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Doctoral dissertation 2004-03-26
Department of Meteorology
Stockholm University
SE-106 91 STOCKHOLM, Sweden
ABSTRACT
Two types of mesoscale wind-speed jet and their effects on boundary-layer structure were
studied. The first is a coastal jet off the northern California coast, and the second is a
katabatic jet over Vatnajökull, Iceland. Coastal regions are highly populated, and studies of
coastal meteorology are of general interest for environmental protection, fishing industry,
and for air and sea transportation. Not so many people live in direct contact with glaciers but
properties of katabatic flows are important for understanding glacier response to climatic
changes. Hence, the two jets can potentially influence a vast number of people.
Flow response to terrain forcing, transient behavior in time and space, and adherence to
simplified theoretical models were examined. The turbulence structure in these stably
stratified boundary layers was also investigated. Numerical modeling is the main tool in this
thesis; observations are used primarily to ensure a realistic model behavior.
Simple shallow-water theory provides a useful framework for analyzing high-velocity flows
along mountainous coastlines, but for an unexpected reason. Waves are trapped in the
inversion by the curvature of the wind-speed profile, rather than by an infinite stability in
the inversion separating two neutral layers, as assumed in the theory. In the absence of
blocking terrain, observations of steady-state supercritical flows are not likely, due to the
diurnal variation of flow criticality.
In many simplified models, non-local processes are neglected. In the flows studied here, we
showed that this is not always a valid approximation. Discrepancies between simulated
katabatic flow and that predicted by an analytical model are hypothesized to be due to nonlocal effects, such as surface inhomogeneity and slope geometry, neglected in the theory. On
a different scale, a reason for variations in the shape of local similarity scaling functions
between studies is suggested to be differences in non-local contributions to the velocity
variance budgets.
 Stefan Söderberg
ISBN 91-7265-812-6, pp 1-45.
Printed by PrintCenter, Stockholm University, Stockholm, Sweden, 2004.
List of papers
This thesis consists of the present summary and the following papers. In the
summary, the papers are referred to by their Roman numerals.
SUPERCRITICAL CHANNEL FLOW IN THE COASTAL ATMOSPHERIC BOUNDARY LAYER:
IDEALIZED NUMERICAL SIMULATIONS.
S. Söderberg, and M. Tjernström
I
Journal of Geophysical Research, 106, 17811-17829, 2001.
DIURNAL CYCLE OF SUPERCRITICAL
ALONG-COAST FLOWS.
S. Söderberg, and M. Tjernström
II
Journal of the Atmospheric Sciences, 59, 2615-2624, 2002.
THE TURBULENCE STRUCTURE OF THE STABLE ATMOSPHERIC BOUNDARY LAYER
AIRCRAFT OBSERVATIONS AND MODELLING RESULTS.
AROUND A COASTAL HEADLAND:
I. M. Brooks, S. Söderberg, and M. Tjernström
III
Boundary-Layer Meteorology, 107, 531–559, 2003.
NUMERICAL SIMULATIONS AND ANALYTICAL ESTIMATES OF KATABATIC FLOW OVER A
MELTING OUTFLOW GLACIER.
S. Söderberg, and O. Parmhed
Submitted to Boundary-Layer Meteorology, 2004.
The idea for Paper I was proposed by the second author. I set up and performed the
numerical experiments, and analyzed the model results. The text was jointly written
with the co-author; I wrote approximately 75% of the paper, including most of the
text describing the model results, the discussion, and the summary. The idea for
Paper II was jointly discussed with the co-author, while I set up the numerical
experiments, analyzed the results and wrote most of the text. In Paper III, my
contribution was to set up and perform the model experiment. I also performed the
initial analysis of the model results and contributed with parts of the text, related to
the model set up and model results. The first author put all the text together into its
final form. In Paper IV, I was responsible for the model set up, the analysis of all
the model results and the related text, and wrote most of the discussion. Additional
preliminary material, concerning the turbulence structure of a boundary layer
dominated by a katabatic wind-speed jet, is included in the summary (see section 5).
For these results, I am solely responsible.
IV
Contents
1. Introduction
1.1 Motivation
1.2 General considerations
3
4
5
2. Tools used in the study
7
2.1 Measurements
2.2 Numerical models
7
8
2.1.1 MIUU Model
TM
2.1.2 COAMPS
8
8
3. Background and internal flow dynamics in
low-level jets
3.1 The coastal jet
3.1.1 Mean MABL structure
3.1.2 Adjustment of the MABL to the coastline
geometry
3.1.3 Transient behavior of supercritical MABL
flows
3.2 The katabatic jet
9
9
9
11
18
20
4. Boundary-layer characteristics in the
presence of low-level jets
25
5. Turbulence structure in stable boundary
layers
28
6. Concluding remarks and future outlook
39
Acknowledgements
41
References
42
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
1. Introduction
In the atmosphere, wind-speed jets exist on many scales, both in time and space. Two
examples are the nocturnal boundary-layer jet, and the midlatitude jet streams. The former is
an inertial oscillation of the wind, typically found a few hundred meters above ground and
initiated close to sunset. Super-geostrophic magnitudes of the wind speed are due to the
rapid reduction of turbulence and the associated momentum flux during the evening
transition, causing an imbalance of forces. The latter jet is geostrophically balanced, and a
result of temperature differences between different air masses, as part of the general
circulation of the atmosphere. Wind speeds of as much as 75 m s -1 are quite typical in the jet
core, which is found at around 10 km height.
In the present thesis we will focus on two other types of jet, both dependent on horizontal
temperature differences in the atmosphere. The first is a coastal jet driven by cross-coast
baroclinicity, which arises from the differential heating of land and sea. The geographical
area in focus in these studies is the west coast of the United States (Figure 1). The second
type is a katabatic jet over a melting glacier. This jet is driven by negative buoyancy, which
acts also in the horizontal due to a temperature difference between the surface and the air at
the same height above sea level, but away from the surface. To study the katabatic jet, we
turn to Breidamerkurjökull, an outlet glacier - part of the Vatnajökull icecap in Iceland
(Figure 2).
While coastal jets are strongly influenced by complex coastal terrain features such as capes,
but not dependent on them for their existence, a sloping terrain is necessary for the
formation of katabatic jets. The adjustment of boundary-layer flows to complex terrain is
therefore one of the issues inherently included in this thesis. Another issue explored here is
the turbulence structure of the stably stratified boundary layer in which these wind-speed
jets exist.
Both observations and numerical simulations are part of the study. The observations have
been used mainly to verify that the numerical models are capable of simulating the physical
properties and processes within the studied flows, and to ensure a realistic behavior of the
model results. Hence, although numerical modeling has been the main tool, observations
constitute one of the fundaments on which the conclusions in this thesis rely. The present
executive summary largely consists of findings from Papers I-IV; however, additional
preliminary material is included in section 5 to broaden the picture. Some of the questions
addressed in the thesis are:
• Is hydraulic theory a valid analog for describing high-velocity flows along mountainous
coastlines, even though results have cast doubts on its applicability? (Papers I and II)
• How well can an analytical model predict katabatic flow characteristics from simulated
background parameters? (Paper IV)
• Can a physical explanation be found for the differences in the form of the local similarity
scaling functions between several studies? (Paper III and preliminary material)
3
STEFAN SÖDERBERG
50
45
Cape Blanco
Latitude
40
H
Cape Mendocino
San Fransisco
L
Point Sur
Point Conception
35
Los Angeles
30
25
−135
−130
−125
−120
Longitude
−115
Figure 1. Map over the U.S. west coast, H and L indicate approximate summertime positions of the North
Pacific High and a continental thermal low, respectively. Terrain elevation greater than 400 m is shaded. The
arrows indicate the prevailing wind directions along the coast.
1.1 Motivation
A classical approach in atmospheric studies is to divide the atmosphere into a number of
spheres. The lowest sphere, approximately 10 km deep, is called the troposphere and
constitutes the larger part of the atmospheric mass. The troposphere is in turn divided into a
turbulent boundary layer, separated from the free atmosphere above, often by an inversion;
there is a large contrast between the two layers in terms of temperature, wind speed, and
humidity. The boundary layer is here the part of the atmosphere, which is directly
influenced by Earth's surface and where all energy input by the sun finally is dissipated to
heat. The atmospheric boundary layer is also the part of the atmosphere where most people
spend most of their time. Still, our current knowledge about the boundary layer, for example
the stably stratified boundary layer, is to some extent inadequate. Although these facts in
themselves motivate studies of the atmospheric boundary layer in general, additional
arguments for studying coastal jets and katabatic flows in particular, are given below.
A reason for studies of coastal meteorology in general, is that a large part of the Earth's
population lives in coastal areas. Typical coastal phenomena, such as sea breezes, may be of
great importance for environmental protection in certain areas. Shipping activities and air
traffic at coastal airports are also severely limited by rapid transitions from clear to foggy
conditions, typical for many coastal regions.
Interest in the persistent northerly or northwesterly boundary layer flow along much of the
U.S. west coast in late spring through early fall, was initiated partly because of its effect on
oceanic upwelling. Along the entire coast, surface waters are forced to flow offshore and is
replaced by cold and nutrient rich water from below, a process driven by Ekman pumping.
In regions along the coast where high wind-speeds frequently are observed, in particular
4
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
69
65
68
67
Grid 1
∆x = 27 km
43x34x40
1000
1500
64.5
66
Latitude
Latitude
1000
65
64
64
1500
1000
500
500
Breidamerkurjökull
Vatnajökull
63.5
63
62
Grid 2
∆x = 9 km
88x70x40
61
Grid 3
∆x = 3 km
91x85x40
63
(a)
60
−30
−25
−20
−15
Longitude
−10
(b)
−5
−18
−17
−16
−15
Longitude
−14
−13
Figure 2. Horizontal model domains for the real-data COAMPS case study in Paper IV and for the
additional material included in section 5. Terrain elevation is contoured every 250 m, black bold lines is the
coastline: (a) Coarse mesh (grid 1) and the two inner nests (grid 2 and 3), gray bold lines indicate outer rim
of each grid; (b) The innermost nest (grid 3) with a horizontal resolution of 3 km.
downstream of capes and headlands, a close correlation between the spatial distribution of
the surface wind stress and that of the coldest sea surface temperatures (SSTs) are found
(e.g., Beardsley et al. 1987; Rogers et al. 1998; Dorman et al. 2000). Thus, there is a direct
economic interest in increasing the knowledge about this coastal boundary-layer flow,
maybe not so much for increasing the number of sunbathing tourists in coastal resorts, as for
the fishing industry.
In contrast to coastal areas, not so many people live in direct contact with glaciers.
Nevertheless, the current interest in climate change and related issues such as glacier
melting and sea-level rise, have highlighted the need for a better understanding of the mass
balance of ice caps. During summer, stable boundary layers are often formed over melting
glaciers and as pointed out by many authors, stable boundary layer are still poorly
understood and not well described in numerical weather and climate models (e.g., Poulos et
al. 2002). Moreover, over sloping melting glaciers, katabatic flows are frequently found, and
drive a turbulent exchange of heat and momentum between the surface and the free
atmosphere. Since glacier melting is most sensitive to changes in long-wave radiation and
turbulent heat flux (e.g., Oerlemans 2001), the properties of katabatic flows are important
for the understanding of glacier response to climatic changes. In addition, katabatic flows
can form over cool sloping surfaces in general and thus can be of importance for the surface
energy budgets in areas where stable boundary layers forms over land, on a regular basis.
1.2 General considerations
Important to note already here is that in atmospheric sciences, a “wind jet” typically stands
for a wind-speed maximum in the vertical, regardless of cross-flow dimensions, and not a
nossle jet like in classical fluid dynamics. This has some implications for the aspect ratios of
the studied jets as will be discussed later. First a few comments on the range of scales of
5
STEFAN SÖDERBERG
atmospheric motions and surface characteristics influencing the studied wind-speed jets are
given.
The background marine atmospheric boundary layer (MABL) flow condition, in which the
coastal jet exists, is synoptically determined. The MABL flow, roughly within a Rossby
radius of deformation (~50-100 km), then adjusts to capes and points, and gaps in the
coastal mountain ridge (Papers I and II). These coastal terrain features typically have a
horizontal scale of the order of ~10-50 km. SST variations due to oceanic upwelling are
found along most of the U.S. west coast. Within a few kilometers, the SST can vary several
degrees. This can lead to the formation of internal stable boundary layers, which influence
the MABL turbulence structure (Paper III).
As will be discussed in Paper IV, katabatic jets over melting glaciers appear to be relatively
insensitive to motions on the synoptic scale. Instead, the length and width of the glacier set
the upper limit of the horizontal scale affecting the flow. In the present thesis, the length of
the glacier is of the order of ~20 km while the width is of the order of ~10 km. However,
note that katabatic jets in principle have no cross-flow limits other than the width of the
slope.
The coastal jet has a wind-speed maximum with vertical extent of the order of ~100 m,
while the cross-flow width of the jet core is ~20-50 km. Although the katabatic layer is
much shallower with a wind-speed maximum typically found ~10-20 m above the glacier
surface, the width of Breidamerkurjökull results in aspect ratios of the order of ~10-3 for
both the coastal and the katabatic jet. However, the glacier is anything but smooth and
surface roughness elements of the order of ~1 m are common. Features on that scale are not
resolved in the numerical simulations performed here, but certainly have an effect on the
observed turbulence structure, since wake turbulence is likely to be found in the immediate
surroundings of these obstacles. Moreover, surface roughness elements with a size of ~1 m,
corresponding to 5-10% of the jet height, will affect also the observed mean vertical
structure. The ocean surface on the other hand is smooth. However, obstacles in the form of
capes, force the boundary-layer flow to either go around, or cross the obstacle. In fact, the
vertical extent of the local terrain at Cape Mendocino (see Figure 1) corresponds to more
than half of the typical MABL depth upwind of the cape. Instead of wake turbulence, as in
the katabatic flow, this obstacle triggers a lee-wave, which in some cases may break.
Considering time scales, the coastal jet found along the northern California coast and the
katabatic flow over Vatnajökull, are mainly summertime phenomena. Observations have
shown that they are persistent features interrupted only for short periods of time. While the
strength of the coastal jet varies with the diurnal cycle (Paper II), the katabatic jet shows no
clear diurnal cycle. This is because the temperature of the coastal landmass varies with solar
insolation, while the glacier surface temperature is typically constant at its melting point
(Paper IV).
Stratiform clouds are common in the two geographical areas in focus here (e.g., Brost et al.
1982; Kaltenböck and Obleitner 1999), and are certainly important for the radiation balance
of the atmosphere and the underlying surfaces. Possible effects of clouds on the jets are not
covered in the present thesis for the reasons given below. Marine stratocumulus within the
6
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
MABL can affect the strength of the inversion through radiational cooling at the cloud top.
However, the coastal jet is mainly driven by a thermal wind caused by the slope of the
inversion towards the coast, rather than by the strength of the inversion. Moreover, the
background conditions used in the simulations in Papers I-III were in fact based on a cloud
free day. The glacier surface temperature, which largely controls the strength of the
katabatic flow, is during summertime limited on the high side by the melting point of snow
and ice; this is typically below the ambient atmospheric temperature. Thus, it is not likely
that the clouds through their effect on the radiational balance at the surface have a direct
influence on the katabatic flow studied in Paper IV, although clouds certainly have an effect
on the rate of the glacier melting itself.
Another feature that lies outside the scope of this thesis is what is usually referred to as a
coastally trapped disturbance or wind reversal. Several times each summer, the persistent
northerly or northwesterly MABL flow off the U.S. west coast is interrupted by a period of
southerly winds, caused by a northward propagating disturbance in the MABL. This
disturbance has in the literature been described as a Kelvin wave (Dorman 1988), internal
bore (Klemp et al. 1997), or a mixed Kelvin wave-bore (Ralph et al. 2000). An overview of
the different views is given by Nuss et al. (2000).
2. Tools used in the study
2.1 Measurements
Although detailed analyses of observational data have not been part of the author's work
they constitute a basis for this thesis. Observational data have been used to verify a realistic
behavior of the numerical models employed and to verify that they are capable of simulating
the physical processes important in the flows. When the general characteristics of the
simulated flow correspond to observations, the model results can be a good help in
understanding physical processes behind features we are observing. Moreover, numerical
simulations can give vital information on what should be monitored, the next time a field
experiment is planned.
The observational data used in the present thesis have been collected from two field
experiments. The Coastal Waves 1996 experiment was set up to map the MABL structure
off the California coast in summer conditions. The primary measurement platform was the
National Center for Atmospheric Research C-130 Hercules aircraft. Several instrumented
sites were also deployed along the California coast. Among the questions addressed in the
experiment, were the supercriticality and the turbulence structure of the MABL. Details can
be found in Rogers et al. (1998).
The main goal of the 1996 glacio-meteorological field experiment on Vatnajökull, Iceland,
was to understand how the energy used in the melting of snow and ice is transferred to the
surface. Several meteorological stations were operated both on and off the ice cap. A
concentration of stations was put on Breidamerkurjökull, an outlet glacier flowing down to
the Atlantic Ocean. To obtain vertical profiles of the boundary layer, a tethered balloon and
radiosonds were used. An overview of the experiment is given by Oerlemans et al. (1999).
7
STEFAN SÖDERBERG
2.2 Numerical models
Two different numerical models have been used in the present thesis. In Papers I-III the
Department of Meteorology Uppsala University (MIUU) mesoscale model was applied to
the coastal jet along the northern California coast. In two preceding studies the same model
was successfully applied to this area and we therefore chose to use the same tool to answer
the scientific questions posed in Papers I-III. The model utilized in Paper IV is the Coupled
Ocean/Atmosphere Mesoscale Prediction System (COAMPSTM) version 2.0 atmospheric
model (Hodur 1997).
2.1.1 MIUU Model
The MIUU mesoscale model is a hydrostatic non-linear primitive equations model. The
vertical coordinate is transformed to a terrain-following sigma-z vertical coordinate system.
To achieve high resolution in the center of the model and locate the lateral boundaries far
from the area of interest, a horizontally expanding grid can be used. The vertical grid is
expanding, log-linearly, towards the model top. The turbulence closure is a modified
“Level-2.5” closure (Mellor and Yamada 1982), including a correction for non-realizable
second-order moments, inherent in this type of closure, and an improved formulation for the
pressure redistribution terms in the turbulent kinetic energy (TKE) equation, the “wall
correction” (Andrén 1990). The model allows for an easy experimental control over the
surface forcing and initial conditions, facilitating both analyses of model results and
sensitivity tests. The MIUU model has been used in a variety of applications, including orographic (e.g., Enger and Grisogono 1998) and coastal flows (e.g., Tjernström and Grisogono
2000). More detailed descriptions are found in Tjernström (1987a, b) and Enger (1990).
2.1.2 COAMPS
TM
The Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPSTM) version 2.0
atmospheric model, developed at the U.S. Naval Research Lab, Monterey, CA, is a
nonhydrostatic compressible model, with terrain-following sigma-z vertical coordinate
system. Nested grids can be used in idealized and real-case simulations allowing high
horizontal resolution for a given area. Physical parameterization schemes include long- and
shortwave radiation (Harshvardan et al. 1987), explicit moist physics (Rutledge and Hobbs
1983), cumulus convection (Kain and Fritsch 1990), and “Level-2.5” turbulence closure
(Mellor and Yamada 1982). The ground surface temperature is computed using a surface
energy balance scheme. Initial and lateral boundary conditions were provided using
ECMWF analyses in the real-case study. This allows us to compare the simulated katabatic
flow in realistic atmospheric conditions to measurements undertaken during the field
experiment on Vatnajökull described above. A more complete model description is found in
Hodur (1997).
8
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
3. Background and internal flow dynamics in low-level jets
Low-level jets are by definition found close to the ground and therefore directly influenced
by the surface properties. In fact, some low-level jets are even formed due to sudden
changes in surface properties. An example of this is the frequently observed low-level jet
over the Baltic Sea (e.g., Smedman et al. 1995). When warm air from land is advected over
the cold sea, this result in a frictional decoupling and the formation of an inertial oscillation
due to an imbalance of forces - a spatial analog to the nocturnal jet briefly described in the
introduction. Below the jet-types specifically studied in this thesis are presented in more
detail, namely coastal jets established along coastlines with elevated terrain and katabatic
jets formed over melting glaciers.
3.1 The coastal jet
3.1.1 Mean MABL structure
In late spring through early fall, the near-surface air flow over the eastern North Pacific is
dominated by the North Pacific high, located ≈ 1000 km west of the California coast near
40° north, and a thermal low over the southwestern U.S. continent (Figure 1). These
synoptic-scale features set the stage for persistent northerly or northwesterly boundary-layer
flow along much of the U.S. west coast. The MABL is typically cool, moist and well mixed
by turbulence; the air temperature is largely controlled by the SST with colder SSTs closer
to the coast than offshore due to upwelling. A strong temperature inversion, typically of the
order of 10 °C in potential temperature, forms a boundary between the MABL and the
subsiding dry and warm air in the free atmosphere above. Due to the decreasing MABL
temperature towards the coast, the inversion slopes gently from west to east. Close to the
coast, the slope of the inversion increases significantly since the land surface is usually
warmer than the air within the MABL. A typical near-shore MABL depth is 200-300 m
while the height of the coastal terrain for much of the U.S. west coast exceeds 400 m (see
Figure 1). The inversion will therefore intersect the coastal mountain range, which acts as a
barrier to the flow. This means that air with an initial component of motion toward the
barrier eventually must turn and flow along the barrier; thus the MABL flow is channeled.
In general, even moderate terrain can block onshore flow if it is hydrodynamically steep
(Grisogono and Tjernström 1996; Tjernström and Grisogono 1996).
Although the channeling of the flow by itself can lead to the formation of coastal jets, the
main contributor to the acceleration of the near-shore flow along straight sections of the
U.S. west coast is a thermal wind. The thermal wind is a result of the sloping inversion,
which give rise to a horizontal temperature gradient directed towards the coast. An
approximate form of the thermal wind can be written as (e.g., Stull 1988):
∂u g
∂z
∂v g
∂z
≈−
g ∂T
,
fT ∂y
(1)
≈+
g ∂T
,
fT ∂x
(2)
9
STEFAN SÖDERBERG
Figure 3. Conceptual model of average lower atmosphere over eastern North Pacific during periods of
persistent northerly or northwesterly winds in summer. (From “Local atmospheric forcing during the coastal
ocean dynamics experiment. 1. A description of the marine boundary layer and atmospheric conditions over
a northern California upwelling region,” R. C. Beardsley, C. E. Dorman, C. A. Friehe, L. K. Rosenfeld, and
C. D. Winant, J. Geophys. Res., 92: 1481, 1987.)
where ug and vg are the geostrophic wind components, g is gravitational acceleration, f is the
Coriolis parameter, and T is temperature. Here u, v, x, y, and z have their usual definition; u
is parallel to the x-axis (directed towards east), v is parallel to the y-axis (directed towards
north), and z is the vertical axis. A sloping inversion towards the coast like off the U.S. west
coast will give a positive RHS in (2). Taking a finite difference over a layer ∆z therefore
yields:
v g ,U − v g ,L > 0
.
(3)
Here U stands for the upper level and L for the lower level. In a northerly flow, (3) implies
that vg,L must be more negative than vg,U, i.e., a stronger northerly flow at the lower level.
Thus, off the U.S. west coast the thermal wind gives rise to an increasing wind speed with
decreasing height. Within the MABL the effect of surface friction also comes into play,
which leads to a jet shaped wind profile with a wind-speed maximum just below the
inversion. It has been suggested that a small thermal wind below the inversion, caused by
the SST gradient towards the coast, also contributes to the northerly flow (Zemba and Friehe
1987). However, numerical simulations have shown that the SST distribution has only
marginal effects on the character of the along-coast flow (Burk and Thompson 1996;
Tjernström and Grisogono 2000). The above description of the MABL off the U.S. west
coast can be summarized by the conceptual model of Beardsley et al. (1987) shown in
Figure 3. Stratus and fog are indicated below the inversion in the intermediate and nearshore
regions but even if marine stratocumulus and fog are common along the entire U.S. west
coast, they are not a topic studied in this thesis for reasons previously discussed. Moreover,
numerical simulations have shown that the cloud field along the coast is determined almost
entirely by local flow dynamics (Tjernström 1999); observations also show a preferred
clearing in the lee of capes and points with prominent local terrain features (e.g., Rogers et
al. 1998; Dorman et al. 2000).
10
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
The momentum balance in coastal jets along the U.S. west coast has been investigated both
from observational data and from numerical model results by several authors (e.g., Zemba
and Friehe 1987; Samelson and Lentz 1994; Ström et al. 2001; Burk et al. 1999; Tjernström
and Grisogono 2000). In strong northerly flow along straight sections of the coastline, the
flow is geostrophic in the cross-coast direction, while the pressure gradient force is balanced
by turbulent friction in the along-coast direction. Downstream of points and in the lee of
capes protruding into the flow, the situation is more complex. Advection of momentum
becomes important, both in the cross-coast and along-coast momentum budgets.
Furthermore, as the MABL depth decreases, friction also becomes important in the crosscoast direction; the increased effect of friction on the flow as the MABL depth decreases
was also demonstrated in Paper I by theoretical considerations of the shallow-water
equations. Variations in the MABL connected to topographic features in the coastal terrain
are not included in the conceptual model described above, since it assumes a straight
coastline with a smooth coastal range. The sensitivity of along-coast flow to terrain forcing
is addressed in Paper I and discussed in the next section.
3.1.2 Adjustment of the MABL to the coastline geometry
As a consequence of the strong capping inversion above the well-mixed MABL, it has been
suggested that the dynamics of the coastal flow off the U.S. west coast can be approximated
as a single-layer reduced-gravity (shallow water) flow past a blocking sidewall (Winant et
al. 1988). Hydraulic theory then predicts that the Froude number, Fr, the ratio of the flow
speed U to the linear gravity wave phase speed c of the waves propagating on the interface
between the MABL and the free atmosphere above, determines the behavior of the flow:
Fr =
U
,
c
(4)
where
c = (g' H )0.5 .
(5)
The depth of the layer is H and g' is the reduced gravity, here expressed using the potential
temperature θ:
g' = g
∆θ
,
θ0
(6)
where θ 0 is the temperature of the lower layer and ∆θ is the temperature jump over the
capping inversion. When the flow is supercritical, Fr > 1, the flow becomes sensitive to
changes in coastline orientation (Ippen 1951). Physically supercritical flow means that the
mass and wind field upstream of a local perturbation cannot adjust to this, since the phase
speed of the waves responsible for the adjustment process is lower that the flow speed. One
way to interpret this is that the information is “swept downstream” by the flow. As the
vertical boundaries of the channel expand or contract, expansion fans or hydraulic jumps
will appear when the flow is supercritical. A simple sketch of a supercritical flow in a
widening channel based on the theory by Ippen (1951) is shown in Figure 4. The blocking
11
STEFAN SÖDERBERG
Figure 4. Schematic sketch of supercritical channel flow based upon the theory of Ippen (1951). U is the
flow velocity and H is the fluid depth. See text for a definition of the angles α, β1, and β2.
coastline turns away from the flow at an angle α to the upstream flow direction. In response
to this the depth of the MABL decreases and the flow accelerates. An expansion fan forms
downstream of a characteristic wave, which makes an angle β1 to the upstream flow given
by:
sin(β1 ) = Fr −1 ,
(7)
beyond which no information can propagate. As the Froude number increases downstream
of the initial characteristic, successive characteristics diverge confining the expansion fan
between the leading wave and the trailing wave, which makes an angle β2 to the downstream
coastline.
If the hydraulic theory holds true for the MABL along the U.S. west coast it would be a
useful analog and explain the observed “patchiness” of the low-level wind fields. High wind
speeds are frequently observed downstream of capes and points and sudden decelerations of
the flow, are also found where it encounters topographic features protruding into the flow
(e.g., Dorman et al. 2000). Numerous experiments with shallow water models have been
reasonably successful in describing the main characteristics of the observed flow (e.g.,
Samelson 1992; Rogerson 1999; Edwards et al. 2001). One of the main results in Samelson
(1992) was the demonstration of the effect of friction. The region with increased wind
speeds resembled a “bulge” rather than a fan when friction was included. Thus, the inclusion
of friction in the theory led to a much better agreement between shallow-water model results
and observed flow structures. But, reality is by its very nature three-dimensional and
therefore it is well motivated to test the hydraulic theory in such a model.
In Paper I the three-dimensional, hydrostatic, non-linear MIUU mesoscale model was used
in a number of sensitivity tests focusing on the effect of terrain forcing on along-coast
supercritical flow. A smooth mountain barrier was generated by fitting a simple parabolic
function to an ensemble of east-west cross-sections of the real terrain north of Cape
Mendocino. A piece-wise linear convex bend in the idealized coastline was then introduced,
making an angle α = 24° to the upstream coastline, similar to the real terrain south of Cape
Mendocino. The simplified terrain thus represents the real terrain forcing of the northern
12
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
(b)
1.5
1
0.5
0
150
Height (km)
Height (km)
(a)
100
100
No
rth
50
−S
0
ou
th
−50
(km
−100
)
1.5
1
0.5
0
)
50
m
(k
0
st
−50
Ea
t−
−100
es
W
−150
rth
1
0.5
0
150
Height (km)
Height (km)
0
th
−50
(km
−100
)
)
50
m
(k
0
st
−50
Ea
t−
−100
es
W
−150
(d)
1.5
100
rth
50
−S
ou
(c)
100
No
150
100
100
No
50
−S
ou
0
th
−50
(km
−100
)
0
−50
t−
−100
es
W
−150
km
t(
s
Ea
1
0.5
0
150
100
100
No
)
50
1.5
rth
50
−S
)
50
ou
0
th
−50
(km
−100
)
0
−50
t−
−100
es
W
−150
km
t(
s
Ea
Figure 5. Examples of the idealized terrain used in Paper I: (a) control run (Ctrl), also used in Paper II; (b)
z1_Cc24; (c) z4_α24; and (d) z1_cape500. Maximum height of the idealized terrain is 1.2 km.
California coastline while ignoring small changes in coastline orientation and terrain height.
By varying this terrain in a simple manner, testing of hypothesis related to coastline
geometry was facilitated. Three sets of tests were performed, focusing on coastline shape,
terrain height variations, and the effects of a cape perpendicular to the flow, respectively.
Examples of different idealized terrain configurations are shown in Figure 5. In Figure 5a
the terrain used in the control run (Ctrl) is shown, 5b shows a curved coastline (z1_Cc24),
5c shows a sloping coastal barrier (z4_α24), and in 5d a coastline with a cape representing
the real terrain at Cape Mendocino is shown (z1_cape500). In Paper I, all model results were
extracted at 1500 local standard time, 21 hours into the simulations; some of the results are
presented below. The flow is always from the north so upstream (downstream) means to the
north (south) of the change in coastline orientation. Detailed descriptions of the experiments
are found in Table 1 in Paper I.
In Figure 6 the MABL depth, maximum MABL wind speed, and Fr from Ctrl are shown.
Downstream of the bend in the coastline, a fan-like depression of the MABL depth extends
from the coast (Figure 6a). In response to the decreased MABL depth, the flow accelerates
and attains wind speeds exceeding 22 m s -1 within a bulge-like wind-speed maximum
(Figure 6b). These results are basically in agreement with results from studies utilizing
shallow-water models. However, even though the flow is supercritical (Figure 6c), there is a
gradual increase of the flow velocity along the upstream coastline. This is clearly not in
agreement with the hydraulic theory described above. This unexpected feature was present
13
STEFAN SÖDERBERG
30
(b)
25
13
50
20
19
0
20
11 17 19
20
22
−50
10151136
21
−100
15
18
18
South − North (km)
100
10
−100
−50
0
50
West − East (km)
400
(a)
100
100
150
500
350
−150
450
South − North (km)
400
350
50
300
250
200
350
−50
100
200
400
450
0
150
50
100
50
300
−100
−150
−100
100
150
2.5
1
(c)
0
−50
0
50
West − East (km)
1.1
50
2.0
1.3
1.5
2
21.5
1.5
1.0
1.3
−50
1
0.9
1.5
2
1.3
0
1.1
South − North (km)
100
−100
0.5
−150
−100
−50
0
50
West − East (km)
100
150
Figure 6. Contour plots from the control run of: (a) MABL depth (m); (b) maximum MABL wind speed
(m s -1); and (c) Froude number.
also in Tjernström (1999) and was one of the questions Paper I was set up to resolve. In fact,
a gradual acceleration of the flow along the upstream coastline was present in all idealized
simulations but the ones with capes. Burk et al. (1999) also found an increase in MABL
winds along the upstream coastline; however, this was in a transcritical simulation with
subcritical flow upstream of the bend. For the moment, we will leave this issue and return to
it in the next section. Instead we will take a look at the simulated vertical structure of the
along-coast MABL flow in Ctrl.
Vertical cross-sections of potential temperature and scalar wind speed taken from west to
east are shown in Figure 7. Upstream of the coastline bend, an oval-shaped wind-speed jet is
found near the inversion base, aligned with the maximum slope of the inversion close to
14
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
2.5
311
4
311
(b)
8
303
3014
6
299
8
295
10
289
1.0
6
297
8
10
285
19
16
−100
0.5
14
−50
0
50
West − East (km)
100
0
150
−150
303
301
4
299
293 295
291
289
14
12
287
16
285
21
20
18
6
18
−150
305
4
12
12
14
16
0.5
1.5
6
1.0
0
4
Height (km)
305
307
2
307
297
Height (km)
307
309
2
2.0
1.5
8
2
309
2
2.0
6 4
(a)
−100
8
2.5
−50
0
50
West − East (km)
100
150
Figure 7. East-West vertical cross sections of potential temperature (K) (dashed) and scalar wind speed
(m s -1) (solid) from the control run: (a) upstream at y ~ 75 km; and (b) downstream at y ~ -50 km.
30
(a)
10
13
22
23
24
−100
23
22
23
19
19
24
−50
0
50
West − East (km)
−50
20
12
15
10
12
20
−100
0
12
15
13
−150
25
21
22
−100
12
0
10
11 15 21 2
−50
20
11
5
14 1
17
20
20
50
13 6
1
0
South − North (km)
20
25
21
11
10
50
30
(b)
100
13
14
South − North (km)
100
10
10
100
−150
150
−100
−50
0
50
West − East (km)
100
150
Figure 8. Contour plots of maximum MABL wind speed (m s -1) from: (a) z1_Cc24; (b) z4_α24.
coastline (Figure 7a). These model results are in most aspects representative of the balanced
state with a sloping inversion towards the coast as described by the conceptual model of
Beardsley et al. (1987, see Figure 3). Downstream of the bend, the slope of the inversion
steepens considerably and in the near-shore region the MABL collapses entirely (Figure 7b).
High momentum air is found at low altitudes within the wide wind-speed jet, which also tilts
considerable towards the coastline as a result of the steeply sloping inversion.
The change of orientation of the real terrain along northern California is however not as
abrupt as in Ctrl. Tjernström (1999) suggested that the gradual curvature of the main coastal
mountains north of Cape Mendocino may be sufficient to excite an expansion fan. To test
this hypothesis, numerical simulations with curved coastlines were performed. Moreover,
the terrain elevation also varies along the coast with the highest terrain north of Cape
Mendocino; therefore were the effects of terrain height variations on the MABL
characteristics investigated. In Figure 8 maximum MABL wind speed from two of these
experiments are shown, z1_Cc24 with a curved coastline (Figure 8a) and z4_α24 with a
decreasing terrain height along the coast (Figure 8b). The simulations with curved coastlines
clearly illustrated that the change in coastline orientation does not have to be abrupt to excite
15
STEFAN SÖDERBERG
an expansion fan, a curved coastline is sufficient. Terrain height variation along the coast
had a significant impact on the MABL flow. An additional along-coast acceleration of the
flow took place and the marine air was able to stretch farther in over land due to the reduced
blocking. The higher magnitudes of the wind speed can be explained by an intensified
secondary coastal circulation due to the lowering of the height of the coastal barrier,
increasing the slope of the inversion towards the coastline further.
Previous studies have indicated that the local terrain at Cape Mendocino has a distinct effect
on local flow characteristics. In our experiments with simplified capes, this was clearly
demonstrated. The terrain of the cape protruding into the flow caused a significant blocking
of the upstream flow, even when the height of the cape was only half of the upstream
MABL depth. As an example, model results from z1_cape500 are shown in Figure 9.
Upstream of the cape, the MABL depth is less than the height of the cape, which is 500 m
(Figure 9a). As the flow passes the cape, it accelerates rapidly and forms a wind-speed
maximum similar in shape to that in Ctrl, but farther away from the main coastal mountain
range (Figure 9b). At the tip of the cape, the flow also becomes supercritical (Figure 9c);
thus the flow is transcritical, a clear distinction from the control run. Furthermore, along the
downstream coastline, the near-shore wind speeds are also weaker than in Ctrl. Note in
particular the relatively low wind speeds found in the lee of the cape where the MABL
collapses entirely.
The effect of the cape on the MABL properties is further appreciated in Figure 10a, showing
vertical cross sections of potential temperature and scalar wind speed taken from west to
east downstream of the cape. A substantial warming of the air below 500 m is found in the
near-shore region and the collapse of the MABL is apparent. Note also the skewed structure
of the jet, tilting towards the coast, and the wedge of low-momentum air brought down from
aloft to low altitudes in the lee of the cape; this is another clear distinction from the results
found in Ctrl (cf. Figure 7b). In fact, observed flow features around Cape Mendocino and
findings in real-case studies with different numerical models are in many respects
reproduced in this idealized simulation (e.g., Rogers et al. 1998; Ström et al. 2001; Burk and
Thompson 1996; Tjernström and Grisogono 2000). It is therefore not too far-fetched to
conclude that the terrain forcing induced on the MABL flow by the simplified terrain used
in z1_cape500 is representative of the real terrain forcing along the northern California
coast. In Paper I it was also found that the terrain at the cape triggers a lee-wave, as
suggested from observations (Ström et al. 2001) and earlier model studies (Burk and
Thompson 1996; Tjernström and Grisogono 2000; Tjernström 1999). Moreover, consistent
with the extended flow regime diagram of Ólafsson and Bougeault (1996), gravity wave
breaking was found on the lee-side of the cape. A north-south cross section through the cape
in z1_cape500 is shown in Figure 10b. Note in particular the isotherms sloping steeply
downward in the along-flow direction and then rising quickly on the lee-side of the cape,
where the low-level flow accelerates to more than 16 m s -1. Above the wind-speed
maximum, a local maximum in TKE is also found (not shown). In fact, a lee-wave was
triggered in all three experiments with capes. Thus, it is likely that terrain features such as
the local terrain at Cape Mendocino, to a large extent determine MABL characteristics in
high-velocity along-coast flows.
16
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
600
(a)
100
550
450
500
500
South − North (km)
450
50
400
40
0
350
300
300
0
250
450
0
10
10
50
200
−50
200
150
100
−100
50
0
−150
−100
−50
0
50
West − East (km)
100
150
25
(b)
16
20
17
10
50
12 10
18
10
19
15
14
12
16
0
10
South − North (km)
100
−50
18
20
10
10
19
−100
5
−150
−100
−50
0
50
West − East (km)
100
150
2.5
(c)
0.
8
0.82
1
1.10.9
1.1 2
1 1
1.
1
1.3
50
0
2
0.7
1.5
1.3
1.1
1.3
2.0
1
1.5
1.5
South − North (km)
0.9
100
2
−50
1
1.5
1.3
2
1
1.5
−100
1.0
0.9
0.5
−150
−100
−50
0
50
West − East (km)
100
150
Figure 9. As Figure 6 except for z1_cape500.
The simulations performed in Paper I with simplified terrain allowed us to test hypothesis
related to coastline geometry in a simple manner. We found that the model results in many
ways conformed to the predictions of supercritical flow response, although differences
between shallow-water and three-dimensional models were illustrated. When a simplified
cape protruding into the flow was introduced, the flow characteristics agreed well with
observations along the northern California coast, suggesting that capes and points with
prominent terrain largely determine the MABL characteristics. However, an unexpected
feature appeared in all simulations but the ones with simplified capes. Along the upstream
coastline a gradual acceleration of the flow occurred which appears to violate the hydraulic
theory if the flow is truly supercritical. Paper II was set up with the aim to resolve this
puzzle.
17
STEFAN SÖDERBERG
311
1.50
(a) 4
2.0
309
2
1.25
305
303
6
301
299
303
301
299
8
12
297
16
2
291
289 10 4 2
93
287
285
20
−150
−100
6
291
8
0.75
4
289
10
16
10
287
12
10
0.25
8
18
293
14
6
12
299 297 295
0.50
295
0
4
301
1.00
Height (km)
Height (km)
1.5
0.5
303
1
10
307
1.0
(b)
1
8
2
2
2.5
−50
0
50
West − East (km)
100
150
0
0
10
20
30
South − North (km)
40
50
Figure 10. (a) East-West vertical cross section at y ~ -50 km of potential temperature (K) (dashed lines) and
scalar wind speed (m s-1) (solid lines) from z1_cape500. (b) North - South cross section at x ~ -5 km of
scalar wind speed (m s-1) (solid lines) and, potential temperature (K).
3.1.3 Transient behavior of supercritical MABL flows
In Paper I it was observed that the only experiments that did not feature a gradual
acceleration of the flow along the upstream coastline were the experiments with simplified
capes inserted perpendicular to the main coastal mountain barrier. Another significant
feature in the simulations with capes was the transition from subcritical to supercritical flow,
when the flow passes the cape. Thus, the results in Paper I suggests that the absence of a
gradual upstream acceleration in the observations from Cape Mendocino may be due to the
presence of blocking terrain at the cape and not the supercriticality of the flow. One
hypothesis offered in Paper I as the cause of the upstream acceleration of the flow was based
on the fact that the initial profile used in all simulations is actually subcritical. The
supercriticality of the flow then develops during the dynamic initialization of the model,
which would allow the gradual acceleration of the flow to be established along the upstream
coastline during the subcritical phase of the simulation. The transient behavior of the flow
was therefore studied in Paper II.
If we take a step back and consider the diurnal variation of the MABL in the near-shore
region described by Beardsley et al. (1987), a plausible explanation for the deviation from
the original hydraulic theory emerges (see Figure 11). During night when the temperature
contrast between land and the MABL is small or negligible, the slope of the inversion
towards the coast is also gentle. Close to the coast, the MABL winds are therefore relatively
weak. In the morning when the sun heats the land surface, the air destabilizes, allowing for a
penetration of the MABL air over land. At the same time, the slope of the inversion towards
the coast increases. Above the eroded inversion over land, a weak return flow helps
depressing the near-shore inversion further. As a result, a strong low-level jet is often found
adjacent to the coastal mountain range.
18
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
Figure 11. Conceptual model of lower atmosphere over the nearshore zone during night (top) and day
(bottom). (From “Local atmospheric forcing during the coastal ocean dynamics experiment. 1. A description
of the marine boundary layer and atmospheric conditions over a northern California upwelling region,” R. C.
Beardsley, C. E. Dorman, C. A. Friehe, L. K. Rosenfeld, and C. D. Winant, J. Geophys. Res., 92: 1482, 1987.)
This diurnal variation of the MABL is reproduced in Paper II; descriptions of the
experiments performed are found in Table 1 of Paper II. Early in the morning a relatively
weak jet is found far offshore below a gently sloping inversion (Figure 12a). The wellmixed MABL close to the coast is about 400 m deep. In the afternoon the depth of the
MABL have decreased notably. The jet is now attached to the main coastal mountains, and
is considerable stronger than during night, as a consequence of the increased slope of the
inversion close to the coast (Figure 12b). Two factors that have a direct effect on the flow
criticality are here easily identified: a low flow velocity and a deep MABL results in a
subcritical flow during the night, while a high flow velocity and a shallow MABL produces
a supercritical flow during the day. The resulting diurnal variation of the flow criticality is
illustrated in Figure 13. Thus, during nighttime and early in the morning when the flow is
subcritical, the gradual acceleration of the flow can take place along the upstream coastline,
without violating the hydraulic theory. This feature will then disappear only very slowly,
once the flow becomes supercritical. Together with results the from Paper I this suggests
that daytime supercritical conditions may not prevail sufficiently long for a true quasisteady-state supercritical flow to be established in the absence of blocking terrain protruding
into the flow. Furthermore, Paper II shows that the gradual acceleration of the flow already
upstream of the change in coastline orientation is not violating the hydraulic theory and that
the flow criticality in MABLs similar to what is found off the U.S. west coast does not only
vary spatially, but also temporally.
19
STEFAN SÖDERBERG
6
(a)
2.5
309 4
309
307
2.0
2.0
303
301
8 297
295
293
1.0
10
14
16
17
285
8
6
−100
−50
0
50
West − East (km)
4
303
6
8
1.0
291
301
150
0
4
14
16 287
18
100
29
7
12
0.5
12
14
−150
305
305
1.5
8
10
0.5
8
4
299
Height (km)
305
1.5
6
307
305
299
Height (km)
307
0
(b)
4
4
16
14
−150
−100
19
2
2.5
12
−50
0
50
West − East (km)
100
150
Figure 12. Vertical cross-sections of potential temperature (K) (dashed lines) and scalar wind speed (m s-1)
(solid lines) from exp_restart taken upstream of the bend at y ~ 75 km: (a) 0600 LST; and (b) 1500 LST.
3.2 The katabatic jet
Katabatic flows, also known as drainage or gravity flows, are formed when the air adjacent
to a sloping surface cools more than the air at the same elevation, but away from the surface.
This triggers a low-level downslope flow since the negative buoyancy will act also
horizontally, the so-called katabatic forcing term. In general downslope flows can form on
any sloping surface. However, the katabatic forcing is usually smaller than the other terms
in the momentum budget, such as the synoptic pressure gradient. Thus, katabatic flows are
typically only observed during clear sky conditions, when nighttime radiative cooling of the
surface peaks. Over icecaps on the other hand, the katabatic forcing term in most cases
appears strong enough to overcome the background pressure gradient term. Persistent
katabatic flows over glaciers have been reported in numerous studies, some of which are
referred to in the present thesis (e.g., Gruell et al. 1994; Oerlemans et al. 1999).
The simplest theory for katabatic flows is the classic Prandtl model (Prandtl 1942), in which
the advected background temperature lapse rate and the buoyant acceleration are balanced
by the divergence of the turbulent fluxes of heat and momentum, respectively. In a
coordinate system with its axes aligned with the slope, the equations read:
∂ (w' θ ' )
= 0,
∂z
θ
∂ (w' u' )
=0.
g sin(α ) −
θ0
∂z
γ u sin(α ) +
(8)
(9)
where γ is the background potential temperature lapse rate, u is now the downslope wind
speed and w the velocity perpendicular to the slope, while α is the slope of the surface. Here
θ is the potential temperature perturbation, θ0 a reference temperature (in the present thesis
the potential temperature of the ambient air). Overbars indicate an averaging operation and
primed terms are deviations from the average. Using K-theory and assuming constant values
20
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
16
24
Phase speed (ms−1)
15 06
14
18
(start)
13
12
18 (end)
12
11
10
12
13
14
15
16
17
−1
Mean MBL wind speed (ms )
18
19
Figure 13. Diurnal cycle of phase speed and mean MBL wind speed for exp_restart. The numbers along the
solid line show local standard time while the dashed line indicates where Fr = 1.
of the eddy diffusivities for momentum and heat, Km and Kh, the equations are analytically
solvable with the solution:
u (z ) = −Cµe − δ z sin(δ z ) ,
θ (z ) = Ce
−δ z
cos(δ z ) ,
(10)
(11)
in which C is the surface temperature deficit, θ (0 ) = C < 0 , and
 g 

µ = 
 γθ 0 Pr 
0.5
 N sin(α ) 

δ = 

 2 K h Pr 
,
(12)
0.5
,
(13)
where Pr is the eddy Prandtl number defined as the ratio between Km and Kh, and N is the
buoyancy frequency
N=
γg
.
θ0
(14)
One of the main deficiencies of the Prandtl model is that the near-surface gradients of the
calculated quantities are too weak. This is because the eddy diffusivities are assumed
constant and do not decrease when the surface is approached. Another deficiency is that the
height of the jet in the Prandtl model does not increase with the strength of the jet
maximum, which is found in observations (e.g., Oerlemans and Grisogono 2002).
To overcome the latter problem, Oerlemans and Grisogono (2002) defined scales that when
put into the Prandtl model, characterize a katabatic state. In their analytical model, they
ended up with three equations predicting steady state jet strength (um), jet height (zm), and
surface sensible heat flux (Fh), from background parameters:
21
STEFAN SÖDERBERG
0.5
k2  g 
 ,
C
k1  θ 0 γPr 
k k
C
,
zm = − 2
k3 γ sin(α )
um = −
Fh =

−kk22C 2 
g 

θ
 0 γPr 
(15)
(16)
0.5
.
(17)
However, the empirical constants k, k1, k2, and k3, are yet to be determined, and as pointed
out by Oerlemans and Grisogono (2002), the currently available observational data does not
allow for a determination of these constants.
In Paper IV the aim was to compare the predictions of the analytical model described above
with katabatic flow characteristics as simulated and modeled by a “state of the art”
mesoscale numerical model. Since the constants in (15)-(17) are unknown, the test had to be
done in three steps.
• First a realistic behavior of COAMPS was verified by comparing results from a real-data
case study to observations; the horizontal model domains are shown in Figure 2.
• Then a series of idealized numerical simulations were set up with conditions similar to
those of the analytical scaling. This allowed us to find estimates of the undetermined
constants.
• Finally, the analytical model was applied to the real-data simulation of the katabatic flow
over Breidamerkurjökull, and its predictions of um, zm, and Fh from simulated
background parameters, were compared to the corresponding quantities, as simulated
and modeled by COAMPS.
The location of some of the observational stations put on Breidamerkurjökull during the
1996 Glacio-Meteorological field experiment (Oerlemans et al. 1999), are shown in Figure
14. Also shown is the simulated near-surface flow over Breidamerkurjökull. From the top of
the glacier and down towards the ocean, the flow accelerates and turns into the glacier fall
line. To further illustrate the spatial variation of the simulated flow over
Breidamerkurjökull, Figure 15 shows vertical cross sections of temperature and the
meridional wind component, taken along B-C in Figure 14. Even though this cross-section is
not along a trajectory, it provides a good view of what an airparcel may experience on its
way from the top of Breidamerkurjökull down towards the ocean. In the upper part of the
glacier, the temperature deficit is small. As one moves down the glacier, the atmospheric
temperature increase adiabatically, while the surface temperature is constant at melting
point; the temperature deficit experienced by an airparcel as it progresses down the glacier,
therefore will increase. In response to the increased katabatic forcing, the flow accelerates
and forms a wide low-level jet, covering a large part of Breidamerkurjökull. Another
consequence of the adiabatic heating is that the buoyancy force will continuously increase
along the trajectory, and in this sense, the katabatic flow drives itself.
22
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
64.35
C
2
64.3
1
64.25
64.2
I6 2
1
Latitude
64.15
A5
64.1
64.05
2
1
3
4
A4
U3
U2
64
63.95
1
2
63.9
63.85
1
B
63.8
−16.6 −16.55 −16.5 −16.45 −16.4 −16.35 −16.3 −16.25 −16.2 −16.15 −16.1
Longitude
Figure 14. Scalar wind speed (dashed) and wind vectors at 11 m height 7 July 12 UTC in the immediate
surroundings of Breidamerkurjökull. Terrain elevation is contoured every 100 m; coastline is bold.
Observational stations and their positions are indicated by dots (Oerlemans et al., 1999). Stars show grid
points used in the estimates of surface sensible heat flux and katabatic jet height and strength in section 3.2.
The cross-sections shown in Figure 15 are taken along B-C.
A principal agreement between observed and simulated flow-characteristics over
Breidamerkurjökull was found in Paper IV, ensuring a realistic behavior of COAMPS.
Scaling arguments also provided support for classifying the simulated low-level flow as a
shooting flow, within the dynamical regimes for downslope gravity flows organized by
Mahrt (1982). This type of flow is defined as a three-term momentum balance between the
buoyancy term, downslope advection and turbulent transport. Our scale analysis also
revealed that cross-slope advection can be of importance in the downslope momentum
equation. Encouraged by these results, we continued the work in finding estimates of the
unknown empirical constants in the analytical theory. Model results from the idealized
experiments, emulating the conditions of the analytical scaling, were used to calculate the
constants. The maximum simulated downslope wind-speeds plotted versus um from the
idealized simulations and from the real-data simulation are shown in Figures 16a and 16b,
respectively. The simulated wind speed and predicted wind speed from the analytical model
line up quite nicely around the one-to-one line, providing the value of the constant. In the
real-data simulation, a reasonable agreement between simulated and estimated values of the
wind speed is found. However, some discrepancies are apparent, primarily in the lower part
of the glacier and where the simulated jet height is above 10 m; here the simulated wind
speed have a magnitude much higher than that predicted by the analytical model. Similar
observations were also made for the jet height and the downward directed surface sensible
heat flux.
23
Elevation (km)
STEFAN SÖDERBERG
1.5
1.0
0.5
0
63.8 63.85 63.9 63.95
64
64.05 64.1 64.15 64.2 64.25 64.3 64.35
1 0
200
180
1
140
Height agl (m)
272
0
160
−1
120
100
282
80
2
60
280
−2
2
278
40
276
−2
282
−1−2
20
1 0
0
63.8 63.85 63.9 63.95
−1
64
−3
−4
274
0
−1
1
64.05 64.1 64.15 64.2 64.25 64.3 64.35
Latitude
Figure 15. Cross-section along B-C of Figure 14 of the meridional wind component (solid) and temperature
(gray dashed) at 12 UTC 7 July. Vertical axis is in m above ground level. Terrain elevation in km above sea
level along the cross-section is also shown. Negative values of the meridional wind component are here
directed close to the glacier fall line.
From the observations described above and the fact that the wind-speed maximum in the
idealized experiments was found at 10 m height or below, one possible reason for the
analytical underestimation of the wind speed in the real-data experiment, can be the
proximity of the jet to the surface. In Paper IV it was hypothesized that local effects, such as
surface inhomogeneity and slope geometry, absent in the idealized experiments but certainly
present in the real-data simulation, lifts the jet to a higher height than it would have been
found at, in the pure katabatic flow conditions assumed in the analytical model. As a result,
the magnitude of the wind speed will also increase since the effect of friction decreases with
height. In the real-data simulation, the analytical model therefore will underestimate the jet
height, the wind speed and also the surface sensible heat flux, since the latter depends on the
katabatic velocity scale.
24
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
8
8
0
α=2
0
α=4
0
α=6
0
α=8
6
upper
mid
lower
7
Maximum downslope wind (m s−1)
Maximum downslope wind (m s−1)
7
5
4
3
2
6
5
4
3
2
1
1
(a)
(b)
0
0
0
1
2
3
4
5
6
7
−1
u (m s )
m
0
1
2
3
−1
um (m s )
4
5
6
Figure 16. Maximum simulated wind speed plotted versus the analytical estimate, um from (a): Ideal
simulations, the markers represent different slope angles according to legend; solid line is a one-to-one line.
(b) Real-case simulation, the markers indicate in which third of Breidamerkurjökull along the fall line the
values are taken. Black markers indicate grid points with a wind maximum below 10 m height; grey markers
indicate a wind maximum between 10 and 30 m height; 825 data points are plotted.
4. Boundary-layer characteristics in the presence of low-level jets
In the presence of a low-level jet, the boundary-layer structure is often dominated, or at least
significantly influenced, by the vertical wind-shear. This is because shear-generated
turbulence acts to mix boundary-layer properties such as temperature and humidity
vertically. If the boundary layer is stable enough, the mixing can however be suppressed.
Moreover, the boundary layer structure can be significantly altered if the low-level jet
encounters complex terrain.
As an example, Figure 17 displays two characteristic MABL profiles of observed and
simulated potential temperature and wind speed in the vicinity of Cape Mendocino. One
profile is taken upstream of the cape and the other within the expansion fan downstream of
the cape (see Paper III for details). In addition to the effects of the wind-speed jet, the SST
variation along the coast, also influences the boundary-layer structure. This can be seen in
the observed potential temperature profile upstream of the cape, indicating a stable internal
boundary layer in the lowest 300 m of the MABL. The model does not capture the internal
boundary layer in detail although the wind-speed profile is fairly well reproduced.
Downstream of the cape the vertical structure of the MABL is more intricate. The MABL
flow has accelerated and the MABL depth has decreased significantly in response to the
altered terrain forcing and the lee-wave induced by the cape; still the model does a fair job
in reproducing the observed features.
Another example of a boundary layer dominated by a low-level jet, is found over
Breidamerkurjökull and studied in Paper IV. Figure 18 shows observed and simulated
vertical profiles of temperature and wind speed from the upper and lower part of
Breidamerkurjökull. One of the most striking differences between the MABL and the
katabatic layer is that the jet is found below the inversion in the former and within the
inversion-layer in the latter. Due to the proximity of the jet to the surface, the near-surface
25
STEFAN SÖDERBERG
900
800
800
700
700
600
600
Altitude (m)
1000
900
Altitude (m)
1000
500
400
500
400
300
300
200
200
100
100
(a)
0
278
280
282
284
286
288
290 292
θ (K)
294
296
298
300
302
(b)
0
304
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Wind speed (ms−1)
Figure 17. (a) Profiles of potential temperature upstream of Cape Mendocino (dark line) and within the
expansion fan (gray line). The dashed lines show profiles from the model for all the grid points spanned by
the aircraft slant-profile. The top of the internal boundary layer is indicated by an arrow. (b) The mean wind
speed for the same locations as in (a). See Paper III for details.
100
(a)
90
90
80
80
70
70
60
60
Height (m)
Height (m)
100
50
40
50
40
30
30
20
20
10
10
0
(b)
0
0
1
2
3
4
5
6
7
8
9
10
m s−1; oC
0
1
2
3
4
5
6
7
8
9
10
m s−1; oC
Figure 18. Observed scalar wind speed (diamonds) and temperature (circles) from profile mast. Simulated
scalar wind speed (solid) and temperature (dashed) from: (a) station A5; and (b) station A4. Also shown in
(b) is a temperature profile from the model grid point due south of the one closest to station A4 (dashdotted). The difference in terrain elevation between the two gridpoints is ~310 m. Scalar wind speed (solid
gray) and temperature (dashed gray) from the tethered balloon sent up from station U3 are also shown in (b).
vertical wind-shear is substantial. Moreover, the shear is more or less confined to the
katabatic layer. In contrast to this, strong vertical wind-shear is also found at higher altitudes
around the inversion in the MABL. When comparing the simulated flow characteristics to
observations, it is apparent that the simulated inversion layer is too deep compared to the
observations; the simulated jet height and magnitude are also too high. In Paper IV it is
speculated that this might be due to a too vigorous vertical mixing close to the surface. One
difficulty in directly comparing simulations to observations in complex terrain was also
highlighted here. Due to the finite horizontal resolution in the model, the difference in
terrain elevation between station A4 and closest the grid point was in fact ≈ 225 m. The
observed temperature profile agrees better with the simulated temperature profile from a
model grid point where the terrain elevation agrees better with the actual height of station
A4 above sea level.
26
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
One may ask why the linear hydraulic theory presented in section 3.1.2 has proven
successful in describing the flow off the U.S. west coast, in spite of the apparent differences
discussed in Papers I-II. Among the assumptions in the theory is a single flow velocity
within the layer, while the actual MABL features a wind-speed jet. For the singe-layer
reduced gravity theory to be applicable, the MABL must also be separated from the free
atmosphere above, since the theory assumes that all gravity-wave energy resides within an
infinitesimally thin inversion. Burk et al. (1999) proposed that wave trapping could be the
process responsible for the separation of the MABL from the free atmosphere above. They
based their conclusion on calculations of the Scorer parameter L2, which within linear theory
is defined as (e.g., Nappo 2002):
L2 =
N 2 1 ∂ 2U
.
−
U 2 U ∂z 2
(18)
Burk et al. (1999) found small and often negative values of L2 above the coastal jet,
implying that vertically propagating buoyancy waves are not possible; instead the waves
decay exponentially with height. Small or negative values of L2 are due to that the curvature
term in (18) becomes large compared to the buoyancy term, a characteristic of the jet itself.
This conclusion is not only interesting as a physical explanation for the observed
supercritical flow-response around coastal bends, but also in a more philosophical way. That
is, the presence of a wind-speed jet below the inversion, instead of a uniform layer-velocity
as assumed in the hydraulic theory, may in fact be the very reason for the success of the
hydraulic theory, in describing observed characteristics of the MABL flow along the U.S.
west coast.
Since wave trapping is common when wind-speed jets are present (Nappo 2002), this points
to a generality for boundary layers dominated by low-level jets, namely that they are
effectively shielded from the free atmosphere above. From observations over
Breidamerkurjökull, Parmhed et al. (2004) hypothesized that wave trapping could be
important for the persistence of katabatic flows. In Paper IV we calculated L2 from model
results and a contour plot typical for the lower part of Breidamerkurjökull is presented in
Figure 19. The deep layer with negative values of L2 is a clear indication of persistent wave
trapping above the katabatic jet and a decoupling of the near-surface flow from the ambient
flow aloft. Although several observational studies have pointed out that katabatic flows are
persistent over glaciers, even at locations like Vatnajökull in the middle of the north Atlantic
stormtrack (e.g., Smeets et al. 1998; Oerlemans et al. 1999), none have given a physical
explanation for this. Our results suggest that wave trapping effectively separate the jet from
the flow aloft. The trapping is a consequence of the vertical structure of the jet itself, and
therefore will katabatic flows over glaciers be persistent, as soon as they are established.
27
STEFAN SÖDERBERG
200
0
180
0
160
0
Height (m)
140
0
120
0
100
80
0
0
60
−0.1
40
20
0
−0.1
−0.2
−0.1
0
1 0
0
−0.1
0.5
011
0.5
3
0
0.5
110.5 0
1
1
1
1.5 0.5
0.5 01.5
6
9
12
Time (UTC)
15
18
1.5
0.5 1.5
21
24
Figure 19. Contour plot of the Scorer parameter, L2 .10-2, from the model grid point closest to station A4.
Bold lines are 0-isolines.
5. Turbulence structure in stable boundary layers
Whenever the surface is cooler than the air, the boundary layer usually becomes stably
stratified. One physical process leading to the formation of stably stratified boundary layers
is radiative cooling, which often is the case at night over land. Another process is advection
of warmer air over a cooler surface. This can lead to the formation of an internal stable
boundary layer like in Paper III. Although stable boundary layers are common, they are still
poorly understood and not well described in numerical weather and climate models (e.g.,
Poulos et al. 2002). One of the reasons for that may be that the stable boundary layer has
been less widely studied than its neutral or convective counterparts. Other reasons can be
that turbulence in stable boundary layers is weak and often co-exists with gravity waves
(e.g., Nappo and Johansson 1999). Measurements at a single location may also not be
representative of the area as a whole due to spatial heterogeneity. Moreover, turbulence in
stably stratified boundary layers can also be sporadic and patchy. This is because sheargenerated turbulence, or mechanically generated turbulence, is suppressed by negative
buoyancy. Depending on the magnitude of these two competitors, stably stratified boundary
layers can range from well mixed to essentially nonturbulent. This sets a limit for how well
an ensemble-average numerical model can simulate atmospheric processes in stable
boundary layers since this type of closure, by its very nature, cannot describe sporadic
turbulence. Furthermore, in ensemble-average numerical models, the turbulence is modeled
rather than simulated.
28
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
3
2.5
z / zjet
2
1.5
1
0.5
0
0.05
0.2
0.5
1
2
5
20
50
200
Ri
g
Figure 20. Dimensionless height z / zjet plotted versus Rig from 00 to 24 UTC 7 July 1996. Profiles are taken
from the grid points marked with stars in Figure 14.
The stability range is often the discussed in terms of the gradient Richardson number Rig
(e.g., Stull 1988):
Rig =
g ∂θ v
θ v ∂z
  ∂u  2  ∂v  2 
  +    ,
  ∂z   ∂z  


(19)
where u and v are now the along-wind and cross-wind components, and θv the virtual
potential temperature. The z-axis is now again aligned with the gravity vector, as is the
vertical velocity, w. Note that in general ∂v ∂z ≠ 0 even if v ≡ 0 , except in the surface layer
where the wind direction is usually assumed constant. It is often assumed that when Rig
exceeds a critical value, turbulence will cease to exist.
Near a local wind-speed maximum, the shear-term becomes small and the stability therefore
increases. This is illustrated in Figure 20 showing the relation between Rig and the jet height
from the simulated katabatic flow over Breidamerkurjökull. The values of Rig are here taken
directly from the turbulence closure scheme utilized in the numerical model. It is clear that
the highest stabilities are found close to the wind-speed maximum. Above the jet there is
also a tendency for the scatter among the Rig profiles to increase. These model results agree
well with observations of katabatic flow over the Pasterze glacier in Austria (Smeets et al.
2000), although the Rig profiles appear smoother here. A reason for this may be
uncertainties in the calculations of the wind-speed gradient near the wind-speed maximum,
both in observations and in the numerical model. It may also be an artifact, since Smeets et
al. (2000) plotted individual data points from fixed instruments, above and below the windspeed maximum, while we instead plot continuos profiles, connecting associated data points
across the wind-speed maximum.
29
STEFAN SÖDERBERG
3
2.5
2.5
2
2
i
z/z
z/z
jet
3
1.5
1.5
1
1
0.5
0.5
(b)
(a)
0
−1.2
−1
−0.8
−0.6
−0.4
−0.2
<uw> /
0
0.2
0.4
0.6
0.8
0
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
<uw>/u2
*
u2*
Figure 21. Simulated momentum flux normalized by the square of the surface friction velocity plotted
against normalized altitude from (a): Breidamerkurjökull, 00 to 24 UTC 7 July 1996, from the grid points
marked with stars in Figure 14. (b) The expansion fan in the lee of Cape Mendocino. The solid line in (a)
and (b) is the profile suggested by Lenshow et al. (1988) for stable stratification.
To further illustrate the vertical turbulence structure of stable boundary layers in which
wind-speed jets are present, profiles of the modeled momentum flux over
Breidamerkurjökull and from within the expansion fan in the lee of Cape Mendocino are
shown in Figure 21. The height of the jet, zjet, has been used to normalize z in Figure 21a,
while in 21b we instead have used the MABL depth zi, defined as the height at which the
momentum flux falls to 1% of its surface value. Note that although the parameters used to
scale z are not identical, the momentum flux profiles scale in a similar way. This highlights
the difficulty of defining one unique length scale for the stably stratified boundary layer, as
will be discussed below. However, in the present case the two length scales converge since
u' w' approaches zero close to the wind-speed maximum.
A striking feature in the profiles of normalized momentum flux, are the high magnitudes of
upward directed flux above the wind-speed jets; this is distinct from stable boundary layers
in general. Above the katabatic jet, upward momentum flux reaches 40% of its surface
value; the high values are due to a reduction in static stability in the katabatic layer, while
the wind shear remains relatively strong. Above the MABL wind-speed maximum, even
higher values of the normalized momentum flux are found, reaching magnitudes up to 80%
of the surface values. Interesting to note is that the high values are all from the coastal side
of the jet. In Paper III it was speculated that this is due to advection of warm inland air
offshore, as a result of the secondary circulation around the jet observed in Paper I. This
reduces the static stability to a point where the Richardson number becomes subcritical and
turbulence can be sustained above the jet. Profiles from the offshore side of the coastal jet
have in fact relatively low magnitudes of momentum flux above the jet, not exceeding 5%
of their surface values; this is also less than for most of the profiles above the katabatic jet.
In Figure 21 we have also plotted the normalized profile suggested by Lenshow et al. (1988)
for stable stratification (thick solid line). Yet another difference between the two boundary
layers is here apparent. In the katabatic boundary layer, the profiles are more concave than
the analytical line. This is due to an increased momentum flux at the surface as the low-level
30
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
flow continuously accelerates on its way down the glacier. The proximity of the jet to the
surface also contributes to the concave profile because the magnitude of the momentum flux
must decrease rapidly, as the wind-speed maximum is approached from below. In the
MABL, the normalized momentum flux profiles are either more concave or convex than the
profile suggested by Lenshow et al. (1988). This is a result of a combination of changes in
MABL depth, wind speed, wind shear, and momentum flux at the surface, in the along-coast
flow. When the flow accelerates in the upwind portion of the expansion fan, the momentum
flux at the surface also increases, giving rise to a concave profile; this is similar to what is
found over Breidamerkurjökull. Where the flow speed is the highest, the combined effect of
a rapid decrease in MABL depth and an enhanced mixing due to an increased vertical windshear acts toward a more linear profile. As the flow speed decreases in the downwind
portion of the expansion fan, so does the momentum flux at the surface, which produces a
convex profile of the normalized momentum flux.
It was pointed out above, that one of the difficulties in studies of stable boundary layers lies
in defining a representative length scale of turbulence. Turbulent variables have traditionally
been scaled in terms of the surface-layer fluxes as a function of z / h, where h typically is the
boundary layer height. However, in stable conditions vertical motions are restricted and
turbulent eddies can for that reason not extend over the whole boundary layer. The use of h
as a characteristic length scale is consequently not always the correct choice in stable
boundary layers. Moreover, the height h itself is not uniquely defined in the stable boundary
layer. It is sometimes defined using the temperature profile, sometimes using the TKE, and
sometimes at the wind-speed maximum. All these heights may be different in the stable
boundary layer. To circumvent this problem, Niewstadt (1984) instead introduced the local
scaling hypothesis, with scales defined in a manner analogous to the Monin-Obukhov
scales. These scales, however, depend on local turbulence quantities at the actual height,
rather than surface values; one can consider Niewstadt's theory as an extension of the ideas
of Monin-Obukhov similarity theory above the surface layer. The local similarity scales
reads:
(
2
u L = w' u' + w' v'
)
2 0.25
,
θ L = − w' θ ' u L ,
LL =
− u L3θ v
(κg w' θ 'v ) ,
(20)
(21)
(22)
where κ is the von Kármán constant (taken as 0.40), uL local friction velocity, θL local
temperature scale, and LL the local Obukhov length.
As the stability parameter z / LL becomes large, the theory of Niewstadt (1984) predicts that
locally scaled quantities should approach a constant value. This is also what has been found
in several observational studies in different environments, although with a not insignificant
scatter between studies (e.g., Sorbjan 1986, 1987; Horst and Doran 1988; King 1990;
Brooks and Rogers 2000). However, in most studies the range of stabilities has been limited
and general applicability has not yet been demonstrated. Shao and Hacker (1990) included a
wider range of stabilities in their study and found that instead of coming close to a constant
31
STEFAN SÖDERBERG
value as predicted by theory, the scaled variances actually increased in a well-ordered
manner at high stabilities. Pahlow et al. (2001) and Al-Jiboori et al. (2002) have
subsequently reported similar results, although with different stability dependencies.
In Paper III local scaling was applied to the standard deviation of the velocity components
taken from observations off the California coast. Figure 22a shows the scaled standard
deviation of vertical velocity variance plotted against z / LL. A division of the individual
estimates have been made into those from within the expansion fan (circles) and those from
outside the fan (triangles). Although the scatter is significant in both sets, there is no
systematic difference between them. The heavy solid line shows the best-fit curve to our
observations; for comparison curves found by other investigators are also plotted. Modeled
vertical velocity variance scaled in the same way as the observations are shown in Figure
22b, along with the best fit to the observations. An excellent agreement between the
empirical function and the model results is evident from here; moreover, no systematic
difference in scaling behavior between regions with distinct flow dynamics can be
discerned. Similar results were also found for the observed and modeled horizontal velocity
variances (not shown).
These findings point to some significant results. The successful scaling of the velocity
variances in a highly perturbed environment, such as in the MABL studied in Paper III,
suggests that local scaling is a robust feature and can be expected to apply widely. The
successful scaling of the modeled velocity variances also provides a strong validation for the
turbulence closure scheme utilized in the model. At the same time it indicates a generality in
the observational results since the model results depend on a turbulence closure derived
from completely independent experimental data. However, although the empirical functions
found in Paper III have a similar form to those observed by several previous studies (Shao
and Hacker 1990; Pahlow et al. 2001; Al-Jiboori et al. 2002), some differences are
significant, in particular at high stabilities. This poses the question as to what can be the
cause of such substantial differences in the observed scaling functions. Moreover, the
departure from the constant values predicted by Nieuwstadt's original theory has not yet
been explained.
One of the assumptions in the local similarity theory is horizontal homogeneity, an
assumption widely used in the search for suitable parameterizations of atmospheric
turbulence processes, since it simplifies the problem substantially. In Paper III this evidently
is not a valid assumption. Another assumption is that the turbulence transport terms are
small in stable conditions and therefore negligible. We hypothesize that the form of the
scaling functions are controlled by non-local transport terms, such as advection, and
turbulent and pressure transport terms, in the TKE budget, and that the curves represent the
controlled breakdown of true local similarity. Since non-local processes may differ
significantly between different data sets, the scaling functions will differ between studies.
Changes in non-local transport may also be responsible for some of the scatter in the
observations within one data set.
32
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
6
out of fan
in fan
5
σ /u
L
4
w
3
2
1
0
−3
10
(a)
−2
10
−1
0
10
10
1
2
10
10
3
10
z/LL
10
9
8
7
σw/uL
6
5
4
3
2
1
0
−3
10
(b)
−2
10
−1
0
10
10
z/L
1
2
10
10
3
10
L
10
9
8
7
w
σ /u
L
6
5
4
3
2
1
0
0
10
(c)
1
2
10
10
z/L
L
Figure 22.. Scaled standard deviations of the vertical velocity component from the MABL off northern
California, normalized by the local velocity scale and plotted against the stability parameter z / LL (a):
Observed turbulence quantities; the thick solid line is a best fit to our entire data set, the dashed line is the
empirical function found by Pahlow et al. (2001), the dotted line the function found by Shao and Hacker
(1990), and the thin solid line the function obtained by Al-Jiboori et al. (2002). (b): Modeled turbulence
quantities; the heavy dashed line is the best-fit curve from the observations, see (a). (c) As (b) but for points
above the top of the modeled boundary layer. The heavy dashed line is the same as above while the dotted
line is the constant value approached at near-neutral conditions. The circles are observational data obtained
in the turbulent region above the expansion fan.
33
STEFAN SÖDERBERG
The modeled vertical velocity variance is plotted also in Figure 22c, however this time, the
data points are taken from above the jet in regions with sustained turbulence (see Figure
21b). Also shown are observations from the turbulent region above the expansion fan
(circles). The scaled variances show a significant scatter and the well-defined functional
relationship with stability appears to break down. Nevertheless, the normalized variance
values are limited on the high side by the empirical function and on the lower side by a
constant value near the neutral limit of the scaled variance. If our hypothesis that the form of
the scaling functions to a high degree are controlled by non-local transport terms is true,
then the scatter in the figure could be explained by differences in non-local processes in the
volume sampled. The densest population of points lies close to the constant defined by
neutral conditions, representing conditions where turbulent processes are truly local. Other
groups of points appear to line up along curves of similar shape, and may represent regions
with similar non-local transport. The break down of the scaling, and the limitations in the
scatter, is also observed for the two horizontal velocity variances (not shown).
To test our hypothesis that non-local processes control the form of the scaling functions, we
turn to a different environment. Local similarity scaling has previously been applied to
observations over Breidamerkurjökull (van der Avoird and Duynkerke 1999; Smeets et al.
1999). In both these studies the scaled variances were close to the constant values suggested
by Nieuwstadt's (1984) original theory. However, the stability ranges covered were limited,
only up to z / LL = 1 in van der Avoird and Duynkerke (1999), and up to z / LL = 0.2 in
Smeets et al. (1999). Horst and Doran (1988) also applied local scaling to katabatic flow
with success, yet again within a limited stability range. Moreover, they only considered
turbulent quantities above the wind-speed maximum. In this thesis, turbulence quantities
have been diagnosed from the COAMPS simulation of katabatic flow over
Breidamerkurjökull; the scaled velocity variances σu, σv, and σw are shown in Figure 23
along with the empirical functions found in Paper III. The scaled variances have also been
partitioned into groups by normalized height according to the legends.
At first sight, the velocity components appear to follow a functional relationship similar to
the empirical functions found in Paper III (solid lines). Nevertheless, some substantial
differences in scaling behavior between the two environments must be pointed out. Note
first the absence of data points at the near-neutral side of the stability range, or weakly stable
cases as defined by Mahrt et al. (1998). The curvature of the data-point cluster for the alongwind component also appears less steep than the empirical function; at the highest stabilities
the points are on the lower side of the empirical function (Figure 23a). More apparent are
the differences for the cross-flow velocity variance (Figure 23b). Most of the data points lie
well above the empirical scaling function and the curvature of the data-point cluster does not
rise as quickly as the empirical function at high stabilities. In fact, the scaled cross-flow
velocity variances group closer to the empirical function for the along-flow velocity
variance. Since the velocity variances are diagnosed from COAMPS model output, this
could be an artifact of this procedure. On the other hand, it could also be a manifest of a
significant wind direction shear through the jet. Another striking feature is that the
magnitude of the scaled vertical velocity variances in many cases is unusually low, well
below the minimum value of the empirical function (Figure 23c). These points are primarily
found above the katabatic jet. A functional relationship between the scaled vertical velocity
variance and stability is also not as clear as for the two horizontal components.
34
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
10
9
8
7
z <= 0.5z
jet
0.5zjet < z <= 1.0zjet
1.0zjet < z <= 1.5zjet
1.5z < z <= 2.0z
jet
jet
2.0z < z <= 5.0z
jet
jet
u
σ /u
L
6
5
4
3
2
1
0
−3
10
(a)
−2
10
−1
0
10
10
z / LL
1
10
2
10
3
10
10
9
8
7
z <= 0.5z
jet
0.5zjet < z <= 1.0zjet
1.0zjet < z <= 1.5zjet
1.5z < z <= 2.0z
jet
jet
2.0z < z <= 5.0z
jet
jet
v
σ /u
L
6
5
4
3
2
1
0
−3
10
(b)
−2
10
−1
0
10
10
z / LL
1
10
2
10
3
10
10
9
8
7
z <= 0.5z
jet
0.5zjet < z <= 1.0zjet
1.0zjet < z <= 1.5zjet
1.5z < z <= 2.0z
jet
jet
2.0zjet < z <= 5.0zjet
w
σ /u
L
6
5
4
3
2
1
0
−3
10
(c)
−2
10
−1
10
0
10
z/L
1
10
2
10
3
10
L
Figure 23. Modeled turbulence quantities over Breidamerkurjökull 00 to 24 UTC 7 July 1996, scaled in the
same way as in Figure 22; (a) along-flow component, (b) cross-flow component, and (c) vertical velocity
component. Solid lines in each panel are the empirical functions found in Paper III. The scaled variances are
partitioned into groups by normalized height according to legends.
35
STEFAN SÖDERBERG
5
5
(a)
4.5
4
3.5
3
L
3
σ /u
L
4
3.5
2.5
2.5
σ
v
w
/u
(b)
4.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0.1
0.2
0.5
1
2
5
z / zjet
0
0.1
0.2
0.5
1
2
5
z / zjet
Figure 24. Normalized standard deviations of (a): the vertical component, and (b): the cross-flow
component, over Breidamerkurjökull, 00 to 24 UTC 7 July 1996, as a function of z / zjet.
From Figure 23a and 23b it is clear that the highest values of the scaled variances are found
near the wind-speed maximum, where the stability also is the largest. Above and below the
jet, the magnitudes of the velocity variances and the stability are more moderate. Since uL
appears both in the scaled standard deviation and in LL, there is a risk of self-correlation
between the scaled standard deviations and z / LL (Mahrt et al. 1998). Moreover, Smeets et
al. (2000) argued that near the wind-speed maximum, where the shear production of
turbulence should cease, uL becomes small and can therefore not represent a proper length
scale while at the same time LL cannot be a proper length scale. To illustrate this, Smeets et
al. (2000) plotted σw scaled with uL against z / zjet and found that the values become
unusually large near the wind maximum. A comparable figure based on our results from
Breidamerkurjökull is shown in Figure 24a. It is immediately clear that there is a difference
in scaling behavior between the observed katabatic flow over Pasterze glacier and the
simulated katabatic flow over Breidamerkurjökull. In contrast to the results in Smeets et al.
(2000), where values become large due to that uL becomes much smaller than σw near the
wind maximum, no such general tendency can be discerned in Figure 24a. In fact, in many
cases our results show the opposite for z / zjet > 0.5, although it appears that σw in most cases
decrease at the same rate as uL when zjet is approached. Moreover, in Smeets et al. (2000) all
three velocity components varies at the same rate with normalized height, as implied by the
constant ratios σu /σw and σv /σw. Here σv actually becomes relatively large compared to uL
close to the wind maximum (Figure 24b). In addition, σv is reduced at a slower rate than σu
near the wind-speed maximum (not shown); this is another indication of directional shear
through the jet as suggested by the unusually large values of the scaled cross-wind velocity
variance in Figure 23b.
From these observations we can only speculate as to why the three velocity variances
behave in such a different manner in the katabatic flow above the two glaciers. One possible
explanation may arise from differences in local flow features. Over Breidamerkurjökull, the
katabatic flow aligns with the glacier fall line on its way down towards the Atlantic Ocean
(see Figure 14). In the lower part of the glacier, cross-slope advection can also be of
36
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
importance for the momentum budget (Paper IV). The exact details of the katabatic flow
over the Pasterze Glacier is presently not known except for that the fetch for the
experimental site which Smeets et al. (2000) collected data from, was relatively uniform.
Differences in local flow characteristics between the two glaciers may therefore lead to
significant differences in the contributions of non-local transport terms to the variance
budgets.
In COAMPS, second order moments are obtained from steady-state analytical expressions
while TKE is a prognostic variable. In order to investigate possible non-local contributions
affecting the form of the scaling functions, we will therefore have to study the TKE budget.
In a coordinate system aligned with the mean flow u, the TKE (e) equation used in
COAMPS reads (Hodur 1987):
∂e
∂e ∂ 
∂e 
∂u
∂v g
+ u −  leSe  = −u' w'
− v' w' + w' θ 'v − ε ,
∂t
∂x ∂z 
∂z 
∂z
∂z θ v
(23)
where the two first terms on the LHS are the tendency and advection of TKE, respectively.
The third term on the LHS is the turbulent diffusion of TKE, and represents both turbulent
and pressure transport. As will become apparent later, this term is vital in the turbulence
closure for studies of wind-speed jets. On the RHS in (23) the first and the second terms are
the shear production of TKE; the third term is the buoyancy production term; and the fourth
term is the dissipation term.
Over flat terrain, the stable TKE budget is essentially a balance between shear production
and dissipation, while buoyancy is a small sink term. Turbulence transport terms have
traditionally been neglected since they are much smaller than the other terms in the TKE
budget; this is also one of the assumptions in the local scaling theory (Niewstadt 1984). In
slope flow TKE budgets, the buoyancy term can also contribute to the balance. A balance
between the production, dissipation and buoyancy terms was also what Smeets et al. (1999)
found at the lower end of Breidamerkurjökull. This conclusion was, however, based on
measurements during neutral or slightly stable conditions, and when the wind-speed
maximum was above the highest level of the mast. Of importance for the present thesis is
that turbulent transport may be significant in the region of the wind-speed maximum, where
turbulence production is small and a local minimum in TKE often occur (e.g., Arrit and
Pielke 1986; Horst and Doran 1988). Moreover, from measurements in katabatic flow over
Breidamerkurjökull, van der Avoird and Duynkerke (1999) found that the contributions by
non-local sources and sinks in the TKE budget, increase with stability.
Figure 25a shows the simulated TKE budget from the lower part of Breidamerkurjökull;
corresponding profiles of downslope wind-speed (black solid), cross-slope wind (dashed),
scalar wind-speed (gray solid), and potential temperature perturbation (dash-dotted) are
shown in Figure 25b as a reference. The layer experiencing katabatic forcing (θ < 0) reaches
up to z / zjet ≈ 3; note that even though the downslope wind-speed dominates the scalar windspeed, there is a directional shear over the jet. In the region of the wind-speed maximum, a
local minimum is found in the TKE (not shown).
37
STEFAN SÖDERBERG
5
5
(a)
buoy
shear
diff
vert mix
4.5
4
U
θ
4
3.5
3
z / zjet
jet
3
z/z
(b)
c
3.5
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
−5
u
d
v
4.5
−4
−3
−2
−1
0
1
2
2
−3
TKE budget (m s )
3
4
5
−3
x 10
0
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
(m s−1; oC)
Figure 25. Normalized profiles from the model grid point closest to station A4 at 12 UTC, 7 July 1996 of
(a): TKE budget, and (b): downslope wind component, cross-slope component, scalar wind speed, and
potential temperature deficit.
The profiles of shear (solid with circles), dissipation (solid with triangles) and buoyancy
(thin solid) shown in Figure 25a are close to what we can expect to find in slope flows.
Some distance away from the wind maximum, large magnitudes of the shear and dissipation
terms are found, while distinct minima in both terms are centered at the wind maximum.
The magnitude of the buoyancy term is smaller than the other two terms, but it is not
negligible, and acts as a sink in the TKE budget. The advection term in the TKE budget is
often neglected since its magnitude is small compared to the other terms (e.g., Stull 1988).
According to our results, advection is a source term in the TKE budget below the wind
maximum while it is a sink term above. However, the magnitude of the advection term is
less than 5.10-5 m 2 s -3 and not included in Figure 25a. We therefore conclude that non-local
contributions to the TKE budget by advection are negligible in the simulated flow over
Breidamerkurjökull.
A more likely non-local contributor to the TKE budget is the vertical mixing term (thick
solid). This is clearly illustrated by the local maximum in the vertical mixing profile in the
region of the wind maximum. In fact, near the wind-speed maximum, local import of TKE
is the main source in the TKE budget. These results also agree with the findings by Denby
(1999) who used a more sophisticated turbulence closure than utilized here, with prognostic
equations for second-order moments. Moreover, assuming stationarity and horizontal
homogeneity, Smeets et al. (2000) found that turbulence and pressure transport terms import
turbulence towards the wind maximum in the katabatic jet over the Pasterze glacier.
Neglecting vertical transport of turbulence in katabatic flows is therefore not a valid
assumption; it is likely that this assumption is invalid also in studies of wind-speed jets in
general.
Recall that the largest discrepancies between the empirical scaling functions found in Paper
III, and the scaled velocity variances from Breidamerkurjökull, are found close to the windspeed maximum. Since non-local contributions to the TKE budget are evident near the
wind-speed maximum, it can very well be so, that the form of the scaling functions are
controlled by non-local transport terms, in this case in the vertical, as hypothesized in Paper
III. This issue cannot be completely resolved in the present thesis but must be further
investigated. A re-assessment of local similarity is required to investigate the dependency of
the scaling functions on non-local properties of boundary layer flows.
38
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
6. Concluding remarks and future outlook
The underlying goal for the studies included in the present thesis, has been to increase our
understanding of mesoscale flow and turbulence characteristics in complex atmospheric
environments. To achieve this goal, the approach has been to combine “observations” from
“the best out of two worlds”: measurements and numerical simulations. Admittedly, a
numerical model is nothing more than just a model, i.e., a more or less crude simplification
of the real world. On the other hand, a model with a physically sound formulation is
dynamically consistent. This implies that if a model is able to reproduce the general flow
characteristics found in measurements then it is likely that these model results can also be
used to draw general conclusions about physical properties and processes within the same
type of flow. This is of importance since measurements generally are sparse, in both time
and space. Combining measurements and numerical modeling may therefore lead to synergy
effects; basically that “1+1 > 2,” or, that measurements and numerical modeling combined
produce greater results than either could separately.
In the present thesis, two types of mesoscale wind-speed jet and their effects on boundarylayer structure have been studied. The first is a coastal jet off the northern California coast,
and the second is a katabatic jet over Vatnajökull, Iceland. Numerical modeling has been the
main tool. However, observations have been used to verify that the numerical models are
capable of simulating the physical properties and processes within the studied flows and to
ensure a realistic behavior of the model results. Hence, observations constitute a basis on
which the conclusions in this thesis rely. The flow response to terrain forcing, the transient
behavior in time and space, and agreement with simplified theoretical models have been
examined for each of the jets. Moreover, the turbulence structure has been investigated in
these stably stratified boundary layers; local similarity scaling was applied to turbulence
quantities. Among the findings in the thesis are:
• The simple shallow-water model provides a useful framework for analyzing highvelocity flows along mountainous coastlines, but for an unexpected reason. Wave energy
is trapped in the capping inversion by the curvature of the wind-speed profile rather than
by an infinite stability in the inversion separating two neutral layers, as assumed in the
theory. The observed supercritical flow response in coastal MABLs, downwind of capes
and points with prominent terrain features is, however, probably a consequence of the
blocking terrain, rather than the supercriticality of the flow. In the absence of blocking
terrain, observations of steady-state supercritical flow are not likely.
• A reasonable agreement between simulated katabatic flow characteristics and those
predicted by an analytical model is found. Nevertheless, some discrepancies are
apparent, primarily in the lower part of the glacier. It is hypothesized that these are due
to non-local effects neglected in the theory, such as surface inhomogeneity and slope
geometry.
• A possible explanation for the different forms of the local similarity scaling functions
between different studies, may be that non-local transports terms significantly
contributes to the velocity variance budgets but not to the stress. Since non-local
processes may differ significantly between different data sets, the scaling functions will
also differ between studies. This issue awaits analysis of a more extensive data set,
covering a wide range of stabilities and flow types, to be settled.
39
STEFAN SÖDERBERG
The results listed above all are examples of how modeling efforts can shed new light on
observed flow and turbulence characteristics: the impact of the local terrain at Cape
Mendocino on the MABL properties was clearly revealed in the numerical simulations
performed here; possible effects of non-local processes, often neglected in simplified
theoretical models, were also illustrated. Important to remember though, is that a numerical
model alone cannot shed new light on atmospheric processes, which are poorly understood
on a more basic level.
The general trend in numerical modeling today is that we are steadily moving towards
higher horizontal and vertical resolutions, primarily as a consequence of the increasing
computer capacity available to the community. This gives us the opportunity to study
atmospheric processes on a finer and finer scale. However, although the resolution in
numerical models steadily increases, we will still need parameterizations of physical
processes in the atmosphere. Even in large-eddy-simulations numerous sub-grid scale
processes must be parameterized; for computational reasons, direct numerical simulations
will likely never be used in numerical weather forecasting. On the other hand, in the early
stages of the ongoing computer-era, Tomas Watson, chairman of IBM said in 1943: “I think
there is a world market for maybe five computers.” Another well-known quote from 1981 is:
“640K of memory should be enough for anybody.” The (in) famous Bill Gates however
states, “I've said some stupid things and some wrong things, but not that” and claims that he
was misquoted.
Nevertheless, an increased resolution in the numerical models also implies that non-local
processes become more important. Including effects like these in parameterizations of
atmospheric processes is therefore one upcoming task for the community if we are to
increase the skill in regional weather predictions. To develop physically sound
parameterizations is probably even more important when they are to be applied in general
circulation models for studies of our present and future climate.
40
TOPOGRAPHICALLY FORCED LOW-LEVEL JETS
Acknowledgements
So, finally the day has come when it is time to present what I've been up to for the last
couple of years. As always when you are about to wrap things up and are just a little bit
short of time, you tend to wonder how all those days just went by. Nevertheless, here I am
with a thesis in my hand and for this, I owe a number of people a lot of gratitude. Some will
specifically be mentioned here, but there is not room to include all of you, so you will have
to trust me when I say that I have you in my mind.
I'm grateful to my supervisors Michael Tjernström and Branko Grisogono. Without your
support, all knowledge you've been willing to share, and enthusiasm, it would have been so
much harder to complete this thesis. Except for being excellent supervisors, you are also two
really nice guys and I want to thank Michael and his family, and Branko for all nice dinner
parties and pastries. I also have to give Michael some extra credit for taking me on as a
student. The opportunities you have given me to attend conferences and present my work
are also highly appreciated; I have learned a lot from these trips.
Another person that has been (still is, and will be in the future) invaluable to me is my wife
Malin. Taking care of our son Martin (who's also invaluable), yourself, and on top of this
me, the last couple of months has been an achievement few people could have managed.
Without you, life just wouldn't be the same. It is hard to find words for how nice it will be
when this thesis finally has been defended, and I can spend more time with the both of you.
However, we have to wait a few more days until we can enjoy some “lazy days” together, at
least lazy relative to today, so here are some more people who work or have been working at
MISU over the years, that I want to thank. My one and only roommate Måns Manilow, also
known as Barry Håkansson, did his best teaching me all he knows about "jazzträsket". To
his despair, I still think Neil Young has written some really nice songs. Nevertheless,
sharing office with you is a privilege to anyone; you are simply one of the nicest and kindest
persons that I've ever met! Following in your footsteps, I spent almost two months in the
Arctic Ocean during the summer of 2001. Special thanks goes to Caroline Leck, who gave
me the opportunity to go there, transformed into a chemist. Patrick Samuelsson is
acknowledged for being such a nice guy and an excellent traveling companion; Mark Zagar
is thanked for the same reasons. Another person who deserves a special recognition is Eva
Tiberg, without you I would have been totally lost some days! Oskar Parmhed is thanked for
a fruitful collaboration; working together is more fun than sitting alone “in the darkness.”
All fellow Ph.D. students are also thanked for various social events making life at and off
work more pleasant.
I have met many inspiring people during my studies. In particular I want to mention Steve
Burk, William Thompson, Tracy Haack, Ola Persson, Carmen Nappo, and Ian Brooks.
Thanks a lot for being such nice guys whenever and wherever we have met. Douglas Adams
and Gary Larson are also acknowledged for their good sense of humor, and so is Leif Enger,
one of Uppsalas most talented chefs.
Finally, I want to thank all my friends and my family for support and distraction from work
on occasions when that have been called for. To all of you, be prepared for here I come,
soon I will be in a place near you for a social visit!!!
41
STEFAN SÖDERBERG
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