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OCR for page 221
Strategy for Future Measurements of
Very-Low-Frequency Sea-Level Change
14
INTRODUCTION
Tide-gauge records of long duration (on the order of 1
century) appear to be dominated by two processes: (1)
noisy fluctuations with time scales on the order of 1 dec-
ade and root-mean-square (rms) amplitudes of 50 mm, and
(2) a long-term trend on the order of 1 mm/yr (Figure
14.1~. The decade fluctuations have horizontal coherence
scales commensurate with the size of ocean basins (Figure
14.2), and they appear to be associated with what is loosely
called El Nino. The long-term trend appears to be global
and has been thought to be associated with climate changes
of glacial periods. Table 14.1 summarizes some possible
numbers, and Figure 14.3 sketches an associated spec-
trum.
We have taken a = 30 m for the rms amplitude of the
ice-age-like fluctuations and typical period scales of 10,000
yr. This corresponds to a frequency of f = 10 ~ cycles per
year (cpy) and an rms rate of 2 fa = 20 mm/yr, which is 20
times the present measured rate. It is hard to account for
this 20:1 discrepancy. The geologic record of sea level
and of glaciation prevents us from taking a much smaller
amplitude than 30 m. Perhaps the current rate of rise is
unusually small because we are approaching a climatic
optimum; in fact, there is some evidence that the rate of
sea-level rise was an order of magnitude larger around
10,000 yr ago.
221
WALTER MUNK, ROGER REVELLE, PETER WORCESTER, and
MARK ZUMBERGE
Scripps Institution of Oceanography
DETECTABILITY OF LONG-TERM TREND
There have been many discussions about the long-term
1-mm/yr sea-level rise. The question is asked whether this
is now accelerating or decelerating. The following very
simple calculation shows the difficulty in making such a
prognosis. The difficulty has to do with the large ampli-
tude of the decadal fluctuations. Suppose we form two
averages, one over each half of the past century, and
compare the two numbers for an indication of the long-
term trend. For any single measurement of sea level, the
high-frequency "noise" associated with the decadal fluc-
tuations is 50 mm rms. The band width is such that we can
form independent samples every 5 yr, or 10 samples in 50
yr. The rms error in a 50-yr average is then 50 mm/~101°5
= 15 mm. The rms error in the difference between the two
50-yr averages is 15 x (2~°.s = 21 mm. The expected
difference in sea level associated with the long-term rise is
50 mm, about twice the noise level. This rough calcula-
tion is in general accord with what is shown in Figure
14.1.
There are a number of ways for improving the estimate
of the long-term rise. The usual way is to form a global
average. From Figure 14.2 we estimate that one can get 10
spatially independent samples, and in this way the decadal
noise in the difference between the two 50-yr averages is
reduced from 21 mm to 21/~101°5 = 7 mm, as compared to
OCR for page 222
222
can
· 0
I ' ' i i 1 ~' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' 1
Period
rms amplitude
rms rate
1910 1920 1930 1940 1950
Current rate
FIGURE 14.1 Honolulu mean sea level after removal of all tidal
components and all noise above 2 cpy (after Munk and Cartwright,
1966~.
the long-term trend of 50 mm. This gives a quite respect-
able signal to noise ratio of 17 db, but the existence of
pronounced local biases malces it difficult to realize a noes
. . . .
gain rom spaclal averaging.
CRUSTAL NOISE
The trouble with the above procedure is that the "solid"
crust to which the tide gauges are anchored moves up and
down at rates that are comparable to the rate of sea-level
rise. Such being the case, the "crustal noise" rather than El
120° 150° E 180° W
Boo - .
N
Do -
n° - ...
S
-
30°;
l5oo
1 20°
1 1 1 ~ , 1 1
_ DEC 1977
Y_
at_
. _
_
..
:::::::: :~::: :`
, ~ .!
_
Am_ 9
1 1 1 1 1 1 1 1 1 1 1 1 ,~
. ~ 150° E 180° W 150° 120° 90°
FIGURE 14.2 Maps of sea-level anomaly (from Wyrtki and
Nakahoro, 1984) for December 1975 and December 1977.
Contours show sea-level anomalies in millimeters after removal
of seasonal cycle. The two cases were selected for their great
contrast.
WALTER MUNK et al.
TABLE 14.1 Periods, Amplitudes, and Rates of
Sea-Level Fluctuations
Climatologic al El Nino - like
Fluctuations Fluctuations
1O,OOO yr
30 m
20 mm/yr
1 mm/yr
10 yr
50 mm
20 mm/yr
Nino-like noise becomes the limiting factor in a global
estimate. If the crustal movement could be independently
measured, then the global estimates of sea-level rise could
be vastly improved.
THERMAL EXPANSION VERSUS CHANGES OF
OCEAN MASS
In considering fluctuations in sea level hats, we distin-
guish between external processes involving variations in
the total mass per unit area and those associated with
internal changes in the density distribution (without much
altering the total mass). The former processes are associ-
ated with hydrostatic pressure fluctuations (pgh) on the
seafloor; the latter, steric processes, give bottom pressure
fluctuations that are smaller by 2 or 3 orders of magnitude
or are absent altogether.
A warming by 1°C of the entire water column would
raise sea level by about 0.2 m. A warming by 10°C would
One
lo4
m2
too
Cpy
1
10-;
1 0
I ce Age -l ike
fluctuation
-
_
10 ~10-4 10-3 10 '
cycles per year
El Ntno -l ike
fluctuation
\ ~
I 1 1 1 1
10-1 1
FIGURE 14.3 A cartoon of the sea-level spectrum. The
climatological (ice-age-like) oscillations are taken to have an
rms amplitude of 30 m and a time scale of 10,000 yr. The El
Nino-like events have an rms amplitude of 50 mm and a time
scale of 10 yr. The inferred rms rate of sea-level change is 20
mm/yr for the climatological oscillations (20 times the present
rate) and also 20 mm/yr for the El Nino-like fluctuations. There
is no firm evidence for the spectral gap at 10-2 cpy, or for the
drop-off below 10-4 cpy; the ice-age-like and E1 Nino-like
fluctuations may be plateaus on a monotonic red spectrum.
OCR for page 223
STRATEGY FOR FUTURE MEASUREMENTS OF VERY-LOW-FREQUENCY SEA-LEVEL CHANGE
raise sea level by about 8 m (much more than 10 x 0.2 m
because the coefficient of thermal expansion increases
with temperature). Implication is that a 30-m variation in
sea level cannot be attributed to thermal expansion. The
conclusion is that the rise over the last few thousands of
years must have been largely due to external processes,
mainly owing to the decay of continental ice sheets.
The situation is not clear with regard to the rise by
approximately 1 mm/yr during this century. The glacio-
logical evidence (Meter, 1984) seems to rule out a contri-
bution by much more than 0.3 mm/yr from mountain and
alpine glaciers; an 8 percent decrease in the current glacial
volume would correspond to a 50-mm rise in global sea
level. A 50-mm rise could also be brought about by a 0.3
percent decrease in the volume of the Antarctic ice sheet
(corresponding to a lowering of the ice-sheet surface by 6
m). Most Antarctic glaciologists, however (Robin, 1985;
National Research Council, 1985), do not believe that
such a change is taking place.
At the same time, there is evidence of a warming by
0.4°C of global surface temperature during the twentieth
century (Gornitz et al., 1982), and this yields a steric rise
of perhaps 0.6 mm/yr when interpreted in terms of a model
of vertical ocean diffusivity. But the evidence is not
convincing, and the model is demonstrably poor. In
summary, the ice-age-like fluctuations are probably largely
external, but the associated long-term trend over the last
50 yr may include an important steric component.
Now with regard to the decadal fluctuations, Roem-
mich and Wunsch (1984) demonstrated that they are largely
steric. Independent measurements could be used to sub-
tract the steric (El Nino-like) noise from the sea-level
measurements and so obtain a better estimate of the exter-
nal component in the long-term trend.
MEASUREMENT OF g
The position of the sea surface relative to a benchmark
on land is affected both by sea-level change and by verti-
cal motions in the crust beneath the benchmark. Tide-
gauge measurements alone may be insufficient to monitor
true changes in sea level the heights of the tide gauges
themselves must be monitored with respect to a worldwide
reference frame.
One technique that can be brought to bear on the prob-
lem is the absolute measurement of g, the acceleration due
to the Earth's gravity. As an observer moves away from
the center of the Earth, g decreases at the fractional rate of
3 parts in 10~°/mm. The local value of g can currently be
determined by portable instruments (Zumberge et al., 1982)
to within an uncertainty of about 1 part in 108, providing a
sensitivity to height changes on the order of 30 mm. Thus,
periodically making accurate determinations of g near tide
223
gauge installations can provide a cost-effective means of
separating sea-level change from tectonic motion.
The nature of the method used to make absolute gravity
measurements makes them well suited to a search for
very-low-frequency signals such as those related to sea-
level change. In a modern absolute gravity meter, a mass
is made to fall freely in a vacuum while its position as a
function of time is determined by a laser interferometer.
By calculating the acceleration of the falling mass, g is
determined in terms of the wavelength of a stabilized laser
and the frequency of an atomic time standard. Both are
referenced to absolute standards having stabilities surpass-
ing 1 part in 109. Other factors limit the accuracy obtained
to about 1 part in 108, but we believe that this can be
improved by a factor of 2 or 3 in the next decade (Faller et
al., 1983~. The important point is that g is determined
absolutely, and thus the data are not likely to become
contaminated by instrumental drift.
Incorporating gravity data into an analysis of a tide-
gauge record can be a complicated problem. A rise in sea
level will affect gravity because of the additional water
mass and through deformation of the crust by the added
load. Conversely, a local expansion of the underlying
crust will move the observation point and displace sea
water, both of which affect g.
An order-of-magnitude assessment of the contribution
to change in g from the two mechanisms sea-level change
as opposed to tectonic motion relies on two simple
numbers. The first (mentioned above) is the free air gradi-
ent of gravity. The fractional change in g for a height
displacement z is
~/g =3 x 10 /mm. (14.1)
This will be roughly the size of the gravity signal accom-
panying a vertical displacement caused by crustal defor-
mation. The other value we need is the gravitational
attraction of a laterally infinite layer of water having
thickness t. Relative to the nominal value of g (about 9.8
m/s2), the attraction is
~g/g =4 x 10 /mm. (14.2)
If we ignore the crustal loading effect (its sign and magni-
tude are comparable to the infinite layer effect) (War-
burton and Goodkind, 1977; Goad, 1980), we see that the
change in gravity accompanying an apparent change in sea
level due to crustal deformation is an order of magnitude
larger than the gravity change caused by a real variation in
sea level. Thus repeated gravity measurements can be
useful in distinguishing one mechanism from the other.
Other geodetic methods can be used to monitor the
vertical positions of tide gauges, and any strategy de
OCR for page 224
224
signed to address the sea-level problem should take advan-
tage of them. For example, electromagnetic distance
measurements to satellites using radio waves or lasers can
determine relative geodetic heights with an uncertainty
ranging from 20 to 50 mm between stations separated by
100 km or more (Christodoulidis and Smith, 1983; Strange,
1984~. Over intercontinental distances, very-long-baseline
interferometry can determine heights to 100 mm or better
(Shapiro, 1978; Herring, 1984~. As these methods im-
prove, they are bound to exert a profound influence on the
strategy for assessing global sea level.
PRESSURE ON THE SEA FLOOR
The best long-term measurements of seafloor pressure
made to date have relied on a careful characterization of
the drift characteristics of the pressure sensor from labora-
tory measurements so that the pressure record could be
corrected for sensor drift after completion of the experi-
ment. Wearn (Wearn and Baker, 1980; Wearn and Larson,
1982), for example, believed that he was able to correct
data obtained using Paroscientific quartz-crystal pressure
sensors with sufficient precision to reduce the residual
drift to between 1 and 3 mbar (approximately 10 to 30
mm) per year at an ambient pressure of about 500 dbar
(approximately 500 m). This corresponds to a relative
accuracy of 0.002 to 0.006 percent, or 2 x 10-5 to 6 x 10-s.
To determine long-term trends in sea level, one would like
to be able to make measurements with substantially greater
precision; an increase of 1 mm in 4000 m of water, for
example, represents a change of 2.5 x 1O-7. One does not
want to make measurements at much shallower depths
because the pressure change then has both barotropic and
baroclinic components. In order to measure such trends,
one therefore needs to either (1) devise a pressure sensor
with lower drift or (2) calibrate the sensor against a pri-
mary standard in situ. A pressure sensor with significantly
lower drift than the Paroscientific quartz sensor used by
Wearn does not exist; a transducer with lower drift would
therefore have to employ an entirely new technology that
is unknown at this time. In the remainder of this section
we will therefore discuss the precision that is achievable
with in situ calibrations.
One could imagine configuring the system in two dis-
tinct ways.
1. A valve could be used to switch the input port of the
pressure sensor between ambient ocean pressure and a
reference pressure used to calibrate the sensor. This tech-
nique requires a sensor with adequate short-term stability,
which may be a problem. The reference pressure would
need to be close to ambient pressure in order to minimize
hysteresis effects in the sensor.
WALTER MUNK et al.
2. A reference pressure could be used with a differen-
tial pressure sensor of much lower full-scale pressure than
required to directly measure ambient ocean pressure. This
configuration takes advantage of the fact that drift is typi-
cally proportional to the full-scale pressure of the sensor,
whereas the absolute size of the fluctuating signal is the
same for either absolute or differential sensors.
In either case, the best candidate for providing the refer-
ence pressure is a high-quality, oil-operated, piston-gauge
deadweight tester. In such a unit a calculable pressure is
generated in a fluid by placing a known mass on a piston
of known diameter. The absolute accuracy achieved with
a high-quality piston gauge is 0.01 percent, or 1 x 10~.
But it is important to remember that what is required is
relative, not absolute pressure changes. The important
parameter is the reproducibility or stability of the pressure
generated by a piston gauge. The National Bureau of
Standards (now National Institute for Standards and Tech-
nology NIST) routinely finds short-term repeatabilities
at the 1 ppm (1 x 10-6) level when intercomparing high-
quality piston gauges, but long-term (years) differences in
some cases have significantly exceeded this level (D. R.
Johnson and C. Tilford, NIST, personal communication).
NIST believes that 10 ppm is a more realistic expectation
for long-term stability; they are not able to verify the
absolute stability at any better level since 10 ppm is the
irreproducibility of the dimensional measurements of the
pistons and cylinders.
This is a rather disappointing state of affairs. A relative
stability of 1 x 10-5 corresponds to 40 mm out of 4000 m.
This is a factor of 2 to 6 better than Wearn claims to have
achieved by measuring and removing the drift of his pres-
sure sensors, but is substantially less stable than desired.
It is conceivable that one might achieve better than 10 ppm
stability in the deep-sea environment; the temperature is
relatively stable (fluctuations are measured in millidegrees
centigrade in the deep Pacific), the instrument would not
be handled, and the piston weights would never be changed.
Testing this speculation would require a substantial re-
search project in which a number of piston gauges would
be maintained in a simulated deep-sea environment for
periods in excess of a year. Such a test would reveal noise
that is uncorrelated between units. Systematic errors can
be determined only by comparison to a better standard; no
such standard exists. The best that one would be able to do
is compare piston gauges from several manufacturers.
An alternative approach would be to deploy the sensor
at significantly shallower depths, decreasing the fractional
accuracy required. At shallower depths one might also
expect the drift problem to be more tractable, since one
would be dealing with lower pressures. The disadvantage
would be that the pressure signal would then have a sig
OCR for page 225
STRATEGY FOR FUTURE MEASUREMENTS OF VERY-LOW-FREQUENCY SEA-LEVEL CHANGE
nificant baroclinic component. However, even at 100 m,
corresponding to 1-mm precision for a stability of 1 x 10-5,
one would expect the baroclinic component to be substan-
tially attenuated relative to that measured by a surface tide
gauge (since the largest ocean temperature changes occur
in the upper ocean). If lower drift could be achieved at
pressures corresponding to a few hundred meters depth,
one might be able to move the sensor somewhat deeper.
But this is clearly not the most desirable approach.
If one were to pursue the use of piston gauges on the
seafloor for long time periods, a number of practical prob-
lems would need to be addressed. Some of the less obvi-
ous ones include the following:
1. Although NIST is of the opinion that a high-quality,
oil-operated piston gauge would function continuously for
a year, there is no evidence of anyone that has done it for
longer than two weeks (D. R. Johnson and C. Tilford,
NIST, personal communication).
2. The axis of the gauge must be aligned to the vertical
to better than 1 milliradian.
3. The gauge would have to be installed on the seafloor
in such a way that it would settle by less than 1 mm/yr.
4. The external pressure applied to the piston by the
gas in the pressure vessel used to house it must either be
measured or eliminated by housing it in a vacuum.
5. The temperature of the piston and cylinder must be
monitored to allow compensation for the approximately 9
ppm/°C temperature coefficient of the effective area.
Many other detailed considerations would be required for
making high-precision measurements; the ones given above
are only examples. Success with this approach would
prompt the desire to construct instruments to be deployed
for multiyear periods, with real time data readouts on
shore. Fiber-optic technology currently under active de-
velopment should make it economically feasible to con-
nect the instruments via cables to shore-based data record
ers.
INVERTED FATHOMETER
Consider a fathometer looking upwards from the
seafloor. The one-way acoustic travel time is Sh, where S
= 1/C is the sound slowness and h is the total water thick-
ness. In a homogenous ocean, the departure in travel time
iS lit = S(Z - z ~ as a result of departures z and z
surface c ' surface c
in the surface and crustal elevations, respectively. We
define z - t/S SO that z = z - z .
IF IF surface c
For an external contribution to the water budget OF =
z - z . The effect of steric disturbances is twofold: it leads
b c
to an additional component, Is, in the surface elevation,
and it changes the sound speed in the water column.
In considering the effect of a long baroclinic wave in a
225
two-layer ocean, the upper ocean is characterized by a
thickness hi, density pi, temperature 0~, and salinity so,
with a corresponding h2, P2, 02, and s2 for the lower ocean.
With a change in layer thickness from h. to (h. + ah. ),
j = 1,2, but leaving Hi and si unchanged, the condition of
constant mass per unit area requires that piths + p2bh2 = 0,
or
Gh2 = -(P1/P2)6h1 (14.3)
The steric change in sea levels equals
z = Gh + Oh = 2 1 ah .
s 1 2 P2 ~
This can be written
The change in acoustic travel time (one-way) is
fit = Sldhi + S26h2 - SOZF- (14.5)
St = S2 dh lo- 5 A. (14.6)
The first term is S2ZS' which is very nearly the increase in
travel time associated with the steric rise in sea level.
Typical values are (P2 - P1~/P2 = 0.001 and (S2 - S14/S2 =
0.01 so that the second term dominates. In terms of the
temperature differential be, we have bp/p =-a~p and bS/
S = -ocCp, with a _ 0.13 x 10-3/°C and a = 3 x 10-3/°C.
Thus the ratio of the second to the first term in brackets is
p = a/a _ 23. (14.7)
The result is that the second (negative) term dominates. A
thickening of the upper layer (positive Chic raises the sea
level by zs = t(P2 - p~/P23Gh~ and thus increases the one-
way travel time by Szs; but the relative thickening of the
warm (high speed) upper layer decreases travel time, and
the latter effect dominates. We can write the result of a
purely steric disturbance in the form
ZIF Zs(l 2), p >> 1. (14.8)
TOMOGRAPHY
The inverted fathometer method is associated with a
steep acoustic path and by its nature depends on a combi-
nation of surface elevation and the variable interior field
of sound speed. Tomography depends on near-horizontal
refracted paths that do not intersect the surface and so can
be used to estimate Is directly. An advantage of the
method is that it provides a spatial average (order 1000
km). This reduces the mesoscale noise (100 km, month
scale) in the estimate of the decadal fluctuations (104 km,
decade scale). Munk and Wunsch (1985) estimated an rms
OCR for page 226
226
noise level of 5 mm in the determination of the mean
annual steric sea level, using tomographic transmissions.
INTERPRETATION
What could be learned when some of these measure-
ments are combined with the tidal recording?
Let Zc designate the crustal height relative to some
suitable reference; changes in Zc can be estimated from the
gravity measurements.
Let Is be the steric sea level; an increase in Is can arise
from thermal expansion and haline contraction. The for-
mer effect is dominant. (The annual tide, with a typical
amplitude of 10 cm, is largely the result of such tempera-
ture changes.) An important consideration is that the
coefficient of thermal expansion increases appreciably with
increasing temperature. Accordingly, most of the changes
in Is are associated with the wand ocean water above the
thermocline, both because the shallow temperature changes
are relatively large and because the coefficient of thermal
expansion of the upper ocean waters is relatively large. A
distinction can be made between the effects of a net change
in heat content (e.g., due to radiation, evaporation) and the
local change in temperature associated with flow diver-
gence, as in baroclinic waves.
Let Zb designate the barotropic sea level. Here a dis-
tinction can be made between the effects of a global change
in the ocean's mass arising from glacial melting, and the
local changes associated with flow divergence. An ex-
ample of the latter is the buildup of the gyre center at the
expense of the flanks, with the net gyre mass remaining
unchanged.
We consider the records obtained from a tide gauge, a
bottom-pressure recorder, an inverted fathometer, and a
grav~meter:
ZTG Zb s c'
ZBP b c'
ZIF - Zb + Z (1 - p) -Z.
ZGR c
Without bottom pressures, which are difficult to measure,
there are three equations in the three unknowns Zb' Zs' and
Zc With bottom pressure, the problem is overdetermined.
The system of equations is oversimplified. As an ex-
ample, a rise in global sea level due to global melting of
ice will lead to an increase in (due to the added water mass
beneath the gravimeter) and a further increase due to the
compression of the crust from loading, in addition to any
effect from tectonic crustal uplift.
The system of equations is intended as an illustration of
how a particular set of observations can be used. There
are many other ways too. Repeated hydrographic surveys
yield a direct measure of the steric level (the classical
WALTER MUNK et al.
dynamic height). Changes in the crustal elevation can also
be surmised by the perturbation of satellite orbits.
A PROPOSED STRATEGY
We propose that two existing tide stations be instru-
mented to include a gravimeter, a bottom-pressure gauge,
and an inverted fathometer.
There is much to be said for combining this new instru-
mentation with a program proposed by NOAA (1985) as a
contribution to the World Ocean Circulation Experiment
(WOCE). This program consists of three principal com-
ponents: (1) new tide gauges that record digitally and have
a high degree of linearity, (2) very-long-baseline interfer-
ometry (VLBI), and (3) the Global Positioning System
(GPS). VLBI and GPS provide a highly accurate (+1 cm)
global reference frame with a role similar to that of the
gravimeter.
We propose Hawaii and Bermuda as the two sites for
developing the advanced concepts of sea-level measure-
ments. Both stations are relatively free of meteorological
noise (Brown et al., 1975), and have long-time series of
tidal measurements. Hawaii is a site for the proposed
NOAA sea-level program. Bermuda has been the site of
the Panulirus measurements of steric sea level since 1954
(see Roemmich, Chapter 13, this volume).
As a second phase of the proposed program of en-
hanced sea-level measurements, one should establish a
limited global network using the experience gained at the
prototype stations (Hawaii and Bermuda). If local adjust-
ment for crustal movement and steric sea-level fluctua-
tions is successful, then presumably the steric component
of the long-term trend can be estimated from measure-
ments at a relatively small number of stations. The steric
component (if any) of the long-term trend will be difficult
(14 9) to extract from the 50-mm rms background noise of the E1
Nino-like fluctuations. For 10 spatially independent sta
tions, each forming 5 independent samples (this takes 25
yr for a 5-yr decorrelation time), we thus have 50 degrees
of freedom in the global mean, or +7 mm standard devia
tion. During this 25-yr period we can expect a change of
10 to 25 mm in steric level resulting from the long-tenn
trend. The problem is made even more difficult by the
expectation that the long-term trend itself will change
during this 25-yr interval.
REFERENCES
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Christodoulidis, D. C., and D. E. Smith (1983~. The Role of
Satellite Laser Ranging through the 1990's, NASA/Goddard
OCR for page 227
STRATEGY FOR FUTURE MEASUREMENTS OF VERY-LOW-FREQUENCY SEA-LEVEL CHANGE
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Failer, J. E., Y. Guo, J. Gschwind, T. Niebauer, R. Rinker, and J.
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Representative terms from entire chapter:
thermal expansion
