GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L05604, doi:10.1029/2008GL036849, 2009
Internal wave induced dispersion and mixing on a sloping boundary
Mark E. Inall1
Received 30 November 2008; revised 10 January 2009; accepted 16 January 2009; published 7 March 2009.
[1] A fluorescent dye-tracer was released in a stratified
water column above a sloping boundary revealing profound
dispersive asymmetry. Initially the dye spread symmetrically
in the vertical until the deeper portion became influenced by
the bottom boundary layer (BBL). The shallower portion then
spread rapidly up-slope in a thin boundary layer influenced by
highly non-linear isopycnal distortions. The deeper, downslope portion of dye subsequently became detached from the
boundary and intruded into the basin interior in three distinct
layers along isopycnal surfaces. The intrusions result from
the gravitational collapse of mixing patches created by
baroclinic flow over an irregular boundary. The vertical
separation of the layers matches the inverse topographic
wavenumber. Up-slope displacement of the dye is consistent
with an advective/diffusive balance near the sloping boundary,
controlled by slope angle and roughness. Horizontal
diffusivities increased from 1.4 m2s1 to 18.4 m2s1 during
the 4-day experiment. Citation: Inall, M. E. (2009), Internal
wave induced dispersion and mixing on a sloping boundary,
Geophys. Res. Lett., 36, L05604, doi:10.1029/2008GL036849.
1. Introduction
[2] The role of internal tides in mixing the ocean interior
has attracted much attention in recent years [Egbert and
Ray, 2000; Ledwell et al., 2000; Munk and Wunsch, 1998;
Sjoberg and Stigebrandt, 1992]. Energy is fed into the
oceanic internal wave field primarily at the semi-diurnal
frequency through the interaction of stratified fluid with
topography under the action of a barotropic tide. A significant portion of the semi-diurnal internal tidal energy
propagates great distances away from generation regions
[Klymak et al., 2006]. Ultimately this energy dissipates
either in the basin interior or at the sloping boundaries,
raising the potential energy of the water column with some
process-dependent efficiency.
[3] It has been speculated that the type of interactions
between internal waves and sloping topography seen in
numerical and laboratory experiments [McPhee-Shaw and
Kunze, 2002; Stigebrandt, 1976] may be manifest in natural
flows, for example the widespread reports of intermediate
nepheloid layers near shelf breaks [Thorpe and White, 1988].
The motivation for this study was to directly image the
interaction of an internal tide with a sloping boundary. It is
the first such detailed study made in the natural environment.
[4] Fjords present excellent natural laboratories to test
our understanding of the interaction of stratified flow with
topography. Loch Etive was selected in this study for two
reasons: 1) a progressive, semidiurnal internal tide prop1
Scottish Association for Marine Science, Dunstaffnage Marine
Laboratory, Oban, UK.
Copyright 2009 by the American Geophysical Union.
0094-8276/09/2008GL036849
agates from the entrance sill toward the sloping boundary at
the head of the fjord, where little, if any, back-reflection
occurs and 2) weak barotropic tidal currents at the head of
the fjord indicate minimal baroclinic tidal generation on the
upper slopes [Inall et al., 2004, 2005; Stashchuk et al.,
2007]. Both facts make Loch Etive an ideal location to
study the details of internal wave boundary mixing and
dispersion into an interior basin.
2. Observations
[5] A solution of 5 kg Rhodamine WT was released on
the sq = 19.59 surface at a point 15 m above the bed (mab)
in water of 60 m, then tracked for four days with closely
spaced CTD casts as it dispersed along the axis of the basin
(Figure 1). The dye was detected using a Chelsea Instruments Aquatracka III Rhodamine WT fluorimeter on a
Seabird 9 – 11+ CTD system operating at 24 Hz. Shear
micro-structure measurements were made using an MSS90
free-fall microstructure profiler [Prandke and Stipps, 1998]
to 10 mab, and not into the BBL. Vemco thermistors placed
at 6 m intervals in the vertical from 8 to 70 mab in 90 m of
water sampled every 120 s throughout the experiment
(Figure 2). Basin bathymetry is known to a horizontal
resolution of approximately 10 m [Howe et al., 2001].
Calm conditions prevailed allowing the CTD to be lowered
to within 0.3 mab; also surface mixing and seiching due to
wind stress may be neglected. A CTD section was completed before the dye release to obtain background fluorescence values, yielding a dye detection limit of 0.09 mgl1.
3. Dye Distribution and Moments
[6] Maps of dye concentration in the xz-plane, where z is
depth and x is the along-basin axis, were obtained by taking
all 24 Hz dye concentration data for each day, subtracting
the predicted background level, setting to zero all estimated
absolute dye concentrations of less than 0.09 mgl1 and
gridding by distance-weighted averaging in the xz-plane,
with Dx = 350 m and Dz = 0.125 m (Figure 3). Isopycnal
displacement maps were obtained from the 1 m vertically
binned CTD data gridded in the same manner (overlaid in
Figure 3).
[7] The evident asymmetry in the dye patch evolution
can be quantified by calculating the moments of the dye
maps, which are defined as,
Z
Mpq ¼
xp zq C ð x; zÞdxdz
ð1Þ
where C(x, z) is the dye concentration map as described
above [Ledwell et al., 1998]. The zeroth moment (M00)
corresponds to the total amount of tracer, the first moments
(M01 and M10) give the x and z positions of the centre of
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Figure 1. (a and b) Location maps and (c) a bathymetric cross-section showing dye release, mooring positions and the
maximum horizontal extent of the dye patch after 4 days.
mass, respectively (M01/M00 and M10/M00), and the second
moments (M20 and M02) lead to the x-variance and zvariance of the dye relative to the respective centres of
mass (s2x = (M20 M210)/M00, s2z = (M02 M201)/M00).
Coefficients for Fickian horizontal and vertical eddy
diffusivities are defined as,
Kn ¼ Ds2n =ð2Dt Þ
ð2Þ
where n = x or z and Dt is the time difference between
the centre time of successive dye maps. Table 1 lists the
statistics of the dye patch during its evolution and the
following paragraph describes the main features.
[8] Day 1 (t = 2 hrs): the point release is tightly confined
to the injection isopycnal (sq = 19.59). Diapycnal diffusivity at 15mab from microstructure measurements is Kz =
2.5 104 m2s1. Day 2 (t = 24 hrs) and 2.84 hours earlier
in tidal cycle compared to Day 1: dye has been drawn into
higher density classes through interaction with the bottom
boundary (dye-based Fickian Kz = 9.2 103m2s1). The
dye is diffusing horizontally with KH = 1.38 m2s1. Day 3
(t = 48.5 hrs): apparently little change in dye distribution
from Day 2, however the total dye captured (65%) is the
lowest of all the surveys. Day 4 (t = 72.5 hrs): dye is now
advecting and diffusing rapidly up-slope in a thin bottom
boundary layer (H 5 m) and layers are beginning to
form in the off-slope direction. Day 5 (t = 97 hrs): upslope of the release point the dye has mixed upward into a
weakly stratified layer of about 30 m thick, and in the
down-slope direction it has detached from the boundary in
three distinct layers (profiles in Figures 3 and 4). Onesided horizontal diffusivities illustrate the dramatic difference between the up-slope (KHU = 27.9 m2s1) and downslope (KHD = 2.83 m2s1) change in horizontal dye
variance. These are estimated separately using (1) and
(2), with calculations confined to x < 0 (up-slope, KHU)
and x>0 (down-slope, KHD) of the dye-release position
(x = 0) between days 4 and 5. The dye patch labelled A
is thought to be a remnant of dye spilled during release.
4. Discussion
[9] There is a qualitative difference between the up-slope
and down-slope evolution of the dye, in this section we
address the following: up-slope versus down-slope evolution; and vertical and horizontal scales of the intrusions.
[10] Up-slope the maximum dye concentration on all
vertical profiles is found on the boundary, whilst downslope the dye maximum has separated from the boundary
and spread into the fluid interior in three distinct layers, with
vertical scales and separations of 10 m (Figures 3 and 4).
We consider three candidate mechanisms for the observed
Figure 2. Isotherm displacement from moored thermistors. Heavy vertical lines correspond to the centre-times of
the five dye surveys, arrows indicate the direction and
magnitude of the near-bed flows associated with a mode one
semi-diurnal baroclinic wave.
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Figure 3. Dye concentration in mgl1: (a) Two hours after release, (b) 24 hours after release, (c) 48 hours after release,
(d) 72 hours after release, and (e) 97 hours after release. Isopycnal contours evenly spaced in black, with injection isopycnal
shown as white in Figures 3b– 3e. (f) Profiles of dye concentration and sq through the vertical section shown as a heavy
white line in Figure 3e.
down-slope fine structure: 1) gravitation collapse of a
mixing patch (GC), 2) high vertical mode-number internal
wave shear combined with shear dispersion (HM), and 3)
double diffusive interleaving (DD). DD can be ruled out
because neither salinity nor temperature gradients are unstable in the vertical. Although HM is found to be important
on the boundary in tank experiments [DeSilva et al., 1997;
McPhee-Shaw and Kunze, 2002], it is unlikely to be the
case here for three reasons. Firstly, there is no observational
evidence for high vertical mode-number internal waves (see
Figure 2). Secondly, the observed dye layers do not cross
isopycnal surfaces (Figure 3), as high mode shear layers
would. Thirdly, under the HM scenario, dye would concentrate in regions of maximum shear and low stratification,
i.e., higher Froude numbers (inverse square root of the
gradient Richardson number). Here, higher Froude numbers
are observed around the edges of the dye-rich layers,
indicative of conditions conducive to shear instability
around the edges of the intruding layers. Taken together,
the above arguments strongly suggest that gravitational
collapse of partially mixed patches of fluid is the most
likely explanation for the intrusion of fluid into the interior.
[11] We now consider the vertical scales of the intrusion.
Baroclinic wave forcing of near-bed flows will produce a
semi-diurnal fluctuation in the thickness of the bottom
boundary layer. During up-slope flows the boundary layer
will thicken as near bed stratification reduces, whilst the
opposite will occur during down-slope flow. It is likely that
variations in bottom roughness will result in spatial variations in boundary mixing, with more mixing over regions of
greater roughness. Vertical isopycnal spacing, averaged
over a tidal cycle, will be increased over regions of greater
topographic roughness leading to horizontal pressure gradients that will draw fluid into the interior centred on such
topographic variations. For the region of the slope below to
the layered intrusions the topographic wave-number spec-
Table 1. Dye Moments and Fickian Diffusivities as Defined in the Text
Day
Mass
sq-CoM
X-CoM (km)
sx2 (km2)
1
2
3
4
5
20
20
13
15
19
19.59
19.72
19.76
19.76
19.78
0.22
0.28
0.05
0.44
1.05
0.102
0.320
0.426
2.05
5.30
KH (m2s1)
1.38
0.54
9.45
18.4
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sxU2 (km2)
KHU (m2s1)
sxD2 (km2)
KHD (m2s1)
0.099
0.365
0.412
2.803
7.720
27.9
0.103
0.157
0.363
0.833
1.333
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Figure 4. Froude number, dye and density profiles of the
layered intrusions at x = 2 km on 28th June (Figure 3e).
trum, transposed onto the vertical axis, shows a clear peak at
0.1 m1 (not shown). Whilst this does not provide causative
evidence for the mechanism behind the spacing of the dyerich layers it seems to provide a consistent and plausible
explanation. Further, the Ozmidov scale over the depth
range of the intruding layers, calculated from microstructure
profiles, is only of order of 0.3m. Laboratory experiments in
which stratified fluid is agitated with uniform grid-generated turbulence and without sloping boundaries tend to
develop into layers with a vertical scale and separation of
the order of the Ozmidov scale [Thorpe, 1982]. Reasons for
self organisation on this scale are not clear, although the
mechanism whereby perturbations to a uniform density
stratification lead to layers of diffusive turbulent flux
convergence and divergence is well described [e.g., Thorpe,
2005, p. 194]. It is therefore likely that topographically
induced mixing with a vertical scale of 10m is occurring
here, and that self organisation on the Ozmidov scale in not
shaping the vertical structure seen some distance from the
sloping boundary. This may be compared and contrasted
with smooth boundary laboratory experiments where near
critical slope-induced wave breaking and overturning sets
the scale for the mixing patches which then undergo
gravitational collapse into the interior [DeSilva et al.,
1997; McPhee-Shaw and Kunze, 2002].
[12] If the intrusions are a result of gravitational collapse
of boundary mixing, then it follows that the observed
horizontal spreading rate of the intrusions, U , might yield
information about the efficiency of mixing on a sloping
boundary. The following expression may be applied
[McPhee-Shaw and Kunze, 2002]:
Rf ¼
1
1 r0 N 2 L2
U
ð1 Cr Þ 12
fi
ð3Þ
where, Rf is the flux Richardson number, Cr a reflection
coefficient, r0 a reference density, N the buoyancy
frequency, L the vertical scale of the mixing causing the
intrusion and fi in incident wave energy flux. Setting N2 =
2.5 104s2, L = 10 m, U = 0.035 ms1 and fi =
3.4 Wm2 [Stashchuk et al., 2007], and Cr = 0.5 (from
calculations based on data presented by Inall et al. [2004])
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yields Rf = 0.044. This gives an upper bound on Rf, since
Cr = 0.5 is an upper estimate of the energy forward
reflected from the sub-critical slope. Rf = 0.044 is a low
value compared with canonical values of between 0.1 and
0.2 [Osborn, 1980]. It is closer to values found in other
fjords [Stigebrandt and Aure, 1989], and concurs with
physical arguments which suggest a low mixing efficiency
for this type of process [Arneborg, 2002; Garrett et al.,
1993]. Further, this is not simply a reflection of low
mixing efficiency in a low turbulent Reynolds number
regime, since e/uN2 200 in the intrusions.
[13] Considering finally the up-slope penetration of the
dye, two of the dye statistics in particular merit explanation:
Firstly the up-slope movement of the centre of mass and
secondly the rapid increase in the horizontal variance from
day 3 (Table 1). Candidate mechanisms can be identified for
each: 1) a residual up-slope flow driven by the pressure
gradient which results from downward sloping isopycnals
surfaces near the boundary and 2) breaking of bore-like
baroclinic waves of tidal period followed by shear dispersion.
[14] Residual Flow: a zero buoyancy flux condition at a
sloping boundary demands that isopycnal surfaces are
locally normal to the boundary. An estimate of the nearbed residual up-slope directed flow (u) can be made by
assuming a steady state balance between the downward
diffusive buoyancy flux into a near bed layer and the upslope buoyancy advection of the up-slope directed, pressure
gradient driven flow (averaged over a tidal cycle). If the
boundary layer has thickness H and the slope has angle f,
then, for f 1, it can be shown that [Garrett et al., 1993]
u0 ¼
Kz
=Hf
ð4Þ
Realistic values are, Kz = 2.5 104 m2s1 (estimated from
microstructure measurements above the boundary layer),
f = 0.016, and H5 m. The last parameter is estimated
from the dye distribution up-slope of the release position.
Substituting these values into equation (4) gives u0.3 cms1,
a small but significant up-slope velocity. If we assume
this mechanism drives the dye up-slope from days 2 to 5
(Figures 3b to 3e), then the predicted up-slope position of the
dye centre of mass would be x = 0.78 km. This closely
matches the observed up-slope movement of the centre of
mass (x = 1.05 km, Table 1), to within the likely error
bounds associated with the values used in estimating u. This
quantitative estimate has parallels with mid-ocean ridges,
where the effects of up-slope residual flows on deep ocean
mixing is beginning to receive attention [St Laurent and
Thurnherr, 2007].
[15] Shear Dispersion: The initial dye injection at sq =
19.59 coincides with the depth of the zero crossing of the
first baroclinic mode, however the draw-down of the dye by
the bottom boundary layer into a higher density region
rapidly brought the vertical centre of mass of the dye below
the zero crossing, giving a dye derived vertical diffusivity
between days 1 and 2 (equation (2)) of Kz = 0.9 102 m2s1. Some caution should be used in applying a
modal approach where the depth varies significantly over a
horizontal distance less than the wavelength of the first
baroclinic mode. Nevertheless, previous studies have shown
a clear mode 1 structure to the baroclinic wave field in this
system [Inall et al., 2004]. Therefore during off-slope flow in
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the upper layer, dye centred in the lower layer is pushed upslope in the turbulent bottom boundary layer. Under this
scenario one might expect the maximum dye concentration
to be above the bottom up-slope of the x = 0 km rather than at
the bottom, since the up-slope velocity must tend to zero at the
bed. However, all the surveys were carried out during nearmaximal down-slope flow in the lower layer, maximum dye
concentrations would therefore be expected at the bottom.
5. Conclusions
[16] This is the first reported in situ study of the vertical
and horizontal dispersion arising from the interaction of
baroclinic tidal period waves with sub-critically sloping
topography. We have shown that the spectral characteristics
of the small scale topographic roughness elements likely
play a role in shaping the evolution of a passive tracer, as do
the slope of the topography and the thickness of the bottom
boundary layer (and, by implication, the bed roughness and
the nearbed baroclinic energy). The robustness of the results
reported here with respect to variations in stratification,
baroclinic structure and energy, and topographic slope and
roughness requires further investigation, and will be the
focus of a series of numerical experiments. If shown to be
robust these results will add to a growing body of evidence
that our inadequate knowledge of the ocean’s topography at
the sub kilometre scale is hampering our understanding of
how the ocean is mixed. Despite relatively high turbulence
Reynolds number (200), the observed intrusion spreading
rate implies a low flux Richarsdon number for the boundary
mixing process (Rf0.04). Estimated isopycnal diffusivities
reported here (in the range of 1.4 m2s1 to 18.4 m2s1) are
comparable to dye-based estimates in coastal and shelf
studies carried out on similar spatial scales [Dale et al.,
2006; Ledwell et al., 2004]; to the best of our knowledge
these are the first reported estimates in the interior of a
fjordic system. The dispersive asymmetry introduced by the
sloping boundary is profound, as evidenced by the order of
magnitude difference between horizontal diffusivities within
the boundary-influenced layer (KHU = 27.9 m2s1) and
intruding layers (KHD = 2.83 m2s1).
[17] Acknowledgments. Thanks go to Colin Griffiths and Paul
Provost for expert technical assistance. The review by Erika McPhee-Shaw
and discussions with Andy Dale and Tim Boyd were greatly appreciated.
Funded by NERC under Oceans 2025 Theme 3.
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M. E. Inall, Scottish Association for Marine Science, Dunstaffnage
Marine Laboratory, Oban, Argyll PA37 1QA, UK. ([email protected])
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