Nonlinear Processes in Geophysics

Nonlinear Processes in Geophysics
Nonlinear Processes
in Geophysics
Open Access
Nonlin. Processes Geophys., 21, 87–100, 2014
www.nonlin-processes-geophys.net/21/87/2014/
doi:10.5194/npg-21-87-2014
© Author(s) 2014. CC Attribution 3.0 License.
Tidally induced internal motion in an Arctic fjord
E. Støylen1,* and I. Fer2
1 Department
of Geosciences, University of Oslo, Oslo, Norway
Institute, University of Bergen, Bergen, Norway
* now at: Norwegian Meteorological Institute, Oslo, Norway
2 Geophysical
Correspondence to: I. Fer ([email protected])
Received: 7 June 2013 – Revised: 14 October 2013 – Accepted: 2 December 2013 – Published: 10 January 2014
Abstract. The internal response in a stratified, partially enclosed basin subject to semi-diurnal tidal forcing through a
narrow entrance is investigated. The site is located above the
critical latitude where linear internal waves of lunar semidiurnal frequency are not permitted to propagate freely. Generation and propagation of tidally induced internal Kelvin
waves are studied, for baroclinically sub- and supercritical
conditions at the mouth of the fjord, using a non-linear 3-D
numerical model in an idealized basin and in Van Mijenfjorden, Svalbard, using a realistic topography. The model results are compared to observations of hydrography and currents made in August 2010. Results from both the model
and measurements indicate the presence of internal Kelvin
waves, even when conditions at the fjord entrance are supercritical. The entrance of Van Mijenfjorden is split into two
sounds. Sensitivity experiments by closing each sound separately reveal that internal Kelvin waves are generated at both
sounds. When the conditions are near supercritical, a wave
pulse propagates inward from the fjord entrance at the beginning of each inflow phase of the tidal cycle. The leading crest
is followed by a series of smaller amplitude waves characterized as non-linear internal solitons. However, higher model
resolution is needed to accurately describe the influence of
small-scale mixing and processes near the sill crest in establishing the evolution of the flow and internal response in the
fjord.
1
Introduction
Internal waves in the Arctic regions have received increasing scientific interest recently because of their role in vertical mixing and influence on the regional and large-scale heat
budget and ice cover. Wind-induced internal waves are important during ice-free conditions in seasonally ice-covered
regions, as demonstrated in the northern Chukchi Sea by
Rainville and Woodgate (2009). The positive effect of reduced ice cover on internal wave forcing, however, may be
offset by increased stratification by meltwater which amplifies the negative effect of boundary layer dissipation on internal wave energy (Guthrie et al., 2013). Another source of
internal waves is the action of tidal flow over topography; the
Yermak Plateau is noted as an important region for enhanced
internal wave activity (e.g. Padman and Dillon, 1991; Fer et
al., 2010). In Arctic fjords short period internal waves are
observed under ice (Marchenko et al., 2010; Morozov and
Marchenko, 2012), and longer period internal Kelvin waves
are documented in the Kongsfjorden–Krossfjorden system
(Svendsen et al., 2002).
Internal waves in fjords are typically forced by changing
winds and the barotropic tide. For fjords that are wide with
respect to the internal (baroclinic) Rossby radius, internal
Kelvin waves may arise, propagating cyclonically around the
fjord. Such waves induce a mean current in the wave propagation direction (Støylen and Weber, 2010), which may lead
to an accumulation or deposition of pollutants and biological material in certain areas along the coastline. As discussed
in Cottier et al. (2010), fjords in the Arctic are typically wide
with respect to the baroclinic Rossby radius. Thus, given sufficient forcing, internal Kelvin waves are to be expected in
many stratified Arctic fjords.
In the present work we consider a particular wide, tidally
forced Arctic fjord, namely Van Mijenfjorden in Svalbard
(Fig. 1). The entrance of this fjord is partly covered by an
island, which restricts water inflow to two narrow sounds,
and thus makes the fjord a good “laboratory” for process
studies (e.g. Widell, 2006; Fer and Widell, 2007). The energy extracted from the barotropic tide, partitioning to tidal
jet flux and baroclinic jet flux, as well as the modal contributions to kinetic energy and horizontal shear are discussed
Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.
88
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
Fig. 1. Map of Van Mijenfjorden in Svalbard, situated at 77.8◦ N,
15.5◦ E. The insets show (top left) a blow-up of the fjord mouth
with the island of Akseløya, and (bottom right) the location of Van
Mijenfjord in Svalbard. Lines and dots denote CTD measurement
sections (A, B, C, and D) with start and end stations indicated. Triangles marked TS1–TS6 are the time series stations (Table 1).
in Fjellsbø (2013). Our aim is to describe the generation
and propagation of internal waves induced by tides in this
fjord, by use of recent observations and a 3-D numerical
model. In particular, we investigate the possibility of internal
Kelvin waves, as suggested in Støylen and Weber (2010) and
Skarðhamar and Svendsen (2010). The sensitivity of internal wave generation to hydraulic conditions through the fjord
entrance is investigated numerically. Finally, as the fjord entrance consists of two sounds we demonstrate numerically
the effect of closing each sound on the respective wave field.
This study is organized as follows: in Sect. 2 we present
a theoretical background for internal Kelvin waves in Arctic fjords. Observations from Van Mijenfjorden are given in
Sect. 3. In Sect. 4 we treat the numerical problem in an idealized geometry, before applying realistic bottom topography
of Van Mijenfjorden. Finally, we discuss our findings and
provide concluding remarks in Sect. 5.
2
Theory
In this section we briefly discuss what types of internal motion we may expect in an Arctic fjord, and describe a basic
theory regarding the internal Kelvin wave. Where appropriate, we will assume a two-layer system with constant densities ρ1 and ρ2 for the upper and lower layer, respectively.
Other variables have similar subscripts. The Cartesian coordinate system (x, y, z) with z as upward vertical direction has
corresponding current components (u, v, w).
Internal waves span a broad range of spatial and temporal scales. In a fjord basin there may be propagating Poincaré
waves (Brown, 1973; Farmer and Freeland, 1983) and soliton
trains (Helfrich and Melville, 2006). Along the coast there
may possibly be propagating edge waves (Llewellyn Smith,
2004; Weber and Støylen, 2011) trapped by the sloping bottom, or the internal Kelvin wave trapped by rotation.
Nonlin. Processes Geophys., 21, 87–100, 2014
The theory behind internal Kelvin waves is well established in the literature (e.g. Gill, 1982). Nonlinearity modifies this solution, as demonstrated in laboratory experiments
by Maxworthy (1983), noting a curvature in the cross-wall
wave front as well as a modification of the transverse scale
ci /f , where f is the Coriolis parameter and ci is the baroclinic phase velocity. Later, others confirmed these findings
(e.g. Renouard et al., 1987); see Helfrich and Melville (2006)
for a comprehensive review. Full 3-D numerical efforts on
fjord scale internal Kelvin waves arose during the late 1990s,
with a primary focus on closed inland lakes. Beletsky et al.
(1997) were among the first to conduct 3-D simulations on
this scale, concerning the response in a closed stratified basin
(the Great Lakes) to changes in the large-scale wind field.
Later numerical studies with increased spatial resolution include Hodges et al. (2000) and Gómez-Giraldo et al. (2006)
for Lake Kinneret, Israel.
There are several mechanisms for internal wave generation. In an Arctic fjord, the dominant sources of forcing are
the changing wind fields and the barotropic tide interacting
with topographic features at the fjord entrance. We will restrict our attention to the tidal case which is particularly relevant for an Arctic fjord that is ice covered. When the crosssectional area of the fjord entrance is narrow, as is the case
in Van Mijenfjorden, the tidal current is intensified here. If
the water is stratified, one can determine whether conditions
are sub- or supercritical with respect to the first baroclinic
mode. For a two-layer system, a densimetric Froude number is defined as FD = |us | /ci , where us is the upper layer
current. FD < 1 indicates subcritical conditions, and favours
generation of long internal waves. When FD > 1, the current enters the fjord as a jet, and there will be a hydraulic
jump just inside the entrance. In this case the generation of
long waves may be prohibited; we will discuss this further
in Sect. 5. For completeness, a derivation of the linear internal Kelvin wave in a two-layer reduced gravity system is
given in the following. If the fjord width is large compared
to the baroclinic Rossby radius, a = ci /f , wave solutions of
Kelvin type may exist. In a two-layer reduced gravity system
with a deep lower layer, the velocities in the lower layer are
neglected. The balance of forces in the lower layer may thus
be written as
0
gηx + PSx /ρ1 = −g ξx
0
gηy + PSy /ρ1 = −g ξy .
(1)
Here g 0 = g (ρ2 − ρ1 ) /ρ2 is the reduced gravity, PS is surface pressure, η and ξ are surface and interface displacements, respectively, and subscripts x and y denote partial
derivation. The baroclinic phase velocity is (Gill, 1982)
s
ρ2 − ρ1 H1 H2
g
,
(2)
ci =
ρ2
H1 + H2
where H1 and H2 are the upper and lower layer thicknesses, respectively. If we take a straight coast at y = 0, the
www.nonlin-processes-geophys.net/21/87/2014/
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
89
Table 1. Overview of observation sections and stations where CTD and LADCP profiles were taken. Names A–D denote horizontal sections,
TS1–TS6 time series. A–1 to A–3, B–1 and B–2, and C–1 and C–2 are the re-occupations of the corresponding sections. Time is date of
August, and hour [DD/HH]. Depth [m] is min–max for sections, average for time series.
Name
B-1
A-1
C-1
TS1
TS2
C-2
TS3
Time
No. of samples
Depth
09/20–10/00
10
42–87
10/02–10/04
10
61–117
10/06–10/09
11
41–110
10/10–10/23
18
60
11/00–11/13
27
60
11/13–11/16
11
42–114
11/17–12/06
27
84
Name
TS4
A-2
TS5
A-3
B-2
TS6
D
Time
No. of samples
Depth
12/07–12/20
27
79
12/20–12/23
10
31–112
12/23–13/12
27
79
13/12–13/15
10
32–113
13/17–13/19
11
34–88
13/20–14/09
27
66
14/09–14/15
15
33–113
first-order current component v1 is zero everywhere. Inserting from Eq. (1), the linear first order momentum balance in
the upper layer becomes
0
u1t = g ξx + ν∇h2 u1 + νu1zz
0
f u1 = g ξy ,
(3)
where ν is the kinematic viscosity coefficient, and ∇h2 is the
horizontal Laplacian operator. The corresponding linearized
continuity equation is obtained by assuming |η| |ξ |:
∂
ξt =
∂x
ZH1
u1 dz.
(4)
0
For the sake of simplicity we neglect the effect of friction here. We consider waves of constant frequency ω in accordance with steady tidal forcing. A solution to Eqs. (3)–
(4) is obtained by assuming internal motion of the form
ξ = ξ0 e−y/a ei(kx−ωt) with wave number k, and near-coast
wave amplitude ξ0 . Letting the real part denote the physical
solution, we obtain
ξ = ξ0 e−y/a cos (kx − ωt)
(5)
with the corresponding dispersion relation ω2 = g 0 H1 k 2 .
Equation (5) describes the linear internal Kelvin wave, in accordance with Støylen and Weber (2010). The wave is travelling along the positive x axis on the Northern Hemisphere,
the interface displacement is largest near the coast, and is exponentially damped seaward. The frequency of the wave is
the forcing frequency; in our case the lunar semi-diurnal M2
tidal component. Retaining the friction term leads to the resulting wave being further damped along the coast (Støylen
and Weber, 2010).
Linear wave solutions for free internal Poincaré waves are
restricted at the critical latitude where ω = f for the respective forcing frequency (Vlasenko et al., 2003). For the M2
tidal component the critical latitude is φ = 74.5◦ . This restriction does not apply to internal Kelvin waves, however,
and they may exist in Arctic regions above the critical latitude (Farmer and Freeland, 1983).
www.nonlin-processes-geophys.net/21/87/2014/
3
Observations
Our study site is Van Mijenfjorden in Svalbard; see Fig. 1.
This fjord is an interesting “laboratory” for the study of propagating baroclinic waves of tidal periodicity. At the outer region of the fjord the near-coast bathymetry is quite steep. The
entrance of the fjord is partly covered by an island, Akseløya.
This restricts the water exchange into two narrow sounds,
Akselsundet in the north, and Mariasundet in the south. The
typical tidal amplitude outside Akseløya is in the range from
0.3 to 0.8 m (see Fig. 2), and currents measured in Akselsundet may exceed 2 m s−1 (Bergh, 2004). During summer and
autumn the water in the fjord is stratified as a result of glacial
melting. It is during this period that we expect baroclinic
tidal activity to be most pronounced. Indeed, using a 22 h
CTD time series near Blixodden in July 1996, Skarðhamar
and Svendsen (2010) observed a vertical displacement of the
pycnocline of 20 m, which they argued was caused by a passing internal Kelvin wave.
Our measurements were performed in the period between
9 and 14 August 2010 during a cruise of the Research Vessel
(R/V) Håkon Mosby. Profiles of hydrography and horizontal currents were collected using a rosette equipped with a
Sea-Bird Electronics 911+ CTD (conductivity, temperature,
depth) system and a set of up- and down-looking 300 kHz,
RD Instruments LADCP (lowered acoustic Doppler current
profiler). Data were processed using well-established routines and averaged vertically in 1 and 4 m-thick bins for CTD
and LADCP, respectively. Weather data were collected from
the ship’s meteorological mast. Figure 1 shows a map of the
measurement locations; see also Table 1. Labels A–D denote
horizontal sections, and TS1–TS6 are time series. For each
time series station, a profile was obtained approximately every 30 min for a station occupation period of 13 h, thus encompassing the M2 tidal period of 12.4 h. The location of the
stations and the sampling frequency in time were chosen so
as to best capture a potentially propagating internal Kelvin
wave, which is expected to have its largest amplitude near
the coast (see Sect. 2), and propagating cyclonically around
the basin. Due to the steep topographic slope near the fjord
Nonlin. Processes Geophys., 21, 87–100, 2014
a
Pressure (dbar)
TS2−2
D
TS6
A−3
B−2
TS5
A−2
TS4
TS3
C−2
TS2
a
0
360
b
270
10
180
5
90
0
10
0
50
631
632
633
634
635
D5
D3
D1
5.5
4.5
26.5
2.5 T( °C)
27
100
0.5
D
−1.5
50
26.5
31
10
11
12
13
14
Day of August 2010
15
16
100
b
10
20
Distance (km)
c
30
d
C9 C11 C1 C3 C5 C7
26
C9 C11
A5
24
A7 A10
27
80
26.5
27
B
27
B-2
0
25
26
26.5
50
27
27
B11
26
60
26
27
B7
40
100 A-2
0
25
26
B3 B5
26
20
27
C-2
B8 B10
25
26.
5
50
f
B1 B3 B5
24
26
27
28
40
e
A1 A3
26
50
50
S
27
C1 C3 C5 C7
Pressure (dbar)
09
33.6
33
26
0
Pressure (dbar)
Ta [°C]
5
628
630
D7
26
100 C
0
629
D9
25
c
0
D11
24
Pressure (dbar)
U15m [m s−1]
−1
20
15
D15
0
Direction [°]
η [m]
1
TS1
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
B
A
C
90
20
26
40
60
27
100
80
Fig. 2. (a) Tidal surface amplitude outside Van Mijenfjorden in100
0
5
10 0
0
4
8
0
4
8 0
4
8
4
8
ferred at the location of TS1 from the AOTIM-5 model (Padman
Distance (km)
638
Figure
2: a) Tidal2004).
surfaceAlternating
amplitude outside
the location
of TS1
and Erofeeva,
thickVan
greyMijenfjorden
and black inferred
portionsat rep639 and black
from
thethe
AOTIM-5
(Padman
and sections
Erofeeva, and
2004).
resent
time of model
occupation
of the
theAlternating
time seriesthick
sta-gray
Fig. 3. Temperature and salinity (colour) and density (black conFigure
salinity (color)
and (b)
density
(black
contours)
sections
D, b)
tions indicated
bytime
letters.
Several of
sections
wereandrepeated,
e.g. stations
A-2 640 indicated
portions
represent the
of occupation
the sections
the time series
tours)3:byTemperature
for CTD and
sections
(a) D,
C-1,
(c)
C-2, for
(d)CTD
A-2,
(e)a)B-1,
641 C-1,
c)
C-2,
d)
A-2,
e)
B-1
and
f)
B-2.
Orientation
of
the
figures
are
west-east
in
a),
and
southcorresponds
to
the
second
occupation
of
section
A.
Time
series
of
and
(f) B-2. Orientation of the figures is west–east in (a), and south–
letters. Several sections were repeated, e.g. A-2 corresponds to the second occupation of
section
in b)-f).
See FigureSee
1 andFig.
Table11 and
for reference.
hourly averaged (b) wind direction and speed, and (c) air tempera- 642 north
north
in (b)–(f).
Table 1 for reference.
A. Time series of hourly-averaged b) wind direction and speed, and c) air temperature,
ture, measured from the meteorological mast onboard R/V Håkon 643
measured from the meteorological mast onboard RV Håkon Mosby at 15 m height.
Mosby at 15 m height.
636
637
entrance and strong tidal forcing, in addition to the long internal Kelvin waves, short non-linear solitary waves are expected (see e.g. Farmer and Armi, 1999; Grue et al., 1999;
Cummins et al., 2003). These short wavelength and highfrequency phenomena are, however, not resolved with our
sampling scheme, and will be treated further in the numerical analysis (Sect. 4). The narrow sound combined with swift
currents posed severe navigational difficulties
and hindered a
23
detailed, high-time and spatial resolution sampling near the
sill region.
A general view of the hydrography in the basin is presented in Fig. 3. The stratification is typical for this season.
Surface water is relatively fresh as a result of summer melt
and runoff from glaciers, and warmer than the water below
due to surface heating; see Fig. 2 for air temperature measurements. The colder and saltier deep water seen in Fig. 3a)
originates from outside the fjord. During high tide this water
is lifted over the sill as can be inferred from the D-section; the
outmost D-profile was the last sample in the section, and was
thus captured during high tide; see Fig. 2. The repeated sections B and C indicate a strong variability on relatively short
time scales throughout the entire fjord basin (Fig. 3). This is
in accordance with Skarðhamar and Svendsen (2010), where
a more detailed description of the short-term hydrographic
variability in Van Mijenfjorden can be found. In the present
work we restrict our attention to the internal waves.
Nonlin. Processes Geophys., 21, 87–100, 2014
Time–depth contours of velocity and density for each time
series station are presented in Fig. 4. Again we refer to Fig. 2
for comparison with the tide. Outside Akseløya (TS1) the
dominant features of the tide entering
and leaving Akselsun24
det are distinguishable in the upper 40 m. Just inside Akselsundet (TS2) we observe intense mixing associated with the
peak tidal inflow. The small-scale overturning and mixing associated with this hydraulic jump will influence the generation of long waves in the lee of the sill; see discussion in
Sect. 5. An intensified inflow may also contribute to generation of short non-linear solitons, as we observe in the numerical experiments reported in Sect. 4. Further south, along
Akseløya north of Mariasundet (TS3), the flow field is dominated by a southward current that is related to the time of
maximum tidal inflow through Akselsundet. Assuming a typical current velocity of 35 cm s−1 as measured at TS3 and a
distance between Askelsundet and the TS3 station of 7.5 km,
we obtain a travel time of about 6 h in agreement with the
time of tidal inflow inside Akselsundet. We note the displacements of the isopycnals which weakly indicate oscillations
of tidal periodicity. Along the southern coast (TS4) the currents are in-fjord during the entire tidal period. This station
is potentially influenced by inflow from both sounds, thus a
more complex current structure is expected. Interestingly, the
isopycnals are displaced vertically by more than 30 m during
the time series, accompanied by a clear vertical shift in the
horizontal current. A similar displacement is not observed
along the northern coast (TS5). The observations from the
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E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
91
Fig. 4. u and v velocities (colour) and density (σtheta , black contours) for each time series station. (a)–(f) correspond to TS1–TS6 respectively. Axes are rotated so that u is aligned eastward along the coast where applicable (TS3–TS6), or along a direct line inwards through
Akselsundet (TS1–TS2).
time series stations suggest the presence of an internal Kelvin
wave propagating in-fjord from Akseløya along the southern coast, and dissipating before returning outward along the
northern side. The role of Mariasundet in this process is difficult to assess (Sect. 4). The main outflow along TS5, and the
www.nonlin-processes-geophys.net/21/87/2014/
dominant eastward flow along the southern slope at TS4 and
TS6 are consistent with Kelvin waves and the mean circulation in Van Mijenfjorden induced by the tidal forcing (Bergh,
2004).
Nonlin. Processes Geophys., 21, 87–100, 2014
92
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
Time series of horizontal velocity and density profiles collected at stations TS1 to TS6 are used to calculate the baroclinic energy flux for the semi-diurnal signal
F E (z) = hu0 p0 iϕ
(6)
where u0 is the baroclinic perturbation velocity vector, p0
is the pressure perturbation, and averaging is over the M2
phase, ϕ. The calculations are made following the methods
detailed in Nash et al. (2005). The pressure anomaly is inferred from the density profiles assuming hydrostatic balance, after removing the full depth average to satisfy baroclinicity. The perturbation velocity is calculated from the
LADCP profiles after removing the depth and time average.
The semi-diurnal components for the pressure and velocity
perturbations are then isolated by harmonic analysis of time
series at each depth level. Using the amplitude and phase
obtained from the harmonic analysis, one cycle of a sinusoidal semi-diurnal wave is constructed, and time averaging
is done over one complete phase. The depth integrated fluxes
are shown in Fig. 5. The semi-diurnal baroclinic energy is
generated at the fjord sill, in both in-fjord and out-fjord directions. The in-fjord energy propagates cyclonically along
the slope, decaying in magnitude, presumably due to friction. The observations of the baroclinic semi-diurnal energy
flux are consistent with Kelvin waves.
The depth of the isopycnal σθ = 26 at TS2 gradually decreases from about 11 m to approximately 59 m, close to the
seabed, which then abruptly rises to 19 m in 2 h. The details
of this transition are not resolved by our observations. The
evolution of the stratification at TS2 is complex, and a twolayer approximation is not possible. Nevertheless, between
2 and 7 h into the record, when the isopyncals are relatively
smooth, the depth of the pycnocline, inferred from the depth
of the maximum in vertical density gradient, is 15±4 m (±one
standard deviation); the depth of the σθ = 26 surface during
this period is approximately 22 m. Assuming that the pynocline depth is representative of the upper layer thickness in
a two-layer flow, this vertical displacement corresponds to a
normalized excursion (vertical displacement divided by the
upper layer thickness) of 2.7, or 1.8 when normalized by the
mean depth of the isopycnal.
At TS4, large vertical isopycnal displacements occur in the
relatively weakly stratified portion of the water column. This
is expected through the WKB scaling, since less energy is
required to vertically displace the isopycnals in weaker stratification. The vertical excursion of σθ = 26.5 surface is 37 m
in approximately 2 h. This isopycnal is located at 40 m depth
on average, corresponding to a normalized excursion of 0.9.
The stratification is strongest in the upper layers, and the pycnocline depth is estimated at 6 ± 2 m using all the profiles.
The typical vertical displacement in the pycnocline, using
the 25 isopycnal, is 4–6 m, yielding a normalized excursion
close to 1. The exact period of the wave at TS4 (or at other
stations) cannot be inferred from the time series with confidence because of the short duration of the record. While the
Nonlin. Processes Geophys., 21, 87–100, 2014
Fig. 5. Depth-integrated semi-diurnal baroclinic energy flux vectors
inferred from the time series stations. The scale is shown on the
bottom right.
semi-diurnal period fits explain up to 50 % of the total variance, the wave period inferred from the first zero crossing of
the autocovariance function is 8 h for the σθ = 26 and 26.5
isopycnals, and increases to 5 h for the isopycnals in the pycnocline (σθ = 25 and 25.5). A similar analysis for the alongshore component of the baroclinic velocity suggests consistent periods (5.2 h at 8 m depth, and 9 ± 0.6 h between 12 and
68 m). The pattern is similar at TS3; the σθ = 26.5 isopycnal
oscillates at 8 h, and the oscillation of the rotated baroclinic
velocity component is between 7 and 10 h throughout the water column.
4
4.1
Numerical simulations
Model and the setup
The model utilized is the MITgcm model (Marshall et al.,
1997; Adcroft et al., 2004). This is a finite volume, nonlinear z coordinate model with non-hydrostatic capabilities.
The model has been widely used for study of internal waves
(Legg and Adcroft, 2003; Vlasenko and Stashchuk, 2007;
Xing and Davies, 2007; Boegman and Dorostkar, 2011). Bottom topography in MITgcm is represented by the use of
shaved cells (Adcroft et al., 1997). For our model setup we
disregard the effect of diffusion by setting small constant
values for the horizontal and vertical diffusivities of temperature and salt1 . We use a vertical turbulent viscosity of
Az = 0.001 m2 s−1 . The horizontal viscosity is of Smagorinsky type with value 2.2 along with a small biharmonic viscosity factor (viscC4smag = 1) as suggested by Griffies and Hallberg (2000). No-slip conditions are employed at side walls,
along with a quadratic bottom drag coefficient of 0.0025. The
time step used is 5 s, the horizontal grid size is 100 m, and we
employ 32 non-uniformly spaced vertical levels with the lowest spacing of 0.75 m where the density gradient is largest,
and up to 14 m spacing below 100 m depth. We let the model
run hydrostatically, as the grid should be too coarse for nonhydrostatic effects to be observable (Berntsen et al., 2009).
1 K = 1 × 10−6 , K = 1 × 10−7 , K =1×10−8 , K = 1 ×
zT
hT
hS
zS
10−9 m2 s−1
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E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
In our simulations we ignore the effect of wind and freshwater runoff. Initially, the system is at rest, with horizontally
constant hydrography. The hydrography of Van Mijenfjorden
varies considerably on short time scales (Sect. 3). Performing a realistic simulation of the entire fjord system requires
detailed information on local winds and freshwater runoff,
and detailed initial and boundary conditions on hydrography
and currents. In this study, we focus on the generation and
propagation of internal waves. Thus, it is instructive to idealize the problem. In the following we describe the idealized
box runs, and subsequently, the relatively realistic Van Mijenfjord runs.
93
Fig. 6. Bottom topography test box run. The western boundary is
the only open boundary.
Salinity
Density
Temperature
−10
−20
Idealized box runs
−30
We shall consider the Van Mijenfjord topography in
Sect. 4.3, but in this section we use a simplified semienclosed box with the topography shown in Fig. 6. The
the long side, x axis, is aligned in the west–east direction,
and the short side, y axis, along south–north. The western
boundary is open, whereas all other boundaries are closed.
At 20 km in-fjord, we place a constriction resembling Akseløya. In this simplified setup we only consider one sound
(recall that Van Mijenfjorden has two sounds), and place it
on the southern side in order to avoid the complicating fac-658
tor of the wave making a 90°turn in the southwestern corner.
659
The sound is 20 m deep and 1300 m wide, giving a crosssectional area similar to Akselsundet. The sloping bottom to-660
ward the eastern coast is introduced as a crude wave damper
to minimize reflections. The open boundary condition is of
the form u = u0 sin(ωt), where u0 is constant across the
open boundary and ω = 1.4 × 10−4 s−1 corresponds to the
M2 semi-diurnal tide. The interior solution is relaxed toward
the boundary over a 64-grid point wide sponge layer. The relaxation time scale increases linearly with distance from the
open boundary, up to 16 h at the inner border of the sponge
layer. The model is initiated with a forcing amplitude of u0 /2
in the first 6 h to ensure a smooth spin-up, and run for 48 h.
The solution is dependent on the choice of the forcing amplitude u0 and the vertical hydrography profile. In the following, we use our idealized setup to test the internal response
of the system by varying these parameters.
661
4.2.1
Box run 1: Van Mijenfjord hydrography
662
663
In the first test, we set forcing and hydrography to resemble664
the conditions in Van Mijenfjorden. From Fig. 2, we want
surface tidal amplitudes close to 0.5 m. If we integrate over
the domain this leads approximately to a boundary forcing of
u0 = 3 cm s−1 (verified a posteriori). Regarding the hydrography, we neglect the effect of horizontal variation, and consider only the outer part of the fjord. Representative profiles
of temperature, salinity, and density are obtained by averaging the CTD measurements from the time series stations
TS2–TS5, and the sections A and C (Fig. 7).
www.nonlin-processes-geophys.net/21/87/2014/
−40
Depth [m]
4.2
−50
−60
−70
−80
−90
−100
−110
31
32
psu
33
1
2
3
4
°C
1025
1026
1027
kg m−3
Fig.
box
Figure7.7:Initial
Initial hydrography
hydrography forfor
box
runrun
1. 1.
In order to visualize the horizontal distribution of the vertical isopycnal displacements, we introduce a perturbation potential energy per unit area, defined as
Zη(t)
Z0
1PE(t) = PE(t)−PE0 =
ρ(t)gzdz−
ρ (t = 0) gzdz . (7)
−H
−H
From this definition PE is negative, so a positive 1PE indicates a mean depression in the water column. Plot of the normalized 1PE 41 h into the simulation is shown in Fig. 8. The
region shown is from 20 to 45 km in-fjord, i.e. the sound is
immediately in the southwestern corner, and the eastern-most
region with the sloping bottom is omitted. We clearly see
that the most energetic displacements occur near the boundFigure 8:
Horizontal
plot of normalized
perturbation
potential energy
aries.
These
displacements
propagate
cyclonically
aroundPE
theafter 41 hours,
basin,
similar
to
internal
Kelvin
waves.
The
radius
of
the
disrun 1. The plotted region corresponds to the first 25 km of the inner basin, see Figure 6. Red
placement
signal isand
2–3
km, and
by comparing
plots
blue indicate depression
elevation
respectively,
see Figuresimilar
9.
at different times we obtain a propagation velocity of 40–
45 cm s−1 (not shown). A similar velocity
can be inferred
27
from λ/TM2 = 42 cm s−1 , using a wavelength of λ = 19 km
obtained from Fig. 8, and the wave period TM2 = 12.42 h.
Here, ci = 42 cm s−1 is taken as the typical internal long
wave speed for this model configuration.
In addition to the internal Kelvin wave signal, a second
feature is evident in Fig. 8. In front of the coastal wave, there
is a narrow pulse, a non-linear soliton that propagates radially
Nonlin. Processes Geophys., 21, 87–100, 2014
94
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
Fig. 8. Horizontal plot of normalized perturbation potential energy
1PE after 41 h, box run 1. The plotted region corresponds to the
first 25 km of the inner basin; see Fig. 6. Red and blue indicate depression and elevation respectively; see Fig. 9.
Fig. 10. Cross-sectional velocity [m s−1 ] at time of maximum inflow (a) and outflow (b) through the sound at 20 km east for box
run 1. Left–right in the figures correspond to south–north.
665
−20
26.5
10
526
5.5
10
102
102
26
6.5
25
1010
25.5
1027.5
1026
1027
1026.5
1026.5
1026.5
10
−40
Density [kg/m³]. Time 41 hours
10
1025
25
1025.5
.5
1026
10251.5
02
6
4.2.2
Box run 2: Two-layer stratification
Depth [m]
Having obtained the expected baroclinic phase velocities (ci )
from
a realistic stratification, we set up a case with two-layer
102
7
1025.5
stratification. We preserve the approximate location of the
−80
1025
pycnocline while keeping the typical value of ci . For sim−100
plicity, we take temperature constant at 2◦ C, and a two-layer
1027.5
1027.5
1027.5
1024.5
20
25
30
35
40
45
salinity profile as shown in Fig. 11. The resulting density
km
is also shown, in which a small correction for pressure can
FigureFig.
9: Vertical
sectionsection
of density
southern
coast
20 and
45 km
east after 41
9. Vertical
ofnear
density
near
thebetween
southern
coast
between
be seen. The phase velocity ci is calculated according to
-3
hours20
simulation,
1. Contour
is 0.1 kg mbox
. run 1. Contour interval
and 45box
kmrun
east
after 41interval
h simulation,
Eq. (2), which yields ci = 0.42 m s−1 using the appropriate
−3
is 0.1 kg m .
values from Fig. 11.
Plot of 1PE for the two-layer stratification after 41 h simulation is shown in Fig. 12. The structure is comparable to
u/ci at sound, t=38 hours
u/ci at sound, t=44 hours
the previous case, with most of the energy near the south
0
coast, and a similar pulse propagating into the basin. A verinto the basin in all directions from the sound. By inspection
−2
tical section near the south coast (Fig. 13) reveals that most
of the circular shape of the leading wave crest, we infer that
−4
of the displacement occurs near the pycnocline, and a steepthe propagation velocity is approximately constant in all di−6
ened structure similar to Fig. 9 is seen. The normalized verections. Model output from successive time steps indicates
−8
locities through the sound (not shown) are very similar to
that the velocity of the pulse is similar to the internal long
−10
the previous case in structure, with the maximum values of
wave speed ci (not shown). From a vertical section of density
−12
3.61 (1.52 m s−1 ) and −3.07 (−1.29 m s−1 ), during the innear the southern wall, at y = 100 m (Fig. 9), we see more
−14
flows and outflows, respectively.
clearly the steep front of the depression, coinciding with the
pulse. The vertical displacement exceeds 20
−2.5 m over a 600 m
−16
3.5
−2
.2
horizontal
distance.
We
discuss
this
pulse
more
thoroughly
4.2.3 Box run 3: Test forcing
−18
−1.9
3.2
2.6
in
Sect.
5.
−1.3 1
1117421
−20
07
0.1
0.3
0.5
1.1
1.3
0.1
0.3
0.5
0.7
0.9
1.1
We
now
turn0.7our0.9attention
to the
velocities
through
the 1.3 In the next set of tests we consider the sub-critical scekm from south
km from south
nario by reducing the forcing amplitude. We perform two
sound. A vertical cross section of normalized maximum veof maximum inflow
(left)
andDuring
outflow (right) runs using the two-layer hydrography from box run 2, with
Figurelocity
10: Cross-sectional
velocity
s-1] at time
us /ci across
the[msound
is presented
in Fig.
10.
−1figures
through
the sound
20 km eastpeaks
for boxat
run3.69
1. Left-right
on sthe
to south- u0 = 0.45 cm s−1 and 0.9 cm s−1 , or 15 % (a) and 30 % (b) of
inflow
theatcurrent
(1.55 m
) in acorrespond
jet at apnorth.proximately 16 m depth. The outflow is relatively homogethe original forcing, respectively. 1PE for the two runs are
presented in Fig. 14. We clearly see the difference between
nous across the sound, peaking at −3.07 (−1.29 m s−1 ). This
28
the two cases; at case a, the pulse in front of the coastal wave
asymmetry is expected in accordance
with tidal choking theis almost absent and the shape is quite similar to what we
ory (Stigebrandt, 1980). Further, these values relate directly
expect from a Kelvin wave (i.e. Eq. 5). At case b, however,
to the definition of the Froude number (Sect. 2). FD ≈ 3 is
the pulse is more visible, and the shape is reminiscent of the
a clear indication of supercritical conditions, reinforcing the
result for the box run 2.
jet-type behaviour shown in Fig. 10.
−60
1027
1026.5
1027
1026
1027
027
16.5
102
02.56
125
1025
10
670
3.2
−2.2
−1.9
2.6
−0.1
−2.8
0.2
−1
−0.4
−0 1
0.8
3.2 3.5
.8
−2
−1.6
0.2
0.8
1.4
2 2.6
2.9
2.3 2
1.4
Nonlin. Processes Geophys., 21, 87–100, 2014
−0.7
−1.6
−2.2
Depth [m]
−0.1
−0.4
−1.3
−1.9
−2.5
−2.8
−2.8
2.9
0.2
674
−1.6
2.6
0.8
672
673
0.2
0.5
1.1
1.7
2.3
2.3 2
1.4
671
−2.5
669
−1
668
−0.1
−0.4
667
2.9
666
www.nonlin-processes-geophys.net/21/87/2014/
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
675
Density
Density [kg/m³]. Time 41 hours
.5
Salinity
95
10
25
6
1026
102
Depth [m]
Depth [m]
−80
−100
−60
1026
−60
26
20
1026.6
1026.4
6.5
102
.5
1026.2
1026
1025.8
1025.6
1025.4
25
30
35
40
45
1025.2
km
685
Fig. 13. Vertical section of density near southern coast after 41 h
686
simulation,
run
Contour
interval
is 0.1
Figure
13: Vertical box
section
of 2.
density
near southern
coast
afterkg
41m
hours.simulation, box run 2.
687
−100
.5
1026
10
684
−80
1025
1026.5
1026
1025.5
−40
−40
.5
26
10
−20
1026
.5 26.5
102510
−20
−3
-3
Contour interval is 0.1 kg m .
688
31
676
32
psu
33 1025
1026
1027
a) Box run 3a, perturbation potential energy. Time 45 hours
kg m−3
1.0
10
0.75
8
Fig. 11. Initial hydrography for box run 2. Temperature is constant
0.5
677
Figure 11: Initial hydrography for box run 2. Temperature6 is constant at 2°C and not shown.
at 2 ◦ C and not shown.
km
0.25
0
4
678
-0.25
2
-0.5
679
20
25
30
35
40
45
km
b) Box run 3b, perturbation potential energy. Time 45 hours
1.0
10
0.75
8
0.5
0.25
6
0
4
-0.25
2
-0.5
20
Fig. 12. Normalized 1PE (similar to Fig. 8) after 41 h
box run 2.
689
simulation,
25
30
35
40
45
-0.75
km
690
Figure 14: Normalized PE (similar to Figure 8) after 45 hours simulation, box runs 3. Forcing
691
at 15% (upper, case a) and 30% (lower, case b) of original forcing.
Fig. 14. Normalized 1PE (similar to Fig. 8) after 45 h simulation,
box run 3. Forcing at 15 % (a) and
The normalized velocities through the sound have maxi30 30 % (b) of original forcing.
mum values of −0.57 (−0.24 m s−1 ) and 0.67 (0.28 m s−1 )
for case a, and of −1.07 (−0.45 m s−1 ) and 1.33 (0.56 m s−1 )
and the shallow, secondary inner basin on the east of Van
for case b. The velocity structure across the sound in the latMijenfjorden is omitted from the simulation. As our point of
ter case is similar to box run 2 with a jet during inflows and
interest is close to Akseløya, this should not influence our
a more barotropic distribution during outflows. The jet is not
solution significantly. When a sound is closed, we simply set
as visible in case a, however; here the structure is rather sim680
the depth to zero in a three grid point-wide band across the
ilar for in- and outflow, in accordance with the subcritical
respective sound. Model parameters are similar to what is
conditions reflected by the normalized velocities.
681
Figure 12: Normalized PE (similar to Figure 8) after 41 hours simulation, box run 2.
used in Sect. 4. As a result of varying depth, 1PE (Eq. 7)
is not easily visualized; instead, contours of constant density
4.3 Van Mijenfjord runs and comparison with
682
are shown.
observations
Results from the first run with realistic hydrography and
683
topography (VMrun1) are shown in Fig. 16. The left column
Having obtained some experience with the idealized model
runs, we now apply a topography similar to Van Mijenfjorshows the density distribution at three different depths after
29
45 h simulation, whereas the right column shows the strucden. As this fjord has two sounds, it is of interest to isolate
ture at −16.9 m depth at different times. As before, we identhe effect of each sound. We also investigate the influence of
tify the internal Kelvin wave pattern along the southern coast.
the realistic hydrography in comparison with the two-layer
structure we applied in Sect. 4.2.2.
It is quite visible at all three depths, but most pronounced at
−16.9 m. Snapshots at different times suggest that the priWe set up four model runs with different topography and
initial hydrography; see Table 2. The bottom topography mamary internal wave generation occurs at Akselsundet. We
trix is shown in Fig. 15. The leftmost part is the sponge layer,
also note the pulse at all depths. Current magnitude and
www.nonlin-processes-geophys.net/21/87/2014/
Nonlin. Processes Geophys., 21, 87–100, 2014
96
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
Table 2. Setup of the four Van Mijenfjorden model runs.
692
Topography:
Hydrography:
VMrun1
VMrun2
VMrun3
VMrun4
Realistic
Realistic
Mariasundet closed
Realistic
Akselsundet closed
Realistic
Realistic
Two-layer
Bottom topography matrix [m]
0
16
−20
14
12
−40
km
10
8
−60
6
−80
4
−100
2
5
10
15
20
25
30
km
35
40
45
50
55
−120
Fig. 15. Bottom topography of Van Mijenfjorden.
693
694
695
696
direction at −16.9 m depth are shown in Fig. 17 at the time
outflow (b) through
Akselsundet. We see the wave along the southern coast propagating inward at all times through the tidal cycle. Near Akselsundet during inflow, the current has a dominantly southeastern direction along Akseløya. During outflow the current
is toward the sound from a much wider region, in accordance
with tidal choking theory discussed earlier. Current amplitudes through Akselsundet reach 1.4–1.6 m s−1 , consistent
with observations from moored current meters (Bergh, 2004;
Fer and Widell, 2007; Fjellsbø, 2013).
As a means of comparing the realistic model run with the
measurements, we plot model density and along-coastal current at the time series locations TS4 and TS5 (Fig. 18). For
the station near the southern boundary (TS4), the largest currents coincide with depression of the isopycnals (low-density
values) in the upper layer (see also Figs. 16 and 17). The excursion of the 25.5 isopycnal, located at 14 m on average,
is 13 m over the tidal period, corresponding to a normalized excursion of 0.9. We also see a local current maximum
around −60 m depth along with elevation of the isopycnals.
This is also observed in the measurements (Fig. 4d). The cur31
rent magnitude and the maximum
isopycnal displacement are
lower than what was measured. At the northern station TS5
(Fig. 18b), we see only small oscillations, consistent with the
observations. The strong mean westward flow in the upper
layer in the measurements at TS5 (Fig. 4e) is not present
in the model because of model shortcomings and simplifications; see Sect. 5.
The next two model runs are presented in Fig. 19. Here we
experiment with closing Mariasundet (VMrun2, left column)
and Akselsundet (VMrun3, right column) separately. The result when the Mariasundet is closed is quite similar to what
was found in the realistic run (Fig. 16), suggesting that Akselsundet is the dominant generator for internal waves. However, as we see in VMrun3, closing off Akselsundet reveals
a substantial contribution from Mariasundet. Comparing the
location of the largest depression from the two runs, we in-
Figure 15:
topography
Van Mijenfjorden.
ofBottom
maximum
inflow
(a) and maximum
Nonlin. Processes Geophys., 21, 87–100, 2014
fer that waves in VMrun2 are only about 2–3 km ahead of
those in VMrun3. Thus the full picture in VMrun1 is close
to a superposition of two internal Kelvin waves. Regarding
the pulse we note that it is clearly visible in all the plotted
depths, propagating radially from each respective sound.
As a last test, we apply the two-layer hydrography with
the realistic topography (VMrun4). The result is shown in
Fig. 20 for −16.9 m depth. Again, a Kelvin-type signal can
be identified, along with several pulses. At −10.3 and −28 m
depths, the density surfaces are approximately undisturbed
(not shown).
5
Discussion and concluding remarks
Through a series of simulations and measurements, we have
shown strong indications of tidally induced internal wave
motion in Van Mijenfjorden, an Arctic fjord in Svalbard. As
expected, a major part of the internal wave energy is in the
form of an internal Kelvin wave, propagating cyclonically
around the fjord. Numerical tests using a two-layer stratification show that the propagating internal wave emerges when
conditions at the sound are baroclinically sub- or supercritical. This is in agreement with observations from Loch Etive,
Scotland (Inall et al., 2004; Stashchuk et al., 2007), which
is a typical “jet-type” fjord following the definition of Stigebrandt and Aure (1989). According to Stigebrandt (1980),
generation of long waves in a jet fjord is prohibited, although
he notes that for weakly supercritical conditions wave generation cannot be ruled out.
Our results indicate that both sounds in Van Mijenfjorden
serve as internal wave generators; the dominant waves propagate from Akselsundet, whereas the contribution from Mariasundet can be substantial. According to the model results, a
realistic summer hydrography leads to waves generated from
the two sounds that are approximately in phase. As propagation velocity is directly affected by the stratification, waves
from the two sounds are likely to be in phase occasionally,
because of the considerable variability in hydrography. Consequently, the contribution from each respective sound may
be masked in measurements. It is important to note that Mariasundet is relatively narrow (0.7 km wide) compared to the
grid size (100 m), and the representation of this sound in the
topography matrix is coarse. Increasing the resolution and
applying a more accurate representation of the topography
would thus contribute to a more realistic wave field.
Increasing the resolution and including non-hydrostatic
terms in the model equations can help resolve the
www.nonlin-processes-geophys.net/21/87/2014/
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
97
Fig. 16. Horizontal plot of density [kg m−3 ], Van Mijenfjorden (VMrun1). Left: 45 h simulated; depth 10.3 m (a), 16.9 m (c), 28 m (e). Right:
depth 16.9 m; 39 h (b), 41 h (d), 43 h (f).
Fig. 17. Current magnitude [m s−1 ] and direction for VMrun1 at
16.9 m depth. Simulated 39 h (a) and 45 h (b).
non-hydrostatic, small-scale overturning and mixing near the
sills (Berntsen et al., 2009), as observed just inside Akselsundet. Our observations do not resolve the near field mixing processes in the vicinity of the sill. It is expected that
www.nonlin-processes-geophys.net/21/87/2014/
Fig. 18. Along-coastal current [m s−1 ] (colour) and density
[kg m−3 ] (black lines) at locations TS4 (a) and TS5 (b) from VMrun1, 35–48 h simulated.
the energy lost to mixing will be unavailable for generation
of the long internal wave, influencing the resulting in-fjord
wave field. Farmer and Armi (1999, 2001), using detailed
Nonlin. Processes Geophys., 21, 87–100, 2014
98
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
Fig. 19. Horizontal plot of density [kg m−3 ]. Left: Mariasundet closed (VMrun2). Right: Akselsundet closed (VMrun3). (a) and (b) 10.3 m,
(c) and (d) 16.9 m, (e) and (f) 28 m.
Fig. 20. Horizontal plot of density [kg m−3 ] after 45 h simulation,
16.9 m depth. Two-layer hydrography profile (VMrun4).
observations, explain the time-dependent evolution of the
stratified flow response over a sill that is typically not captured by the numerical simulations because of unresolved
mixing. Rapid flow over the sill separates the streamlines into
a fast deep current immediately downstream of the crest, beneath a spreading, weakly stratified, nearly stagnant intermediate wedge susceptible to overturning and further mixing.
Small scale instability and boundary layer separation act in
concert to determine the time evolution of the flow. Numerical studies, e.g. Berntsen et al. (2009), examine the resulting
lee waves as a result of varying grid sizes, and thus the ability of the model to resolve the aforementioned mixing. They
report a distinct difference in the lee wave field moving from
Nonlin. Processes Geophys., 21, 87–100, 2014
100 to 12.5 m resolution; as the resolution increases, waves
become shorter in scale and their amplitudes increase. For
the smaller scales, the vortices in the wedge zone increase
in intensity. In all their experiments, Berntsen et al. (2009)
reported boundary layer separation, a well-mixed wedge and
internal waves in the lee of the sill; however, those in the
hydrostatic and relatively coarser resolution runs were a result of artificial mixing, whereas the high-resolution nonhydrostatic models resolved the mixing processes. In our
simulations, it would be of interest to apply a nested grid,
where resolution is increased near the sounds, in order to resolve the mixing.
One must take care when directly comparing the observations to the model results, due to the numerous and crude
assumptions involved. Firstly, we neglected wind. Thus, the
modelled near-surface turbulence distribution and currents
are bound to differ from the measured data. Another important aspect of the wind is that it may set up internal
waves upon changing direction. Wind measurements during
the cruise (Fig. 2) show that the wind direction in Van Mijenfjorden is unidirectional for long periods of time, which sets
up a tilting interface that when released (as a result of wind
changes), will propagate along the coast as an internal Kelvin
wave. A second aspect is that our model is started from rest
and is run for 48 h in an attempt to isolate the effect of internal wave propagation on an otherwise undisturbed system. In
www.nonlin-processes-geophys.net/21/87/2014/
E. Støylen and I. Fer: Tidally induced internal motion in an Arctic fjord
order for a mean circulation pattern to be set up, one would
need a spin-up time of several days, possibly months. In addition, the model does not resolve the estuarine circulation
properly, and large-scale pressure gradients from outside the
fjord region, bar the tidal contribution, are also absent in the
simulation. These shortcomings are all likely candidates for
explaining why the observed mean currents in Fig. 4e) are
not captured in the model. However, this might just as well
be another example of insufficient resolution not resolving
small-scale overturning and mixing from e.g. wind and background shear.
The wave pulse observed in the model results demands
some discussion. When the conditions are near super-critical,
a wave pulse propagates inward from the fjord entrance at the
beginning of each inflow phase of the tidal cycle. This pulse
of steep front is not dominantly affected by rotation, propagates with the long internal wave speed, and the leading crest
is followed by a series of smaller amplitude waves. The train
of waves develops as a result of dispersion, since the group
velocity of the shorter waves is smaller than the long wave
speed. The waves are characterized as non-linear internal
solitons. The process is similar to that for free surface waves
which are comparatively less non-linear. The latter has been
illustrated comparing fully nonlinear computations and KdVformulation (Grue et al., 2008), and similar generation process for internal waves was computed using fully nonlineardispersive calculations in Grue (2005). The vertical velocity
profiles of the two-layer solitary wave are described in detail
in Grue et al. (1999). The upper and lower layer velocities are
in opposite directions, with an abrupt change across the interface. This may be compared to our observations, i.e Fig. 4d),
where a similar separation of velocities is seen. It should
be kept in mind, however, that the KdV-theory is applicable
to two-layer solitons with non-dimensional excursions of up
to 0.4 (Grue et al., 1999; Grue, 2005). In our study, results
from both observations and the numerical simulations indicate normalized excursions up to 1, exceeding the validity
of the KdV-formulation. It must be emphasized that the hydrography in Van Mijenfjorden varies considerably, and the
profile at TS4 cannot be characterized by a two-layer system. Thus, caution must be exercised in the idealized theoryobservation comparison.
The generation process for internal solitons as explained
in Cummins et al. (2003) and Grue (2005) is as follows: during intense outflow, an internal depression forms over the sill
crest, accompanied by an elevation more upstream. When the
outflow relaxes, the depression propagates upstream into the
fjord. Interestingly, as flow is supercritical during both inand outflow through Akselsundet, we might expect solitons
emerging on both sides of the sound at opposite phases of
the tidal cycle. The dispersive degeneration of the solitary
front into smaller waves is not properly resolved. An additional test with non-hydrostatic dynamics turned on yielded
little difference in the results, which is expected due to insufficient horizontal resolution.
www.nonlin-processes-geophys.net/21/87/2014/
99
Acknowledgements. The authors are grateful to Jarle Berntsen and
Jiuxing Xing for valuable input on the model configuration, and
the captain and crew of RV Håkon Mosby during the field work.
This study is partly supported by the Research Council of Norway,
through the “Internal hydraulic processes in an Arctic fjord”
project. The comments from two reviewers helped to improve an
earlier version of the manuscript.
Edited by: Y. Troitskaya
Reviewed by: J. Berntsen and J. Grue
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