whereas Roβ over the Atlantic
increased noticeable to 3.6. In the Omega close to the west coast of North America
the flow is subcritical with the main transport in the northern branch. The next
figure (Fig. 9c) still exhibits a blocking pattern, which looks more like a high-overlow situation in this case, or even like a split jet (regarding the right graph in the
lower right corner). Again, the PU and A sections have not changed significantly
compared to the previous figure, although φc over the Pacific is far south. The
subcritical cross section (Roβ = 0.73) shows a double jet configuration, again with
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 41
Figure 9a,b: Same as Fig. 4a for (a) 1 February 2003 and (b) 5 February 2003 with the
Pacific sector and the western part of the Atlantic only. The cross sections in the lower
right corner are the same as for Fig. 4a, displayed are in each case the upstream and
downstream section of the Pacific.
a higher mass flux in the northern branch than in the southern one. In Fig. 9d
the blocking pattern is not so obvious but still apparent. The section over the East
Pacific is still subcritical. The upstream value of Roβ has decreased again to a value
of 5.2, whereas V is at a high level of 1.6·108 m2 s−1 . Going on to the next figure (Fig.
9e), a high-over-low situation can be observed once more with the main part of V in
the northern branch. The position of the block is still the same and the acceleration
further downstream is close to the blocking pattern. Figure 9f demonstrates the
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 42
Figure 9c,d: Same as Fig. 9a,b for (c) 9 February 2003 and (d) 21 February 2003.
decay of the blocking period at the end of March. Roβ at PU is still supercritical
but the zonal extent of the jet is short. At the same time the flow at both PD
and A is nearly critical or slightly supercritical. This is to say that the flow is still
decelerated before reaching the continent of North America but no blocking pattern
is obvious now. The blocking situation described in this section lasts for nearly two
months. A summary of the whole period is displayed in Fig. 10.
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 43
Figure 9e,f: Same as Fig. 9a,b for (e) 3 March 2003 and (f) 27 March 2003.
3.6.2
Jump results
With the help of Fig. 10 we want to verify whether blocking can be explained as
a hydraulic jump in the atmosphere. As explained in section 3.2.1, the development of a block is determined by the conditions downstream of the hydraulic jump.
Therefore, it is necessary to evaluate Roβ and the width of the jet for the section
downstream of the block as well. That is what we have done in Fig. 10. At the
beginning the Atlantic jet is weak (low Roβ ) and no block has formed yet. On
3 February Roβ starts to increase and at PD concomitantly a block forms. That
state persists almost uninterruptedly until the end of March. During this time the
Froude/Rossby number over A is always supercritical but most of the time subcrit-
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 44
Pacific upstream (PU)
Pacific downstream (PD)
Atlantic (A)
←
03/23
03/16
03/09
←
03/02
02/23
←
02/16
←
02/09
←
02/02
←
01/26
0
5
10
0
1
Roβ
2
3
0
Roβ
5
10
Roβ
Figure 10: Blocking situation during winter 2002-03 from the end of January to the
end of March. Displayed are the Froude/Rossby numbers (Roβ ) from the upstream (PU)
and downstream (PD) Pacific section as well as the West Atlantic (A) region. The stars
indicate Roβ (P D) ≤ 0.8 and the Atlantic Roβ (A) ≥ 1.3. The arrows on the r.h.s. indicate
the positions of the individual cases described in section 3.6.
Pacific upstream (PU)
Pacific downstream (PD)
Atlantic (A)
03/23
03/16
03/09
03/02
←
02/23
02/16
02/09
←
02/02
01/26
0
5
Roβ
10
0
1
2
Roβ
3
0
5
10
Roβ
Figure 11: Same as Fig. 10 for winter 1994-95.
ical further upstream in the PD region.
A comparison of February/March 1995 and 2003 in Fig. 11 shows similar Roβ at
PU, the farthest upstream region. Yet the blocking period in 1995 was much shorter.
The key seems to be Roβ over the Atlantic. The flow in section A is supercritical
at the beginning of the period and there is a block immediately upstream in the
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 45
Figure 12a,b: Same as Fig. 9a,b for (a) 5 February 1995 and (b) 27 February 1995.
PD region. Figure 12a is an example from this period. Both the Pacific jet and the
block are very strong (Roβ = 0.25). Further downstream the jet accelerates again
and forms an Atlantic jet with a Roβ of 3.8. But in the middle of the month Roβ
all of a sudden drops to values close to one. Figure 12b is an example from the
unblocked state. The Pacific jet is still strong but the Atlantic jet has decelerated
reaching a Roβ of 0.92 which indicates near-critical flow. Without an Atlantic jet,
it seems that a block over the western part of North America cannot form. As a
result of this, we suggest that the block in the PD section shows a similar behavior
as a hydraulic jump in open-channel flow. In section 3.2.1 it is explained that a
jump is determined by a control acting further downstream of the jump. Applied
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 46
5
4.5
4
ting
illa
osc
y2/y1
3.5
3
ak
we
2.5
un
du
la
r
2
1.5
1
1
2
3
4
5
6
7
Roβ (Fr2)
8
9
10
11
12
Figure 13: Relation between Roβ (P U ) and ratio of PD (Pacific downstream) to PU
(Pacific upstream) width. The solid line shows the theoretical relation evaluated by Chow
(1959) for single layer open-channel hydraulics. The stars indicate Roβ (P D) ≤ 0.8 and
Roβ (A) ≥ 1.3 (downstream of the block).
to the atmosphere, this means that the PD block is determined by a control acting
between this block and the Atlantic. Hence, the question is weather one can conclude
from the succession of supercritical-subcritical-supercritical (PU-PD-A) flow that the
block in the PD section is a hydraulic jump. Moreover, this would imply that, if the
Atlantic does not exceed critical flow, no jump (block) can form.
For further comparison, we want to compare our results with the classification
of hydraulic jumps in single layer open-channel hydraulics described for example
in Chow (1959). Equation 3.10 shows the relation between the width ratio h2 /h1
and the upstream Froude number F r1 . This is displayed in Fig. 13. Furthermore,
we took the results from our blocking cases (from Fig. 10 and 11) and computed
the width ratio of the jet stream versus the Froude/Rossby number Roβ which are
display in the same figure. Apart from a slight bias between the theoretical curve
and the measurements we have made for the atmospheric jet stream, especially for
higher Froude/Rossby numbers, most of the jumps are in the range of weak or
oscillating jumps. Only few of them are really strong jumps.
3.7
Discussion and conclusions
The prediction of atmospheric blocking still poses big problems in medium range
forecasts. Knowing the initiating mechanism would help to enhance the forecast
skill. In this paper we introduced a new dynamical index, which primarily shall
Draft of the article ”Beta-plane Hydraulics of the Northern Hemisphere Jet Stream” 47
help to describe the flow state of the atmospheric jet stream. Furthermore, we also
tried to use the index for the verification whether blocking can be explained as a
hydraulic jump in the atmosphere similar to its open-channel-flow analogue (Rossby,
1950).
A total of 11 winter periods has been analyzed with the help of ECMWF ERA40 and T511 analysis data, two examples are presented in this paper. We took these
two periods as they show a similar behavior in the Pacific jet stream but are totally
different in the occurrence of blocking over the the western part of North America.
Furthermore, they are similar in the ENSO cycle.
In the first part, a new dynamical index for the description of the Pacific jet is
presented. Compared to other measures like height indices, our index is not fixed in
latitude and accounts for the seasonal variability in the latitudinal position of the
jet stream. A further application to the Atlantic jet would be desirable.
The results from section 3.6, the second part of this work, suggest that blocking
over North America could be explained as a hydraulic jump like phenomenon. A
comparison between February/March 1995 and 2003 shows that during the former
period, when the part downstream of the blocked region is nearly critical or even
subcritical, the blocking frequency is very low. On the other hand, in the latter
case with a mostly supercritical jet downstream of the block the blocking persists
almost uninterruptedly. The problem is that for a hydraulic jump (block) the exact
downstream conditions have to be determined, in this case the conditions over the
Atlantic. Our analysis consists only of 300 hPa data, which is not fully consistent
with the theory. It would be very important to use data on isentropic levels since this
would ensure the conservation of energy, momentum and mass. Through a hydraulic
jump mass and momentum are conserved and energy is lost. In our analysis we have
a loss of energy and mass as well, which is primarily due to the data we used.
3.8
References
See references at the end of the thesis.
References
Acheson, D.J., 1992: Elementary Fluid Dynamics. Clarendon Press. 397pp.
Armi, L., 1989: Hydraulic control of zonal currents on a β-plane. J. Fluid Mech.,
201, 357-377.
Berggren, R., B. Bolin, and C.-G. Rossby, 1949: An aerological study of zonal
motion, its perturbations and break-down. Tellus, 1, 14-37.
Charney, J.G., and J.G. DeVore, 1979: Multiple Flow equilibria in the atmosphere
and blocking. J. Atm. Sci., 36, 1205-1216.
Charney, J.G., J. Shukla, and K.C. Mo, 1981: Comparison of a barotropic blocking
theory with observations. J. Atm. Sci., 38, 762-779.
Charney, J.G., and D.M. Straus, 1980: Form-drag instability, multiple equilibria
and propagating planetary waves in baroclinic, orographically forced, planetary wave systems. J. Atm. Sci., 37, 1157-1176.
Chen, T-C., and J-H. Yoon, 2002: Interdecadal variations of the North Pacific
wintertime blocking. Mon. Weather Rev., 130, 3136-3143.
Chow, V.T., 1959: Open-channel hydraulics. McGraw-Hill Book Company. 680pp.
D’Andrea, F., S., Tibaldi, M., Blackburn, G., Boer, M., Déqué, M.R., Dix, B.,
Dugas, L., Ferranti, T., Iwasaki, A., Kitoh, V., Pope, D., Randall, E., Roeckner, D., Strauss, W., Stern, H., Van den Dool,and D. Williamson, 1998: Northern Hemisphere atmospheric blocking as simulated by 15 atmospheric general
circulation models in the period 1979-1988. Climate Dyn., 14, 385-407.
Egger, J., 1978: Dynamics of blocking highs. J. Atm. Sci., 34, 1788-1801.
Garriott, E.B., 1904: Long Range Forecasts. U.S. Weather Bureau Bulletin, 35.
Geer, I., 1996: Glossary of Weather and Climate. AMS. 272pp
Henderson, F.M., 1966: Open Channel Flow. Macmillan. 522pp.
References
49
Hoskins, B.J., 1997: A potential vorticity view of synoptic development. Meteor.
Appl., 4, 325-334.
Hugoniot, H., 1886: Sur un théorème relatif au movement permanent et à
l’écoulement des fluids. C.r.hebd. Séanc. Acad. Sci., Paris, 104, 1178-1181.
Kundu, P.K., and I.M. Cohen, 2002: Fluid Mechanics. Second Edition. Academic
Press. 730pp.
Lejenäs, H., and H. Økland, 1983: Characteristics of northern hemisphere blocking
as determined from long time series of observational data. Tellus, 34A, 350362.
Liepmann, H.W., and A. Roshko, 1957: Elements of Gasdynamics. John Wiley
and Sons. 456pp.
Mauritsen, T., and E. Källén, 2004: Blocking prediction in an ensemble forecasting
system. Tellus, 56A, 218-228.
Molteni, F., R. Buizza, T.N. Palmer and T. Petroliagis, 1996: The new ECMWF
ensemble prediction system: methodology and validation. Q. J. R. Met. Soc.,
122, 73-119.
Mullen, S.L., 1989: Model Experiments on the impact of Pacific Sea Surface Temperature Anomalies on blocking frequency. J. Climate, 2, 997-1013.
Pelly, J.L., and B.J. Hoskins, 2003a: A new perspective on blocking. J. Atm. Sci.,
60, 743-755.
Pelly, J.L., and B.J. Hoskins, 2003b: How well does the ECMWF Ensemble Prediction System predict blocking? Q. J. R. Met. Soc., 129, 1683-1702.
Prandtl, L., 1994: Führer durch die Strömungslehre. 2. Auflage. Vieweg, Braunschweig, 384pp.
Rex, D.F., 1950a: Blocking action in the middle troposphere and its effect upon
regional climate. I. An aerological study of blocking action. Tellus, 2, 196-211.
Rex, D.F., 1950b: Blocking action in the middle troposphere and its effect upon
regional climate. II. The climatology of blocking action. Tellus, 2, 275-301.
Rex, D.F., 1951: The effect of Atlantic blocking action upon European climate.
Tellus, 3, 100-112.
Reynolds, O., 1886: On the flow of gases. Phil. Mag(5), 21, 185-199.
References
50
Rossby, C.G., 1950: On the dynamics of certain types of blocking waves. J. Chinese
Geophys. Soc., 2, 1-13.
Sabersky, R.H., A.J. Acosta, E.G. Hauptmann, and E.M. Gates, 1999: Fluid Flow.
A first course in fluid mechanics. Fourth Edition. Prentice-Hall. 606pp.
Schneider, S.H., 1996: Encyclopedia of Climate and Weather. Volume 1. Oxford
University Press. 476pp.
Simmons, A.J., and J. K. Gibson, 2000: The ERA-40 project plan. ERA-40 Project
Report Series, 1, 62 pp.
Tibaldi, S., and F. Molteni, 1990: On the operational predictability of blocking.
Tellus, 42A, 343-365.
Tibaldi, S., E. Tosi, A. Navarra, and L. Pedilli, 1994: Northern and Southern
Hemisphere seasonal variability of blocking frequency and predictability. Mon.
Weather Rev., 122, 1971-2003.
Tung, K.K., and R.S. Lindzen, 1979: A Theory of Stationary Long Waves. Part I:
A Simple Theory of Blocking. Mon. Weather Rev., 107, 714-734.
Wiin-Nielsen, A., 1979: Steady state and stability properties of a low-order,
barotropic system with forcing and dissipation. Tellus, 31, 375-386.
Appendix
A
Impact of the different datasets (T319/T511
vs. ERA-40)
ECMWF analysis from three different model set-ups were used: ERA-40 (T159L60),
T319L60 and T511L60 (Tab. A.1). Since not only the horizontal and vertical mesh
width differ, but also the data assimilation procedures, a comparison was carried out
between ERA-40 and T319L60 for two overlapping years (1998-99 and 1999-2000).
Data of the whole study were retrieved on a 1◦ ×1◦ grid. Figures A.1 and A.2 show
the differences in the four jet properties used in the auto jet analysis. The difference
of the Froude/Rossby numbers ∆Roβ is high, especially at the beginning and end
of each winter period the discrepancy is high (Fig. A.1 and A.2). The computation
of the mean absolute error (MAE) results in high values of 18.4 percent for the
winter 1998-99 and an even higher value for 1999-2000 with 27.5 percent. This
error can be explained with the longitudinal position of the jet maximum found in
the automatically generated analysis. As described in section 3.3.2, Roβ is highly
dependent on the jet width. Therefore, a shift in the longitudinal position also leads
to different outer boundaries of the jet. In Fig. A.1d one can see that the shift
sometimes reaches values of 10 degrees or even more. This is to say that the jet
maximum is found in a location which differs approximately 1000 kilometers or more.
If the MAE is computed for the core of the winter period (beginning of December
until the end of February, period of a stable jet stream), the error is reduced to 10.9
percent in each case. Since the volume flux is less dependent on the outer regions
(compare Fig. 3.3 in section 3.3.2) than Roβ the MAE reaches lower values of 7.1%
Year
Data/Model type
Approx. grid distance (km)
1994-1998
1998-2000
2000-2003
ERA-40 (T159L60)
T319L60
T511L60
∼ 120
∼ 60
∼ 40
Table A.1: Types of data used.
52
Appendix
(a)
(c)
(b)
(d)
05/01/99
04/01/99
03/01/99
02/01/99
01/01/99
12/01/98
11/01/98
10/01/98
09/01/98
−10
−5
0
∆ Roβ
5
−0.5
−0.25
0
0.25
−5
−2.5
∆ V (108 m2s−1)
0
2.5
∆ φ c/deg.
−20
−10
0
10
∆ long/deg.
20
Figure A.1: Test of the differences in the auto jet analysis due to the use of different
datasets (T511 vs. ERA-40). Displayed are the differences between (a) the Froude/Rossby
number ∆Roβ , (b) the volume flux ∆V , (c) the center latitude of the volume flux ∆φc
and (d) the longitudinal extent ∆long for the winter period 1998-99.
(8.1%) for 1998-99 (1999-2000). This behavior is also found in φc , the errors due to
the different dataset is low with 2.0% and 3.1% respectively. This also demonstrates
the sensibility of the computation of Roβ , whereas V and φc are robust measures.
For further studies using Roβ as dynamical index the automatic computation of Roβ
needs to be improved. One possibility is the use of a mean wind field about several
degrees of longitude.
B
Additional analyzed winter cases - Comment
on the automatical analysis
This section includes additional automatically analyzed winter cases and can be seen
as a short-term climatology of the Pacific jet stream during the ”winter period”.
Furthermore, in the following two paragraphs some additional comments serve as a
supplement to the procedure of the automatical analysis explained in section 3.3.2.
As explained in section 3.3.2, we have chosen the area between 120◦ to 180◦ W
53
Appendix
(a)
(c)
(b)
(d)
05/01/00
04/01/00
03/01/00
02/01/00
01/01/00
12/01/99
11/01/99
10/01/99
09/01/99
−10
−5
0
∆ Roβ
5
−0.5
−0.25
0
0.25
∆ V (108 m2s−1)
−5
−2.5
0
2.5
∆ φ c/deg
−20
−10
0
10
∆ long/deg
20
Figure A.2: Same as Fig. A.1 for the winter period 1999-00.
for the automatically generated analysis, since this is the region with low blocking
frequency. In order to compute the jet properties the algorithm searches for a
cross section through the jet with highest mean mass flux. Than, this region is
divided into two parts, from 120◦ to 150◦ W and 150◦ to 180◦ W. The next step is to
evaluate the mean zonal velocity along constant latitude for each part. The resulting
latitude with the highest mean velocity (for both areas) is searched for the absolute
maximum of the zonal wind speed and then the north-south cross section is chosen
along constant latitude. As explained before, the northern and southern boundary
is determined by a drop of the zonal velocity below 20 percent of the maximum
value.
In the automatically analyzed winter cases we have also added the zonal extent
of the jet stream. This is an interesting feature of the Pacific jet stream, since in
some periods the extent spans nearly 180 degrees longitude and it is also a measure
for the strength of the jet stream. We define the zonal jet stream as the region where
the zonal wind speed exceeds 50 ms−1 (section 3.3.2). The limit of the zonal extent
is reached if either the wind speed decreases below this value, or the jet crosses the
northern or southern boundaries determined by the northerly and southerly limit of
the cross section used for the computation of the jet properties.
54
Appendix
(a)
(c)
(b)
(d)
05/01/93
05/01/93
04/01/93
04/01/93
03/01/93
03/01/93
02/01/93
02/01/93
01/01/93
01/01/93
12/01/92
12/01/92
11/01/92
11/01/92
10/01/92
10/01/92
09/01/92
09/01/92
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2.5 20
30
2 −1
Volume flux per m (10 m s )
40
50
latitude/deg.
−120
−180
120
longitude/deg.
60
Figure B.1: Same as Fig. 5 for Winter 1992/93.
(a)
(c)
(b)
(d)
05/01/94
05/01/94
04/01/94
04/01/94
03/01/94
03/01/94
02/01/94
02/01/94
01/01/94
01/01/94
12/01/93
12/01/93
11/01/93
11/01/93
10/01/93
10/01/93
09/01/93
09/01/93
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2 −1
Volume flux per m (10 m s )
2.5 20
30
40
50
latitude/deg.
−120
Figure B.2: Same as Fig. 5 for Winter 1993/94.
−180
120
longitude/deg.
60
55
Appendix
(a)
(c)
(b)
(d)
05/01/96
05/01/96
04/01/96
04/01/96
03/01/96
03/01/96
02/01/96
02/01/96
01/01/96
01/01/96
12/01/95
12/01/95
11/01/95
11/01/95
10/01/95
10/01/95
09/01/95
09/01/95
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2.5 20
30
2 −1
Volume flux per m (10 m s )
40
50
latitude/deg.
−120
−180
120
longitude/deg.
60
Figure B.3: Same as Fig. 5 for Winter 1995/96.
(a)
(c)
(b)
(d)
05/01/97
05/01/97
04/01/97
04/01/97
03/01/97
03/01/97
02/01/97
02/01/97
01/01/97
01/01/97
12/01/96
12/01/96
11/01/96
11/01/96
10/01/96
10/01/96
09/01/96
09/01/96
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2 −1
Volume flux per m (10 m s )
2.5 20
30
40
50
latitude/deg.
−120
Figure B.4: Same as Fig. 5 for Winter 1996/97.
−180
120
longitude/deg.
60
56
Appendix
(a)
(c)
(b)
(d)
05/01/98
05/01/98
04/01/98
04/01/98
03/01/98
03/01/98
02/01/98
02/01/98
01/01/98
01/01/98
12/01/97
12/01/97
11/01/97
11/01/97
10/01/97
10/01/97
09/01/97
09/01/97
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2.5 20
30
2 −1
Volume flux per m (10 m s )
40
50
latitude/deg.
−120
−180
120
longitude/deg.
60
Figure B.5: Same as Fig. 5 for Winter 1997/98.
(a)
(c)
(b)
(d)
05/01/99
05/01/99
04/01/99
04/01/99
03/01/99
03/01/99
02/01/99
02/01/99
01/01/99
01/01/99
12/01/98
12/01/98
11/01/98
11/01/98
10/01/98
10/01/98
09/01/98
09/01/98
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2 −1
Volume flux per m (10 m s )
2.5 20
30
40
50
latitude/deg.
−120
Figure B.6: Same as Fig. 5 for Winter 1998/99.
−180
120
longitude/deg.
60
57
Appendix
(a)
(c)
(b)
(d)
05/01/00
05/01/00
04/01/00
04/01/00
03/01/00
03/01/00
02/01/00
02/01/00
01/01/00
01/01/00
12/01/99
12/01/99
11/01/99
11/01/99
10/01/99
10/01/99
09/01/99
09/01/99
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2.5 20
30
2 −1
Volume flux per m (10 m s )
40
50
latitude/deg.
−120
−180
120
longitude/deg.
60
Figure B.7: Same as Fig. 5 for Winter 1999/00.
(a)
(c)
(b)
(d)
05/01/01
05/01/01
04/01/01
04/01/01
03/01/01
03/01/01
02/01/01
02/01/01
01/01/01
01/01/01
12/01/00
12/01/00
11/01/00
11/01/00
10/01/00
10/01/00
09/01/00
09/01/00
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2 −1
Volume flux per m (10 m s )
2.5 20
30
40
50
latitude/deg.
−120
Figure B.8: Same as Fig. 5 for Winter 2000/01.
−180
120
longitude/deg.
60
58
Appendix
(a)
(c)
(b)
(d)
05/01/02
05/01/02
04/01/02
04/01/02
03/01/02
03/01/02
02/01/02
02/01/02
01/01/02
01/01/02
12/01/01
12/01/01
11/01/01
11/01/01
10/01/01
10/01/01
09/01/01
09/01/01
2
4
6
8 10
Roβ
15 20
30 0
0.5
1
1.5
2
8
2 −1
Volume flux per m (10 m s )
2.5 20
30
40
50
latitude/deg.
−120
Figure B.9: Same as Fig. 5 for Winter 2001/02.
−180
120
longitude/deg.
60
59
Appendix
C
Mathematical Derivations
In this part the momentum equation for the flow in a rectangular open channel
(Eq. 2.4) and the equation for the Froude/Rossby number for atmospheric flow on
a β-plane (Eq. 2.28) are derived in more detail.
C.1
Momentum equation
The rate of change of fluid momentum is equal to (c.f. Henderson, 1966; Kundu and
Cohen, 2002)
(qρU )2 − (qρU )1
(C.1.1)
where subscript 1 denotes the momentum at section A (c.f. Fig. 2.3) and subscript
2 at section B. This change is balanced by the difference in the hydrostatic pressure
exerted on each cross section. The resulting equation is as follows:
gρh21 gρh22
−
= (qρU )2 − (qρU )1
2
2
(C.1.2)
After cancelling ρ and rearranging the terms one obtains
gh21
gh2
+ qU1 = 2 + qU2 .
2
2
(C.1.3)
Now, U can be substituted by U = q/h, resulting in
h21
q2
h2
q2
+
= 2+
2
gh1
2
gh2
(C.1.4)
or using q = hU to get the same form as in Eq. 2.4:
h21 h1 U12
h22 h2 U22
+
=
+
2
g
2
g
C.2
(C.1.5)
Froude/Rossby number Roβ
This section explains the derivation of Eq. 2.28. In order to get to this form, one
has to start with Eq. 2.27:
G=
Z +a
1
|
−a
2
3
u(y) dy − β
Z +a
−a
u(y)
Z y
{z
−a
u(y 0 )y 0 dy 0 dy +
}
1
p−a V
− f0 V 2
ρ
2
(C.2.1)
G0
The flow to be considered is steady with a total volume transport V, thus a stream
function can be defined with
ψ(y) =
Z y
−a
u(y 0 )dy 0 ,
ψ(−a) = 0,
ψ(a) = V.
(C.2.2)
60
Appendix
Therefore, G0 in Eq. C.2.1 can also be written as
Z a
Z a
Z y
1 03
0
G0 =
ψ (y)
ψ 0 (y 0 )y 0 dy 0 dy.
ψ (y) − β
−a
2
−a
−a
(C.2.3)
If the outer integration is performed by parts and a symmetric velocity distribution
assumed, Eq. C.2.3 can be written as
G0 (V, a) =
Z a µ
1
2
−a
¶
ψ(y)03 + βψ(y)ψ(y)0 y dy
(C.2.4)
In order to apply the specific energy flux to arbitrary zonal velocity distributions, a
velocity distribution function g(η) is defined (c.f. Henderson, 1966):
u ≡ um g(η),
η=
y
a
(C.2.5)
where um is the maximum velocity of the zonal current. Furthermore, coefficients
for energy (α) and pressure (α0 ) are introduced:
α≡
α0 ≡ −
Z 1
−1
Z 1
3
−1
g
g dη/
Z η
−1
µZ 1
−1
¶3
gdη
g(η 0 )η 0 dη 0 dη/
(C.2.6)
Z 1
−1
g 3 dη
(C.2.7)
With a Froude/Rossby number defined by
Roβ =
um
α0 βa2
(C.2.8)
the specific energy flux per mass (Eq. C.2.4) can be written as
"
#
(βV )2/3 αα02/3
G0 (V, Roβ ) =
V 5/3 [(Roβ )2/3 + 2(Roβ )−1/3 ]
2
|
{z
(C.2.9)
}
(1)
Dividing by part (1) results in the non-dimensional specific energy flux
G00 = (Roβ )2/3 + 2(Roβ )−1/3 .
(C.2.10)
Acknowledgement/Danksagung
An erster Stelle möchte ich mich bei meinen Eltern Evelyn und Max Riffler dafür
bedanken, dass sie mein Studium in den vergangenen Jahren stets gefördert haben.
Ich hoffe, sie sind zufrieden, dass ich ihre Investition mit unversicherbarem Risiko
ohne Schadenfälle zu Ende bringen konnte.
Ebenso wichtig ist mir der Dank an Andrea für den Rückhalt und ihre
Herzenswärme. Sie hat mich immer motiviert und letztendlich zum Durchhalten
während Flauten oder stürmischen Zeiten angetrieben.
Ein großes Dankeschön richte ich an meinen Betreuer Georg Mayr, der mein
Interesse an der Atmosphären- und Fluiddynamik geweckt hat. Er hatte stets offene
Ohren für meine Anliegen und stand mir mit wertvollen Ratschlägen zur Seite.
I am deeply grateful to Eva and Larry Armi who enabled me a great stay in
San Diego and an unforgettable time in California. I want to express my gratitude
especially to Larry for the interesting visit of the Scripps Institution of Oceanography
(SIO) at the University of California, San Diego (UCSD) and for the many useful
lessons on fluid mechanics he gave to me. He is a great teacher and mentor and
always willing to share his enormous knowledge. I am also indebted to SIO for the
possibility of participating in two courses for free.
Danken möchte ich auch der gesamten Arbeitsgruppe vom 9. Stock und Sabine
für die angenehme Zeit auf und außerhalb der Universität. Die Erlebnisse mit Philip
und Esther haben für die nötige Portion Aufheiterung und für das Verschwinden
von Sorgen rund um Blockinglagen gesorgt. Und ohne die Hilfe von Alexander
Gohm wäre so manches Matlab-Problem ungelöst geblieben. Schließlich hat Gery
Keller meinen Studienabschluss dadurch gefördert, dass er meine unzähligen Sonderwünsche bei der Einteilung der Dienstpläne berücksichtigt hat.
Curriculum Vitae
Name
Michael Riffler
Geburtsdatum
14. April 1977
Geburtsort
Schruns, Österreich
Ausbildung:
1984–1987
Volksschule Schruns
1987–1997
Bundesrealgymnasium Bludenz
Juni 1997
Matura
1997–1998
Präsenzdienst
seit 10/1998
Diplomstudium der Meteorologie und Geophysik an der Universität Innsbruck
09/2003–12/2003 Studienaufenthalt an der ”Scripps Institution of Oceanography”,
University of California, San Diego
Berufserfahrung:
1/2003–6/2004
Meteorologe beim Alpenvereinswetterdienst, ZAMG Innsbruck
seit 6/2004
Meteorologe bei der meteomedia ag, Schweiz
Meteorologischer Trainingskurs, Feldexperiment:
6/2003
”Use and interpretation of ECMWF products”, ECMWF (European Centre for Medium Range Weather Forecasts), Reading,
GB.
7/2003
Gaudergrat Experiment (GaudEx) - High resolution observations of separated flow, Davos, Schweiz. Arbeitsfeld: Instandhaltung der Wetterstationen und Radiosondenaufstiege
62

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