Supplemental notes to Tape (2001):
Oceanic tides: A response to differential force
throughout the earth’s history
Author’s note
I would appreciate advice for improvements or any corrections.
Carl Tape
[email protected]
Last modified: July 27, 2008
1
Table of Contents
List of Figures
2
List of Tables
2
1 Additional tides references
3
2 Cross-bedding: A record of sand dune migration
4
3 Tidal harmonics and tidal sedimentation
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Previous work in the Upper Mississippi Valley . . . . . . . . . . . . . . . . . . . .
3.3 Tides background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Background on tidal sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Tidal harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Kwajelein Island water level data (modern), compared with the Cambrian Jordan
sandstone data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References
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List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
The relationship between dune migration and cross-bedding . . . . . . . . . . . .
Modern eolian dunes (Coral Pink Sand Dunes State Park, Utah) . . . . . . . . .
The source of the daily and monthly tidal cycle . . . . . . . . . . . . . . . . . . .
Sand and mud transport and deposition during one ebb-flood cycle of a strongly
asymmetrical tide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of dune migration during one ebb-dominated ebb-flood cycle . . . . .
Tidal-bundle sequences and the neap-spring-neap tidal cycle . . . . . . . . . . . .
Source of the semiannual tidal cycle . . . . . . . . . . . . . . . . . . . . . . . . .
The semiannual tidal cycle (equinoctial tides) . . . . . . . . . . . . . . . . . . . .
Tidal spectrum for modern tidal data (Kwajelein Island) . . . . . . . . . . . . . .
Tidal spectrum for Jordan sandstone foreset thickness data . . . . . . . . . . . .
Zoom-ins of tidal spectrum for modern tidal data (Kwajelein Island) . . . . . . .
Comparison of frequency spectra from a modern semidiurnal tidal system with
spectra from the Jordan sandstone . . . . . . . . . . . . . . . . . . . . . . . . . .
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List of Tables
1
2
3
4
5
6
7
Fundamental periods of the earth’s and the moon’s orbital motion
Fundamental periods of the earth’s rotational motion . . . . . . . .
Tidal harmonics of the earth’s oceans . . . . . . . . . . . . . . . .
Continuation of Table 3 . . . . . . . . . . . . . . . . . . . . . . . .
Tidal bundles per sequence for a Cambrian tidal system . . . . . .
Different types of months . . . . . . . . . . . . . . . . . . . . . . .
Tidal cycles in the rock record versus modern measurements . . . .
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1
Additional tides references
Wells et al. (2005) presented results from a numerical model of tides in vast, shallow seas. It
would be nice to apply this model to the Carboniferous or Cambrian North America setting, for
where there are documented tides in the geological record (e.g., Kvale et al., 1999; Archer , 1998;
Tape et al., 2003).
3
2
Cross-bedding: A record of sand dune migration
Cross-bedding can be observed in sedimentary rocks in the rock record or in cross-sections of
sand pits where modern sand deposition is occurring. Cross-bedding is a record of the migration
of dunes under the influence of a flowing fluid. If the fluid is air, then the process is called eolian
or subaerial; if the fluid is water, then the process is called marine or subaqueous. In either case,
the basic process of cross-bedding formation is the same and is shown in Figure 1A. We see
from this illustration that the paleohorizontal is given by the bounding surfaces of the cross-sets;
at the scale of 100 m, or with a slightly tilted base level, the planes in Figure 1A will appear
horizontal. Figure 1B shows a slightly more complicated (and realistic) version, which results in
trough cross-bedding in any plane oblique to the primary flow direction. The properties of the
fluid and granular medium are primary controls on the geometry of the dunes (we’ll assume the
velocity of the fluid is constant).
Here we note a couple key terms; others are listed in Figure 2A. A foreset is a lamina of
sediment that is deposited on the slipface of a dune, either via fallout from the air (or water) or
via avalanching, after the slipface at the crest exceeds the angle of repose. In many cases, there is
a hierarchy of lamina. Cross-bedding refers to the sedimentary record left by migrating dunes. A
cross-bedding set or cross-set is the collection of foresets, bounded above and below by bounding
surfaces and interpreted to represent an uninterrupted time interval of a single dune’s migration.
The toesets or toes refer to the lower reaches of the foresets, which typically run tangential to
the lower bounding surface, giving a good indication of the paleohorizontal. It is important to
distinguish terms describing dunes and dune migration from terms describing cross-bedding.
Some nice examples of computer simulations of cross-bedding formation can be found at the
USGS bedform sedimentology site: http://walrus.wr.usgs.gov/seds/.
Figure 2 shows an example of a modern, slowly migrating, eolian sand dune. The sand has
been weathered from the nearby cliffs of the Lower Jurassic (ca 190 Ma) Navajo Sandstone, which
preserve spectacular examples of the migration of ancient eolian sand dunes. Thus, the modern
dunes serve as a nice analogue for the ancient ones. It is clear from Figure 2 that it is possible to
create very large eolian dunes from the Navajo sands. In fact, we know the ancient dunes reached
heights in excess of 20 m, the maximum observed height of the cross-bedding sets.1
1
It turns out that the foresets in the Navajo Sandstone may record evidence of ancient decade-scale climatic
cycles (Chan and Archer , 2000). Kocurek (2003) presents an interesting comparison between the Early Jurassic
Navajo Sand Sea and the modern Sahara.
4
A
B
Figure 1: The relationship between dune migration and cross-bedding, with arrow indicating
direction of flow (i.e., air or water). What is seen at the outcrop will clearly depend on the angle
of the outcrop face with respect to the dune’s migration. Figures from Reineck and Singh (1980).
A. Simplest scenario for dune migration. With large distances between dunes, the bounding
surfaces of the cross-bedding sets are approximately horizontal. B. A slightly more complicated
depiction. Compare with the modern eolian dune pictured in Figure 2.
5
e
sid
lee face)
p
(sli
PREVAILING
WIND
EOLIAN
DUNE
stoss
Navajo
Sandstone
side
A
A
e
ac
slip
f
B
Figure 2: Modern eolian dunes being formed at Coral Pink Sand Dunes State Park, not far from
the east entrance to Zion National Park in southwestern Utah. Here the ancient sands of the
Navajo Sandstone (ca 190 Ma) have eroded, and new dunes are being formed in the windy stretch
between two bluffs of sandstone. In essence, these sands are forming a second generation of eolian
dunes, separated by about 200 million years. A. Sketch with photo showing a typical profile of
a dune, which exhibits an asymmetry (lee side steeper than stoss side) due to a prevailing wind
direction. Note the cliffs of the Navajo Sandstone in the background. The height of this dune at
the crest is approximately 25 m. B. View from above, showing the crest of one of these dunes.
Note the circled person for scale. (This photo was taken by a camera attached to this person’s
kite, which was tied to a nearby tree.) Also note that plants, which stabilize the dune, were
probably only present during the wetter climatic periods during the time of the Navajo Sand Sea.
6
3
Tidal harmonics and tidal sedimentation
3.1
Introduction
The information here is a combination of notes and background relevant to our (Carl Tape, Clint
Cowan, and Tony Runkel) paper on the Jordan Sandstone, “Tidal-bundle sequences in the Jordan
Sandstone (Upper Cambrian), southeastern Minnesota, U.S.A.: Evidence for tides along inboard
shorelines of the Sauk epicontinental sea” (Tape et al., 2003). Some of the figures here are from
earlier versions of our article, some are modified from my physics comps, and several are new.
The main new information regards the data analysis of modern water level data, which is based
upon (predicted) tidal records at Kwajelein Island, an island in the western Pacific. The spectral
analysis is applied to the Jordan foreset thickness data, and the results are then compared.
3.2
Previous work in the Upper Mississippi Valley
Previous studies of Lower Paleozoic quartz arenites in Wisconsin and Minnesota have reported
sedimentary features that are generally considered indicative of peritidal conditions. These peritidal features include desiccation cracks, flaser bedding and shale drapes, herring-bone crossstratification, and reactivation surfaces (Driese et al., 1981; Dott Jr. et al., 1986; Runkel, 1994;
Byers and Dott Jr., 1995). Specifically, Driese et al. (1981) and Terwindt (1988) attributed the
consistent occurrence of shale drapes on pause planes in the Upper Cambrian Mt. Simon Formation in Wisconsin to tidal influence. Dott Jr. et al. (1986) identified flaser bedding in outcrops
of the Wonewoc Formation in Wisconsin and implicated tides as a plausible mechanism. Runkel
(1994) studied the upper part of the Jordan Sandstone at Homer, Minnesota, and recognized
shale drapes, reactivation surfaces, and reversely dipping ripple cross-strata, leading him tentatively to infer a tidal influence. His photograph of a cross-bedding set at Homer, Minnesota,
served as the inspiration for Tape et al. (2003). Byers and Dott Jr. (1995) reported reactivation
surfaces and shale drapes in the Jordan Sandstone, but noted that “no confirming tidal bundling
has been observed” (p. 295).
3.3
Tides background
Tidal cycles are due to astronomical variations (Archer , 1998). The earth’s monthly revolution
about the earth–moon center creates a differential force on the oceans. The ebb-flood cycle is due
to the earth’s 24-hour rotation under this differential force. The neap-spring-neap cycle is due to
the combined gravitational effects of the sun and the moon over the course of the ≈28-day lunar
cycle. The strongest tides, the spring tides, occur when the moon, earth, and sun are aligned
(syzygy); the weakest tides, the neap tides, occur when the moon, earth, and sun form a right
angle (quadrature).
The earth’s daily rotation generates two components of tides from both the moon and sun, the
semidiurnal tide component (modern period of about 12 hours) and the diurnal tide component
7
(modern period of about 24 hours) due to the moon and sun (Marchuk and Kagan, 1989). Either
component or a combination of the two may dominate a particular coastal area, so that the period
of the ebb-flood cycle may be approximately 12 hours (semidiurnal system) or approximately
24 hours (diurnal system) or something more irregular (mixed system) (Garrison, 1998). In
semidiurnal systems, one of the semidaily ebb-flood cycles is generally greater in magnitude than
the other, a feature known as the diurnal inequality (de Boer et al., 1989). This is due to the
nonzero declination of the moon, which tilts the bulges with respect to the earth’s equator.
In most tidal systems, either the ebb or the flood dominates sediment transport and deposition
in a particular area. A tidal bundle is the sediment deposited on the slipface of a migrating dune
during the dominant current stage of an ebb-flood tidal cycle (about 12 to 24 hours) (Boersma
and Terwindt, 1969; Visser , 1980). A tidal bundle sequence is the sediment deposited by a
migrating dune during one complete neap-spring-neap tidal cycle (about 14 days). The bundles
within a sequence, therefore, normally vary in thickness; relatively thin bundles are deposited
during neap tides, relatively thick bundles are deposited during spring tides.
Visser (1980) is perhaps the seminal study to describe the neap-spring-neap cycle in sediment.
Boersma and Terwindt (1981) give a detailed description of the sedimentary structures and
patterns associated with neap-spring-neap tidal bundle sequences. Our study is similar to several
studies that have used tidal bundle thickness measurements (i.e., thickening toward spring tides,
thinning toward neap tides) to identify neap-spring-neap cycles in sediment (e.g., Visser , 1980;
Boersma and Terwindt, 1981; Allen, 1981; Allen and Homewood, 1984; Richards, 1986; Driese,
1987; McKie, 1990; Singh and Singh, 1992; Eriksson and Simpson, 2000). The age of the tidally
deposited sediments in these studies ranges from Archean (3.2 Ga of Eriksson and Simpson
(2000)) to Recent (Visser , 1980).
3.4
Background on tidal sedimentation
The flow of water in and out of coastal areas is driven by the gravitational forces of the sun,
moon, and earth. The volume of sediment transported and deposited, V (t), is approximately
proportional to the cube of the velocity of the current (Allen, 1985):
V (t) = [U (t) − Usand ]3 ,
(1)
where U (t) is the speed of the current and Usand is the threshold velocity of sand movement
(Figure 4). Thus, in coastal areas dominated by tidal currents, we would expect the thickness
of the sedimentary layers to correlate to the changes in the driving gravitational forces, known
generically as “tidal forces”. (We use the term “tidal forces” to refer to the net effects of the of
forces due to the sun, moon, and solid earth on the ocean.)
Background information on tidal rhythmites and tidal bundles can be found in Williams
(2000); Kvale et al. (1999); Visser (1980) and references therein. In essence, tidal rhythmites are
deposited in quieter, offshore settings; the sedimentary record consists of thin (mm-thick), nearly
8
horizontal lamina. Tidal bundles are deposited by migrating dunes in generally intertidal or subtidal settings; the sedimentary record consists of “bundles” of mm- or cm-thick foresets, sometimes
bounded by shale laminae. Compared with bundles, rhythmites generally contain a much longer
and more continuous record of lamina thickness variation. See Tape (2001, Figure 33).
Tidal cycles are astronomical variations in tidal forces. The most familiar tidal cycle is the
ebb-flood cycle (Figure 3A), whereby a particular area experiences approximately one or two lowtide/high-tide/low-tide cycles per day. The ebb is the tidal current going in the direction away
from land; the flood is the current coming toward land. Figure 4 shows a schematic of the ebbflood cycle and of the predicted volume of sediment volume deposition. Figure 5A shows sediment
transport and deposition of a migrating dune during one-ebb flood cycle. Figure 5B shows a
slightly more complicated version, one that we envision for the formation of the sedimentary
structures observed in the Jordan Sandstone at Homer, Minnesota.
Another tidal cycle that many are familiar with is the neap-spring-neap tidal cycle (Figure 3B).
In most cases, the neap-spring-neap cycle is due to changes in the position of the moon with
respect to the sun, i.e., the phase of the moon. Figure 6 shows how the neap-spring-neap cycle
influences the sediment deposition of a migrating subaqueous dune. In essence, spring tides
correspond to thicker foresets and neap tides correspond to thinner foresets.
3.5
Tidal harmonics
The study of tidal harmonics involves analyzing the spectral content of the water height data from
coastal tide gage stations from around the world. The location of the peaks, i.e., the frequencies
(or periods), in a tidal spectrum should reflect the astronomically driven fluctuations in tidal
forces, while the amplitude (power) of the peak may be controlled by local affects, such as the
shape of a nearby basin, the coastline geomorphology, and the latitude of a particular station. A
compilation of tidal spectra from around the world should give a “global response” of the world’s
oceans, which can be presented in terms of tidal harmonics (Tables 1–4).
The values in the Table 1 are for orbital motion and have nothing to do with the rotational
motion of the earth, which is manifested in the values in Table 2. The tidal harmonics for the
earth’s oceans are combinations of the fundamental frequencies in Tables 1 and 2. The tables
listing the fundamental astronomical periods and the tidal harmonics (Tables 1–4) are compiled
from among the following sources: Bartels (1957); Platzman (1971); Marchuk and Kagan (1984).
Analyzing a frequency spectrum of a water-level data set from a tide-gage station with an appreciable tide reveals peaks — tidal harmonics – at beautifully predictable frequencies, assuming
the astronomical periods are known (Tables 1 and 2). The quality of the peaks will depend on
a number of factors, such as the time length of the data set, the amount of noise (e.g., storms,
missing data), and the time increment of sampling (e.g., hourly or daily). In most areas, it is
possible to make accurate (noise-free) predictions of the tides, since the astronomical forcing is
known (and essentially noise-free). Analyzing predicted tides will reveal the frequencies used to
generate the model predictions.
9
3.6
Kwajelein Island water level data (modern), compared with the Cambrian
Jordan sandstone data
Kvale’s reviews of our Jordan paper (Tape et al., 2003) prompted me to look into some of the
water-level data from modern tides. So I took the 1999–2001 data set of the Kwajelein Island2
from the NOAA website. I obtained two data sets: (1) the hourly measurements of the water
level and (2) the high tide events. The figures here are of the high tide events data only, since this
more analogous to the sedimentary layers (dominant current deposition only). We can convert
the high-tide events data to time, since we know the exact time interval of the data set (3 years
= 1096 days). The conversion factor C is simply given by
C =
number of high tide events in time t
number of days in time t
(2)
This value is used to give a quantitative estimate of the strength of the semidiurnal constituents
of the particular tidal system (Kvale et al., 1995). For predominantly diurnal systems, with one
high-tide event per day, C ≈ 1; for semidiurnal systems, with approximately two high-tide events
per day, C ≈ 2. For the Kwajelein Island data over a three-year period,
CKwaj =
2118 high tide events
= 1.92 ,
1096 days
(3)
indicating that the tidal system at Kwajelein Island is predominantly semidiurnal. It should
be noted that this quantitative measure C is different from the F -value measure of semidiurnal
character in Table 5.
With the rock record, we do not have the luxury of simply extracting an exact interval of
time; indeed determining the length of time over a particular interval of tidal bundles is in essence
the goal of the study. YOU NEED TO IDENTIFY MULTIPLE CYCLES TO DETERMINE
THE LENGTH OF TIME; IT IS NOT ENOUGH TO SAY IT IS SIMPLY TIDAL.
My main objective in using the modern water-level data was to check that our spectral analysis
program was working properly before examining the foreset-thickness data. With the modern
data, we know what to expect, and we should be able to identify the constituent peaks of the
tidal spectrum for the particular data set.
My first objective was to make sure that our spectral analysis program was able to identify
a diurnal inequality if there was one, whether using the modern data or the Jordan data. Thus,
I analyzed the Kwajelein data, which has known diurnal cycles. The diurnal spectra are indeed
present (Figure 11A), along with several other expected peaks. Thus, running the Kwajelein data,
we see the diurnal peaks (Figure 9); running our Jordan data, we do not (Figure 10). Secondly,
I looked at the approximate sediment volume data. I took the high-tide event data, took the
gradient,3 and then cubed the data, as suggested in Archer et al. (1995) (see also Figure 4).
2
I used Kwajelein Island, because it is analyzed in Kvale et al. (1995) and Kvale et al. (1999).
The high-tide event data measure the water level height, sampled only at the peaks. Taking the gradient of
this data is a measure of the rate of change of the water level, which is also a proxy for the velocity of the tidal
3
10
What I found is shown in Figure 11B.
Figure 11A shows various zoom-ins of the power spectrum for the high tide events at Kwajelein
Island, over the time interval 1999–2001. The tidal harmonic peaks are labeled as shown in
Table 3. The high-tide event data have been converted to time using the conversion factor C;
thus the periods are in days. Figure 11B shows various zoom-ins of the power spectrum for the
approximate sediment transport at Kwajelein Island (see text). Comparing with Figure 11A,
we see that the peaks are generally not as well defined. The yearly peak is not prominent,
although the semiyearly peak is relatively stronger (see upper left). The half-tropical month is
not distinguishable from the half-synodic month. Note the three diurnal peaks remain. Also it
should be noted that we did the same analysis on the hourly data set over the same time interval,
and the resulting spectra are similar.
Let us examine a particular frequency range—one that covers the monthly and semi-monthly
periods—and look at the spectra from (1) the high-tide events, (2) the approximate sediment
volume transport, and (3) the Jordan foreset thickness data (Figure 12). As stated in the caption
of Figure 12B, it appears that the additional peaks are “amplified” due to the interference of
the three dominant peaks, the half-synodic month (S), the anomalistic month (A), and the halftropical month (T). The interference gives rise to peaks at S+A, S+T, A+T, S-A (“evectional” cycle),
and T-A. All of the peaks are combinations of eight fundamental frequencies shown in Tables 1
and 2.
Figure 12B might represent a best-case scenario for what you’d get in the rock record deposition at Kwajelein Island. It shows that in a system with a strong semidiurnal signal, additional
peaks should be expected at certain multiples. The purpose of this figure is similar to the purpose
of Figure 5 of Kvale et al. (1995); both show how to predict the rock data spectrum from the
water data spectrum.
Additional References
Some additional tides references, not cited otherwise. See references in Tape et al. (2003) as well.
Ashley and symposium members (1990); Bunker et al. (1988); Dalrymple and Rhodes (1995);
Datta et al. (1999); Friedman and Sanders (1978); Hallam (1981); Hughes and Hesselbo (1997);
Lochman-Balk (1971); Runkel et al. (1999); Sloss (1980, 1988)
currents, assuming that the water level changes over the same time interval from one high tide event to the next.
11
Table 1: Fundamental periods of the earth’s and the moon’s orbital motion. The five angular
frequencies, ωi , are used in defining the tidal cycles in Tables 3 and 4. The perihelion is the solar
perigee.
Period, T
Angular frequency, ω
360◦ /T (deg/day)
(rad/s)
27.321 582 days
365.242 199 days
8.847 years
18.613 years
20 940 years
13.176 397
0.985 647
0.111 410
0.052 954
0.000 047 070
ω1 = 2.6617 × 10−6
ω2 = 1.9911 × 10−7
ω3 = 2.2505 × 10−8
ω4 = 1.0697 × 10−8
ω5 = 9.5084 × 10−12
Nomenclature
Period
Period
Period
Period
Period
of
of
of
of
of
lunar declination
solar declination
lunar perigee rotation
lunar node rotation
perihelion rotation
Table 2: Fundamental periods of the earth’s rotational motion. ω0 is the sidereal rotation
velocity and is defined as ω0 ≡ ωM + ω1 = ωS + ω2 .
Period, T (hours)
23.934
24
24.841
360◦ /T
Angular frequency, ω
(deg/hour)
(rad/s)
15.041 07
15
14.492 05
ω0 = 7.2921 × 10−5
ωS = 7.2722 × 10−5
ωM = 7.0259 × 10−5
12
Nomenclature
lunar-solar declinational
basic solar
basic lunar
Table 3: Tidal cycles, measured from the tidal spectrum of the earth’s oceans. The table is
divided into semidiurnal cycles (1–4), diurnal cycles (5–8), and long-period cycles (9–15). Periods
are calculated from the expressions listed in Table 4. Names reflect the preferred usage of tidal
sedimentation studies (e.g., Kvale et al., 1999).
Name
Label
declinational
basic solar
basic lunar
elliptical (to M2 )
declinational
basic solar
basic lunar
elliptical (to O1 )
tropical (declinational to M0 )
synodic (variational, lunar)
anomalistic (elliptical to M0 )
evectional
semiyearly (declinational to S0 )
yearly (solar elliptical)
nodal (lunar)
K2
S2
M2
N2
K1
P1
O1
Q1
T (M f )
S
A (M m)
E
SY (SSa)
Y
N
Period, T = 2π/ω
days
hours
0.4986
11.967
0.5000
12.000
0.5175
12.421
0.5274
12.658
0.9973
23.934
1.0028
24.066
1.0758
25.819
1.1195
26.869
13.661
163.932
14.765
177.180
27.555
330.660
31.812
381.744
182.621 2191.452
365.260 4383.120
18.613 years
Table 4: Continuation of Table 3. The frequencies ωi are listed in Tables 1 and 2. Amplitude
coefficients are listed for c > 0.05.
Label
K2
S2
M2
N2 = M2 − A
K1
P1
O1
Q1 = O1 − A
T (M f )
S
A (M m)
E =S−A
SY (Ssa)
Y
N
Angular frequency, ω
Expression
deg/hour
rad/s
2ω0
30.08214
1.458452 × 10−4
2ωS
30
1.454441 × 10−4
2ωM
28.98410
1.405144 × 10−4
2ωM − (ω1 − ω3 ) 28.43973
1.378835 × 10−4
ω0
15.04107
7.292259 × 10−5
ωS − ω2
14.95893
7.252261 × 10−5
ωM − ω1
13.94304
6.759864 × 10−5
ωM − 2ω1 + ω3
13.39866
6.495940 × 10−5
2ω1
1.0980330 5.323333 × 10−6
2(ω1 − ω2 )
1.0158958 4.925300 × 10−6
ω1 − ω3
0.54437
2.639639 × 10−6
ω1 − 2ω2 + ω3
0.4715211 2.285994 × 10−6
2ω2
0.08214
3.982130 × 10−7
ω2 − ω5
0.0410667 1.990967 × 10−7
ω4
0.00221
1.069665 × 10−8
13
Coefficient, c
0.1151
0.4229
0.9081
0.1789
0.5305
0.1755
0.3769
0.0722
0.1564
xxx
0.0825
xxx
0.0729
xxx
0.0655
Table 5: Expected number of tidal bundles per sequence for a Cambrian tidal system. The tidal
character F -value determines the type of tidal system: F = (K1 + O1 )/(M2 + S2 ), where K1 and
O1 are diurnal tidal constituents and M2 and S2 are semidiurnal tidal constituents (see Table 3).
After Lisitzin (1974).
tidal system
tidal character
F -value
ebb-flood cycles
per fortnight
tidal bundles
per sequence*
semidiurnal
semidiurnal/mixed
diurnal/mixed
diurnal
F < 0.25
0.25 < F < 1.5
1.5 < F < 3.0
F > 3.0
30
30 (variable)
15 (variable)
15
30
15–30
15–30
30
*Assumes preservation of every dominant current stage of the ebb-flood cycles that exceeds the
threshold velocity of sediment transport (Nio and Yang, 1991). For the mixed systems, 15 and
30 are the maximum possible values for the ends of the range.
Table 6: Different types of months. A month is the period of the revolution of the moon around
the earth. Tabulated from Mineta et al. (2000). These values may differ slightly from those
published in Bartels (1957).
name of month
synodical
anomalistic
sidereal
tropical
nodical
moon’s revolution
measured relative to...
sun
perigee
fixed star
vernal equinox
ascending node
modern value
29.530
27.554
27.321
27.321
27.212
588 days
440
661
582
220
Table 7: Selected paleoastronomical values determined from Neoproterozoic rhythmites in Australia, in comparison with modern astronomical vales (after Williams, 2000). Value G, the period
of the earth’s rotation about the sun, is assumed to be the same during the Neoproterozoic as it
is today.
Parameter
A
B
C
D
E
F
G
Length of solar day (hours)
Lunar days per synodic month
Lunar days per neap-spring-neap cycle (= B/2)
Solar days per synodic month
Synodic months per year
Solar days per year (= D × E)
Hours per year (= A × F = constant)
Neoproterozoic
(∼620 Ma)
21.9 ± 0.4
29.5 ± 0.5*
14.8 ± 0.3*
30.5 ± 0.5
13.1 ± 0.1*
400 ± 7
8766.7
*Primary value determined directly from rhythmite data (Williams, 2000).
14
Modern
24.00
28.53
14.27
29.53
12.37
365.24
8766.7
A
Source of the (Semidiurnal) Ebb-Flood Tidal Cycle
B
Source of the Neap-Spring-Neap Tidal Cycle
SPRING TIDES
Full Moon
syzygy
lunar orbit
tidal bulge
tidal bulge
NEAP TIDES
Last Quarter
quadrature
NEAP TIDES
First Quarter
quadrature
Earth
Earth
Moon
Ocean covering
SPRING TIDES
New Moon
syzygy
lunar cycle
= about 29 days
semidiurnal ebb-flood cycle
= about 12 hours
C
Sun
The neap-spring-neap tidal cycle = about 14 days
NEAP
TIDES
Sun
SPRING
TIDES
NEAP
TIDES
Sun
Sun
SPRING
TIDES
Sun
Figure 3: The source of the daily and monthly tidal cycles. A. The tidal force on the earth due
to the moon causes two bulges of the oceans. Most coastal areas, spinning through the bulges,
experience two tides per day. Dashed line sphere is the ocean covering without the gravitational
effect of the moon (or sun). B. The sun also produces a tidal force on the earth, which, over the
course of the lunar cycle, produces the neap-spring-neap tidal cycle. The sun and the moon’s
tidal effects are constructive in syzygy (spring tides) and destructive in quadrature (neap tides).
The lunar month defined by the phase of the moon is called the synodic month.
15
ebb
ebb current
0
Stage 1
l
eve
rl
te
a
w
Stage 2
Stage 3
Stage 4
flood current
flood
current
velocity, v(t)
A
time, t
B
current speed, U(t)
volume of sediment transported = [U(t) - Usand]
ebb current
U(t)
3
flood current
Usand
Umud
0
t1
t3
t2
Stage 1
t4
t5
Stage 2
t7
t6
Stage 3
t8
t9
Stage 4
time, t
Figure 4: Sand and mud transport and deposition during one ebb-flood cycle of a strongly
asymmetrical (ebb-dominated) tide. A. Velocity profile and relative water level profile. As
indicated by the dashed line, low tide occurs during Stage 2, and high tide occurs during Stage 4.
B. Current speed profile and sediment volume transport profile. Sand is transported as bedload
when the current speed U (t) exceeds the threshold velocity of sand movement Usand . The volume
of sand transported is proportional to the stippled area under the curve [U (t) − Usand ]3 . Due
to the asymmetry of the tidal currents, much more sand is transported by the peak ebb current
(Stage 1) than by the peak flood current (Stage 3). Mud is deposited when the current speed drops
below the threshold velocity of mud deposition Usand . Periods of mud deposition are represented
by the bold segments on the x-axis (Stage 2 and Stage 4). Sand is not transported, nor is mud
deposited, in the time intervals between stages. After Nio and Yang (1991); Allen (1982).
16
A
B
ebb current
ebb current
A
Stage 1: Dune migrates with peak ebb current
Stage 1: Dune migrates with peak ebb current
Low tide
Low tide
Stage 2: Mud drape during slackwater after ebb current
Stage 2: Mud drape during slackwater after ebb current
flood current
flood current
C
B
Stage 3: Flood current ripples and erosion
Stage 3: Flood current ripples and erosion
High tide
High tide
Stage 4: Mud drape during slackwater after flood current
Stage 4: Mud drape during slackwater after flood current
ebb current
Stage 1: ... and the cycle continues...
Stage 1: ... and the cycle continues...
Figure 5: A. Schematic of dune migration during one ebb-dominated ebb-flood cycle, assuming
the dune is subaqueous at all times. Only one foreset is shown to be formed during the dominant
(ebb) current; in actuality, a number of foresets may be preserved during a single stage (Visser ,
1980). B. Schematic of dune migration during one ebb-flood cycle, based on the sedimentary
structures observed in the cross-set of interest. During the peak ebb current stage (Stage 1), the
dune migrates, and back-flow ripples form on the lower slipface (A in figure). During the peak
flood current stage (Stage 3) ripples (B) migrate across the bottomset of the dune, and the mud
and some of the sand from the upper slipface of the dune is eroded to form a reactivation surface
(C). When the ebb current resumes (bottom of diagram, repeat Stage 1) at the start of a new
ebb-flood cycle, mud is removed from the upper reaches of the dune.
17
A
1 ebb-flood cycle
mean
water level
HIGH TIDE
Idealized
Semidiurnal
Tidal
Fluctuation
0
LOW TIDE
LUNAR
PHASE
Neap
B
Neap
a tidal bundle
Spring
Neap
Spring
Neap
~1m
composite
shale drape
tidal bundle
thickness
composite
shale drape
C
Tidal bundle thickness (cm)
TIDAL BUNDLE SEQUENCE
4.5
Spring
4.0
3.5
3.0
Neap
Neap
2.5
2.0
1.5
1.0
0.5
0.0
Tidal bundle number
Figure 6: Tidal-bundle sequences and the neap-spring-neap tidal cycle. A. Diagrammatic tidal
fluctuation for Jordan time (latest Cambrian). The tidal range is greatest during spring tides and
smallest during neap tides. We have not depicted a diurnal inequality of the tides, because there
is not convincing evidence for diurnal inequality in the rocks of this study. The bold portion
of the curve, indicating the ebb-flood cycle, is schematically shown in Figure 4A. B. Schematic
of a tidal-bundle sequence at the outcrop near Homer, Minnesota. A tidal-bundle sequence is
the sediment deposited by a migrating dune during one complete neap-spring-neap tidal cycle
(about 14 days), whereas a tidal bundle consists of the sediment deposited during a single ebb
stage. The bundles within a sequence, therefore, normally vary in thickness; relatively thin
bundles are deposited during neap tides, relatively thick bundles are deposited during spring
tides. C. Histogram (actual data) of bundle thicknesses. Thicker bundles generally correspond
to deposition during spring tides. The thick-thin alteration of the bundle thickness is possibly
indicative of a diurnal inequality.
18
1. fall
autumnal equinox
tic
ip
l
c
E
Earth
2. summer
summer solstice
0°
-23°
+23°
Sun
2. winter
winter solstice
0°
solar
declination
2. summer =
weaker tides
+23°
1. spring
vernal equinox
1. equinox =
stronger tides
-23°
Earth
2. winter =
weaker tides
Sun
Figure 7: Source of the semiannual tidal cycle. This half-year cycle (SY in Tables 3 and 4) is
due to the declinational changes of the sun. See Figure 8.
19
The Earth without the effects of the Sun and the Moon: a
spherical covering of water. The ocean-covering can be thought
of an equipotential gravitational surface; under the Earth's gravity
alone, it is spherical. The depth of the ocean-covering is highly
exaggerated here for illustrative purposes. The tidal range (shown
here for zero-latitude for convenience) is the difference between
the high water level (HWL) and the low water level (LWL). In this
case, HWL = LWL always; thus there are no tides anywhere.
1. Equinox: Maximal tidal effect due to the zero-degree
declination of the Sun. Under the gravitational effect of the
Sun, combined with the (centripetal) acceleration of the Earth
toward the Sun, the ocean-covering departs from its spherical
shape (dashed) and forms an ellipsoidal equipotential surface,
with two "tidal bulges" on each end. The bulges are observed
with the twice-a-day tides on the Earth (HWL every 12 hours for
a given point on the Earth). The stronger-than-average tides
around the equinoxes are known as equnioctial tides.
tidal bulge
NP
NP
"high" water
level (HWL)
high water
level (HWL)
tidal bulge
Earth
Earth
"low" water
level (LWL)
low water
level (LWL)
Sun
solar declination
= 0 degrees
(equinoxes)
Ocean-covering
The other extreme case: 90-degree solar declination. In this
case, the Sun is in the direction of the North Star, and there is NO
tidal range (HWL = LWL for every point on the Earth), since the
Earth's spin axis is precisely the long-axis of the ellipsoid. The
maximal declination of the Sun with respect to the Earth over the
course of the year is 23 degrees.
Sun
2. Solstice: Minimal tidal effect due to the maximal 23degree declination of the Sun. Compared with the zerodegree case (equinox), here the HWL is less, the LWL is greater,
making the tidal range less. Thus the tidal effect due to the solar
declination is maximal at the equinoxes and minimal at the
solstices.
solar declination
= 90 degrees
(never occurs)
solar declination
= 23 degrees
(solstices)
Sun
NP
NP
high water
level (HWL)
"high" water
level (HWL)
Earth
Earth
low water
level (LWL)
"low" water
level (LWL)
Figure 8:
Figure 7.
Equinoctal tides, marking the maximum range for the semiannual tidal cycle. See
20
1999-2001: 2118 high-tide levels in 1096 days
1.0
water level (m)
A
0.5
0
-0.5
-1.0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
high tide number
1.0
rel spectral power
B
27.74
0.5
51.87
2.09
0
0
0. 1
0. 2
0. 3
0. 4
1.93
0. 5
0. 6
0. 7
f (1/[high tide number])
approx sed transport
1999-2001: 2118 high-tide levels in 1096 days
C
0.10
0.05
0
-0.05
-0.10
0
200
400
600
800
1000
1200
1400
1600
1800
2000
high tide number
1.0
D
rel spectral power
2.08
1.92
0.5
29.10
0
0
1.87
2.17
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
f (1/[high tide number])
Figure 9: Tidal spectrum for modern tidal data (Kwajelein Island). A. Water level at high tide
at Kwajelein Island, 1999–2001. B. Power spectrum of A, with the dominant peaks labeled by
xxx. C. Approximate sediment transport, computed from A. D. Power spectrum of C, with the
dominant peaks labeled by xxx. Periods around 2 reflect the diurnal inequality of a semidiurnal
tidal system, with every other high tide event corresponding to the relatively stronger (“dominant”, see Tape (2001, Figure 35) tidal event. It is clear from B and D that the Kwajelein Island
record has a diurnal signal.
21
Jordan foreset thickness measurements -- Left Section
6.0
A
thickness (cm)
4.0
2.0
0
-2.0
-4.0
-6.0
0
20
40
60
80
100
120
140
160
180
200
foreset number
1.0
1.0
B
relative spectral power
relative spectral power
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
32.3
43.9
0.6
0.4
0.2
0
0.7
C
15.3
26.4
0.8
0
0.05
f (1/foreset)
0. 10
f (1/foreset)
Jordan foreset thickness measurements -- Right Section
6.0
D
thickness (cm)
4.0
2.0
0
-2.0
-4.0
-6.0
0
50
100
150
200
250
300
foreset number
1.0
1.0
E
relative spectral power
relative spectral power
0.8
0.6
0.4
0.2
0
F
181.9
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
41.1
22.7
0.4
0.2
0
0.7
f (1/foreset)
71.5
0.6
0
0.05
0. 10
f (1/foreset)
Figure 10: Tidal spectrum for Jordan sandstone foreset thickness data. A. Jordan foreset
thickness data (Left Section). B. Power spectrum of A. C. Zoom-in of B, showing the longer
period peaks. D.-F Same as A-C, only for the Right Section. It is clear from B and E (see
arrows) that the Jordan data do not contain a diurnal signal.
22
A
semiyearly and yearly periods
0.03
semimonthly and monthly periods
semimonthly and monthly periods
1.0
Y
0.10
A
S 14.74
370.17
177.56
0.01
0.5
A
T
27.61
0
0
0.01
0
0.02
0.04
0.06
f (1/day)
semimonthly and monthly periods
S
rel spectral power
0.99617
0.04
0.02
P1
0
0.075
0.9
24-hour periods
K1
P1
4
2
0
0.99
1
1
1.01
f (1/day)
semimonthly and monthly periods
0.20
8
x 10-3
semimonthly and
monthly periods
S
S
SY
183.5
rel spec power (AST)
rel spec power (AST)
x 10-3
f (1/day)
semiyearly and yearly periods
0.06
0.04
0.02
14.7867
0.15
0.10
0.05
0.08
1.0033
0.95
f (1/day)
0.06
K1
1.0753
1.07
B
6
rel spec power (AST)
rel spectral power
0.06
13.6911
0.07
0.04
f (1/day)
24-hour periods
T
0.05
0.065
0.08
O1
14.7587
0
E
0
f (1/day)
0.10
T
0.05
13.70
rel spectral power
0.02
rel spectral power
rel spectral power
rel spectral power
S
SY
A
14.7
E
6
32.6
A
4
27.6
2
27.8
0.01
0
0.02
0.04
f (1/day)
8
24-hour periods
rel spec power (AST)
rel spec power (AST)
14.11
4
2
0.070
f (1/day)
0.075
0.08
24-hour periods
1.0753
0.5
P1
1.0694
0.9
0.95
f (1/day)
K1
K1
0.99668
1.0028
0
0.065
0.06
1.0
O1
6
0.04
f (1/day)
1.0
S
14.76
0
0
0.08
f (1/day)
semimonthly and
monthly periods
x 10-3
0.06
1.00
rel spec power (AST)
0
0
0.9971
0.5
P1
1.0028
0
0.99
1.00
1.01
f (1/day)
Figure 11: Tidal frequencies associated with modern tidal data from Kwajelein Island, 1999–
2001. The tidal harmonic peaks are labeled according to Table 3. A. Various zoom-ins of the
power spectrum for the high tide events at Kwajelein Island. B. Various zoom-ins of the power
spectrum for the approximate sediment transport at Kwajelein Island. The x-axis ranges here
are the same as in A; only the y-axis ranges are different. Comparing with A, we see that the
peaks are generally not as well defined. See text.
23
relative spectral power
0.10
0.08
0.04
0.02
relative spectral power
S+A
18.61 T+A
17.67
E=S-A
62.18
0.02
0.03
0.04
0.05
f (1/[high tide number])
S+T
13.72
0.06
0.07
S
28.45
S+A
18.61
0.02
A
53.29
E=S-A
63.60
B
S+T
13.72
0.03
0.01
0.08
Spectral analysis of approx. sediment transport interpreted from high tide events at Kwajelein Island
0.04
0
T-A
0.02
0.03
0.04
0.05
f (1/[high tide number])
0.06
0.07
0.08
Spectral analysis of Jordan foreset thickness -- Left Section
1.0
S
0.8
A?
44.5
C
26.8
32.2
15.4
0.6
0.4
0.2
0
0.02
0.03
0.04
0.05
f (1/foreset)
0.06
0.07
0.08
Spectral analysis of Jordan foreset thickness -- Right Section
1.0
relative spectral power
A
0.06
0
relative spectral power
Spectral analysis of high tide events at Kwajelein Island
S
28.45
T
26.34
A 52.67
D
0.8
0.6
73.9
A?
40.8
22.9
0.4
0
12.6
15.9
0.2
0.02
0.03
0.04
0.05
f (1/foreset)
0.06
0.07
0.08
Figure 12: Comparison of frequency spectra from a modern semidiurnal tidal system with spectra from the Jordan sandstone. All plots are normalized with the maximum peak equal to 1. The
range of periods on the x-axis is the same in each case, from 40 to 12 events, which covers the
monthly and semi-monthly periods in a tidal system. A = anomalistic month, T = half-tropical
month, S = half-synodic month; other peaks represent combinations of these periods. A. Spectral analysis of high tide events at Kwajelein Island, 1999–2001 (i.e., about two events per day).
B. Spectral analysis of approximate sediment transport at Kwajelein Island, 1999–2001. The
approximate sediment transport is calculated by cubing the gradient of the high-tide event measurements. The anomalistic month (fA ) is relatively weaker compared to the peaks that are
influences by the half-synodic period. Note the half-tropical month is no longer distinguishable.
C.-D. Jordan sandstone foreset thickness data, Left Section and Right Section, plotted as a power
spectrum.
24
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