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Interior versus boundary mixing of a cold intermediate layer

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1] The relative importance of interior versus boundary mixing is examined for the erosion of the cold intermediate layer (CIL) of the Gulf of St. Lawrence. Based on 18 years of historical temperature profiles, the seasonal erosion of the core temperature, thickness and heat content of the CIL are, respectively, _ T min = 0.24 ± 0.04°C mo −1 , _ d min = −11 ± 2 m mo −1 and _ H = 0.59 ± 0.09 MJ m −3 mo −1 . These erosion rates are remarkably well reproduced with a one–dimensional vertical diffusion model fed with turbulent diffusivities inferred from 892 microstructure casts. This suggests that the CIL is principally eroded by vertical diffusion processes. The CIL erosion is best reproduced by mean turbulent kinetic energy dissipation rate and eddy diffusivity coefficient of ' 2 × 10 −8 W kg −1 and K ' 4 × 10 −5 m 2 s −1 , respectively. It is also suggested that while boundary mixing may be significant it may not dominate CIL erosion. Interior mixing alone accounts for about 70% of this diffusivity with the remainder being attributed to boundary mixing. The latter result is in accordance with recent studies that suggest that boundary mixing is not the principal mixing agent in coastal seas.
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Interior versus boundary mixing of a cold intermediate layer
F. Cyr,
1
D. Bourgault,
1
and P. S. Galbraith
2
Received 2 June 2011; revised 15 September 2011; accepted 6 October 2011; published 21 December 2011.
[1] The relative importance of interior versus boundary mixing is examined for the erosion
of the cold intermediate layer (CIL) of the Gulf of St. Lawrence. Based on 18 years of
historical temperature profiles, the seasonal erosion of the core temperature, thickness and
heat content of the CIL are, respectively,
_
T
min
= 0.24 ± 0.04°C mo
1
,
_
d
min
= 11 ± 2 m mo
1
and
_
H = 0.59 ± 0.09 MJ m
3
mo
1
. These erosion rates are remarkably well
reproduced with a onedimensional vertical diffusion model fed with turbulent
diffusivities inferred from 892 microstructure casts. This suggests that the CIL is principally
eroded by vertical diffusion processes. The CIL erosion is best reproduced by mean
turbulent kinetic energy dissipation rate and eddy diffusivity coefficient of
10
8
Wkg
1
and K 4×10
5
m
2
s
1
, respectively. It is also suggested that while boundary
mixing may be significant it may not dominate CIL erosion. Interior mixing alone
accounts for about 70% of this diffusivity with the remainder being attributed to
boundary mixing. The latter result is in accordance with recent studies that suggest that
boundary mixing is not the principal mixing agent in coastal seas.
Citation: Cyr, F., D. Bourgault, and P. S. Galbraith (2011), Interior versus boundary mixing of a cold intermediate layer,
J. Geophys. Res., 116, C12029, doi:10.1029/2011JC007359.
1. Introduction
[2] Cold intermediate layers (CILs) are common summer
feature of many subarctic coastal seas. Such water masses
are found for example in the Black Sea [Tuzhilkin, 2008],
the Baltic Sea [Chubarenko and Demchenko, 2010], the
Bering Sea [Kostianoy et al., 2004] the Sea of Okhotsk
[Rogashev et al., 2000], the Gulf of St. Lawrence [Banks,
1966] and can also be found in coastal oceans [Petrie et al.,
1988]. At formation, CILs may represent up to 45% of the
total water volume of those systems [e.g., Galbraith, 2006]
and therefore largely control the state and climate of subarctic
coastal systems as well as the marine ecology [Ottersen et al.,
2004].
[
3] The characteristics of CILs are principally governed
by the properties of the surface mixed layer formed during
the previous winter [Gilbert and Pettigrew, 1997; Gregg
and Yakushev, 2005; Galbraith, 2006; Smith et al.,
2006a]. The CIL is formed when this surface mixed layer
becomes insulated from the atmosphere by nearsurface
stratification caused by seaice melt, heat fluxes and increase
runoff at the onset of spring. Other mechanisms such as
horizontal/intralayer convection may also contribute to CIL
formation [Chubarenko and Demchenko, 2010].
[
4] Most previous studies on CILs have focused on for-
mation mechanisms but there is little published information
about the summer deterioration of their properties by mixing
processes, later recalled as erosion. One particularity of
CILs is that since they lie at intermediate depths, away from
surface and bottom boundary layers, one may hypothesize
that their erosion is principally governed by interior mixing
processes. However, CILs also intersect the sloping bottom
around lateral boundaries where turbulent processes may be
much more intense than within the interior. While the
fraction of CIL volume in contact with the sloping bottom
may be small the role of boundary mixing may still be
important if turbulence is sufficiently large. This is analo-
gous to the boundary mixing hypothesis proposed by Munk
[1966] and Munk and Wunsch [1998] for the abyssal ocean
but applied here to coastal seas. The idea of boundary mixing
has been considered for mixing in sill fjords [Stigebrandt,
1976, 1979] and estuaries [Bourgault and Kelley, 2003;
Bourgault et al., 2008] with the hypothesis that breaking
internal waves along sloping boundaries is the main mixing
agent. It has also been proposed that boundary mixing may be
the main contributor to the mixing budget of lakes [Goudsmit
et al., 1997], where windinduced seiches are the principal
driving mechanism, and to that of coastal seas with virtually
no tides such as the Baltic Sea (see Reissmann et al. [2009] for
a review). Other studies have however concluded that mixing
in coastal seas is predominantly driven by interior rather than
boundary processes [e.g., MacKinnon and Gregg, 2003;
Rippeth et al., 2005; Palmer et al., 2008].
[
5] Another interesting aspect of CILs is that they can be
considered as passive tracers when the buoyancydriven
circulation of coastal seas is principally driven by salinity
gradients. In this case, CILs are analogous to the yearly
realization of a largescale dye release experiment. Under this
hypothesis, the CIL fills uniformly the entire sea and can only
be modified, or eroded, by vertical turbulent processes.
1
Institut des Sciences de la Mer de Rimouski, Université du Québec à
Rimouski, Rimouski, Québec, Canada.
2
Ocean and Environmental Science Branch, MauriceLamontagne
Institute, Fisheries and Oceans Canada, MontJoli, Québec, Canada.
Copyright 2011 by the American Geophysical Union.
01480227/11/2011JC007359
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, C12029, doi:10.1029/2011JC007359, 2011
C12029 1of12
[6] The objectives of this study are to examine whether
vertical mixing alone can explain CIL erosion rates and, if
so, to determine the relative importance of boundary versus
interior mixing. To reach these objectives, we examine
the erosion of the CIL in the Gulf of St. Lawrence using
18 years of historical CTD data, new turbulence measure-
ments (892 casts) collected during summers 20092010 and
a onedimensional heat diffusion model. The historical
CTD observations are used to provide statistics on CIL
erosion rates and the turbulence measurements are used to
provide eddy diffusivity values used in the onedimen-
sional model. After assessing the model we examine the
relative importance of interior versus boundary mixing by
comparing the modeled CIL erosion rates with and without
considering diffusivity values measured near boundaries.
The results are then interpreted based on a geometric
scaling from which an effective eddy diffusivity is inferred
and the relative contribution of boundary versus interior
CIL mixing is determined.
2. The Gulf of St. Lawrence
[7] The Gulf of St. Lawrence, including the estuary, is an
area of about 236000 km
2
opened to the Atlantic Ocean
through Cabot Strait and the Strait of Belle Isle (Figure 1).
The bathymetry is characterized by deep channels (>200 m),
large shelves and islands. The main channel, called the
Laurentian Channel, is a submarine glacial valley that runs
from Tadoussac to the continental slope, past Cabot Strait
(Figure 1). The residual circulation in this channel is estu-
arinelike, principally driven by the freshwater discharge of
the St. Lawrence River and other surrounding rivers [e.g.,
Koutitonsky and Bugden, 1991]. In winter, the water column
exhibits a twolayer structure with a 40150 m thick surface
mixed layer with temperature near the freezing point
[Galbraith, 2006] overlying a warmer (16°C) but saltier
(>33 psu) bottom layer of oceanic origin (roughly 150 m
bottom). The rest of the year, the water column is charac-
terized with three layers with the CIL sandwiched between
the warmer surface and bottom layers [Koutitonsky and
Bugden, 1991].
[
8] The CIL is characterized by nearfreezing tempera-
tures (T) and salinities ( S )of3233 psu. Although the Gulf
of St. Lawrence CIL is frequently mentioned in the litera-
ture, its definition varies between authors. For instance,
Lauzier and Bailey [1957] used T 0°C as their definition.
Other authors have used other definitions such as T 1.5°C
[Banks, 1966; Bugden et al., 1982], T 3°C [Gilbert and
Pettigrew, 1997; Smith, 2005] or more recently T 1°C
[Galbraith et al., 2011]. Since the focus here is on the CIL
erosion during icefree months, we adopted the latter defi-
nition which approximately defines the coldest limit of what
remains at the end of our study period, i.e., when the CIL is
replenished the following winter.
[
9] The CIL renewal occurs in winter when the surface
mixed layer deepens following a combination of cold air
temperature, winddriven mixing and, to a lesser extent,
brine rejection due to sea ice formation [Galbraith, 2006].
While the CIL is found throughout the Gulf, its presence at a
given location may be due to horizontal advection from a
remote formation site rather than resulting from local for-
mation. For example, the region roughly located between
Tadoussac and PointedesMonts is too stratified, due to
important freshwater input, to allow for winter convection
and CIL formation [Ingram, 1979; Galbraith, 2006; Smith
et al., 2006b].
Figure 1. Location of 892 VMP casts (black dots) and bathymetric features of the St. Lawrence Estuary
(figure) and the Gulf of St. Lawrence (inset). The black contour lines are the 50, 150 and 250 m isobaths.
Rimouski station is identified with a white circle superimposed over the maximum profile concentration.
The red line is the section referred to in Figure 12 and where sampling was performed during 10 days in
July 2010.
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
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[10] While insights about CIL formation mechanisms were
gained from previous field and modeling studies cited above,
still little is known about CIL erosion mechanisms and rates.
Based on 10 years of data from three regions of the Gulf
(Honguedo Strait, Central Gulf and Cabo t Strait), Banks
[1966] found that from April to November the CIL mini-
mum temperature (T
min
) warms, on average, by 0.2°C mo
1
.
Based on a similar analysis but using 47 years of data from
five regions, Gilbert and Pettigrew [1997] obtained CIL
erosion rates ranging from 0.08°C mo
1
for the Central Gulf
to 0.30°C mo
1
for the Estuary. This study also revisits these
warming rates.
3. Data Sets and Methodology
3.1. CTD Data
[
11] The CTD dataset was collected at a station named
Rimouski and located at 48°40N 68°35W, about 25 km
north of the city of Rimouski (Figure 1). Thereafter, we will
refer to this station as the interior station, i.e. a station in deep
water (>300 m) away from lateral boundaries. This dataset
consists of 418 casts collected by MauriceLamontagne
Institute staff (Fisheries and Oceans Canada, DFO) between
1993 and 2010 and obtained through the DFO Oceano-
graphic Data Management System [Fisheries and Ocean
Canada, 2011]. The station is typically visited once a
week during icefree months [Plourde et al., 2008] and, for
the purpose of this study, only casts sampled between April
and November have been selected. Temperature and salinity
profiles were averaged into 1 m vertical bin size and all
profiles of the same month have been averaged into a single
monthly climatological profile (Figures 2 and 3). The 95%
confidence intervals on the monthly averaged profiles have
been estimated by performing 500 bootstrap replicates of
each monthly sampling [Efron and Gong, 1983].
[
12] Using this climatology, the depthaveraged volu-
metric heat content of the CIL was calculated as
H ¼
1
d
Z
z
b
z
t
c
p
T T
f

dz for T < 1
C; ð1Þ
where d is the CIL thickness, z
t
and z
b
are, respectively, top
and bottom limits of the layer, r is density determined from
the equation of state of the seawater [Fofonoff and Millard,
1983] and c
p
= 4.00 kJ kg
1
°C
1
, the specific heat of sea
water, is considered constant. For convenience, the heat
content is calculated relative to the typical freezing point
temperature of sea water T
f
= 1.8°C.
3.2. Sea Surface Temperature
[
13] Since 2002, a meteorological buoy has been deployed
at Rimouski station by the MauriceLamontagne Institute
(DFO), usually from May to November when it then acquires
data of various type every 15 minutes [Fisheries and Ocean
Canada, 2011]. Among them, sea surface temperature (SST)
is measured at 0.5 m below sea surface with a SBE37SI, a
temperature sensor manufactured by SeaBird Electronics.
Figure 2. Monthly mean temperature profiles calculated from April to November over the period
19932010 from the CTD casts of Figure 4. The gray shadings are the 95% confidence intervals.
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
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In this study, SSTs are used to provide boundary conditions
to a onedimensional model of heat diffusion presented in
section 5.
3.3. Turbulence Measurements
[
14] Turbulence measurements were collected with two
freefall, looselytethered, vertical microstructure profilers
(VMP500) manufactured by Rockland Scientific Interna-
tional (RSI). Together with standard SeaBird Electronics
CTD sensors, the VMPs are equipped with a micro
fluorescence/turbidity sensor, two fastresponse thermistors
and two airfoil shear probes which allow measurements of
microscale vertical shear u
z
. One of the VMP also has a
microconductivity sensor. All microstructure sensors sam-
ple at 512 Hz while the CTD samples at 64 Hz. See Bourgault
et al. [2008] for more details on sensors and probes.
[
15] Our dataset consists of 73 casts from 6 sorties done in
July and September 2009 and 819 casts from 26 sorties done
between May and October 2010. Most of these sorties were
realized opportunistically, depending on weather and boat
availability, except for a 10days survey accomplished in
July 2010. Two small craft boats, each carrying a VMP,
were then mobilized to carry out sampling across the
channel on a section passing through the Rimouski interior
station. Overall, 420 casts out of 892 were realized within
5 km of Rimouski station (Figure 1). All together, these
casts have no bias towards any phase of the M
2
tide cycle
and are slightly biased toward neap tide (not shown).
Uncertainties in mean quantities caused by this bias will be
reflected in our analysis below through 95% confidence
intervals obtained by bootstrapping. All surveys were car-
ried out in relatively calm sea conditions. The wind speed
was generally less than 20 km h
1
and wave heights less
than 1 m.
[
16] Assuming isotropic turbulence, the dissipation rate
() of turbulent kinetic energy was calculated, using standard
procedures [e.g., Lueck et al., 2002], as
¼
15
2
u
z
ðÞ
2
; ð2Þ
where n = f(T) is the kinematic molecular viscosity as
function of temperature and the overline indicates here a 5m
scale spatial average. The shear variance (u
z
)
2
was obtained
by spectral integration to remove random noise. Eddy diffu-
sivity coefficients were calculated as [Osborn, 1980]
K ¼ G
N
2
; ð3Þ
where G is the dissipation flux coefficient and N is the 5m
scale background buoyancy frequency. Using G = 0.2
[Osborn, 1980; Moum, 1996], an upper bound for the eddy
diffusivity coefficient was determined. An estimation of the
95% confidence intervals of the mean turbulence profile was
obtained by performing 500 bootstrap replicates of the sam-
pling set.
4. Observations
4.1. CIL Characteristics and Variability
[
17] The CIL structure and variability for the 19932010
period can be qualitatively appreciated in Figure 4. During
Figure 3. Monthly mean salinity profiles calculated as in Figure 2.
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
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spring, the CIL is roughly 50100 m thick and centered
around 60 m. It persists throughout summer and fall until it
becomes regenerated and replenished the following winter.
The temperature field within the CIL exhibits important
intraseasonal variability. For example, in 2001 isotherms
displacement reached 50 m and the minimum CIL temper-
ature varied by 1°C on monthly timescales. The CIL even
disappeared, according to our definition, for a 2week
period at the beginning of July. A striking feature often
occurs in fall (e.g., 2004, 2007 and 2009), when within
about a week, the CIL suddenly plunges downward by
approximately 50 m. The origin of these intraseasonal var-
iations is unknown and will not be addressed in this study.
[
18] The evolution of monthly averaged profiles reveal
that the temperature (Figure 2) and the salinity (Figure 3) of
the surface layer increases from April to August. While the
salinity of this layer continues to increase from September to
November, the layer rapidly cools and deepens during the
same period. Underneath the surface layer, that is under
50 m, the CIL temperature minimum increases from April to
November while the salinity stays relatively constant. We
interpret this as an indication that the middepth salinity field
is approximately in steady state (
@S
@t
0) during icefree
months, likely due to a balance between longitudinal
advection and vertical diffusion. Unlike the salinity, no
equivalent compensating source of cold water that can feed
the CIL once it has been replenished during winter. The CIL
thus acts like a passive tracer, being slowly mixed by vertical
diffusion.
[
19] The monthly climatology of the temperature distri-
bution within the CIL (Figure 2) can be synthesized in a
single contourplot that reveals the general characteristics of
the CIL and its seasonal erosion (Figure 5a). From April to
November the CIL thickness steadily decreases while the
minimum temperature increases. This erosion can be
quantified by examining the evolution of the climatological
minimum temperature of the CIL T
min
(Figure 6a), which
increases linearly at a rate of
_
T
min
= 0.24 ± 0.04°C mo
1
,as
determined by performing linear best fits to the climato-
logical timeseries. The error reported is the standard error
determined by bootstrap analysis [Efron and Gong, 1983].
The erosion of the CIL can also be quantified by the rate of its
thickness decrease which is
_
d = 11 ± 2 m mo
1
(Figure 6b).
Finally, the seasonal change in CIL mean heat content is
_
H =
0.59 ± 0.09 MJ m
3
mo
1
(Figure 6c). Note that the sudden
fall deepening events of the CIL observed in Figure 4 clearly
Figure 4. Evolution of temperature profiles from April to November, linearly interpolated from 418 CTD
casts that are indicated by lines at the top of each figure. To focus on the CIL, the color scale is saturated at
5°C while summer surface temperature can reach more than 10°C, and 1°C isotherms are highlighted with
a black contour.
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
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show up in the monthly temperature climatology (Figure 5a)
and also in the climatological evolution of CIL core depth
(Figure 6d). The CIL also shows interannual variability of the
warming rate of the CIL core (Figure 7), reaching values as
high as
_
T
min
= 0.30°C mo
1
(1999 and 2004) and as low as
_
T
min
= 0.15°C mo
1
(1997 and 2003). Note that years 2009
and 2010, i.e., the years that turbulence was sampled were
climatologically closetonormal (Figure 7).
4.2. Turbulence
[
20] A typical VMP cast is shown in Figure 8 providing
profiles of temperature T, density r, microstructure shear u
z
,
turbulent dissipation rate and eddy diffusivity K. Turbu-
lence within the CIL (gray intervals in Figure 8) is typically
low compared to patches of much higher dissipation often
found in the top 50 m or so of the water column.
[
21] Turbulence measurements collected in 2009 and
2010 are synthesized in Figure 9. Two series of mean pro-
files are presented, i.e., the average of all available casts
(892), including interior and boundary regions (dark gray,
solid line), and the average of interior casts only (light gray,
dashedline). Globally, highest dissipation rates and diffu-
sivities are found in the top 20 m or so of the water column
with values reaching 10
6
Wkg
1
and K 10
3
m
2
s
1
.
Under 20 m, there is a decreasing trend of the dissipation
and the diffusivity with depth; the increase in dissipation
and diffusivity observed near 130 m likely comes from the
numerous turbulence casts made on the shelf that passed
through the bottom boundary layer at that depth. When
considering only interior casts (420), the mean dissipation
rate and diffusivity underneath the surface layer (20180 m)
are, respectively,
i
= 1.3(0.9, 1.7) × 10
8
Wkg
1
and K
i
=
2.4(1.5, 3.5) × 10
5
m
2
s
1
, where the numbers in paren-
theses are the bootstrap 95% confidence intervals. When
considering all available casts (892), the mean dissipation
and diffusivity for the same depth range are respectively
a
= 1.9(1.3, 2.8) × 10
8
Wkg
1
and K
a
= 4.3(2.6, 6.6) ×
10
5
m
2
s
1
, i.e., respectively 1.5 and 1.8 times higher
than when considering only interior casts.
[
22] These diffusivities are close to an order of magni-
tude smaller than that previously inferred by Bugden
[1991] from least square fit on temperature data in the
Gulf between 200300 m and on the same order than that
inferred by Ingram [1979] near Tadoussac in the 50100 m
depth range.
5. Heat Diffusion Model
5.1. Model Description
[
23] We now examine with a onedimensional heat dif-
fusion model whether the turbulence measured during calm
conditions could explain the observed climatological CIL
erosion rates. With this approach we neglect horizontal
Figure 5. April to November water temperature. (a) Monthly
climatology over the period 19932010 calculated from the
CTD casts of Figure 4. (b) Modeled evolution of tempera-
ture using the observed mean interior diffusivity profile K
i
.
(c) Modeled evolution of temperature using the diffusivity
profile from all available casts K
a
. Figure properties are the
same as Figure 4.
Figure 6. Evolution of CIL properties. Solid lines are
monthly averages over the period 19932010. The shaded
areas are the 95% confidence interval. (a) Temperature of
the CIL core, (b) thickness of the CIL, (c) heat content of
the CIL and (d) depth of the CIL core. Thin short lines in
Figures 6a6c are the mean slopes (erosion rates) calculated
in section 4.1. The mean slope is reported on figures with
the 95% confidence intervals (see Table 1 for comparison
with model results).
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
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advection under the assumption that only vertical turbulent
mixing can redistribute heat within the water column.
[
24] The model numerically solves the following equation
for the temporal evolution of the water column temperature
T(z, t) [e.g., Kundu and Cohen, 2007]:
@T
@t
¼
@
@z
K
@T
@z

; ð4Þ
where K is the eddy diffusivity of heat taken here to be
equivalent to the eddy diffusivity of mass as defined in
equation (3) [Thorpe, 2007]. Eddy diffusivity is assumed to
be variable with depth but constant in time, i.e, K = K(z).
[
25] Equation (4) is solved numerically with a firstorder
Euler scheme on a grid size Dz = 1 m and using a time step
Dt = 100 s. The vertical resolution thus adequately resolve
the different layers and fits the resolution of CTD bins. The
time step respects the CourantFriedrichLevy stability
condition. A noflux boundary condition is applied at the
sea bottom fixed at z = 300 m. A daily climatological sea
surface temperature (SST) is imposed as boundary condi-
tions at the top of the water column. This climatology was
evaluated over the period 20022009 using observations
from the oceanographic buoys [Fisheries and Ocean
Canada, 2011] and interpolated to the model time step Dt.
Equation (4) was initialized with the climatological temper-
ature profile for April (Figure 2).
[
26] Two simulations were carried out with different heat
diffusivity profiles. The first simulation uses the mean interior
diffusivity profile K
i
, while the second uses the mean diffu-
sivity profile of all casts collected across the channel, i.e., K
a
(Figure 9). Since no casts were done deeper then about 180 m,
K
i
and K
a
are extended to 300 m using a constant value equal
to the minimum value of the mean profile.
[
27] Note that this model is designed to examine the
average seasonal evolution of the temperature structure and
Figure 7. Interannual variability of the warming rate of the CIL core, calculated between April and
November for every year over the period 19932010. The warming rate is defined as the slope of
the best linear fit of the evolution of the CIL core temperature as shown in Figure 6a. The dashed
line is the average warming rate (0.24°C mo
1
) and the shaded area is the 95% confidence interval
envelope (±0.04°C mo
1
).
Figure 8. Typical turbulence profiler measurements, from July 21 2009. (ac) Unfiltered profiles
(temperature, sea water density and vertical shear), i.e., sampled at 64 Hz for T and r and 512 Hz for u
z
.
(d and e) Fivem scale TKE dissipation rate, , and eddy diffusivity, K. The CIL has been highlighted in
Figure 8a and its depth is reported on all figures as shaded areas.
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
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not a particular year. For this reason, the model is forced
with quasi climatological boundary conditions. For consis-
tency, a climatological profile of eddy diffusivity should
also be used. However, such statistics are not available. We
therefore work under the assumption that the turbulence
measured in 2009 and 2010 are representative of longterm
mean conditions. This appears to be a reasonable assump-
tion given that years 20092010 were subject to closeto
normal erosion rates (Figure 7).
5.2. Results
[
28] While the model has difficulties to reproduce the
evolution of the surface layer above 50 m (see Doy on and
Ingram [2000] for a similar model of the surface mixed
layer) it reproduces qualitatively well the CIL from April to
November (Figures 5 and 10). Overall, the simulation using
K
a
better reproduces the observations except perhaps for the
month of June where using K
i
offers a slight improvement
(Figure 10). The better performance of the model with K
a
instead of K
i
is also seen when examining a spacetime
contour plot of T(z, t) (Figure 5).
[
29] Quantitatively, the simulation with K
a
reproduces
remarkably well, i.e., within the uncertainties, the CIL ero-
sion rates (Figure 11 and Table 1). In this case, the modeled
and observed erosion rates (i.e.,
_
T
min
,
_
d and
_
H) agree to
within 8% or better. On the other hand, using only the
Figure 9. Mean quantities from all 892 VMP casts for
20092010 (dark gray, solid lines). (a) Mean dissipation rate
of turbulent kinetic energy, (b) mean buoyancy frequency
squared and (c) mean eddy diffusivity coefficient. The gray
shades indicate 95% confidence intervals. Mean quantities
for casts taken at midchannel (420) are also presented
(lightgray, dashedlines).
Figure 10. Details of the evolution of monthly mean temperature profiles for observations and simula-
tions. The observed 95% confidence intervals of the observed mean temperature profile (shaded area) is
compared to two simulations done with respectively the average observed diffusivity at midchannel K
i
(dashed lines) and from all available casts along the section K
a
(solid lines).
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
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interior diffusivity K
i
accounts for about 65% of the
observed erosion rates.
6. Discussion
[30] Our analysis suggests that interior diffusivity during
calm wind conditions (Figure 9) accounts for about 65% of
the midchannel CIL erosion rate (Table 1). However, the
apparent missing mixing is recovered when all profiles
collected throughout the section are included in the analysis.
This suggests that nonlocal boundary mixing processes,
although not dominant, may contribute significantly to mid
channel erosion.
[
31] With this in mind, we now examine the role that
boundary mixing may play in eroding the CIL at mid
channel. We start by assuming that, on monthly timescales,
there is a continuous exchange of properties along iso-
pycnals between boundaries, where the CIL intersects the
sloping bottom and the interior. Inspired by Armi [1978],
Garrett and Gilbert [1988], Garrett et al. [1993] and Toole
et al. [1997], we estimate the effective diffusivity coefficient
using the following geometric scaling:
K
e
¼ K
b
A
b
z; h
b
ðÞ
AzðÞ
þ
~
K
i
A
i
z; h
b
ðÞ
AzðÞ
; ð5Þ
where K
e
is the effective diffusivity coefficient at depth z,
while K
b
and
~
K
i
are, respectively, constant diffusivity
coefficients within the bottom boundary layer and in the
interior, and h
b
the bottom boundary layer thickness.
Similarly,
A
b
A
and
A
i
A
=1
A
b
A
are respectively the fraction
of the CIL inside and outside the bottom boundary layer
of thickness h
b
at depth z (Figure 12). Note that unlike
the previously cited authors, we kept in the formulation
for K
e
the contribution from interior mixing (i.e., the
second term in equation (5)), since we have indications
that boundary mixing is not dominant in front of interior
mixing.
[
32] The thickness of the bottom boundary layer h
b
was
visuallyinferred by inspecting the vertical structure of the
dissipation rates of 150 casts, out of 892, that hit the
bottom out of the 892 (Figure 13). On average, these mea-
surements show an approximate exponential growth of
towards the bottom, starting at about 10 m above the bot-
tom. Based on these measurements we set h
b
=1×10
1
m.
[
33] Considering the climatological CIL limits from our
observations (Figure 5a) and assuming that the depth
spanned by the CIL is uniform across the section, the
fraction of the CIL area within the bottom boundary layer
varies between 23% between April and October (the CIL
Figure 11. Comparis on between observed and modeled
CIL erosion rates: (a) core temperature, (b) thickness and
(c) heat content. 95% confidence interval of the observations
(19932010) are presented, together with the linear best fits
of the monthly means from the model when forced respec-
tively with K
i
(dashed lines) and K
a
(solid lines). All slopes
are provided in Table 1.
Table 1. Slopes of Linear Best Fits for Figure 11
a
_
T
min
(°C mo
1
)
_
d
(m mo
1
)
_
H
(MJ m
3
mo
1
)
Climatology of observations 0.24 11 0.59
Model, K
i
0.16 7 0.46
Model, K
a
0.22 12 0.63
a
The three columns are respectively the CIL co re temperature warming
rate, the CIL thinning rate and the rate of increase in CIL hea t content.
Each is calculated for the climat ology of CTD observations (19932010),
the modeled temperature dif fusion using respectively the interior
diffusivity profile K
i
and the the mean diffusivity profile from all available
casts K
a
.
Figure 12. Details of the geometric scaling used to calcu-
late the apparent eddy diffusivity (equation (5)). (top)
Bathymetric section of the estuary at Rimouski station (see
Figure 1, red line). (bottom) Enlargement of the rectangle in
Figure 12 (top). For both figures, x is the distance from the
South shore. The dashed line corresponds to the upper limit
of the bottom boundary layer of height h
b
(not to scale for
emphasis). A(z) is the channel width at depth z and A
b
(z, h
b
)
is the width of A(z) within the boundary layer.
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
9of12
has disappeared by November according to our climatol-
ogy). In other words,
A
b
z; h
b
ðÞ
AzðÞ

¼
1
z
t
z
b
ðÞ
Z
z
b
z
t
A
b
z; 10 mðÞ
AzðÞ
dz 3%; ð6Þ
where z
t
and z
b
are the climatological CIL limits and the
overline indicates an average in time between April and
October.
[
34] We extracted from our measurements the mean
diffusivity that is within the bottom boundary layer (h
b
=
1×10
1
m) using casts that reached the bottom. This
yielded K
b
= 3.3(2.1, 4.8) × 10
4
m
2
s
1
. Letting
~
K
i
=
K
i
= 2.4(1.5, 3.5) × 10
5
m
2
s
1
, i.e. the measured
interior diffusivity (Section 4.2), equation (5) gives K
e
=
3.3 × 10
5
m
2
s
1
. This value is slightly lower, but within
confidence intervals, than
K
a
, i.e., the depthaveraged
diffusivity of all profiles collected across the section
(Section 4.2) . We note also that both terms in equation (5)
are significant in the mixing budget, but their contribution
to the effective diffusivity is not the same. Our scaling sug-
gests that the interior mixing supplies 70% of the effective
diffusivity while the boundary mixing supplies the remaining
30%. We conclude from this analysis that boundary mixing
may contribute to the CIL erosion observed at midchannel
but that its role is not dominant in front of interior mixing.
[
35] To examine whether this analysis is relevant to a
broader geographical context, we calculated the fraction of
the volume of the CIL found within the bottom boundary
layer throughout the Gulf of St. Lawrence. This calculation
was done using historical CIL data for the months of August
and September between 1995 and 2010 (see Galbraith et al.
[2011] and Tamdrari et al. [2011] for details) and assuming
a10m thick bottom boundary layer throughout the Gulf.
The result suggests that, on average, 6 ± 1% of the Gulfs
CIL is within the bottom boundary layer. This fraction is
about twice as large as the value inferred from our sampling
section. This suggests that boundary mixing may be more
important throughout the Gulf than at our sampling section
off Rimouski. This possibility is highlighted when exam-
ining the area throughout the Gulf where the CIL intersects
the sloping bottom (Figure 14). On average during this
Figure 13. All dissipation rate profiles (150) that hit the
bottom (gray curves) and their mean (thick black curve).
Data are presented relative to height above bottom (hab).
The gray shaded area highlights the visuallyinferred 10
m thick bottom boundary layer.
Figure 14. Occurrence of the CIL that reaches the seabed (1995 to 2010). The white regions are where
the CIL never reached the seabed.
CYR ET AL.: INTERIOR VERSUS BOUNDARY MIXING OF A CIL C12029C12029
10 of 12
period, the CIL is in contact with approximately 30% of the
Gulfs seabed. Regions where the CIL reached the seabed
more often are located in the Southern Gulf (Bradelle and
Orphan Banks), Northeastern Gulf and the bank off of the
western tip of Anticosti Island. These regions may therefore
contribute substantially to CIL erosion.
7. Conclusion
[36] Two goals were pursued in this study: 1) to test a
model of vertical diffusion for the erosion of a CIL and; 2) to
discuss the importance of boundary versus interior mixing on
this erosion. The seasonal erosion of the core temperature,
thickness and heat content of the Gulf of St. Lawrence CIL
off Rimouski are, respectively,
_
T
min
= 0.24 ± 0.04°C mo
1
,
_
d
min
= 11 ± 2 m mo
1
and
_
H = 0.59 ± 0.09 MJ m
3
mo
1
.
These rates are remarkably well reproduced with a one
dimensional diffusion model fed with observed turbulent
diffusivities. This suggests that the CIL is principally eroded
by vertical diffusion processes. Our analysis further suggests
that interior mixing processes contribute to approximately
70% to this erosion with the remaining being attributed to
boundary mixing. This conclusion supports recent studies
[e.g., MacKinnon and Gregg, 2003; Rippeth et al., 2005;
Palmer et al., 2008] that also proposed that boundary
mixing may not be the predominant mixing source in
coastal seas. However, it would be relevant to extend this
study to other regions of the Gulf where boundary mixing
may be more important.
[
37] It is unclear at this point what mixing mechanisms
operate either in the interior or at boundaries. Given that our
analysis is based on observations collected during calm
wind conditions leads us to believe that wind plays a sec-
ondary role in eroding the CIL. Internal shear associated
with the internal tide that can cause up to 20 m isopycnal
displacement is the most likely candidate for producing
turbulence in the interior. Near sloping boundaries where the
CIL intersects the bottom, internal wave breaking and bot-
tom shear stresses are likely at work. It is not possible to
conclude at this point on the modulation of the mixing with
the M
2
and neap/spring tide cycles. Preliminary results (not
presented) however suggest that while the neap/spring cycle
has no incidence on the interior mean diffusivity, it can
modulate the mean diffusivity at boundaries by up to a
factor of two. However, new field experiments are required
to make further progress in those directions.
[
38] Acknowledgments. This work was funded by Le Fonds de
recherche du Québec Nature et technologies, the Natural Sciences and
Engineering Research Council of Canada, the Canada Foundation for Inno-
vation and Fisheries and Oceans Canada and is a contribution to the scien-
tific program of QuébecOcéan. The authors would also like to thank Pierre
Joly and his collaborators for their sampling effort since 1993, now part of
the Atlantic Zonal Monitoring Program, as well as the technicians and stu-
dents who helped in our 2009 and 2010 summer campaigns: Bruno Cayou-
ette, Rémi Desmarais, Gilles Desmeules, Sylvain Leblanc, Camil Hamel,
Joachim Bobinet and Guillaume Turbide. Thanks also to Richard Dewey
for sharing his code for calculating dissipation rates and to Barry Ruddick
for his helpful comments.
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