JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. C9, 3124, doi:10.1029/2001JC001177, 2002
Application of an internal tide generation model to baroclinic
spring-neap cycles
T. Gerkema
Netherlands Institute for Sea Research, Texel, Netherlands
Received 11 October 2001; revised 28 December 2001; accepted 10 January 2002; published 18 September 2002.
[1] A numerical internal tide generation model is used to study the behavior of baroclinic
spring-neap cycles in open basins and channels. The motivation for this study comes
from an observation on tidal currents in Faeroe-Shetland Channel, which is briefly
described; a surprisingly large baroclinic semidiurnal tidal signal is found in the lower
part of the water column above the slope on the Shetland side, concurring with barotropic
neap tides. The numerical model results indeed show that the baroclinic spring-neap cycle
may have a phase shift with respect to the barotropic spring-neap cycle and that the
phase of the baroclinic cycle may vary strongly within short distances. It is also shown
that even small changes in the background conditions (e.g., stratification) can provoke a
large phase shift in the baroclinic cycle; a possible connection to ‘‘intermittency’’ is
INDEX TERMS: 4544 Oceanography: Physical: Internal and inertial waves; 4255
discussed.
Oceanography: General: Numerical modeling; 4219 Oceanography: General: Continental shelf processes;
KEYWORDS: internal tides, spring-neap cycles, intermittency
Citation: Gerkema, T., Application of an internal tide generation model to baroclinic spring-neap cycles, J. Geophys. Res., 107(C9),
3124, doi:10.1029/2001JC001177, 2002.
1. Introduction
[2] In past decades it has been borne out by observations
that internal tides can manifest themselves as beams in
which the energy propagates in both the horizontal and the
vertical direction [DeWitt et al., 1986; Pingree and New,
1989, 1991]. The angle of propagation a (with respect to
the vertical) depends on the frequency s as
tan2 a ¼
N 2 s2
;
s2 f 2
ð1Þ
where N is the buoyancy frequency and f is the Coriolis
parameter [LeBlond and Mysak, 1978]. For N > f the
right-hand side is a monotonically decreasing function of
s. Hence a solar semidiurnal (S2) tidal beam propagates at
a slightly steeper angle than does a lunar semidiurnal (M2)
tidal beam; the beams thus follow different paths. This fact
has not received much attention in the literature; it is the
goal of this paper to explore some of its consequences.
[3] First of all, it may explain why in some regions a
baroclinic S2 signal can be found that is stronger than the
baroclinic M2 signal even though the forcing (by the
barotropic tide) of the latter is stronger; this was observed,
for example, by Gould and McKee [1973] and Wang et al.
[1991]. A further consequence is that the cophase lines of
M2 do not coincide with those of S2, a fact that provides the
key to the interpretation of the results presented in this
paper. As will be derived below (section 3, equation (5)),
Copyright 2002 by the American Geophysical Union.
0148-0227/02/2001JC001177$09.00
the phase of the baroclinic spring-neap cycle depends on the
phase difference between the M2 and S2 baroclinic signals.
Therefore, because of the angle between the respective
cophase lines, the moment of spring tides in the baroclinic
signal will vary throughout the basin. So, in general, this
signal will not be synchronous with the barotropic springneap cycle.
[4] In recent acoustic Doppler current profiler (ADCP)
observations from Faeroe-Shetland Channel (presented in
section 2) it was found that close to the bottom, a strong
baroclinic semidiurnal signal coincides with barotropic
neap tides (first quarter). The time series, however, covers
only one spring-neap cycle and is therefore too short to
warrant the conclusion that the strong baroclinic semidiurnal signal can indeed be interpreted as an occurrence
of baroclinic spring tides (rather than alternative candidates, like an interaction between inertial and semidiurnal
baroclinic signals). The goal of this paper therefore is to
explore, by theoretical means, the possibility of such an
interpretation.
[5] The rest of this paper is organized as follows. In
section 3 a simple internal tide generation model is presented, which will be used to analyze the baroclinic springneap cycle in a half-open basin (Bay of Biscay, section 4.1)
and a channel-shaped basin (Faeroe-Shetland Channel,
section 4.2). We emphasize that the goal of this paper is
to shed light on the behavior of the baroclinic spring-neap
cycle in a general and qualitative way, rather than to attempt
to reproduce observations in much detail. In fact, it will be
argued (section 5) that in certain regions it may be practically impossible to model the baroclinic spring-neap cycle
faithfully because of the ever present ‘‘noise’’ in stratifica-
7-1
7-2
GERKEMA: BAROCLINIC SPRING-NEAP CYCLES
Figure 1. The (a) along-slope and (b) cross-slope current (total signal, i.e., barotropic plus baroclinic) at
19 different depths; the scale on the left-hand side Figures 1a and 1b is indicative of the magnitude of the
current (in cm s1).
tion and background currents to which the baroclinic springneap cycle may respond sensitively, giving the appearance
of intermittency.
described earlier elsewhere in the literature (see the discussion in section 5). In the rest of the paper this issue will
be explored from a theoretical point of view.
2. An Example From Faeroe-Shetland Channel
3. Model Description
[6] As part of the project Processes on the Continental
Slope (PROCS) at Netherlands Institute for Sea Research
(NIOZ), a cruise was made to Faeroe-Shetland Channel in
spring 1999 [van Haren and van Raaphorst, 1999]. On the
Shetland side slope (at position 60.8667N, 3.0834W; local
depth 605 m) a long-range ADCP mooring was deployed,
covering the vertical range from 15 m above the bottom to
94 m below the surface (divided in 128 bins of 4 m),
sampling at a resolution of 300 s, during 13.2 days. Data
from the upper 140 m proved largely worthless because of a
lack of scatterers and have been removed. A presentation
and discussion of the data was given by van Veldhoven
[2000]; here we restrict ourselves to one salient feature in it.
[7] Figure 1 shows the total (i.e., barotropic plus baroclinic) current speeds, decomposed into an along-slope
component and an across-slope component (positive values
indicate a flow directed northeast and southeast, respectively). In both the semidiurnal tide is clearly present. Large
semidiurnal oscillations occur over the whole column at the
beginning and end of the period. This coincides with new
moon (day 106) and full moon (day 120), as one would
expect. However, in the lower 100 m of the water column,
semidiurnal oscillations of the same order (or even larger)
occur around first quarter (i.e., day 112), when one would
expect neap tides (and hence a small signal). Being
restricted to the lower part of the column, this large signal
is evidently baroclinic. In other words, we must conclude
that in these observations, relatively strong baroclinic tidal
currents occur at a time when the barotropic tide is weak
(barotropic neap tides). Phenomena like this have been
[8] In the internal tide generation model used here we
assume along-slope uniformity (i.e., @/@y = 0); we can
therefore introduce a stream function y (expressing the
baroclinic cross-slope and vertical current speeds as u =
@y/@z and w = @y/@x). The linear hydrostatic equations
then become [Gerkema, 2001]
@3y
@v @r
¼ 0;
f
@z2 @t
@z @x
ð2Þ
@v
@y
þf
¼ 0;
@t
@z
ð3Þ
@r
@y
zN 2 Q sin st dh
þ N2
¼
:
@t
@x
½ H hð xÞ2 dx
ð4Þ
Here f is the Coriolis parameter, v is the transverse velocity
component, and r is the density perturbation with respect to
its local static value (multiplied by g/r*, for convenience,
where g is the acceleration due to gravity and r* is the mean
density). The right-hand side of equation (4) represents the
forcing due to the barotropic tidal flow over the topography,
in which Q is the amplitude of the cross-slope barotropic
flux and s is the semidiurnal tidal frequency (either M2 or
S2); the bottom lies at z = H + h(x), where H is the
undisturbed ocean depth; and the upper surface (rigid lid)
lies at z = 0.
[9] In reality the amplitude of the cross-slope barotropic
flux Q would decrease oceanward on the scale of the
(barotropic) Rossby radius of deformation, as well as
7-3
GERKEMA: BAROCLINIC SPRING-NEAP CYCLES
coastward on the scale of the shelf width (see, e.g., the
discussion by LeCann [1990, sect. 4.3]); however, since
both scales are large compared to the width of the shelf
break region, where the main generation of internal tides
takes place, the assumption of a constant Q can be
considered justifiable.
[10] The equations are solved numerically for given
stratification N(z) and topography h(x) (as well as given
values for the constant parameters defined above: f, H, Q,
and s). First, we transform the x, z domain to a rectangular
shape, via the transformation h = 1 + 2z/[H h(x)], where
h denotes the new vertical coordinate. This allows the
application of a pseudospectral method in the vertical, here
involving Chebyshew polynomials. For the horizontal and
time derivatives we use finite difference methods (centered
differences in x and centered differences or third-order
Adams-Bashforth in time; see [Durran, 1999]). At the
outer ends of the domain, sponge layers are used to absorb
the incoming waves. Separate runs are performed for the
M2 case (s = sm) and the S2 case (s = ss). The fluid is
initially at rest; the transients will have left the region of
interest after a certain number of tidal periods (40, say).
The variables, for instance, um (the M2 cross-slope baroclinic component), then will have become periodic in
time:
um ðt; x; zÞ ¼ Am ð x; zÞ sin½sm t þ fm ð x; zÞ;
in which the amplitude Am and phase fm vary in space. An
entirely similar result is obtained for the S2 run, yielding us
(replacing subscripts m by s). Afterward, the two results can
be combined. A particularly convenient way to express the
combined effect of M2 and S2 tides was described by
Simpson et al. [1990]; thus we use the following identity:
u ¼ Am cosðsm t þ fm Þ þ As cosðss t þ fs Þ
¼ Aðt Þ cos½sm t þ fðtÞ;
where
A2 ðtÞ ¼ A2m þ A2s þ 2Am As cos cðt Þ;
tan fðtÞ ¼
Am sin fm þ As sin½fm þ cðt Þ
;
Am cos fm þ As cos½fm þ cðtÞ
cðt Þ ¼ ðss sm Þt þ fs fm :
Spring tides (i.e., A takes its maximum: A(t) = Am + As)
occur when c(t) = 0, and neap tides (i.e., A takes its
minimum: A(t) = jAm Asj) occur when c(t) = p. Hence the
time at which baroclinic spring tides occur is given by
f fs
t* ¼ m
;
ss sm
ð5Þ
which varies spatially via fm(x, z) and fs(x, z).
4. Model Results
[11] In the first example we show results for conditions in
the Bay of Biscay (in terms of stratification and topography). We start with this case because the internal tide beams
travel only away from the continental slope and do not
return; hence the result is fairly transparent, and the beams
can be easily identified. In the second case the conditions
correspond to those in Faeroe-Shetland Channel. Here the
beams reflect several times at the side slopes of the channel
before they escape onto the shelves, and this gives rise to a
much less transparent pattern.
[12] Notice that an a priori estimate can be made of the
difference in steepness between the M2 and S2 beams.
Under the hydrostatic approximation the numerator on the
right-hand side of equation (1) reduces to N 2; hence M2 and
S2 beams traveling over the same horizontal distance travel
over vertical distances (hm, hs) that are related as
hm
¼
hs
2
1=2
sm f 2
:
s2s f 2
For the Bay of Biscay this gives (see values given below)
hm/hs = 0.92, and for the Faeroe-Shetland Channel (where f
is larger) this gives hm/hs = 0.84.
4.1. Open Basin (Biscay)
[13] The stratification N(z) is shown in Figure 2a; it
involves a seasonal and permanent pycnocline and is
representative of the summer conditions in the Bay of
Biscay; the profile and topography are derived from Pingree
and New [1991]. The barotropic tidal cross-slope flux is
taken to be Q = 100 m2 s1, corresponding with current
speeds of about 0.02 m s1 in the deep ocean (comparable
to values given by Pingree and New [1991, Table 1]). The
M2 and S2 tidal frequencies are sm = 1.405 104 s1 and
ss = 1.454 104 s1, and the Coriolis parameter at that
latitude is f = 1.07 104 s1. For these frequencies the
slope (see Figure 3) has two distinct supercritical regions,
namely, one between 8.4 (8.8) and 25.6 (22.8) km and the
other between 29.2 and 30.8 km (the values between
parentheses refer to the values for the S2 beam, when
different from those for the M2 beam).
[14] In the runs, 64 Chebyshew polynomials were used,
and the steps in x and t (centered differences) were 400 m
and 120 s. The results of the separate runs for M2 and S2
can be combined afterward, following the procedure outlined in section 3. In accordance with data from Table 1 by
Pingree and New [1991] we chose the M2 forcing to be 1.6
times stronger than the S2 forcing.
[15] Figure 3 shows the results: the total amplitude Am +
As (the maximum of amplitude A(t)), the relative strength of
M2, and the moment at which spring tides occur in the
baroclinic signal. All these quantities vary strongly spatially.
In Figure 3a one can identify two beams that emanate from
the upper parts of the slope; one of them travels directly
downward into the basin, and the other one travels first
upward to the surface. The former is the beam Pingree and
New [1991] focused on; their mooring 116, which was
located in the region where the reflection at the bottom takes
place, corresponds to x = 70 km in Figure 3. At this position
we find a maximum current speed (near the bottom) of
about 0.14 m s1, which agrees fairly well with the sum of
the observed baroclinic M2 (10.4 cm s1) and S2 (1.7 cm
s1) contributions [Pingree and New, 1991, Table 1]. At
higher positions at x = 70 km in Figure 3a the total
amplitude Am + As first decreases but then takes again larger
7-4
GERKEMA: BAROCLINIC SPRING-NEAP CYCLES
Figure 2. Stratification profiles used in the numerical calculations: (a) the profile in the Bay of Biscay
[after Pingree and New, 1991, Table 1] and (b) a profile (solid line) derived from sections in FaeroeShetland Channel [after van Haren and van Raaphorst, 1999]; the dashed line shows a hypothetically
modified profile.
Figure 3. A numerical calculation of the combined effect of M2 and S2 internal tidal beams in the Bay
of Biscay: (a) Am + As, i.e., the amplitude of the baroclinic cross-slope current speed at spring tides, (b)
the the relative importance of the components, (Am As)/(Am + As), and (c) the spatial distribution of the
time (in days) at which spring tides occur in the baroclinic signal. Figure 3c is cyclic: white (day 0)
denotes the same phase as black (day 14.84).
GERKEMA: BAROCLINIC SPRING-NEAP CYCLES
7-5
Figure 4. A numerical calculation of the combined effect of M2 and S2 internal tidal beams in FaeroeShetland Channel: (a) Am + As, i.e., the amplitude of the baroclinic cross-slope current speed at spring
tides, (b) the the relative importance of the components, (Am As)/(Am + As), and (c) the spatial
distribution of the time (in days) at which spring tides occur in the baroclinic signal. Figure 4c is cyclic:
white (day 0) denotes the same phase as black (day 14.84).
values at depths between 2000 and 500 m; this is in
qualitative agreement with the observations.
[16] Figure 3b shows the relative importance of the two
baroclinic semidiurnal components; values can range from
1 (complete S2 dominance) to +1 (complete M2 dominance). Overall, the M2 component is stronger (not surprisingly, in view of the fact that the M2 forcing is 1.6 times
stronger). However, in some regions the S2 signal becomes
equally (or even more) important, for instance, near the
bottom around 60 km, indicating that the S2 beam reflects
before the M2 beam does (i.e., more to the left); this is in
accordance with the fact (mentioned in section 1) that the S2
beam travels at a steeper angle.
[17] Finally, Figure 3c shows the moment at which spring
tides occur in the baroclinic signal; this moment represents
the shift with respect to the barotropic spring tides, which
occur at day 0 (white) and, one cycle later, at day 14.84
(black). Particularly interesting is the region near x = 68 km
at about 3 km depth, where the phase of the spring-neap
cycle varies strongly in the vertical; however, this would
probably not be very noticeable in a real signal since in this
region the total amplitude is small (see Figure 3a). Near x =
85 km, where the total amplitude takes fairly significant
values throughout the water column, we find phase differences of some 4 days over the vertical. In the main area of
reflection (near the bottom around x = 70 km) the baroclinic
spring-neap cycle lags the barotropic cycle by about 2 days.
4.2. Channel-Shaped Basin (Faeroe-Shetland Channel)
[18] For the stratification we use the cross-channel average
of observed vertical profiles from Faeroe-Shetland Channel
[van Haren and van Raaphorst, 1999] (see Figure 2b (solid
line)); in the upper and lower part of the column, N has simply
been taken as constant since the data show little variation.
The Coriolis parameter is now f = 1.28 104 s1. The
(total) barotropic tidal cross-slope flux is taken to be Q =
60 m2 s1, and the ratio of M2 and S2 forcing strength is
2.0 (these values are based on the observed cross-slope
component, shown in Figure 1, from which the depthaveraged signal was derived and analyzed). The uniformity
in the along-channel direction, assumed in section 3,
means that we ignore possible effects of an along-channel
propagating internal tide generated at the Wyville-Thomson Ridge (observations by Sherwin [1991] pointed to
significant generation at the ridge).
[19] In the calculations, 50 Chebyshew polynomials were
used in the vertical, and horizontal and time steps were
500 m and 70 s (here a third-order Adams-Bashforth
scheme was used for the time integration). A simple
Rayleigh friction term was added, with a damping timescale
of 15 periods (i.e., about 1 week); this weak frictional term
suffices to mitigate the growth near critical regions at the
slope, where the signal would otherwise grow linearly with
time (in accordance with the theoretical analysis by Dauxois
and Young [1999]).
[20] The results from the numerical calculation (obtained
after 100 tidal periods, when the transients have left the
domain) are shown in Figure 4. Relatively strong currents
are found near the upper part of the slope (on both sides of
the channel), between 300 and 600 m depth (see Figure 4)a;
this is in qualitative agreement with observations by van
Raaphorst et al. [2001]. In particular, at x = 215 km
7-6
GERKEMA: BAROCLINIC SPRING-NEAP CYCLES
Figure 5. The shift in the baroclinic spring-neap cycle (in days) due to the change of stratification
shown in Figure 2b (solid line versus dashed line). The plot is cyclic: white and black both denote a zero
shift (or nearly so); medium shading indicates a shift of about 1 week.
(corresponding to the observational site; section 2) the
largest values, between 0.2 and 0.3 m s1, occur (only)
within the lowest 100 m of the water column. This is in
agreement with the observations shown in Figure 1, where
between days 112 and 114 (the reason for selecting these
days is given below when we discuss the phase shifts in the
baroclinic spring-neap cycle) the amplitude in the crossslope component reaches maximum values of about 0.2–
0.25 m s1 in the lowest part of the water column.
[21] The paths along which the internal tides propagate
can be identified but form a somewhat blurred pattern
because the beams traverse the channel several times before
they escape onto the shelves, thus giving rise to a complicated ‘‘web.’’ In these (and other similar) model runs, no
indications are found that closed paths exist, i.e., ‘‘attractors’’ in the sense of Maas and Lam [1995] and Maas et al.
[1997]. A necessary condition for the occurrence of these
attractors is that supercritical regions are present at both sides
of the basin; this condition is satisfied here since four such
regions exist: on one side between 87.0 and 98.5 km and
around 104 km for M2 (for S2: between 87.5 and 92.0 km
and between 94.5 and 98.5 km) and on the other side
between 226.5 and 231.0 km and between 232.0 and 237.0
km (for S2: between 227.0 and 230.5 km and between 232.5
and 236.5 km). Nevertheless, no attractors were found; this
can be ascribed to the fact that the basin is not entirely
closed, so that the beams can (and in fact do) escape onto the
shelves. It should be noted, however, that the structure of the
web depends on the wave frequency and that for internal
waves of a different frequency (i.e., not semidiurnal), but for
the same stratification and topography, attractors may exist.
[22] Figure 4b shows that M2 is dominant in the baroclinic signal, except at a few distinct regions. Figure 4c
shows the day at which spring tides occur in the baroclinic
signal; it varies strongly spatially. However, the regions
where the variation is strongest more or less coincide with
those in which the amplitude (Figure 4a) is small, the same
phenomenon as we saw in section 4.1. Nevertheless, a
significant variation is present; in particular, on the slope
on the right-hand side in Figure 4c (i.e., the Shetland side),
at x = 215 km (the observational site), we find that in the
lowest 200 m baroclinic spring tides lag the barotropic
spring tides by 4 – 7 days. This is in agreement with the
observations shown in Figure 1, where in the lowest part of
the water column strong semidiurnal baroclinic signals
occur some 6 days after barotropic spring tides. (However,
no excessive importance should be attached to this agreement, in view of the points discussed in section 5.)
[23] As we saw above (equation (5)), the moment at which
baroclinic spring tides occur depends spatially on the local
difference in phase between M2 and S2. These phases, in
turn, depend on the trajectories that the beams follow, and
hence on background conditions like stratification N. Thus
one would expect that even a minor variation (in time) in, for
example, N, which changes the paths of the beams, may at
some positions give rise to a drastic shift in the baroclinic
spring-neap cycle. To verify this idea, we carried out a calculation identical to the one that led to Figure 4, except that N
was slightly different, being now given by the dashed line in
Figure 2b. The outcome looks very much like Figure 4 (and
is therefore not shown), but interesting details are revealed if
we plot the local shift in the baroclinic spring-neap cycle
GERKEMA: BAROCLINIC SPRING-NEAP CYCLES
7-7
(see Figure 5). At most positions the (small) change in
stratification gives no significant shift in the cycle, but at
some positions a shift as large as a week is found (notice that
this happens in the regions where the spatial variation shown
in Figure 4c is strong). Clearly, at such positions one would
never expect to find a consistent spring-neap cycle since in
nature, small variations in background conditions (noise) are
always present.
can in principle be described by the simple linear model of
section 3, if the prescribed quantities N (stratification) and Q
(barotropic flux) are subject to noise.
5. Conclusion and Discussion
References
[24] With regard to the behavior of baroclinic springneap cycles, three general conclusions can be drawn from
the numerical results presented in section 4. First, the phase
of this cycle may vary strongly spatially; for example, one
may find baroclinic spring tides at some position and
simultaneously neap tides at a nearby position. Second,
precisely in those regions where such a strong spatial
variability exists, the phase of the baroclinic spring-neap
cycle becomes sensitive to small variations in background
conditions, like the stratification N. Third, this sensitivity to
background conditions, which are always only imperfectly
known from observations, makes it unlikely that the baroclinic spring-neap cycle can be modeled faithfully in such
regions.
[25] These conclusions suggest a link with the occurrence
of ‘‘intermittency.’’ In past decades, abundant evidence has
been gathered of intermittent behavior in internal-tide signals; they ‘‘come and go’’ [Wunsch, 1975]. The classic
example is the observation by Magaard and McKee [1973].
Importantly, as their current meter data show, the intermittency manifests itself not in the semidiurnal signal as such
(this signal is persistent and identifiable throughout the time
series) but rather in its envelope (i.e., at a timescale of
several days to a week), in other words, in the absence of a
persistent spring-neap cycle. Similarly, in current measurements on the northwest African continental slope [Huthnance and Baines, 1982], no persistent spring-neap cycles
can be seen in the current data, although both M2 and S2 are
prominently present in the spectra. At a more southern
position on the same slope [Schott, 1977, Figure 6], current
measurements at deep positions follow the barotropic
spring-neap cycle, while at higher positions the cycle (if
present at all) seems shifted in time. Siedler and Paul
[1991] found from current measurements in the eastern
North Atlantic that the envelope of the semidiurnal baroclinic signal may have timescales about twice as short as a
spring-neap cycle. On the Malin Shelf, horizontal incongruities are found [Sherwin, 1988]; no clear spring-neap
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[26] So, these observations have in common that the
intermittency manifests itself predominantly as a lack of a
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to irregularities in the cross-slope currents due to wind and
upwelling [Sandstrom, 1991]. Either way, the background
conditions, and hence the paths of the internal tide beams,
will vary, perhaps only slightly, but this may cause the
phase of the baroclinic spring-neap cycle to wander all the
time. The results of this paper suggest that this phenomenon
[27] Acknowledgments. This work was carried out within the NIOZ
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T. Gerkema, Netherlands Institute for Sea Research, PO Box 59, 1790
AB Den Burg, Texel, Netherlands. ([email protected])