Effects of small-scale bathymetric roughness on the global
internal wave field
John A. Goff and
Brian K. Arbic
Institute for Geophysics,
JJ Pickle Research Campus,
Goff: phone: (512) 471-0476; fax: (512) 471-0999;
email: goff@ig.utexas.edu
Arbic: phone: (512) 471-0472; fax: (512) 471-8844;
email: arbic@ig.utexas.edu
Award Number: N00014-07-1-0792
http://www.ig.utexas.edu/research/projects/roughness_internalwaves/Goff_Arbic_PR07.htm
The small-scale roughness properties of
the seafloor are increasingly being recognized as critical parameters in
determining important processes in physical oceanography. For instance, in situ observations (e.g., Polzin et
al., 1997) find that mixing levels are greatly elevated in regions of rough
topography. Gille et al. (2000) demonstrate that
mesoscale eddy energy tends to be lower in areas where the bottom is rough
(suggesting the possibility that dissipation of eddy energy takes place in such
areas), and Egbert and Ray (2003) show that substantial tidal dissipation
occurs in such areas. The dissipation is generally thought to arise from the
breaking of internal waves generated by flows over the rough seafloor. On the
time scales of internal waves, mesoscale eddies and the general circulation can
be regarded as steady, while tides are oscillatory. The physics of linear
internal wave generation is different for these two classes of motions (e.g.,
A significant dilemma for physical
oceanographers studying these processes is that the kind of bathymetric
resolution required to model these processes over
entire ocean basins are not available, nor will be any time soon. Acoustic
bathymetric data, which can achieve lateral resolutions of 0.1-0.2 km,
presently cover only a few percent of the ocean floor beyond the exclusive
economic zones in coastal areas. A complete swath survey of all the deep oceans
would take ~200 years of ship time at a cost of billions of dollars (Carron et
al., 2001). The most comprehensive determination of bathymetry world-wide is
the Smith and Sandwell (1994; 1997; 2004) model
derived from satellite altimetry data combined with data from ship soundings,
but the resolution of this product is limited to >10 km in the deep ocean. We seek to resolve this dilemma through a
novel approach of relating the texture of satellite altimeter data to seafloor
roughness characteristics. We will then address issues of importance to the
Navy with tide models that utilize this information, either by directly
resolving internal wave generation over the rough seafloor, or by
parameterizing the dissipation of these internal waves. In particular, we will
investigate internal wave generation in a global baroclinic tide model, and we
will investigate the effect of parameterized dissipation over rough topography
in both barotropic and baroclinic global tide models.
OBJECTIVES
Our main
objectives are to characterize seafloor roughness from satellite altimetry data
and investigate the impact of rough topography on the global internal tide
field. Specific tasks include:
(1) Full
spectral characterization of altimeter noise world-wide and transfer function
for predicting resultant noise in the gravity signal via the processing path.
(2) Advance the
study of the relationship between gravity and abyssal hill fabric.
(3) Generate map
of abyssal hill roughness parameters across the ocean basins.
(4) Determine
the location of dissipation in global baroclinic tide models.
(5) Parameterize
unresolved topographic wave drag in global tide models.
(6) Determine the
impact of better roughness estimates on the resolved generation of internal
tides in a global model.
APPROACH
Satellite
altimetry data and the derived gravitational field (Fig. 1) may make it
possible to infer seafloor statistical parameters over entire ocean basins, and
are the focus of tasks 1-3 noted above. While it is difficult, owing to the
limits of upward continuation of gravity in the deep ocean, for seafloor
features < ~10 km scale to be distinguished individually in the altimetry
data, the aggregate fabric of small-scale features, such as abyssal hill
morphology, can have a quantifiable effect on the gravity fabric (Goff and
Smith, 2003; Goff et al., 2004). Our governing hypothesis, which is partially
confirmed by these prior results, is that an empirical transfer function can be
determined which relates the primary attributes of bathymetric and gravity
roughness (rms height, characteristic horizontal scales, and fabric
orientation) to each other. However, at these limiting scales of altimetry resolution,
process filtering (Smith and Sandwell, 1997), which
is necessary to convert raw altimetry data into coherent gravity field
estimates, and data noise, related both to data uncertainties and oceanographic
variability, will also have a significant effect on gravity fabric. These must
be fully accounted for.
The
Goff and Smith (2003) and Goff (2004) analyses demonstrate clearly the
importance of quantifying altimetry noise (Task 1 above) if we are to extract
abyssal hill roughness properties from the altimetry data set. Altimetric noise
is neither constant nor simple in statistical character. Figure 2 plots the rms
difference in the along-track sea surface slope as computed from the version
15.1 altimetric gravity field model of Smith and Sandwell
and as measured by the Geosat altimeter during its
Geodetic Mission (GM). This rms
difference includes the effects of random errors and (non-tidal) time-varying
dynamic sea surface slopes, and is an error source in the estimation of gravity
(and hence abyssal hill fabric) from altimetry.
The 1 to 4 micro-radian (mm per km) sea surface slope errors in Figure 2
can be expected to produce roughly 1 to 4 milliGal
errors in gravity. The inference from this plot is that noise strength is
regionally variable, and correlated strongly to average sea state and to
currents and attendant mesoscale eddy conditions. Thus, not only will the
contribution of noise to the gravity roughness be regionally dependent, but we
can also expect a non-white statistical character to the altimetry noise,
corresponding to the horizontal scales of oceanographic and atmospheric
heterogeneities, that will complicate the noise response in the gravity map.
Figure 1. Sun-shaded altimetric gravity field (Sandwell and Smith, 1997), emphasizing roughness, over a
portion of the Southeast Indian Ridge (yellow) corresponding to a change in
ridge morphology: from an axial high in the west to an axial valley in the east
progressing into the Australian-Antarctic Discordance (AAD), and a corresponding
change in abyssal hill roughness (Goff et al., 1997). The blue dashed line
marks a visually-determined textural boundary off axis. Boxes indicate areas
chosen by Goff and Smith (2003) for gravity texture estimation, which
demonstrated that the quantitative characterization of gravity texture varied
in concert with the abyssal hill roughness.
To characterize noise in the gravity map,
we must first characterize noise in the raw altimetry data, prior to the
application of the various filtering and processing steps. In our preliminary
analyses, we have devised a means of largely isolating the noise by computing
and comparing the auto- and cross-covariances of overlapping or closely
adjacent parallel altimeter tracks. The overriding assumptions here are that (1)
adjacent tracks will be highly correlated with respect to the influence of
seafloor morphology on altimetry, but (2) highly decorrelated with respect to
noise components, including variations related to mesoscale
eddies. The first assumption is justified based on the short distance
between adjacent tracks (average ~6 km at the equator, less at higher
latitudes) compared to the effective upward continuation distance (~20
km). The second assumption is justified
because the minimum time between adjacent tracks (~17 days) is similar to the
decorrelation time for mesoscale circulation (~15 days; Traon
et al., 1998).
Figure
2. RMS sea surface gradient errors for the altimetry data making up version
15.1 of the Smith and Sandwell gravity model (from David Sandwell, personal
comm.). These results are indicative of noise contribution to the gravity data,
and demonstrate correlation between physical oceanographic features and
altimetry noise.
Approaches for
Tasks 2 and 3 will be detailed in later progress reports.
The spatial
distribution of dissipation in the ocean is a matter of intense interest in the
oceanographic community, including the Navy. Much of the interest stems from
the suggestion by Munk and Wunsch (1998) that the strength of the meridional
overturning circulation is controlled by ocean mixing. In addition, the general
oceanic circulation in models (e.g., Scott and Marotzke 2002) shows a strong
sensitivity to the spatial distribution (in both the vertical and horizontal
directions) of mixing. Mixing diffusivity k is related to energy dissipation e by the relation k = Ge/N2, where G is an efficiency factor of about 0.2 and
N is the Brunt-Vaisala buoyancy
frequency (Osborn 1980). Since mixing is connected to dissipation,
quantification of mixing in the ocean must consider energy sinks, which balance
energy sources in averages taken globally and over long periods of time.
Quantification of the sources and sinks of energy for the deep ocean has been
studied in earnest in recent years. Wunsch (1998) showed that the winds put
approximately 1 TW of energy into the oceanic general circulation, while Alford
(2001) showed that winds put about 0.5 TW of energy into the near-inertial
internal wave field. The details of how these wind energy inputs are eventually
converted into a dissipation are not yet well known. Tides put a total of 3.5
TW into the ocean, and of this about 2.5 TW is dissipated in coastal areas,
where tidal velocities are much larger, while 1 TW is dissipated in the open
ocean, in regions of rough topography (Egbert and Ray 2003). Although the budget
for tidal energy is understood better than for wind-forced motions, important
questions about the tidal energy cycle remain.
We will accomplish
three tasks (4-6) in this part of the proposed work. In task 4 we will use the
global baroclinic tide model of Arbic et al. (2004) to examine whether the
dissipation of internal tides generated in the open ocean takes place in
coastal areas, in the abyssal parts of the open ocean, or in the thermocline of
the open ocean. In tasks 5 and 6 we will use the roughness estimates from our
work on tasks 1 to 3 in global tide models. In task 5 we will examine the
difference that a better roughness estimate makes on our parameterizations of
unresolved topographic wave drag, which are used in both barotropic and
baroclinic tide models. In task 6 we will return to global model of the
resolved internal tide field, as in task 4, but this time with better estimates
of seafloor roughness, which will improve the resolution of high baroclinic
modes. An interim roughness map can be produced if the final roughness
estimates are not ready by the time we need better roughness estimates in our
global tide models.
WORK COMPLETED
This project
began funding in June. Over the summer Goff
began working with the full Geosat altimetry data set
and completed a world-wide analysis of altimetry noise at scales < 50 km
(i.e., scales that are of importance to characterizing abyssal hill
fabric). These results are detailed
below. Arbic
has begun formulating his methodology for incorporating small-scale seafloor
roughness characterization into his tide modeling.
RESULTS
Initial
processing of the altimetry data for noise analysis was focused on removing, as
much as possible, the time-invariant component of the altimetry measurements by
differencing nearest-neighbor track lines.
Given that the effective upward continuation filter scale (~20 km) is
large compared with the average track spacing (~6 km at the equator, less at
higher latitudes), the residual between two adjacent tracks should be dominated
by a time-varying signal. Such
variability will presumably be related to oceanographic and atmospheric
processes that contribute to uncertainty in measuring the static geoid. Figure 3 displays an example of a altimetry residual profile generated by differencing two
tracks that are spaced only ~0.5 km apart.
Variations of up to a meter in amplitude over scales > 100 km are
prominent. Altimeter variations of these
scales were studied extensively by Kuragano and Kamachi (2000), who attributed these features to mesoscale
oceanographic circulation. Such features
are not, however, of interest here because they do not contribute to gravity
roughness on the scales that are important to abyssal hills (i.e., < 50
km). To focus our analysis on those
scales, we high-pass filtered the difference function using a 50-km cos**2 filter (Figure 3).
We found that this filter scale very effectively separated the
time-varying altimeter response into large- and small-scale components; i.e.,
there is very little power in the spectrum at the filter wavelength, but
substantial power at both lower and higher wavelengths. A strong high-frequency noise signal is
evident in the filtered residual.
Figure 4 displays the covariance function estimated from the
high-pass filtered difference profile shown in Figure 3. We observe two distinct components to the
covariance functionality: (1) a “white noise” variance, indicated by the height
of the spike that exists between the 0th and 1st lag points, and (2) a
correleted component that crosses into negative lag values after ~5 km lag, and
then returns to ~0 covariance by ~15 km lag.
Initial investigation of the correlated component suggests that it is
related to the Kalman filter response that is applied by the altimeter to
predictively track the signal. Sudden
changes in the signal, such as going from land to sea, or crossing a storm
cloud, can lead to fluxtuations in the Kalman response.
Figure
3. Residual profile between two
altimetry tracks (black), and after low-pass (blue) high-pass (red)
filtering.
Figure
4. Covariance function computed from the high-pass filtered residual profile
shown in Figure 3, illustrating the two primary functional components observed
broadly.
The functional
structure of these high-pass filtered altimetry difference profiles changes
little from one profile to the next, except for the amplitude of the two
components. The consistancy of form
therefore suggest that small-scale (< 50 km) altimetry noise can be parameterized
rather simply by the variance of the two components; i.e., the “white noise
variance”, which is the height of the 0-lag spike, and the “2nd lag variance”
(Figure 4). Figure 5 displays the
world-wide distribution of white noise variance computed in this way. Measurements are averaged within 2 by 2
degree bins. There are two observations
of note here: (1) that ascending and descending tracks, which are essentially
independent data sets, produce very similar results in terms of geographical
distribution, and (2) the pattern displayed by both plots is very strongly
correlated to significant wave heights (plotted in Figure 6). This correlation is certainly expected, as
waves are the primary source of random fluctuations in altimeter data.
Figure
5. World-wide distribution of white
noise variance measurements from high-pass filtered, differenced altimeter
tracks, averaged in 2 by 2 degree bins, for both ascending (top) and descending
(bottom) tracks.
Figure
6. World-wide distribution of significant wave heights (top) and plot of
significant wave heights versus white noise RMS in the same 2 by 2 degree bins
(bottom). Blue-dashed line illustrates
the highly linear nature of the relationship.
The world-wide
distribution of second-lag variance is presented in Figure 7. As with the white noise variance, the 2nd-lag
variance displays a coherent geographic pattern that is very similar both for
ascending and descending tracks. The
pattern, however, is quite different from the white noise variance. We qualitatively observe two evident
correlations: (1) with sea ice at high latitudes, and (2) with rainfall rates
at mid- and low latitudes (e.g., Ikai and Nakamura, 2003). Both ice and rain storms are likely to
generate the sort of abrupt change in altimetry response that will cause
fluctuations associated with the Kalman filtering.
Figure
7. World-wide distribution of 2nd-lag
variance measurements from high-pass filtered, differenced altimeter tracks,
averaged in 2 by 2 degree bins, for both ascending (top) and descending
(bottom) tracks.
IMPACT/APPLICATIONS
With these
results, we can now quantitatively characterize short-scale (< 50 km)
altimeter noise by region. Our next task
will be to transfer the altimetric characterization into a gravity noise
characterization, which we plan on doing by generating synthetic noise along
all the altimetry tracks, scaled by the relationships shown in Figures 5 and 6,
and then processing that “data” through the Smith and Sandwell
(1997) methodology for deriving the world-wide gravity map. The resulting map can then be analyzed for
global distribution of gravity noise.
This will enable us to distinguish what part of the actual gravity
fabric is due to noise and which part is influenced by seafloor fabric. An accurate map of small-scale seafloor
roughness is expected to have a tremendous impact on oceanographic modeling
efforts to predict critical phenomena such as the generation of internal waves
and mixing by both tidal and non-tidal (i.e., mesoscale eddy) flows.
RELATED PROJECTS
Arbic has a separate contract with the Naval
Research Laboratory (Stennis) to accurately implement
global tides in HYCOM. HYCOM is planned
to be the next-generation Navy operational global model. Arbic has developed
a good working relationship with several of the key HYCOM investigators, and
has achieved significant progress in the implementation of tides. The parameterization of topographic wave drag
on both tidal and non-tidal motions in HYCOM is expected to be enhanced by the
bathymetric roughness work done in this project. In addition, once HYCOM is ready to be used
as a tide model, it as well as the Hallberg Iscopycnal Model (Arbic et al.,
2004) can be used to accomplish tasks 4-6 of this proposal.
REFERENCES
Alford, M.H., 2001. Internal swell generation: The spatial distribution of energy flux from the wind to mixed-layer near-inertial motions. Journal of Physical Oceanography 31, 2359-2368.
Arbic, B.K., Garner, S.T., Hallberg, R.W., Simmons, H.L., 2004. The accuracy of surface elevations in forward global barotropic and baroclinic tide models. Deep-Sea Research II 51, 3069-3101.
Carron, M. J., P. R. Vogt, Jung, W.-Y., 2001. A proposed international long-term project to systematically map the world’s ocean floors from beach to trench: GOMaP (Global Ocean Mapping Program). International Hydrographic Review 2, 49-55.
Cushman-Roisin, B.,
1994. Introduction to geophysical fluid dynamics.
Prentice-Hall,
Egbert, G.D., Ray, R.D., 2003. Semi-diurnal and diurnal tidal dissipation from TOPEX/Poseidon altimetry. Geophysical Research Letters 30, 1907, doi:10.1029/2003GL017676.
Gille, S.T., Yale, M.M., Sandwell, D.T., 2000. Global correlation of mesoscale ocean variability with seafloor roughness from satellite altimetry. Geophysical Research Letters 27, 1251-1254.
Goff, J. A., Smith, W. H. F., 2003. The correspondence of altimetric gravity texture to abyssal hill morphology. Geophysical Research Letters 30, doi:10.1029/2003GL018913.
Goff, J. A., Ma, Y., Shah, A., Cochran, J. R., Cochran, Sempéré, J.-C., 1997. Stochastic analysis of seafloor morphology on the flank of the Southeast Indian Ridge: The influence of ridge morphology on the formation of abyssal hills. Journal of Geophysical Research 102, 15,521-15,534.
Goff, J. A., Smith, W. H. F., Marks K. M, 2004. The contributions of abyssal hill morphology and noise to altimetric gravity fabric. Oceanography 17, 24-37.
Ikai, J, Nakamura, K., 2003. Comparison of rain rates of the ocean derived from TRMM microwave imager and precipitation radar. J. Atmos. Oceanic Tech. 20, 1709-1726.
Kuragano, T. Kamachi, M., 2000. Global statistical space-time scales of oceanic variability estimated from TOPEX/POSEIDON altimeter data. J. Geophys. Res. 105, 955-974.
Munk, W., Wunsch, C., 1998: Abyssal recipes II: energetics of tidaland wind mixing. Deep-Sea Res. I 45, 1977-2010.
Osborn, T.R., 1980. Estimates of the local rate of vertical diffusion from dissipation measurements. Journal of Physical Oceanography 10, 83-89.
Polzin, K.L., Toole, J.M., Ledwell, J.R., Schmitt, R.W., 1997. Spatial variability of turbulent mixing in the abyssal ocean. Science 276, 93-96.
Scott, J.R., Marotzke, J., 2002. The location of diapycnal mixing and the meridional overturning circulation. Journal of Physical Oceanography 32, 3578-3595.
Smith, W. H. F, Sandwell, D. T., 1994. Bathymetric prediction from dense satellite altimetry and sparse shipboard bathymetry. Journal of Geophysical Research 99, 21,803-21,824.
Smith, W. H. F, Sandwell, D. T., 1997. Global seafloor topography from satellite altimetry and ship depth soundings: Evidence for stochastic reheating of the oceanic lithosphere. Science 277, 1956-1962, 1997.
Smith, W. H. F, Sandwell, D. T., 2004. Conventional bathymetry, bathymetry from space, and geodetic altimetry. Oceanography 17, 8-23.
Thurnherr, A.M., Richards, K.J., 2001. Hydrography and high-temperature heat flux of the Rainbow
hydrothermal site (36°14'N, Mid-Atlantic Ridge). Journal of
Geophysical Research 106, 9411-9426.
Thurnherr, A.M., Speer, K.G., 2003. Boundary mixing and topographic blocking on the Mid-Atlantic Ridge
in the
Thurnherr, A.M., Richards, K.J., German, C.R., Lane-Serff, G.F., Speer, K.G., 2002. Flow and mixing in the rift valley of the Mid-Atlantic Ridge. Journal of Physical Oceanography 32, 1763-1778.
Traon, P. Y., Nada, F., Ducet, N., 1998. An improved mapping method of multisatellite altimeter data. Journal of Atmospheric and Oceanic Technology 15, 522-534.
Wunsch, C., 1998. The work done by the wind on the oceanic general circulation. Journal of Physical Oceanography 28, 2332-2340.
Zilberman, N., Merrifield, M., 2006. Internal tide generation over mid-Atlantic ridge topography. Eos Trans. AGU, 87(36), Ocean Sci. Meet. Suppl., Abstract OS23L-03.