Research Papers
Institute for Creation Research
February 2, 2005

JOHN R. BAUMGARDNER, Ph.D.
1965 Camino Redondo
Los Alamos, NM 87544 DANIEL W. BARNETTE, Ph.D.
1704 Sadler, N.E.
Albuquerque, NM 87112


Presented at the Third International Conference on Creationism
Pittsburgh, PA, July 18-23, 1994



ABSTRACT

This paper presents results from a set of numerical experiments that
explore the patterns of ocean circulation that arise when the
earth's continental surface is mostly flooded. The calculations
employ a code that solves the 2-D shallow water equations on a
rotating sphere with surface topography. Several continental
configurations are considered, including that of a single
Pangean-like supercontinent. A surprising yet persistent feature in
these calculations is the appearance of high velocity currents
generated and sustained by the earth's rotation above the flooded
continents. Water velocities in the deeper ocean by contrast are
much smaller in magnitude. The patterns typically include strong
cyclonic gyres at high latitudes with water velocities on the order
of 40-80 m/s. The gyres tend to be compressed against the western
continental margins and produce strong equatorward currents parallel
to the western coastlines. The calculations argue that strong
currents spontaneously arise over flooded continents. They suggest
that accurate observational data on the current directions in the
Paleozoic and Mesozoic rocks coupled with careful numerical modeling
could be extremely fruitful in understanding the origin of much of
the earth's sedimentary record.

INTRODUCTION

A central question in understanding the Flood catastrophe is what
was the hydraulic mechanism that was able to transport millions of
cubic kilometers of sediment, to distribute most of it in widely
dispersed layers--in many cases hundreds to thousands of kilometers
in horizontal extent, and to accomplish such a vast amount of
geological work in only a few months time. Clark and Voss [1] have
suggested that resonant lunar tides might qualify as the primary
mechanism. Such tides, if they were to occur with sufficient
amplitude in the presence of the continents, could indeed erode and
transport huge volumes of sediment and deposit the sediment in
laterally extensive layers. A major difficulty with this idea is
that the resonance condition for a smooth earth without continents
is a water depth of about 8000 m [1,4]. This represents about three
times the volume of water currently in the world's oceans.

Resonance, however, can occur at higher spatial harmonics and
smaller water depths. For a smooth earth the critical water depth
for the next higher harmonic is about 2000 m [1,4], much closer to
the amount of water currently at the earth's surface. But the
strength of the resonance of this mode is much smaller than that of
the fundamental mode. It is not clear if the effects of bottom
friction and other sources of dissipation are small enough for a
resonant tide in this higher mode to arise even on a smooth earth,
much less for an earth with realistic topography. Detailed numerical
calculations are still needed to resolve these uncertainties.

In the process of exploring the possibility of a more localized
tidal resonance in the large bay on the eastern side of the Pangean
supercontinent known as the Tethys Sea, an entirely different
mechanism was identified. It was found that the Coriolis force
arising from the earth's rotation produces strong currents on top of
the flooded continents independent of any tidal forcing. These
currents form closed paths that generally have the same sense of
rotation as the jet streams in the atmosphere, that is, cyclonic, or
counterclockwise in the northern hemisphere and clockwise in the
southern hemisphere. They are stronger above a continent localized
to higher latitudes where the Coriolis force is stronger that for
the same continent at lower latitudes. For continental flooding
depths up to several hundred meters, these currents typically
achieve speeds of several tens of meters per second, which is
sufficiently strong to reduce the water depth to zero in much of the
region enclosed by the cyclonic patterns of flow. Such speeds are
easily adequate to erode by cavitation processes [3] and transport
huge volume of relatively coarse clastic sediment for large
distances.

This paper describes a set of numerical experiments using a code
that solves the shallow water equations on the sphere to explore the
necessary conditions and dynamical characteristics of these
currents. This work is seen as only a beginning effort to understand
this phenomenon. It should be emphasized here that tidal effects in
general would add to rather than compete with the flow produced by
such currents. These Coriolis force driven currents then are almost
certainly only one of several factors responsible for the large
scale sedimentation patterns during the Flood.

MATHEMATICAL FORMULATION

The shallow water equations describe the behavior of a shallow
homogeneous incompressible and invicid fluid layer. On a rotating
sphere these equations may be expressed [8, p. 213]

du/dt=- f k x u - g h (1)
and
dh*/dt=- h* u, (2)

where u is horizontal velocity (on the sphere), f is the Coriolis
parameter (equal to 2W sin q for rotation rate W and latitude q), k
is the outward radial unit vector, g is gravitational acceleration,
h is the height of the free surface above some spherical reference
surface, and h* is the depth of the fluid. If ht denotes topography
on the sphere, then h=h* + ht. The d/dt operator is the material or
substantial or co-moving time rate of change of an individual parcel
of fluid. The operator is the spherical horizontal gradient
operator and the operator is the spherical horizontal divergence
operator. These simple equations are appropriate when the depth of
the fluid is small compared with the important horizontal length
scales. This criterion is satisfied for the problem at hand where
the water depths are typically less than four kilometers while the
horizontal dimensions of ocean basins and continents are measured in
thousands of kilometers. Also note that the fluid density does not
appear in these equations. This means that to the degree the
approximations apply, the same equations describe flow in both the
atmosphere and oceans.

These equations are solved in discrete fashion on a mesh constructed
from the regular icosahedron as shown in Fig. 1. The mesh has 40962
nodes and the spacing between nodes is about 125 km. A separate
spherical coordinate system is defined at each node such that the
equator of the system passes through the node and the local
longitude and latitude axes are aligned with the global east and
north directions. This approach has the advantage that the
coordinates are almost Cartesian and only two (tangential) velocity
components are needed. A semi-Lagrangian formulation [7] of
equations (1) and (2) is used which involves computing the
trajectories during the time step that end at each node. Values for
h and u at the beginning of the time step at the starting point of
each trajectory are found by interpolating from the known nodal
values at the beginning of the time step. Changes in h and u along
the trajectory are computed using (1) and (2). This Lagrangian-like
method eliminates most of the numerical diffusion that is associated
with Eulerian schemes. Second-order accurate interpolation is used
to find the starting point values of the trajectories. This
formulation using the icosahedral mesh has been carefully validated
using the suite of test problems developed by Williamson et al. [8].

RESULTS

A small set of problems was investigated to explore some of the
conditions under which a coherent pattern of strong currents arises
on top of a flooded continent. Fig. 2 is a sequence of snapshots
from a calculation initialized with a Pangean-like distribution of
continent of uniform height flooded to a depth of 500 m. The ocean
depth is also taken to be a uniform value of 3980 such that the
total water volume equals that in the present oceans and ice caps.
The initial velocity field is everywhere zero. Fig. 2 (a) displays
the initial height of the water above the bottom topography. Frames
(b)-(g) in Fig. 2 show the development of the pattern of flow with
time at 10 day intervals. Two closed circulations, one in the
northern hemisphere and one in the southern, emerge from the initial
state. The sense of rotation of these circulations is cyclonic,
which implies low pressure or reduced surface height inside the
circulations. By a time of 30 days the currents are so strong that
the water depth has decreased to zero over portions of the
continental surface inside circulating flows. The current velocities
continue to increase to peak values of about 87 m/s until the 50 day
snapshot after which time there is no significant change in the peak
velocity. The patterns show prominent wavelike structure in the
latitude zone between 20 and 50. In the atmosphere such features
are known as planetary waves. They occur because of the variation of
the Coriolis parameter with latitude and are also referred to as
Rossby waves [5, p.93-95].

The effect of increasing the depth of flooding to a value of 1000 m
is shown in Fig. 3. In this case the time dependence of the pattern
is much stronger, the peak velocities are somewhat lower at about 78
m/s, and the reduction of the surface height inside the regions of
cyclonic flow is approaching 1000 m. Snapshots at 80 and 90 days
reveal how dramatically the flow pattern varies with time. Usually
there is more than one cyclonic gyre in a given hemisphere. Fig. 4.
shows the effect of increasing the depth of continental flooding
further to 1500 m. Under these circumstances the peak velocities are
reduced by almost a factor of two to about 40 m/s and the peak
drawdown of the surface inside the cyclonic gyres is only about 500
m. The time variation of the pattern is still strong and the number
of gyres is somewhat larger. Well-defined anticyclonic gyres are
also evident. This latter case suggests that when the static water
depth over the continent exceeds 1000 m, the strength of these
currents begins to diminish significantly.

The calculations described thus far all used the same Pangean-like
continent. To determine what role the special geometry of this
continent might be playing in producing the observed patterns of
flow, several cases were run with simpler continental geometries.
Fig. 5 shows the results from a configuration consisting of a
circular continent 50 (5560 km) in diameter centered at 45
latitude in each hemisphere. The depth of flooding is 1000 m. The
snapshot at 80 days shows single cyclonic gyres on each continent
with peak velocities of about 40 m/s and reductions in surface
height of about 400 m. Moderate time dependence is present. When
these continents are moved to points centered at 60 latitude, the
character of the solution remains the same but the strength of the
flow increases to yield peak velocities of about 55 m/s and a
maximum reduction in the surface height of about 1000 m. On the
other hand, when the circular continents are moved such that they
are centered at 30 latitude, the peak velocity falls to about 37
m/s and the maximum reduction in surface height is only about 200 m.
These results reflect the fact that the magnitude of the Coriolis
parameter increases with the sine of the latitude. Calculations
using circular continents are thus found to yield the same general
behavior as observed for the Pangean-like continent.

Another issue that was addressed was the minimum horizontal
dimension for a continental region required to obtain the strong
gyre-like flow. Fig. 6 displays results for a configuration
consisting of a circular continent 30 (3330 km) in diameter
centered at 45 latitude in each hemisphere. The depth of flooding
is 250 m. The snapshot at 40 days shows nicely developed gyres on
each continent with a peak velocity of about 40 m/s. The identical
case except for a continent diameter of 20 (2220 km), however,
fails to produce any such feature. From these relatively small
number of tests, it is inferred that there is a minimum critical
dimension on the order of 2500 km below which this phenomenon does
not occur.

DISCUSSION

The tendency for strong cyclonic gyres to form above flooded
continents, particularly at the higher latitudes, appears to be a
robust characteristic of the earth's ocean. Physically, these
patterns of flow arise because of the earth's rotation and are
influenced by the fact that the Coriolis parameter varies with
latitude and that transport of absolute vorticity is a strongly
nonlinear process. The absolute vorticity is defined as z + f, where
z=curl u is the fluid vorticity in the shallow water limit and f is
the Coriolis parameter. To good approximation the transport of
absolute vorticity is described by the vorticity equation [5, p. 95]

d(z+ f)/dt=- (z+ f) u (3)

For the situation of constant water depth, the velocity divergence
u is zero. Equation (3) in this case requires a parcel of fluid
moving equatorward have its vorticity z increase since f decreases
with decreasing latitude, while the opposite holds for a poleward
moving parcel. Thus there is transfer of vorticity from the earth's
rotation to the fluid and vice versa. Because absolute vorticity
itself depends on the local fluid velocity u through z, the right
hand side of (3) is seen to be nonlinear in terms of velocity, and
the dynamics associated with the vorticity therefore are highly
nonlinear. Equation (3) can also be expressed as the conservation of
potential vorticity, which is defined as (z+f)/h*, where h* is water
depth [5, p. 96]. Since potential vorticity is inversely
proportional to h*, the nonlinear dynamics are accentuated where the
water is shallow. This suggests why high velocity currents arise in
the shallower depths above the continents and not in the regions of
deep ocean. For such strongly nonlinear problems, physical reasoning
can only take one so far, and numerical simulation is generally the
most practical means for obtaining deeper understanding of the
system dynamics.

Calculations described above suggest there exists a minimum diameter
for the strong gyres on the order of 2500 km. This seems to account
for the absence of such features today since the present continental
shelves are much narrower than this minimum scale length. On the
other hand, the sedimentary record indicates there have been several
major, apparently global, transgressions of the ocean over the
continents since the beginning of the Cambrian period. In the case
of North America, Sloss [6] has described six such major
transgressions. The basal formation of the second transgression that
occurred during the Ordovician is composed of extremely pure quartz
sand and known as the St. Peter Sandstone [2, pp. 221-225]. It is
readily identified and covered a large fraction of North America.
The continental areas flooded by epeiric sea during such major
transgressions easily exceed the 2500 km dimension. It is difficult
then to imagine how strong currents such as observed in the
numerical experiments did not arise and play a major role in
development of the sedimentary record. Given the fact that
catastrophic cavitation occurs for water velocities generally above
30 m/s [3], one would also expect severe and rapid erosion to be
associated with any major transgression of the continents by the
ocean. These phenomena relating to very fundamental aspects of
sedimentary geology, to the authors' knowledge, have never before
been considered or addressed in the scientific literature.

CONCLUSIONS

Numerical solution of the shallow water equations on a rotating
sphere with parameters appropriate to the earth and flooded
continental topography yields closed patterns of flow with
velocities of 40 to 80 m/s and length scales typically 2500-5000 km
above the flooded continental regions. Such currents would be
expected to arise in the context of a global Flood as described in
Scripture when "all the high mountains everywhere under the heavens
were covered" with water (Gen. 7:19). The ability of such currents,
combined with cavitation, to erode huge volumes of rock and also to
transport the resulting sediment and distribute it over extensive
areas in a short span of time not only helps to satisfy the Biblical
time constraints for the Flood but also appears to be able, in a
general sense, to account for the continent-scale extent of many
Paleozoic and Mesozoic sedimentary formations as well as evidence in
many of these rocks for high energy water transport. Clearly,
calculations with more detailed and realistic topography that is
allowed to change with time are the next step in this research
program. If such calculations prove to be able to reproduce some of
the primary features of the sedimentary record, then confidence that
the Phanerozoic portion of geological history is indeed a
consequence of Noah's Flood should spread substantially beyond its
present bounds.

REFERENCES


[1] M. E. Clark and H. D. Voss, Resonance and Sedimentary Layering
in the Context of a Global Flood, Proceedings of the Second
International Conference on Creationism, R. E. Walsh and C. L.
Brooks, Editors, 1991, Creation Science Fellowship, Inc.,
Pittsburgh, PA, Vol. 2, pp. 53-63.

[2] R. M. Dott, Jr. and R. L. Batten, Evolution of the Earth, 2nd
Ed., 1976, McGraw- Hill, New York.

[3] E. W. Holroyd, Cavitation Processes During Catastrophic
Floods, Proceedings of the Second International Conference on
Creationism, R. E. Walsh and C. L. Brooks, Editors, 1991, Creation
Science Fellowship, Inc., Pittsburgh, PA, Vol. 2, pp. 101-113.

[4] S. S. Hough, On the Application of Harmonic Analysis to the
Dynamical Theory of the Tides, Philosophical Transactions of the
Royal Society, 119(1897), pp. 139-185.

[5] J. T. Houghton, The Physics of Atmospheres, 1977, Cambridge
University Press, Cambridge.

[6] L. L. Sloss, Sequences in the Cratonic Interior of North
America, Geological Society of America Bulletin, 74(1963), pp.
93-114.

[7] A. Staniforth and J, Cote, Semi-Lagrangian Integration Schemes
for Atmospheric Models--A Review, Monthly Weather Review,
119(1991), pp. 2206- 2223.

[8] D. L. Williamson, J. B. Drake, J. J. Hack, R. Jakob, and P. N.
Swatztrauber, A Standard Test Set for Numerical Approximations to
the Shallow Water Equations in Spherical Geometry, Journal of
Computational Physics, 102:1(1992), pp. 211-224.