Analysis of Three Situations of the Foehn Effect over the ...
Contributions to the Dynamics of South Foehn · 2019. 12. 16. · considered in the hydraulic...
Transcript of Contributions to the Dynamics of South Foehn · 2019. 12. 16. · considered in the hydraulic...
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Contributions to the Dynamics of
South Foehn:
A Gap Flow Study during the
Mesoscale Alpine Programme
A dissertation submitted to the
Department of Meteorology and Geophysics,
University of Innsbruck
for the degree of
Doctor of Natural Science
presented by
Alexander Gohm
March 2003
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Proposal for the composition of the examining committee presented to the dean of
the Faculty of Natural Sciences at the University of Innsbruck as required for the
submission of the dissertation under § 62 (7) UniStG:
Reviewing committee:
1. Ao. Prof. Dr. Georg Mayr (advisor), University of Innsbruck
2. Prof. Dr. Christoph Schär, ETH Zurich
3. Prof. Dr. Stephen Mobbs, University of Leeds
Committee for oral examination:
1. Ao. Prof. Dr. Georg Mayr (advisor), University of Innsbruck
Subject: Dynamics of Mountain Flows
2. Prof. Dr. Laurence Armi, University of California, San Diego
Subject: Oceanography
3. Ao. Prof. Dr. Ignaz Vergeiner, University of Innsbruck
Subject: Atmospheric Waves
Date: March 2003
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To my father
wind
heaven
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”... and last, not least, how to choose among the great many
parameters to be varied and tested, how to look at three-dimensional
flow configurations with our restricted cinemascopic abilities, how to
help being swamped by the mass of data?”
(Vergeiner 1975)
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Abstract
Many facets of Alpine foehn winds have been investigated in the past. However,
one which has been neglected so far is the discussion of south foehn as gap flow
through deep incisions in the Alpine chain. The present thesis examines these gap
flows through the Brenner Pass and along the Wipp Valley (Austria/Italy). The
study was motivated by field measurements conducted during the Special Observ-
ing Period (SOP) of the Mesoscale Alpine Programme (MAP) in fall 1999.
The first part examines the applicability of single-layer hydraulic theory to the
Brenner Pass foehn. Numerical shallow-water simulations for a wide range of initial
conditions including shallow and deep foehn cases are discussed and compared with
selected observations of the MAP SOP, such as data from a ground-based scanning
Doppler lidar, an airborne aerosol backscatter lidar, radiosoundings, and automatic
weather stations. Radiosoundings reveal that the average MAP SOP foehn case had
subcritical flow south and nearly critical flow north of the Brenner. The hydraulic
model indicates flow transition to a supercritical state near the pass, and at least
nearly critical conditions in the northern part of the Wipp Valley. Prominent loca-
tions for hydraulic jumps are 2 km north of the pass and near the exit of the Wipp
Valley. The vertical topographic contraction exerts stronger control for the flow at
the Brenner gap than the lateral contraction. The observed and simulated across-
valley asymmetry in the strength of the foehn flow is a result of the complex valley
geometry rather than the influence of upper-level synoptic winds as suggested in a
previous study. Limitations of the applicability of the single-layer hydraulic concept
arise from the fact that foehn cases have been observed with no distinct inversion
or with an inversion depth comparable to the depth of the foehn flow. For such
cases the foehn cannot be described as a flowing layer being separated by a sharp
density step from a passive layer aloft. Further, turbulent entrainment processes
along the inversion layer, which change the reduced gravity across the pass, are not
considered in the hydraulic model. Nevertheless, the model captures essential flow
patterns such as the location of the pressure minimum, wind speed maxima, and
steeply amplified or breaking gravity waves resembled by hydraulic jumps.
The second part presents a case study of a deep south foehn, the MAP IOP 10
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event (24–25 October 1999), based on a detailed comparison and verification of
high-resolution numerical simulations with observations. The simulations were per-
formed with the Penn State/NCAR mesoscale model MM5. The observational data
sources were the same as in the first part and additionally contained a Doppler sodar.
As a first step, the study provides a synoptic-scale and mesoscale overview of the
event. The quantitative agreement between the numerical results and the observa-
tions is discussed in terms of root-mean square error and mean error. Discrepancies
detected at the beginning of the event are related to an unrealistic upstream profile
in the model which leads to the simulation of a spuriously extended shallow foehn
phase prior to the deep foehn stage. Evidence is found for the overestimation of
the mass flux through the lower Brenner gap and subsequent underestimation of
the flow descent on the leeward side of the pass. Nevertheless, the model captures
several of the striking foehn features: Simulated isentropes and aerosol backscatter
measurements both indicate regions of flow descent, across-valley flow asymmetries,
and hydraulic-jump-like features.
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Contents
Abstract v
Contents vii
1 General introduction 1
1.1 Alpine foehn studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Gap flow studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Goals and outline of the thesis . . . . . . . . . . . . . . . . . . . . . . 9
2 Hydraulic aspects of foehn 11
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Topographic environment . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Types of foehn . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Gap flow studies . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Model and measurement description . . . . . . . . . . . . . . . . . . . 20
2.2.1 The shallow-water model . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Weather stations and radiosoundings . . . . . . . . . . . . . . 21
2.2.3 Lidars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Motivation for hydraulic solutions . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Vertical structure of foehn winds . . . . . . . . . . . . . . . . 23
2.3.2 An exemplary case . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Numerical solutions for different upstream conditions . . . . . . . . . 26
2.4.1 Condition upstream of the pass . . . . . . . . . . . . . . . . . 26
2.4.2 Flow pattern along the Wipp Valley . . . . . . . . . . . . . . . 28
2.4.3 Flow pattern in the Wipp and Inn Valley . . . . . . . . . . . . 31
2.4.4 Statistical analysis of the flow structure . . . . . . . . . . . . . 34
2.5 Comparison of model flow fields with observations . . . . . . . . . . . 36
2.5.1 Flow condition up- and downstream of the pass . . . . . . . . 37
2.5.2 Flow pattern along the Wipp Valley . . . . . . . . . . . . . . . 38
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viii CONTENTS
2.5.3 Flow pattern in and across the Wipp Valley . . . . . . . . . . 43
2.5.4 Flow pattern across the Inn Valley . . . . . . . . . . . . . . . 46
2.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 48
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Mesoscale model verification: Case study of a deep foehn 53
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Model and measurement description . . . . . . . . . . . . . . . . . . . 59
3.2.1 The mesoscale model MM5 . . . . . . . . . . . . . . . . . . . 59
3.2.2 Weather stations and radiosoundings . . . . . . . . . . . . . . 60
3.2.3 The NOAA/ETL Doppler lidar . . . . . . . . . . . . . . . . . 60
3.2.4 The ZAMG Doppler sodar . . . . . . . . . . . . . . . . . . . . 61
3.2.5 The NCAR airborne backscatter lidar . . . . . . . . . . . . . . 61
3.3 Overview of the event . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 Synoptic-scale environment . . . . . . . . . . . . . . . . . . . 62
3.3.2 Mesoscale structure . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Comparison methodology . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5 Comparison with observations . . . . . . . . . . . . . . . . . . . . . . 71
3.5.1 Surface data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.2 Vertical profiles of the horizontal wind . . . . . . . . . . . . . 74
3.5.3 Backscatter intensities versus potential temperature . . . . . . 77
3.5.4 Spatial distribution of radial wind velocities . . . . . . . . . . 81
3.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 89
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4 Conclusions and outlook 93
Bibliography 97
Acknowledgments 107
Curriculum Vitae 109
Epilogue 111
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Chapter 1
General introduction
The present thesis is a contribution to the understanding of the dynamics of foehn
in an Alpine valley. It specifically investigates south foehn occurring in the Wipp
Valley near the Brenner Pass region (Austria/Italy). The emphasis is placed on the
discussion of foehn as gap flow through this pass and along the associated valley.
The study was motivated by field measurements during the Special Observing Period
(SOP) of the Mesoscale Alpine Programme (MAP) in fall 1999. The introductory
part contains a review of relevant foehn literature (section 1.1) and of gap wind
studies (section 1.2) and presents the major goals and the outline of the dissertation
(section 1.3).
1.1 Alpine foehn studies
The research on Alpine foehn has now a history of more than hundred years. The
earliest works go back to the middle of the 19th century (e.g. Hann 1866; Wild
1868). An extensive body of literature is available on many aspects of foehn, includ-
ing theories and concepts, synoptics and statistics, observational analyses as well
as climatic, physiological, and neurological effects (see e.g. Lehmann 1937; Ficker
and de Rudder 1943; Kuhn 1989). Referring to a bibliography compiled by Schlegel
(1975), Seibert (1985) mentioned a remarkable number of 139 foehn studies con-
ducted solely within the period of 1945–1975. A more recent foehn bibliography
compiled by Dürr (1999, 2000) contains more than 150 entries.
Foehn winds in the Innsbruck area (Inn and Wipp Valley) have been investigated
based on various approaches and theories. The results and ideas were not applied
exclusively to this region but also to other mountainous areas. Without trying to
be exhaustive, some of these studies are mentioned in the next few paragraphs.
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2 General introduction
In a series of four papers with the title ”Innsbruck foehn studies I–IV” Ficker (1906,
1910) investigated the dynamics of foehn based on observational case studies, Defant
(1907) searched for the explanation of periodic temperature fluctuations observed
at the beginning of foehn and during foehn breaks, and Trabert (1908) studied the
physiological influence of foehn on the human condition.
Worth mentioning are the foehn studies conducted by Ficker (1912, 1913, 1931)
based on several gas balloon flights with Innsbruck as take-off site. He reported
strong descent of the foehn flow to the lee of the mountain range north of Innsbruck
(”Karwendel”) on the order of 1500–2000 m. Frequent up- and downdrafts above
the jagged terrain were a typical feature, whereby the perturbations decayed with
increasing height. With the background of Billwiller’s (1899, 1903) studies and
based on his own observations, Ficker explained the reason for the foehn descent
into the valleys to be a compensating effect for the cold low-level air which drains
off the valleys into the Alpine foreland due to a depression northwest of the Alps (the
”aspirating” effect of the cyclone; see Billwiller 1899, Ficker and de Rudder 1943).
With this theory, he was in heavy dispute with Streiff-Becker (1931b, 1931a), who
explained the reason for the foehn descent as follows: A strong jet crossing the
mountain ridges sucks up air from the underlying stagnant cold layer embedded in
the valleys and consequently has to descend to compensate the mass deficit, i.e. to
fill the ”air hole” (”vacuum theory”; see Streiff-Becker 1931a).
Pibal soundings by Kanitscheider (1932, 1938) during foehn conditions in the vicin-
ity of Innsbruck yielded remarkable results. Based on two sets of ascents with a total
of 147 balloon trajectories, Kanitscheider found regions of flow splitting as well as
strong updrafts near the mountain range north of Innsbruck and flow separation
of the descending foehn winds to the lee of the mountains south of Innsbruck. He
classified some cases as ”shallow foehn” for which the southerly flow was only a few
100 m deep and the flow aloft turned to westerly directions. In a third paper, Kan-
itscheider (1939) investigated the flow pattern in the upper Inn Valley 23 km west
of Innsbruck (Telfs) at a location without a north–south aligned valley extending
towards the main Alpine crest.
A popular concept to explain the warming effect of the foehn flow on the leeward side
of mountains, not solely in the Alps, was the thermodynamic foehn theory. It goes
back to Hann (1866) and was discussed e.g. in Ficker and de Rudder (1943), however,
its origin can be traced back even to Espy (1841) (see Hann 1885). The key point
in this theory is that the difference in temperature is gained by a moist-adiabatic
ascent upstream of the barrier and a dry-adiabatic descent to the lee. This theory
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1.1 Alpine foehn studies 3
was reviewed and vigorously confuted by Seibert (1985, 1990). The concept relies on
the occurrence of precipitation on the windward side, whereas foehn cases have been
observed without an upstream cloud cover. Based on observations collected during
the Alpine Experiment (ALPEX) in 1982, Seibert rehabilitates a different and a
today widely accepted explanation for the temperature difference across the Alps
during south foehn: It is mainly caused by the warming of the air on the leeward
side due to the descent of the flow from the crest level into the valleys, whereas
upstream of the Alpine ridge the stable stratification favors low-level blocking and
the formation of cold pools. A contribution to the warming by latent heat release
may be not unlikely but is of minor importance.
An interesting approach to explain the dynamics of foehn, which is specifically rel-
evant to the present thesis, is the one by Schweitzer (1953). He compared foehn
winds with a hydraulic flow of a single layer with a free surface. In the stratified
atmosphere this free surface would be represented by a temperature inversion, i.e.
an interface forming a density step between a lower and an upper layer. Foehn was
explained as the flow of the lower layer which has a supercritical velocity, i.e. a flow
speed exceeding the velocity of linear shallow-water gravity waves. Rotors occurring
in the flow were interpreted as hydraulic jumps. As one specific example, Schweitzer
mentioned the foehn which originates at the Brenner Pass and develops into a strong
supercritical flow along the Wipp Valley.
Two publications investigated foehn in a theoretical/analytical manner. Mayr (1987)
studied the dynamics of shallow foehn along theWipp Valley cross-section based on a
linearized two-dimensional model with an isentropic coordinate system, whereby the
mass-flux into the Inn Valley was parameterized. Sensitivity studies showed that a
reduction of the stability increases the amount of flow over the mountain range north
of Innsbruck, i.e. decreases the amount of mass deflected into the Inn Valley, but
also increases the strength of the flow descent above the Inn Valley. Vergeiner (1996)
presented a conceptual dynamic model for the foehn penetration into an idealized
two-dimensional valley filled with a cold pool. He derived a regime diagram for the
total flow penetration with two Froude numbers based on the buoyancy frequency
of the free air stream and of the valley atmosphere. Vergeiner noted the limited
suitability of his 2D model for the application to the Inn Valley due to the (highly
3D) gap-like topography of the Wipp–Inn Valley intersection.
During recent years the early conceptual models and theories of Alpine foehn were
supplemented by a series of three-dimensional numerical simulations of the flow
field. Vergeiner (1975, 1976) developed a linearized model for wavelength scales
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4 General introduction
between 10 and 100 km and applied it to the mountain-induced gravity wave flow
in the Innsbruck area. In agreement with the observations by Kanitscheider (1932,
1938), the simulated surface wind pattern reveals flow divergence at the exit of the
Wipp Valley. For various different initial conditions, orographically induced gravity
waves produced flow patterns with topographically fixed features, such as updraft
regions. The simulations revealed typical magnitudes of the vertical velocities of
≈ 5–10 m s−1 at about 1 km above the mountain summits. Although the linearizedmodel provides valuable insight into the foehn dynamics, it obviously has to fail if
the gravity wave amplitudes become large, i.e. if non-linear effects become relevant.
Zängl (1999) used a non-linear mesoscale model to study the foehn structure
based on a highly idealized Wipp Valley–Inn Valley terrain configuration. He noted
the importance of the westerly outflow of cold low-level air from the Inn Valley into
the Alpine forelands in order to observe a foehn breakthrough in the Inn Valley. Fur-
ther, he showed that both surface friction and positive vertical wind shear decrease
the likelihood or intensity of wave breaking and that the shape of the Wipp Valley,
i.e. whether it widens north of the pass or not, considerably influences the gravity
wave pattern above the valley and the strength of the foehn near the pass. Recently,
Zängl (2003) and Zängl et al. (2003b) performed high-resolution numerical simula-
tions for the foehn flow over realistic orography. In the first work, idealized initial
and boundary conditions for shallow and deep foehn were used and the results were
compared with surface observations from the MAP SOP. A gravity wave asymmetry
was found in the Inn Valley with stronger wave amplitudes to the east of Innsbruck
than to the west causing a local pressure gradient along the Inn Valley which seems
to drive the usually observed westerly pre-foehn at Innsbruck. The second paper
is a case study of the MAP Intensive Observing Period (IOP) 10 deep foehn event,
which was conducted in order to assess whether it is possible to reproduce the tem-
poral evolution and the spatial structure of a south foehn with a high-resolution
model having a mesh size of ≈800 m. Model data were compared with observationsfrom automatic weather stations and radiosoundings. The study showed encourag-
ing results but also revealed discrepancies concerning, e.g., the small-scale structure
of the wind field near Innsbruck, the amount of turbulent vertical mixing along the
Wipp Valley, and the intensity of precipitation on the southern side of the Alps.
Flamant et al. (2002) presented a simulation with a non-hydrostatic mesoscale
model as part of an extensive study of the MAP IOP 12 foehn case (see below).
The spatial resolution was rather coarse (≈2 km) in the face of the details of theBrenner gap. Nevertheless, the results were useful to complement the flow picture
drawn from the observational analysis.
Two field experiments, ALPEX and MAP, stimulated several observational studies of
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1.2 Gap flow studies 5
the foehn in the Brenner Pass region. Seibert (1985) presented analyses of potential
temperature and pressure at the surface, isentropic cross-sections, and investigations
of the wind field of several ALPEX foehn cases for a regional scale (Innsbruck area)
and a larger scale (Alps) and conducted statistical foehn investigations for a period
of four years. The main results were summarized in Seibert (1990). For a south
(and a north) foehn case prior to ALPEX, Steinacker (1983) compiled analyses of the
pressure pattern on isentropic surfaces and showed vertical transects of potential and
equivalent-potential temperatures across the Alps near the Brenner Pass. Fine-scale
pressure and wind analyses in the Inn andWipp Valley of an ALPEX deep foehn case
were published by Vergeiner et al. (1982). Hoinka (1985, 1990) presented airborne
observations of the potential temperature and wind field for ALPEX foehn flights.
Since the MAP SOP in 1999, a few observational studies have been published.
Mayr et al. (2002) discussed observations taken with a car-based platform along
the Brenner Pass transect for the 30 October 1999 (MAP IOP 12) foehn case.
These surface measurements revealed locations of hydraulic jumps and were useful
to extend aircraft observations to the ground. The same event was discussed in an
exemplary manner by Mayr et al. (2003a, 2003b) and studied in detail by Flamant
et al. (2002) based on Doppler lidar data, airborne backscatter lidar observations,
airborne in situ measurements, radiosoundings, and data from automatic weather
stations. Durran et al. (2003) compared ground-based Doppler lidar and airborne in
situ wind measurements for three different MAP foehn events and calculated RMS
errors and bias to estimate the differences between the two data sources.
With the background of this bulk of Alpine foehn literature, the legitimate question
arises if further research on foehn is necessary. We have learned from these studies
that the dynamics of foehn has many different facets. Some of them have been
neglected in the past and therefore gaps – especially orographic ones – have to
be ”filled”. The concept of gap flows has been applied to the Wipp Valley foehn
particularly since the planning phase of the Mesoscale Alpine Programme (MAP)
field experiment (Bougeault et al. 1998). Since the major aim of the present thesis
is to discuss foehn winds as a gap flow phenomenon, a short review of its literature
deserves a separate section.
1.2 Gap flow studies
Gap winds have been observed in many parts of the globe – wherever an incision oc-
curs in the orography and allows the flow to pass through it. Examples are manifold:
winds in the Strait of Gibraltar (Scorer 1952; Dorman et al. 1995), the Mistral in the
French Rhône Valley (Pettre 1982), strong winds near Hokkaidō in Japan (Arakawa
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6 General introduction
1969), airflow near Unimak Island in the Aleutian Chain (Pan and Smith 1999),
winds in the Howe Sound of British Columbia (Jackson and Steyn 1994a, 1994b;
Finnigan et al. 1994), winds in straits and gaps formed by the Coast/Cascade
Mountains between British Columbia and Washington State (Reed 1931; Overland
and Walter 1981; Mass et al. 1995; Colle and Mass 2000; Doyle and Bond 2001),
airflow through gaps embedded in the Washington Cascade Mountains (Cameron
1931; Mass and Albright 1995; Colle and Mass 1998a, 1998b; Sharp and Mass 2002),
winds in straits, inlets, and gaps near the coast of Alaska (Lackmann and Overland
1989; Macklin et al. 1990; Colman and Dierking 1992; Bond and Stabeno 1998),
winds near a Swedish lake embedded in high mountains (Smedman and Bergström
1995), the outflow over the Gulf of Tehuantepec in Mexico (Schultz et al. 1997;
Steenburgh et al. 1998), a coastal jet with partly gap-flow characteristics at Spits-
bergen (Sandvik and Furevik 2002), intense upvalley winds in the Himalayan Kali
Gandaki Valley (Egger et al. 2000, 2002; Zängl et al. 2001), and the foehn across
the Brenner Pass in the Alps (Flamant et al. 2002; Mayr et al. 2003a).
In many of these cases, especially for gaps embedded in elongated barriers, the gap
flows are highly ageostrophic and follow the pressure gradient force along the gap axis
(e.g. Overland and Walter 1981). The pressure difference is frequently associated
with a pool of cold air upstream of the barrier (e.g. Sharp and Mass 2002). In other
cases the large-scale geostrophic flow is perpendicular to the mountain ridge and
thus aligned parallel to the gap winds (e.g. Pan and Smith 1999). The theories for
the strength of the gap flow were diverse. In the early days, Reed (1931) explained
the gap winds in the Strait of Juan the Fuca as a Venturi effect due to the horizontal
contraction of the sidewalls. Such an effect would place the strongest wind at the
narrowest section. However, it was shown later that the velocity maximum occurred
further downstream (e.g. Colle and Mass 2000). Gap winds driven by the outflow of
cold air along a channel showed characteristics of hydraulic flow (e.g. Jackson and
Steyn 1994b). Even for the airflow through gaps formed by isolated hills with no
stagnant cold pool upstream, shallow-water dynamics seemed to capture the essence
of the phenomenon (Pan and Smith 1999). In these cases, gap winds extending
downstream of the obstacle occur since they avoid Bernoulli loss and maintain their
velocity whereas the adjacent strong flow over the hills soon becomes subcritical and
forms a wake with weaker winds (see discussion in chapter 2). Related to gap winds
is the phenomenon of channeling such as observed in the German Upper Rhine
Valley. The key feature here is the decoupling of the flow in the valley from the
upper-level flow as the valley winds follow the pressure gradient force imposed by
the geostrophic wind balance (e.g. Wippermann 1984; Kalthoff and Vogel 1992).
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1.2 Gap flow studies 7
The hydraulic aspects of flow through a horizontal contraction have been investi-
gated by several studies in a theoretical framework for continously stratified fluids as
well as single-layer shallow-water fluids. As an extension of Long’s 1954 theory and
Houghton and Kasahara’s 1968 numerical solutions for the shallow-water flow over
an infinitely long ridge, Arakawa (1969) developed the theory for a single-layer flow
in a rectangular channel through a combined lateral and vertical contraction. The
non-dimensional parameter defining the state of the flow is the local Froude number
F = U/√g∗H, with U and H being the fluid velocity and depth, respectively, and
g∗ being the reduced gravity. F describes the ratio between the fluid speed and the
velocity of linear shallow-water gravity waves. If an initially subcritical flow (F < 1),
i.e. a flow with relatively small velocities and substantial fluid depth, impinges on
a topographic contraction, i.e. rising terrain and/or narrowing channel, it can be
brought to a critical state (F = 1) at the highest and narrowest section – the ”point
of control”. The flow will become asymmetric with respect to the control, i.e. the
fluid will continously accelerate and reduce its free surface height across the contrac-
tion and will become supercritical (F > 1) to the lee. A hydraulic jump will occur
further downstream as an adjustment to the far-downstream subcritical condition.
The jump is an dissipative process which destroys kinetic energy and thus reduces
total energy (Bernoulli head). Since the control fixes the layer depth for a given
flow rate, partial blocking will occur on the windward side of the control with a bore
propagating upstream. If the highest and narrowest section of the channel do not
coincide, the location of the control is not uniquely specified by the geometry of the
channel alone but also depends upon the flow rate (Armi 1986).
The extension of the steady single-layer dynamics to the one of a stably stratified
fluid flowing through a contraction was made e.g. by Wood (1968). He studied
the flow from a stagnant reservoir through a horizontal contraction. The bottom
(top) of the lowest (uppermost) flowing layer was bounded by a stagnant fluid and
represented a surface along which the pressure gradient was zero. For two flowing
layers, he found a solution for which the velocity and density distributions are self
similar (which means e.g. that the relative change of the layer depth is the same in
each layer). Critical flow may occur at other points than the narrowest section, i.e.
at a virtual control (see also Armi 1986). For a multi-layered system the accelerating
flow is controlled at the narrowest point in the channel and through a succession of
virtual controls, at each of which the flow passes from subcritical to supercritical with
respect to a particular wave mode (see also Armi and Williams 1993). Økland (1985)
took Wood’s idea and applied it to the atmosphere, i.e. used isentropic surfaces
instead of density surfaces as streamlines. Vergeiner et al. (2003) recently tackled
the problem of adding a vertical constriction to the horizontal one (resembling the
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8 General introduction
geometry of a mountain pass) and allowing for the pressure gradient to be non-zero
at the surface (for a realistic case the pressure decreases along the valley bottom
upstream of the pass). They took Wood’s model only to a certain distance upstream
of the pass, from which on they continued with a single-layer model. In this setup
the acceleration of the flow was enough to reach a critical Froude number near the
pass height and supercritical speeds beyond. As a first step, however, this model
was designed to represent the conditions only upstream of the pass.
Recently, several three-dimensional numerical modeling studies investigated gap
wind dynamics in an idealized framework. They assumed idealized ridge-like to-
pographies having embedded gaps, and idealized initial and boundary conditions.
The influence of ambient rotation on the formation of gap flows based on a back-
ground flow mainly parallel to the ridge was studied by Sprenger and Schär (2001)
and Zängl (2002a). Such a configuration resembles e.g. the one of shallow foehn
in the Alps. Gaberšek (2002) and Zängl (2002c) investigated gap flows evolving
from a background flow which was basically normal to the ridge, i.e. had a major
component parallel to the gap axis. Such a setup might represent e.g. the one
of the Mistral winds in the French Rhône Valley. A more thorough discussion of
theses studies is given in chapter 2. Ross and Vosper (2003) investigated the role
of surface friction in stratified flow for two types of topographies: the flow through
an infinitely long valley (2D) and through a mountain pass embedded in a ridge
(3D). Outside the boundary layer the background flow was along the valley (normal
to the ridge), whereas inside the boundary layer the flow had a cross-valley (ridge-
parallel) component due to surface friction. For the 2D valley setup, separation
of the boundary-layer flow from the valley surface and the formation of a blocked
region in the valley was observed for sufficiently small Froude numbers (based on
the valley depth). For the 3D pass setup and for sufficiently strong flow, potential
vorticity generated within the boundary layer by surface friction was advected into
the interior of the flow downwind of the mountain by a hydraulic jump and thus
provided a mechanism for the formation of potential vorticity banners.
-
1.3 Goals and outline of the thesis 9
1.3 Goals and outline of the thesis
The aims of this dissertation are closely related to two of the five scientific objectives
of the Mesoscale Alpine Programme (MAP) as stated in the MAP Design Proposal
(Binder and Schär 1996; see also Bougeault et al. 2001):
(2a) To improve the understanding and forecasting of the life cycle of Foehn-related
phenomena, including their three-dimensional structure and associated bound-
ary layer processes.
(3) To provide data sets for the validation and improvement of high-resolution
numerical weather prediction, hydrological and coupled models in mountainous
terrain.
Further, the thesis addresses two of the three scientific questions which were formu-
lated in the MAP Science Plan (Bougeault et al. 1998) for the gap flow project P4
in the Brenner Pass region:
(1) To determine the relative importance of gap width versus terrain elevation
changes along the floor axis on deep, continuously stratified flow through real-
istic topography.
(3) To study the vertical and cross-gap distribution of wind speed and thermody-
namic properties. These are controlled by inviscid stratified dynamics together
with surface friction along the valley floor and side walls. The frictional effects
as well as dissipation and mixing of the low-level high speed flow need to be
included in realistic models.
As mentioned in the MAP Science Plan, the flow beneath a strong low-level in-
version may, in a first approximation, be studied using single-layer reduced-gravity
hydraulics. Following this idea, the first part of this thesis (chapter 2) investigates
the hydraulic aspects of the gap winds in the Wipp Valley for south foehn conditions.
The main aim is to determine how well single-layer hydraulics captures the essence
of the foehn dynamics manifested e.g. in the structure of the flow descent and in
the locations of local wind maxima, hydraulic jumps, and flow splitting. A further
aim is to find the reason for across-valley asymmetries of the flow strength and to
determine the relative importance of the vertical versus the lateral contraction of
the Brenner Pass. For this purpose, a numerical single-layer shallow-water model
designed by Schär and Smith (1993a, 1993b) is applied to the foehn flow in the Bren-
ner Pass region for various initial foehn conditions, i.e. different upstream Froude
numbers and fluid layer heights. The model version used in this study assumes a
two-dimensional hydrostatic flow over realistic orography neglecting surface friction
-
10 General introduction
and ambient rotation. The model fields are compared with measurements from the
MAP SOP. The results of this study were submitted to the Quarterly Journal of the
Royal Meteorological Society.
The second part (chapter 3) provides verification of a specific deep foehn simulation
conducted with a high-resolution three-dimensional numerical weather prediction
system, the Penn State/NCAR mesoscale model MM5. The verification is based on
a detailed comparison of the model fields with remote sensing observations (lidar
and sodar), surface measurements, and radiosoundings, which were collected during
the MAP IOP 10. The aims are to assess the quality of a high-resolution simulation
with a state-of-the-art model concerning details of the flow structure, try to quantify
differences between predicted and observed fields based on error measures, as well
as to illuminate the reasons for these discrepancies. The results of this investigation
were submitted to the Monthly Weather Review.
The last part (chapter 4) briefly summarizes the main results from the previous two
studies, whereby more detailed conclusions are given in section 2.6 and 3.6. This
chapter also provides an outlook for future work to address remaining questions.
-
Chapter 2
Hydraulic aspects of foehn
11
-
12
-
Hydraulic aspects of foehn winds in an Alpinevalley ∗
Alexander Gohm † and Georg J. Mayr
Department of Meteorology and Geophysics, University of Innsbruck, Austria
Summary
This study examines the applicability of single-layer hydraulic theory to Alpine foehn
winds for the example of the flow in the Brenner Pass region (Austria/Italy). South foehn
is described as gap flow over the pass and along the associated Wipp Valley. Numerical
shallow-water simulations for a wide range of initial conditions including shallow and deep
foehn cases are discussed and compared with selected measurements collected within the
Mesoscale Alpine Programme (MAP). The observational analyses are based on Doppler
and aerosol backscatter lidar data for a specific foehn case and on surface observations
and radiosoundings for all foehn cases observed within the 70-day MAP Special Observing
Period (SOP).
Radiosoundings at a location up- (south) and downstream (north) of the pass reveal
that the average MAP SOP foehn case had subcritical flow south and nearly critical flow
north of the Brenner. The hydraulic model indicates flow transition to a supercritical
state near the pass, a hydraulic jump 2 km further north, mainly subcritical flow in the
upper Wipp Valley except near the Brenner, and especially for shallow foehn a tendency
for becoming supercritical in the lower Wipp Valley. The model results suggest that the
vertical topographic contraction exerts stronger control for the flow at the Brenner gap
than the lateral contraction. In accordance with Doppler lidar measurements the hydraulic
model captures the decrease of the wind speed in across-valley direction from east to west
in the northern part of the valley. This asymmetric flow pattern is a result of the complex
valley geometry rather than the influence of upper-level synoptic winds as suggested in a
previous study.
For some observed cases the strength of the temperature inversion on top of the foehn
layer decreased across the pass presumably due to entrainment processes. Consequently,
the reduced gravity decreased as well. The less than perfect agreement between the hy-
draulic parameters retrieved from soundings and those provided by the model has two
major reasons: The model assumes a constant reduced gravity and the calculation of hy-
draulic parameters for foehn cases without a strong and sharp temperature inversion is
ambiguous. Essential flow patterns in the simulated fields, such as the location of the pres-
sure minimum and wind speed maxima, compare well with the surface observations along
a valley transect. Indications for a steeply amplified or breaking gravity wave resembled
by a hydraulic jump to the lee of two mountain ridges are found in lidar observations as
well as simulations.
∗Submitted to Quarterly Journal of the Royal Meteorological Society, February 2003†Corresponding author: Department of Meteorology, University of Innsbruck, Innrain 52, A-
6020 Innsbruck, Austria; e-mail: [email protected]
13
-
14 Hydraulic aspects of foehn
2.1 Introduction
The goal of this study is to elucidate the dynamics of south foehn winds in an
Alpine valley by means of single-layer hydraulic theory. We address the question
how well hydraulics captures the features of gap winds using the example of the
foehn flow over the Brenner Pass (Austria/Italy) and along the associated Wipp
Valley. The work is part of extensive research activities which have been carried out
within the framework of the Mesoscale Alpine Programme (MAP). Our investigation
incorporates part of the data set collected during the 70-day MAP Special Observing
Period (SOP) in fall 1999 (Bougeault et al. 2001). The observational analysis
complements the numerical results of a shallow-water model.
2.1.1 Topographic environment
The European Alps act as a major obstacle for the impinging large-scale atmospheric
flow. Instead of being simply an elongated, arc-shaped barrier with a smooth surface,
the Alpine massif has a complex structure with many embedded valleys (Fig. 2.1).
Some of them are aligned towards the main Alpine crest where they form more or less
deep incisions in the crest line. These usually very narrow mountain passes are not
only hot spots for the channelling of transalpine traffic causing air quality problems
and noise pollution, but also for the channelling of airflow leading to the formation of
gap flows with partly severe winds. The Brenner Pass (BRE), located in the central-
northeastern part of the Alps, is at 1373 m MSL one of the deepest incisions in the
Alpine chain. The associated valley is the Wipp Valley which stretches in a south-
north direction from the Italian town Sterzing (STZ) to the Austrian city Innsbruck
(IBK) where it merges with the east-west aligned Inn Valley (Fig. 2.2(a)). To the
north, the Inn Valley is bounded by the mountain range Nordkette. The Wipp
Valley has several tributaries with the largest being the Stubai and the Gschnitz
Valley. The pass has a double-gap structure (Fig. 2.2(b)). The base heights (central
widths) are approximately 1.4 km MSL (2 km) for the lower gap and 2.1 km MSL
(15 km) for the upper gap.
2.1.2 Types of foehn
Describing Alpine south foehn as gap flow is a new concept which has been applied
especially since MAP. However, ”gap flow” and ”foehn” are not synonyms. The
former is a rather distinctive description for the below-crest-level flow through the
Brenner gap and along the Wipp Valley. South foehn includes the low-level gap flow
but is a rather general description for the mountain induced downslope windstorm
to the lee of the Alps. It is therefore not necessarily restricted to the locations of
-
2.1 Introduction 15
Longitude (deg E)
Latit
ude
(deg
N)
Gulf ofGenoa
AdriaticSea
Po Valley
(a)
4 6 8 10 12 14 1643
44
45
46
47
48
49
x (km)
y (k
m)
(b)
BRE
ZUGInn Valley
WippValley
P1
P2
P3
P4
−40 −20 0 20 40
−40
−20
0
20
40
60
Figure 2.1: Topographic map of the European Alps (a) and the MAP Brenner Pass
target area (b). The rectangle in (a) shows the location of the full SWM domain which is
displayed in (b). In (b) the solid rectangle indicates the subdomain of Figs. 2.2(a), 2.6–
2.8 and the two dotted rectangles show the subdomains of Figs. 2.13–2.14. Grey shaded
elevation contours in (a) start at 0 m with increments of 600 m and in (b) at 900 m with
increments of 300 m. The 600-m contour line in (b) is light grey. The dashed line in (b)
denotes the cross-section of Fig. 2.5. The solid line P1P2 indicates one flight leg of the
NCAR Electra aircraft on 20 October 1999 as shown in Fig. 2.15. The solid line P3P4
denotes the cross-section of Fig. 2.12. Filled circles mark the Brenner Pass (BRE, 1373 m)
and the mountain peak Zugspitze (ZUG, 2962 m).
topographic gaps. South foehn is usually divided into a shallow and a deep type
depending on the structure of the above-crest level winds (e.g. Kanitscheider 1932;
Seibert 1990). For the shallow foehn case geostrophic winds have rather westerly
directions and are therefore aligned parallel to the main Alpine ridge or upper-level
winds may be weak at all. The driving force for the cross-Alpine flow through
the gaps is a northward-pointing pressure gradient force which has basically two
potential sources: a synoptic-scale pressure difference due to a depression north of
the Alps and a hydrostatically induced pressure difference on the mesoscale due to
air mass differences north and south of the Alps. The southerly flow is confined to
below the main Alpine crest height at approximately 3 km MSL. For the deep foehn
case the geostrophic wind direction above the crest has a component perpendicular to
the Alpine ridge and the forcing for cross-barrier flow is therefore stronger. The Alps
are on the leading edge of a synoptic-scale pressure trough and southerly winds are
-
16 Hydraulic aspects of foehn
x (km)
y (k
m)
(a)
STZ
IBK
SER
NOS
SAB BRE
ELB GED
STE
PAK
IV
SV
GV
WV
NK
−30 −20 −10 0 10
−10
0
10
20
30
40
−30 −20 −10 0 10
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
Hei
ght (
km M
SL)
x (km)
(b)
SWMHigh Resolution
Figure 2.2: Topographic map of the Wipp Valley region (a) and height of the main
Alpine crest across the Brenner Pass (b). Elevation contours in (a) are shaded as in
Fig. 2.1(b). The white dashed line in (a) indicates the crest line shown in (b). The black
dotted line in (a) denotes the cross-section of Fig. 2.5. The solid and dashed lines in
(b) are the crest lines as represented by the SWM orography and by a 30-m-resolution
orography, respectively. Crosses in (a) indicate automatic weather stations, among others,
at Sterzing (STZ, 944 m), Brenner (BRE, 1373 m), Steinach (STE, 1116 m), Gedeir (GED,
1084 m), Ellbögen (ELB, 1080 m), and Innsbruck (IBK, 609 m). Filled circles in (a) mark
the mountain peaks Sattelberg (SAB, 2107 m), Nösslachjoch (NOS, 2231 m), Serles (SER,
2717 m), and Patscherkofel (PAK, 2252 m). Italic letters in (a) denote the location of the
Wipp Valley (WV ), Inn Valley (IV ), Stubai Valley (SV ), Gschnitz Valley (GV ), and the
northern range Nordkette (NK ).
therefore present at least throughout the lower troposphere. Downslope windstorms
to the lee may be observed even in valleys which have no origin at the main crest,
i.e. which are rather west-east aligned. In the Wipp Valley a low-level gap flow is
embedded in the deep foehn flow.
2.1.3 Gap flow studies
Gap flows occur in many mountainous parts of the world. Pan and Smith (1999)
compiled an extensive list of literature about gap wind observations. They also
reviewed two analytical models, linear theory of continuously stratified flow and
non-linear shallow-water flow, for the application to gap winds.
-
2.1 Introduction 17
Continuously stratified gap flows
Armi and Williams (1993) studied the steady hydraulics of a continuously stratified
fluid flowing through a horizontal contraction in an experimental and theoretical
manner. Their results showed that the flow passes from sub- to supercritical (with
respect to a particular wave mode) at a succession of virtual controls as the flow
accelerates through the contraction. Recently, several numerical studies treated gap
winds as stratified flow past idealised gap-like topographies (Sprenger and Schär
2001; Zängl 2002a, 2002c; Gaberšek 2002). Sprenger and Schär (2001) investigated
the role of earth’s rotation assuming a geostrophically balanced westerly flow par-
allel to an infinitely long ridge with an embedded gap excluding surface friction.
This setup resembles idealized conditions for shallow south foehn in the Alps. The
proposed mechanism for the formation of the gap winds is the decoupling of the flow
within the gap from the geostrophic flow aloft. The southerly gap flow is then driven
by the geostrophic pressure gradient across the ridge. Zängl (2002a) showed that
the replacement of this infinite ridge by an isolated one fundamentally changes the
gap flow pattern. Flow splitting on the upstream edge of the isolated mountain and
Coriolis effects can even result in a weak northerly gap flow since the geostrophic
pressure gradient is compensated or even reversed by potentially cold low-level air
which is piled up along the northern side of the mountain. Simulations with an
arc-shaped mountain similar to the real Alps showed that a southerly component in
the large-scale flow appears to be necessary to obtain southerly gap flows. In such
a case a hydrostatic pressure difference across the Alps forms as potentially cold
low-level air is piled up on the southern side of the Alps. Two studies discussed,
among other details, gap winds arising in a non-rotating environment for large-scale
flow perpendicular to an elongated mountain with an embedded level gap (Zängl
2002c; Gaberšek 2002). Zängl (2002c) noted that in the linear regime wind and
pressure perturbations along the gap are caused by gravity waves radiating from
the mountain towards the gap axis, and by low-level confluence within the gap.
In the non-linear regime, however, confluence becomes negligible due to upstream
blocking and the amplitude of the vertically propagating gravity waves is reduced
due to low-level wave breaking. In this regime the driving force for the gap flow
is the low-level pressure difference across the mountain. Similar to Zängl’s results,
Gaberšek (2002) identified two mechanism responsible for gap flow enhancement de-
pending on the non-dimensional ridge height ε = Nh/U (which describes the degree
of non-linearity), with h being the obstacle height, U the uniform fluid velocity, and
N the buoyancy frequency. The first one is characterized by a mountain lee-wave
(ε ∼ 1) and the second one by upstream flow blocking (ε ≥∼ 5).
-
18 Hydraulic aspects of foehn
Single-layer hydraulic gap flows
The simplified problem of a single-layer fluid with a free surface flowing through a
gap is related to the problem of a single-layer shallow-water flow over a ridge (see
the review of its theory e.g. in Henderson 1966; Baines 1995; Durran 1990). The
continuous acceleration of the flow across the ridge is related to the transition from
a subcritical into a supercritical state at the summit where the local Froude number
F = U/√g∗H becomes unity, i.e. where the speed of the flow U equals the speed
of linear shallow-water gravity waves√g∗H (with g∗ being the reduced gravity and
H the fluid layer depth). Further downstream of the obstacle, a hydraulic jump
causes the transition back into a subcritical state. Dissipation of energy in the jump
reduces the Bernoulli function. The two parameters controlling the flow are the
upstream Froude number F∞ = U∞/√g∗H∞ and the non-dimensional mountain
height M = hm/H∞ (ratio between the ridge height and the upstream depth of
the fluid layer). Arakawa (1969) presented a theory which combines the effects of a
lateral and a vertical contraction of the orography for the flow through a channel.
He applied it to gap winds occurring in Hokkaidō, Japan. A lateral contraction
due to a gap has a similar effect as a vertical contraction due to a ridge or as a
combination of both (rising terrain height and decreasing channel width). All three
configurations favour flow transition from a sub- to a supercritical state at the point
of control which is at the narrowest and highest part of the orography. A more
complicated situation occurs if the highest and narrowest section do not coincide.
In such a case critical flow will occur somewhere in between and the exact location of
the control depends not only on the geometry of the channel but also on the flow rate
(Armi 1986). Applications of the single-layer hydraulic concept with a comparison
to observations include studies of gap winds in the Rhône Valley (Pettre 1982), near
the Howe Sound (Jackson and Steyn 1994b; Finnigan et al. 1994), in the Strait
of Gibraltar (Dorman et al. 1995), and near the Aleutian Chain (Pan and Smith
1999). The shallow-water simulations of Pan and Smith (1999) showed qualitative
agreement with the flow field derived from satellite-borne synthetic aperture radar
(SAR) images. Their numerical results obtained for real terrain geometry (mountain
gaps in Unimak Island) as well as for a family of idealized gap topographies (gap
formed by two bell-shaped hills) showed that critical conditions always occurred first
at the mountain peaks. They concluded that the effect of rising terrain dominates
over the effect of difluence (near a peak) and confluence (near a saddle). The authors
describe gap winds as streams of air that have avoided Bernoulli loss over the terrain
by passing through gaps and thus have maintained their strength in the wake – in
contrast to the airflow over the adjacent hills that has experienced strong Bernoulli
loss in hydraulic jumps leading to weak flow in the wake.
To our knowledge, the first work which discussed the similarity between the
-
2.1 Introduction 19
dynamics of south foehn in the Wipp Valley and the one of a hydraulic flow with a
free surface was by Schweitzer (1953). His interpretation of south foehn was the one
of a shooting flow with supercritical velocity, i.e. with fluid velocities exceeding the
linear gravity wave speed. These gravity waves are excited at an internal interface
formed by a temperature inversion. Such an inversion layer is a frequently observed
foehn property (e.g. Seibert 1990). Schweitzer explained the observed rotors in the
foehn flow as hydraulic jumps. The description of foehn by means of hydraulic theory
experienced a renaissance during the planning phase of the MAP SOP (Bougeault
et al. 1998). It was noted that the flow beneath a strong low-level inversion may
be studied, in a first approximation, using single-layer reduced-gravity hydraulics.
Flamant et al. (2002) followed this idea in a study of a shallow foehn event observed
during the MAP Intensive Observing Period (IOP) 12. A quasi two-dimensional
steady-state hydraulic model was used for the calculation of the Froude number along
the Wipp Valley. Essentially, this model describes the flow through a rectangular
channel having rigid side walls with varying channel width and bottom height, and
with constant fluid layer height across the valley. The underlying concept is similar
to the one used in Jackson and Steyn (1994b) for the Howe Sound, except that the
latter included the effect of an external large-scale pressure gradient and of surface
friction. The change of the channel width along the Wipp Valley was idealized with
a parabolic function. There are several problems connected to this simplified quasi
two-dimensional consideration: The concept neglects the effects of the tributary
valleys, which is justified for cases only where the flow into the Wipp Valley is
weak. The definition of the channel width at locations where tributary valleys
merge with the Wipp Valley seems to be problematic and to some extent arbitrary.
As mentioned also by the authors, the treatment of the flow to be homogeneous in
across-valley direction (which arises from the definition of a rectangular geometry
of the valley) has its limitations since the real valley topography has a complex
structure favouring across-valley variations of the flow. Finally, cases where the
fluid height exceeds the height of the sidewalls cannot be treated and therefore the
quasi two-dimensional concept is limited to very shallow foehn cases.
Our approach of the problem differs from the quasi two-dimensional concept in
the point that we solved the time-dependent hydraulic equations for a truly two-
dimensional single-layer shallow-water flow over realistic orography. It is therefore
similar to the approach in Pan and Smith (1999) and avoids the above mentioned
problems of the quasi two-dimensional concept. The paper is organized as follows:
Section 2.2 introduces the hydraulic model and describes the observational data
set. Section 2.3 gives a motivation for the application of hydraulic theory to foehn
winds in the Wipp Valley. The numerical results for various upstream conditions are
discussed in section 2.4. Section 2.5 presents a comparison of the hydraulic solution
-
20 Hydraulic aspects of foehn
with observations. The study concludes with a summary in section 2.6.
2.2 Model and measurement description
2.2.1 The shallow-water model
The numerical shallow-water model (SWM) used in this work was developed by
Schär and Smith (1993a, 1993b). Applications of the SWM include e.g. the study
of bottom friction for the flow past an isolated obstacle (Grubǐsić et al. 1995), wake
formation by mountains (Smith and Smith 1995; Smith et al. 1997), gap winds
in the Aleutian Chain (Pan and Smith 1999), and the generation of shocks in a
single-layer (Jiang and Smith 2000) and a two-layer (Jiang and Smith 2001a, 2001b)
flow. The reader is referred to these studies for a detailed description of the model.
Our version of the SWM is based on the two-dimensional governing equations for a
single-layer hydrostatic flow without ambient rotation and surface friction:
Dû
Dt̂+∂(ĥ + Ĥ)
∂x̂= 0, (2.1)
Dv̂
Dt̂+∂(ĥ + Ĥ)
∂ŷ= 0, (2.2)
∂Ĥ
∂t̂+∂(ûĤ)
∂x̂+∂(v̂Ĥ)
∂ŷ= 0, (2.3)
with the non-dimensional variables (henceforth generally labelled with hats) û and v̂
as the two horizontal velocity components, Ĥ and ĥ as fluid layer depth and terrain
height, respectively, Ẑ = ĥ+Ĥ as fluid layer height, and t̂ as time. Equations (2.1)–
(2.3) are non-dimensionalised with the following scales: a typical length L for the
horizontal length scale, the initial far-upstream depth of the fluid layer H∞ (with
h∞ = 0) for the vertical length scale, the phase speed of linear gravity waves√g∗H∞
for the velocity scale, and the time scale L/√g∗H∞. The reduced gravity is defined
as g∗ = g∆ρ/ρ with ρ being the density of the fluid and ∆ρ being the density
difference across the free surface. With a typical wind speed of the foehn flow of
U = 15 m s−1, a length scale of L = 50 km (length of the Wipp Valley between
STZ and IBK), and a Coriolis parameter of f = 1.1× 10−4 s−1, the typical Rossbynumber is Ro = U/(fL) ≈ 2.7. Thus, the neglect of Coriolis effects is justified to afirst approximation. The SWM solves Eqs. (2.1)–(2.3) in the momentum flux form.
Its numerical scheme conserves mass and momentum but allows energy dissipation
such as the destruction of kinetic energy in hydraulic jumps. The model has the
capability of handling material surfaces collapsing to zero-mass layers at regions
where the topography pierces the fluid (Schär and Smolarkiewicz 1996). This is
-
2.2 Model and measurement description 21
of particular importance for the simulation of shallow foehn cases where the flow
through the Brenner gap is confined to heights below the main Alpine crest and
therefore mountains may protrude through the top of the fluid.
The total domain of the SWM orography is shown in Fig. 2.1(b). It is centred
near the Brenner Pass and covers the Wipp Valley and major parts of the Inn Valley.
The domain is resolved with 233 (259) grid points in west–east (south–north) direc-
tion and with a mesh size of 500 m. As shown in Fig. 2.2(b) the essentially structure
of the Brenner gap in the Alpine ridge is well resolved in the SWM orography, al-
though the heights of the pass and of individual mountain peaks are off by an order
of 200 m compared to a high-resolution elevation model. The choice of the horizontal
length scale L does not affect the general type of the solutions of the shallow-water
equations. For example, a doubling of the non-dimensional mesh sizes ∆x̂ and ∆ŷ
will only stretch the flow but will not change its regime and general pattern. For
convenience, we chose L = 1 km which gives a grid spacing of ∆x̂ = ∆ŷ = 0.5. The
non-dimensional time step was ∆t̂ = 0.1. The initial flow, having a horizontally
homogeneous layer height and constant southerly velocity, was introduced as an im-
pulsive start from rest. The model was integrated with 3000 iterations (t̂ = 300) to
obtain a quasi-steady state. Near the lateral boundaries the flow was relaxed to the
initial values within a region spanning over 8 grid points. This boundary condition
effectively removes propagating disturbances, such as hydraulic jumps, impinging
on the boundaries from the model domain interior without noticeable reflections.
For an initial fluid layer height Z0 and velocity U0 being horizontally homoge-
neous and for a specific topography, the solution of the flow is determined by Z0
and the initial local Froude number F(0,X) at an arbitrary place X (such as at the
upstream location STZ with F(0,STZ) = U0/√
g∗H(0,STZ)). We performed a total of
35 simulations with the control parameters Z(0,STZ) = 2, 2.5, 3, 3.5, 4 km MSL
and F(0,STZ) = 0.25, 0.5, 0.75, 1, 1.25, 1.5, 2. For the sake of clarity the name of
the simulations include these two parameters, e.g. Z20F025 denoting the simula-
tion with Z(0,STZ) = 2 km MSL and F(0,STZ) = 0.25. At t̂ = 300 two simulations,
Z35F200 and Z40F200, had still supercritical flow at Sterzing (F(300,STZ) > 1; see
Fig. 2.4(a)). Such cases are rather unlikely to be observed (see section 2.3) and they
are therefore skipped in the following discussion. Based on the flow field at t̂ = 300,
the remaining 33 numerical experiments are divided into 10 shallow foehn cases with
Z(300,STZ) < 3.2 km MSL and 23 deep foehn cases with Z(300,STZ) ≥ 3.2 km MSL.
2.2.2 Weather stations and radiosoundings
During the MAP SOP the Wipp and Inn Valley were instrumented with a total
of 35 automatic weather stations. Mayr et al. (2003a) give a detailed description
-
22 Hydraulic aspects of foehn
of the gap flow measurements during MAP. In our study we used the 10-minutes-
average data set from a subset of 16 stations aligned along the Wipp Valley floor
between STZ and IBK (see Fig. 2.2(a)). During south foehn conditions radiosondes
were released, among other places, at STZ and Gedeir (GED) (see Fig. 2.2(a)) to
record the vertical conditions up- and downstream of the pass. From a total of 17
IOPs, soundings were made during 11 IOPs at STZ and during 7 IOPs at GED. We
used the whole data set of these two stations for the statistical analysis of hydraulic
parameters.
2.2.3 Lidars
During the MAP SOP, the National Oceanic and Atmospheric Administration’s
Environmental Technology Laboratory (NOAA/ETL) operated a scanning Doppler
lidar (TEACO2) in the Wipp Valley at Gedeir (GED) about half way between the
Brenner Pass and Innsbruck (see Fig. 2.2(a)). The system is described in Post and
Cupp (1990). Recent analyses of its MAP data set are presented in Gohm et al.
(2003), Flamant et al. (2002), and Durran et al. (2003). Other studies containing
TEACO2 observations involve e.g. the investigation of flows in the Grand Canyon
(Banta et al. 1999), on the Colorado Front Range (Neiman et al. 1988; Darby
et al. 1999), and at the Monterey Bay (Darby et al. 2002b). The lidar emits
pulses of infrared light at 10.59 µm. The signal is backscattered from aerosols
which move with the flow. The Doppler-shifted frequency of the received signal
reveals the radial wind velocity component along the direction of the lidar beam. In
our definition, winds towards (away from) the lidar have positive (negative) radial
velocities. Various types of elevation and azimuthal scans were conducted, typically
within two sectors centred at an azimuth angle up-valley (178◦) and down-valley
(320◦) of the lidar site.
The Scanning Aerosol Backscatter Lidar (SABL) of the National Center for
Atmospheric Research (NCAR) was operated during several MAP foehn cases on
board the NCAR Electra aircraft in a nadir-pointing mode. A brief description of
the system is given e.g. in Rogers et al. (1998). SABL data were recently used in
a study of a deep foehn event which was part of the MAP IOP 10 (Gohm et al.
2003). The lidar emits pulses of light at 532 nm (green) and 1064 nm (infrared) and
detects the signal backscattered from aerosols, air molecules, and hydrometeors.
The backscattered signal describes the aerosol structure in the lower troposphere
along the trajectory of the aircraft. In our study we used the 1-Hz data set of the
green channel with a resolution of 7.5 m along the lidar beam. The top of multiple
aerosol layers can be estimated from backscatter intensity profiles with a simple
gradient-method described by Gohm et al. (2003).
-
2.3 Motivation for hydraulic solutions 23
The analysis of lidar data presented herein focuses on the foehn event of 20
October 1999 (MAP IOP 8). Investigated TEACO2 data include scans performed
within one hour centred at 14 UTC. SABL data are used from one short flight leg
flown near 14 UTC across the Inn Valley at a cruising altitude of 5.2 km MSL (see
Fig. 2.1(b)).
2.3 Motivation for hydraulic solutions
2.3.1 Vertical structure of foehn winds
The vertical structure of Alpine south foehn is usually associated with a more or less
pronounced stable layer (in frequent cases a temperature inversion) on top of the less
stably stratified foehn flow (see e.g. Fig. 2.3). Within this layer winds frequently
change from across-barrier (southerly) direction near its bottom to along-barrier
(westerly) direction near its top. The wind shift is most pronounced for shallow
foehn for which the cross-mountain flow is confined to below the main Alpine crest
level (and therefore restricted to major gaps in the Alpine topography) and where the
prevailing synoptic winds are westerly. Vertically propagating gravity waves excited
by the orography decay as they pass the zone of turning winds due to 3D-critical-
level absorption (e.g. Shutts 1995). The directional wind shear and the density step
(caused by the stability change) across the transition zone decouples the underlying
foehn winds from the upper-level flow. Through the decoupling the influence of
the terrain is essentially restricted to the foehn layer and not transmitted to the
upper-level flow. Therefore, the dynamics of foehn, especially the one of the shallow
type, may be governed to a certain extent by reduced-gravity single-layer shallow-
water hydraulics. Consequently, the free surface height of the single-layer system
would represent the centre height of the stable layer. The former is an infinitely
thin boundary between the lower flowing fluid and some passive fluid aloft, whereas
the latter is a boundary of finite depth between the foehn layer and the upper-
tropospheric flow. For a well-developed deep foehn the shift to westerly winds
rather occurs in the upper troposphere or even near the tropopause and no distinct
temperature inversion may be found. Thus, the decoupling is not as pronounced as
for shallow foehn.
2.3.2 An exemplary case
Figure 2.3 depicts the vertical structure of a foehn event observed during the MAP
IOP 8. In its early stage it was a pure shallow foehn which became deeper in the
afternoon of 20 October 1999. Data were taken from two radiosonde ascents at
Sterzing and Gedeir on 12 UTC 20 October 1999. These two soundings illustrate
-
24 Hydraulic aspects of foehn
0 5 10 15 201
1.5
2
2.5
3
3.5
4
4.5
5
Wind Speed (m/s)
Hei
ght (
km M
SL)
(b)
90 135 180 225 2701
1.5
2
2.5
3
3.5
4
4.5
5
Wind Direction (deg)
Hei
ght (
km M
SL)
(c)
285 290 295 300 305 3101
1.5
2
2.5
3
3.5
4
4.5
5
Potential Temperature (K)
Hei
ght (
km M
SL)
(a)
STZGED
Figure 2.3: Vertical profiles from radiosonde ascents at Sterzing (940 m MSL; solid line)
and Gedeir (1070 m MSL; dashed line) on 12 UTC 20 October 1999. Potential temperature
in K (a), wind speed in m s−1 (b), and wind direction in degrees (c). Stars and crosses
denote the bottom and the top of the foehn flow, respectively. Upward and downward
pointing triangles indicate the bottom and the top of the strongly stable layer, respectively.
hb Z ∆Z H U ∆θ θ g∗ c
Location (m MSL) (m MSL) (m) (m) (m s−1) (K) (K) (m s−2) (m s−1) F
Sterzing 2010 2900 650 890 6 6.4 290 0.22 14 0.4
Gedeir 1110 2390 540 1280 14 6.6 293 0.22 17 0.8
Table 2.1: Hydraulic parameters calculated from two radiosoundings. Radiosonde ascents
at Sterzing and Gedeir on 12 UTC 20 October 1999. See text for further explanation.
the basic features of the foehn flow and the modification which the flow experiences
as it crosses the Brenner Pass. The centre height of the stable layer Z, which
marks the top of the foehn flow, is located at 2900 m MSL at STZ and descends
to 2390 m MSL at GED. The depth of the temperature inversion ∆Z reduces from
650 m upstream to 540 m downstream of the pass. Flow descent and vertical mixing
causes warming of the foehn air and the formation of a jet with velocities exceeding
15 m s−1 to the lee of the pass. In both profiles, winds change across the stable layer
from southerly to westerly directions. Upstream of the pass the bottom height of
the foehn flow hb does not coincide with the ground level. The flow is nearly blocked
to a height of 2010 m MSL which is the base height of the upper Brenner gap (see
Fig. 2.2(b)). For the calculation of the local Froude number of the foehn layer the
following parameters have to be retrieved from the two soundings: the depth of
the foehn layer H defined as the depth between hb and Z, the mean wind speed U
averaged vertically over H , the difference of potential temperature ∆θ across the
stable layer, and the average potential temperature of the foehn layer θ. Based
-
2.3 Motivation for hydraulic solutions 25
hb Z ∆Z H U ∆θ g∗ c
Location Measure (m MSL) (m MSL) (m) (m) (m s−1) (K) (m s−2) (m s−1) F
Sterzing Average 1400 3730 970 2330 7 9 0.29 26 0.3
Minimum 940 2700 90 890 2 3 0.09 11 0.1
Maximum 2040 5590 2870 4550 17 19 0.60 46 0.7
Gedeir Average 1120 2660 630 1540 14 5 0.17 16 0.9
Minimum 1070 2010 180 910 7 2 0.06 9 0.5
Maximum 1400 3510 2070 2440 22 11 0.36 28 1.3
Table 2.2: Statistical analysis of hydraulic parameters. Calculated from 49 radiosonde
ascents at Sterzing and 23 at Gedeir during south foehn conditions within the MAP SOP.
See text for further explanation.
on these parameters the reduced gravitational acceleration g∗ = g∆θ/θ, the local
gravity wave speed c =√g∗H , and finally the local Froude number F = U/
√g∗H
can be derived. Table 2.1 lists these parameters for the two soundings. The observed
flow is clearly subcritical at STZ (F = 0.4). At GED the mean wind speed is close
to the local gravity wave speed and therefore the flow is nearly critical (F = 0.8).
Transition to supercritical flow (F > 1) in between these two locations, i.e. near the
pass, and further downstream of GED are likely to occur as we will see in section 2.4.
2.3.3 Statistical analysis
Table 2.2 provides a statistical analysis of the MAP SOP radiosonde data set col-
lected during south foehn conditions. The analysis is based on 49 soundings from
STZ and 23 from GED. Wind data, and thus local Froude numbers, are available
only for 41 soundings from STZ and 21 from GED. Orographic blocking in the
basin of STZ occurred in many foehn cases. On average, however, the height of the
blocked layer was lower than in the discussed IOP 8 case. It coincided with the base
height of the lower Brenner gap at 1400 m MSL. At GED the foehn flow typically
reached the surface with the exception of a few cases where it was lifted a few 100 m
off the ground. At STZ the centre height of the stable layer Z marking the top of
the upstream foehn flow ranged between 2700 and 5590 m MSL with an average
of 3730 m MSL. The stable layer usually had a depth ∆Z of several 100 m. The
potential temperature step ∆θ across the stable layer (corresponding to a vertical
density step) was on average lower at GED (5 K) than at STZ (9 K). The hori-
zontally varying density step led to a on-average reduction of the reduced gravity
between STZ and GED from 0.29 to 0.17 m s−2 and thus also to a reduction of the
gravity wave speed from 26 to 16 m s−1. Farmer and Armi (1999) observed such a
decrease of the density difference along the flow trajectory for stratified flow over a
sill in the Ocean and discussed its effects on the modification of the hydraulic solu-
-
26 Hydraulic aspects of foehn
tion. The decrease was by a factor of four between an upstream location and the sill
crest. They explained it to be caused by entrainment processes which continuously
eroded the stable layer. It is likely that for south foehn in the Wipp Valley a similar
mechanism occurs. However, the effect is certainly smaller than in the Oceanic case
since the on-average change is less than a factor of two and in cases as the IOP 8
(see Table 2.1) and IOP 12 (see Flamant et al. 2002) even negligible. In these two
events the potential temperature step was basically constant along the Wipp Valley
cross-section. The flow at STZ during MAP SOP south foehn conditions was always
subcritical with an average local Froude number of 0.3. The average wind speed at
GED of 14 m s−1 was close to the value of the gravity wave speed, which means that
the flow was nearly critical on average. However, the maximum of F = 1.3 indicates
that at least in some cases the flow became supercritical at GED. The numerical
simulations in section 2.4 will identify potential locations for supercritical flow.
2.4 Numerical solutions for different upstream
conditions
2.4.1 Condition upstream of the pass
Figure 2.4(a) depicts the flow conditions at the upstream location Sterzing for all
35 simulations at the initial state (t̂ = 0) and the quasi-steady state (t̂ = 300). All
simulations except two (Z35F200 and Z40F200) have subcritical flow upstream of
the pass at t̂ = 300. This is due to the ridge-like geometry of the Brenner Pass
orography, which causes partial blocking of the upstream flow with lifting of the free
surface and a reduction of the fluid velocity (Long 1970). The difference between
the initial and final layer height at STZ is therefore a measure for the amount
of upstream blocking. While the upstream reservoir height is lifted by blocking,
the downstream reservoir height is reduced due to the Bernoulli loss in a series of
hydraulic jumps. This reservoir difference can be interpreted as a pressure gradient
across the Alps. For the 33 simulations with subcritical upstream flow at t̂ = 300,
29 cases had critical flow near the Brenner Pass. In four cases (Z25F025, Z30F025,
Z35F025, and Z40F025) the flow remained subcritical there (Fig. 2.4(b)). In the
former cases the pass is controlling the flow rate through the gap by adjusting the
layer height to a specific value. This explains the non-linear relation between the
final layer height ZSTZ and the final Froude number FSTZ in Fig. 2.4(a)–(b). Such
a non-linear relation was derived by Arakawa (1969) (as an extension of Long’s
1954 result) for the critical non-dimensional mountain height Mc = hc/H∞ (ratio
between critical ridge height and upstream layer depth) of a ridge in a rectangular
channel with variable channel width d. For steady-state conditions a subcritical flow
-
2.4 Numerical solutions for different upstream conditions 27
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5Initial and Final Conditions
Upstream Froude Number
Laye
r H
eigh
t (km
MS
L)
←subcritical supercritical→
(a) InitialFinal
C1
S1 S2
S3 S4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5Final Conditions
Upstream Froude Number
Laye
r H
eigh
t (km
MS
L)
(b) F=1 near BREF
-
28 Hydraulic aspects of foehn
result, since it basically says that the flow over the Brenner Pass is mainly controlled
by the vertical topographic contraction of the gap, i.e. the effect of the lateral
contraction is minor. The effective controlling mountain height is at 1900 m MSL
greater than the actual height of the pass but slightly below the base height of
upper gap (see Fig. 2.2(b)). The slight deviation of the model conditions to the left
of curve C1 in Fig. 2.3(b) might indicate some influence of the lateral contraction
at least for intermediate upstream fluid layer heights (≈3.5–4.5 km MSL). Some ofthe simulations with lower ZSTZ deviate to the right of C1, suggesting that the flow
is controlled by a lower effective mountain height. This might be a result of the
double-gap structure of the pass, which means that for rather deep (rather shallow)
foehn the upper (lower) gap is mainly controlling the flow.
2.4.2 Flow pattern along the Wipp Valley
To illustrate the overall similarities and local differences among the bulk of nu-
merical experiments we show four simulations which are typical for shallow foehn
(Z20F075 and Z20F125 in Fig. 2.5(a)–(b)) and deep foehn (Z25F100 and Z25F150 in
Fig. 2.5(c)–(d)). Table 2.3 lists their initial conditions at STZ and their steady-state
conditions at STZ (see also Fig. 2.4(b)) and GED. For the classification of ”shal-
low” and ”deep” we chose a threshold value for the steady-state fluid layer height
at STZ of ZSTZ = 3.2 km MSL which is close to the maximum crest height in the
Brenner region (see Fig. 2.2(b)). For cases with ZSTZ lower than this value, fluid
can only flow through the Brenner gap. High mountain peaks west and east of the
pass pierce the fluid surface and stay ”dry”. We will see that the features of shallow
foehn change rather smoothly into the ones of deep foehn as ZSTZ increases beyond
3.2 km MSL.
Figure 2.5 illustrates the flow pattern along the axis of the Wipp Valley. A
common feature in all four numerical experiments is an overall decrease of the free
surface height and an increase of the wind speed from south to north. The flow
becomes critical near the pass and stays supercritical for about 2 km, where a
Z(0,STZ) Z(300,STZ) Z(300,GED) Foehn
Simulation (km MSL) F(0,STZ) (km MSL) F(300,STZ) (km MSL) F(300,GED) type Figure
Z20F075 2.0 0.75 2.5 0.17 2.0 0.99 shallow 2.5–2.7
Z20F125 2.0 1.25 3.0 0.34 2.3 0.87 shallow 2.5–2.7
Z20F150 2.0 1.50 3.1 0.35 2.4 0.91 shallow 2.12,2.15
Z20F200 2.0 2.00 3.5 0.50 2.7 0.95 deep 2.12–2.15
Z25F075 2.5 0.75 3.1 0.06 2.5 0.97 shallow 2.12,2.15
Z25F100 2.5 1.00 3.4 0.28 2.7 0.88 deep 2.5–2.7,2.12,2.15
Z25F150 2.5 1.50 4.0 0.34 3.1 0.92 deep 2.5–2.7
Table 2.3: Parameters for selected SWM simulations. See text for further explanation.
-
2.4 Numerical solutions for different upstream conditions 29
Fwspd
0
0.5
1
1.5
2
F, w
spd
Z20F075
(a)
−10 −5 0 5 10 15 20 25 30 35 400
1
2
3
4
Hei
ght (
km M
SL)
Distance (km)
STZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBK
Fwspd
0
0.5
1
1.5
2
F, w
spd
Z20F125
(b)
−10 −5 0 5 10 15 20 25 30 35 400
1
2
3
4
Hei
ght (
km M
SL)
Distance (km)
STZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBK
Fwspd
0
0.5
1
1.5
2
F, w
spd
Z25F100
(c)
−10 −5 0 5 10 15 20 25 30 35 400
1
2
3
4
Hei
ght (
km M
SL)
Distance (km)
STZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBK
Fwspd
0
0.5
1
1.5
2
F, w
spd
Z25F150
(d)
−10 −5 0 5 10 15 20 25 30 35 400
1
2
3
4
Hei
ght (
km M
SL)
Distance (km)
STZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBKSTZ BRE STE ELB IBK
Figure 2.5: Vertical cross-section along the Wipp Valley (indicated in Figs. 2.1(b) and
2.2(a)) for the simulation Z20F075 (a), Z20F125 (b), Z25F100 (c), and Z25F150 (d) at t̂ =
300. Top panel: local Froude number (solid) and non-dimensional wind speed (dashed).
Bottom panel: fluid layer height (solid) and topography (grey-shaded) in kmMSL. Markers
indicate the location of Sterzing (STZ), Brenner (BRE), Steinach (STE), Ellbögen (ELB),
and Innsbruck (IBK).
-
30 Hydraulic aspects of foehn
hydraulic jump causes the flow to become subcritical again. The location of the
jump is related to the widening of the valley immediately north of BRE, but also
to the influence of a smaller ridge protruding into the valley from the eastern side.
This ridge causes the valley to bend westward for about 4 km (see Fig. 2.2(a)). In
the upper part of the Wipp Valley, except for the pass region, the flow along the
valley axis is subcritical at least to the south of Steinach (STE) with only moderate
velocities, however, it is generally stronger for deep foehn cases. While for deep
foehn cases the wind speed in the central part of the Wipp Valley shows weak
variability, for shallow foehn cases the flow continuously accelerates north of STE
and becomes supercritical a few kilometres south of Ellbögen (ELB). In nearly all
simulations the wind speed maximum along the valley axis is located in the northern
part of the Wipp Valley – frequently near ELB. For very shallow foehn cases, such
as shown in Fig. 2.5(a), a hydraulic jump occurs near ELB where the Wipp Valley
widens as it merges with the Stubai Valley. For these cases the flow along the
valley axis north of ELB is weak due to the blocking effect of the mountain range
north of IBK (Nordkette). The mass flux over the Nordkette is zero as the flow
is completely deflected into the Inn Valley. For the shallow foehn case shown in
Fig. 2.5(b) the upstream layer height ZSTZ is high enough so that part of the fluid
is able to flow over the Nordkette. In this simulation the region of supercritical flow
north of ELB extends from the Wipp Valley into the Inn Valley and at least two
hydraulic jumps can be identified between ELB and IBK. Increasing the upstream
layer height only slightly affects the magnitude of the non-dimensional wind speed
in the northern part of the Wipp Valley (see Fig. 2.5(b)–(d)). However, an increase
of the layer depth increases the gravity wave speed and therefore reduces the local
Froude number. Thus, a transition from a rather shallow to a rather deep foehn
changes the flow regime in the northern part of the transect from a supercritical
state into a nearly critical or even subcritical state. At least for the flow along the
valley axis and except for the pass region, hydraulic jumps seem to be less likely for
deep foehn cases. However, local variations of the free surface height (which would
correspond to local up- and downdrafts) are at least as strong for deep foehn cases as
for shallow foehn cases. The effect of the narrowing of the valley immediately north
of STE and the subsequent widening near the next eastern tributary valley (see
Fig.2.2(a)) is much stronger for deep foehn. In the experiment Z25F150 this lateral
contraction causes a sharp descent of the free surface height near STE together with
F > 1, an effect which is basically absent in the shallow foehn simulations.
-
2.4 Numerical solutions for different upstream conditions 31
2.4.3 Flow pattern in the Wipp and Inn Valley
The flow varies not only along the valley axis but also across. Therefore, we now
discuss the flow structure based on plan-view illustrations which cover the whole
Wipp Valley and part of the Inn Valley. Figure 2.6 displays regions with sub- and
supercritical flow. Hydraulic jumps are apparent at locations where the wind vector
crosses the ”F = 1”-contour line pointing to a region with F < 1. Figure 2.7
illustrates the wind field with horizontal wind barbs, whereby the non-dimensional
velocity was scaled with a gravity wave speed of 20 m s−1 which is close to the
average values listed in Table 2.2.
While the flow along the valley axis in the southern part of the Wipp Valley
may stay subcrit