3.6. Simulation of Topographic Cyclogenesis
Vortices are understood in oceanology as dynamic formations characterized by closed lines
of flow. Oceanic vortices range over a great variety of space and time scales, from quasi-stationary
circular rings of a planetary scale to short-lived micro-eddies releasing viscid dissipation of energy.
Many factors may be responsible for vortex formation. This paper discusses the vortices formed in
oceanic flows under the impact of local topographic features of seamounts (Kozlov et al. 19821).
Much more is known about topographic cyclogenesis based on work by theoreticians and
laboratory experimentalists than from instrumental observations in the ocean. The appearance of
column-shaped vortical disturbances in the flow of a rapidly rotating homogeneous fluid was
theoretically prognosticated in 1916 by Proudman (1916), and shortly after that corroborated by
Taylor’s laboratory experiments in 1917 (Taylor 1923).
R. Hide (1961) was the first to note possible geophysical applications of the discovered
effects and to introduce the term Taylor column. Ingersoll (1969) gave a simple analytical solution
for a stationary obstacle shaped as a right circular cylinder. In the case considered by Ingersoll, a
Taylor column of infinitesimally small, and therefore negligible, viscidity degenerated into a
stagnant area. The effect of water stratification was investigated in detail by N. Hogg (1973), who
demonstrated that, given an invariant buoyancy frequency, vortical disturbance takes the shape of a
cone (Taylor cone). Later, V. N. Zyryanov (1985, 1995) used the term Taylor–Hogg cone. The
conditions generating topographic vortices over axially symmetric disturbances in bottom relief
were analyzed for homogeneous and stratified fluids by Huppert (1975). The influence of the beta
effect, accompanied by the formation of leeward Rossby waves, was studied by McCartney (1975)
and Johnson (1977, 1978).The process of formation of a topographic vortex and an accompanying
free vortex in a stratified flow, which originate from a condition of rest and then become steady, was
studied by Happert and Bryan (1976) using numerical and schematic simulation. The stationary
configurations of topographic and free vortices in a homogenous fluid were discussed by Johnson
(1978) and Kozlov (1981).
Although the above is far from an exhaustive account of recent research into topographic
cyclogenesis, these studies showed the principal mechanism of the generation of vortices over
seamounts, based on the law of potential vortex preservation. Detailed information on the theory of
1
Kozlov, doctor of physical and mathematical sciences, authored the theoretical portion of the work set forth in Kozlov
et al. (1982).
Taylor vortices and Taylor–Hogg cones can be found in the monographs by V. F. Kozlov (1983)
and Zyryanov (1985, 1995).
The identification of topographic vortices in the real ocean is not an easy process for two
main reasons: the non-stationary nature of real vortex formations, and a dearth of direct instrumental
measurement data on the three-dimensional structure of flow fields. Various indirect signs, as well
as dynamic computations of
over underwater mountains and in their environs, reveal the
existence of quite complicated vortical structures that are not completely specified by simple
theoretical descriptions. In order to interpret an observed vortex as topographic, theoretical models
are required that are close to the oceanographic situation under observation, in all key parameters.
From this viewpoint, the most successful example of a theoretical analysis was Hogg’s attempt to
explain certain topographic effects observed in the Mediterranean (Hogg 1973).
In theoretical studies of topographic vortices (specifically in the papers mentioned above),
density stratification is commonly characterized by constant buoyancy frequency; the vertical
structure of the accumulating flow’s velocity is approximated by a linear function (in most cases
assumed to be constant); and schematized forms of bottom relief, such as cylinder, cone,
hemisphere, etc., are studied.
Here we discuss the construction of a stationary model of topographic cyclogenesis for a
situation where stratification, velocity of the accumulating flow, and ocean bottom configuration are
prescribed by analytical relations that are closer to the true distributions of oceanological features, as
they are in the model of Type 2 geostrophic motion (Monin et al. 1974) presented by Kozlov and
Sokolovski (1978) and Kozlov et al. (1979). All the key parameters were assessed against field data
obtained from oceanographic survey of a seamount in the Emperor Ridge, Mt. Milwaukee. The
morphological peculiarity of the bottom relief here is the existence of two tops that rise into the
upper baroclinic layer of the ocean. Two-topped mounts and guyots are common all over the Pacific
in various geological units (Description 1989). The results of computation of the pressure, density,
and horizontal velocity fields indicate anti-cyclonic topographic vortices encompassing the feet of
both tops, and also a column-shaped disturbance in the density field above them.
Consider a stationary situation with a small disturbance caused by a given bottom shape that
influences the stratified flow of a rotating fluid with vertical and horizontal displacement of the
velocity. The impact of wind, bottom friction forces, and beta effect are neglected here. The
situation is formulated in dimensionless variables; the z axis in a Cartesian coordinate system is
directed vertically down from the undisturbed oceanic surface.
.
2
In a quasi-geostrophic approximation, the potential vortex preservation equation and vertical
boundary conditions take the forms (Hogg 1973):
J (P, ΔP + 1/B2 [Pz/N2] z) = 0
(1)
J (P, Pz/B2N2)
Z=0
(2)
J (P, Pz/B2N2 - σh)
Z=0
(3)
where P(x,y,z) is dimensionless disturbance of pressure; h(x,y) and N(z) are prescribed
dimensionless elevations of the bottom; and buoyancy frequency, J, and Δ are the Jacobi and
Laplace horizontal operators, respectively. The inferior indices designate the respective partial
derivatives. The three principal dimensionless parameters included in these relationships are
1) the Burger number defining the stratification
B = H*N*/Ω*L*
2) the topographic parameter
σ = h*/H*ε
3) the Rossby number
ε = U*/L*Ω*
expressed in terms of the values of ocean depth H*, Vaisaala–Brent buoyancy frequency N*,
velocity of the accumulating flow U*, horizontal extent of the bottom configuration disturbance L*,
the bottom configuration disturbance h*, and the coriolis parameter Ω*. Strictly speaking, Equation
(3) is based on the assumption that σ = 0 (1), i.e. h*/H* = 0 (ε); the assumption that the current is
quasi-geostrophic suggests that ε « 1.
To assess the distinctive parameters of the situation, we employed standard bathometric
measurement data recorded in the vicinity of Mt. Milwaukee by the Survey Floating Station (SFS)
Equator on December 26–30, 1975. The topography of the bottom and the layout of the stations
taken in the vicinity of the mount are shown in Figure
. The vertical density distributions within
the 0–1000 m layer, measured at 16 stations, were averaged; then with the help of a spline
differentiation of the obtained average profile, the buoyancy frequency N(z) was calculated and
approximated in the functions class (a similar procedure is described in Kozlov and Sokolovsky
1978):
N(z) = 1/1+ γz
(4)
At the scales of H* = 1000 m and N* - 10-2 sec-1, the average buoyancy frequency
determined by Equation (4) with γ = 2.2 was computed from the observed data (Figure 4, solid line)
and model frequency (Figure 4, dotted line); the approximation is assumed to be quite satisfactory,
with the exception of a thin quasi-homogenous surface layer.
3
The dynamic method of computing horizontal velocities relative to a 1000 m surface
revealed a dominant northward flow with which the z axis was aligned (axis y has a westward
direction). The current field had a noticeable transverse displacement and strengthened increasingly
from west to east. For its infinity approximation the relationship of the form
U(y,z) = U(z) (γ0 +γ1y)
was calculated with parameters γ0 and γ1 constant:
U(z) = 1+ γ/1 + γZ
(5)
A monotonic slackening of the flow with depth, defined by Equation 5, reproduces very well the
changes computed with the dynamic method.
Each top was approximated by an axially symmetrical obstacle of the algebraic form
[1 - (r/r0)2]2 with 0 < r < r0 :
h(r) = h0 {
(6)
0
r › r0
where r0 is radius on the artificial bottom at a depth of H* = 1000 m (Figure
). The choice of
such a model bottom relief was conditioned principally by a dearth of hydrological evidence from
below the indicated horizon. .
Let’s assess the key parameters of the situation (apart from the above-mentioned H* and N*)
given the following values: h* = 500 m, L* = 10 km, U* = 10 cm/sec, and * = 2sin 32° = 7.73 
10-5 sec-1. The computation yields  = 0.13,  = 3.85, B = 12.9, and b = βL*2/U* = 1.94  10-2. The
small magnitude of the parameter  justifies the use of the quasi-geostrophic approximation. The
fact that the topographic parameter  noticeably exceeds 1, generally speaking, casts some doubt on
the fact that transport of the boundary conditions to the undisturbed bottom is possible. However,
good results obtained in a situation similar to this one (Hogg 1973) can serve as some justification in
our case. As B2 » 1, the investigated case belongs, according to Hogg’s terminology (Hogg 1973), to
the class of strongly stratified flows. And finally, the small value of the planetary parameter
indicates that neglect of the beta effect proves its worth.
Employing the above equations, computation of pressure, density, and velocity fields was
carried out for a two-topped obstacle modeling the system of Mt. Milwaukee’s two tops,
Barabinskaya and Astronom, hereinafter referred to as Obstacle 1 and Obstacle 2, respectively. The
following dimensionless values of parameters were employed:
Obstacle
Ξ
4
ηi
r0i
h0i
1
-1.234
-1.068
1.5
1.2
2
1.666
1.444
1.1
1.0
The lines were filled out with the final sums corresponding to the first eleven eigenfunctions.
Certain scalar fields were computed for different horizons and vertical cross sections, and printed
out in the form of isolines.
Let us see now in more detail how the computation was done for the case of the
accumulating flow without a sideways displacement, at γ1 = 0 and γ0 =1. Figure presents computed
pressure fields as isolines of the flow of geostrophic currents, and density disturbance fields for the
horizons z = 0.4 and z = 0.8, i.e. higher and lower, respectively, than the tops of the mounts.
The pressure field in the upper layer is scarcely disturbed (Figure
) and only slightly
differs from an undisturbed flow, while in the lower layer closed lines of flow encompass the clearly
larger Obstacle 1 (Figure
) . On the horizon z = 1, a similar closed circulation is clearly seen to
trace around Obstacle 2. The field of density (and consequently, that of temperature) is a more
sensitive indicator of ocean bed disturbances, with closed isopycnals appearing above the tops of
both mounts (Figure ). With the general decline of the density anomaly from left to right (when
facing downstream, the undisturbed flow), one can see areas of relatively cold water that encompass
both obstacles on the sides and over their tops (Figure ).
Figure
gives an idea of the flow’s vertical structure. On the cross sections perpendicular to
accumulating flow, isotachs of longitudinal velocity (A, B) and isopycnals of the density disturbance
field (Б, Г) are seen passing though the tops of the first (А, Б) and second (В, Г) obstacles. It is
clearly seen that around both mounts, powerful anticyclonic vortices have formed, rising up to about
half the height of the mounts. The wavy nature of the isopycnals over the obstacles (Figure
) is
probably explained by errors made in the computation, caused by a slow convergence of the series ρ
= ρ2, and due to the use of a limited number of components during the truncation. However, from
the general character of the isopycnals it is clear that the presence of an obstacle reduces the stability
of the density stratification to its right. Possible loss of stability with a flow in real topographic
vortices is also suggested by Eide (1979).
Figure
shows the vertical profiles of a velocity constituent (U) parallel to accumulating
flow, as computed on the cross section going through Obstacle 1 at the points (ξ1, 0) and (ξ1, -2)
respectively to the left and right of the mountain; one can clearly see the effect of the anti-cyclonic
vortex that strengthens the flow near the bottom on the left (A) and reverses its direction on the right
5
(Б). The wavy nature of the curves may also be explained by inaccurate truncation of the series
representing the velocity constituents.
Similar computations were done for the accumulating flow with a transverse velocity
displacement. In terms of the dynamic topography, it was assumed that γ0 = 1, and γ1 = -0.22, with all
other parameters constant. The results showed that the qualitative character of topographic
disturbance in flow and density fields remained unchanged, with some quantitative changes caused
by the general strengthening of the accumulating flow to the right and its slackening to the left.
The lack of instrumental observations across the area can be offset by indirect measurements
of standard oceanographic variables, which evidence a blocking effect conditioned by topographical
vortices in the vicinity of Mt. Milwaukee. The ring-shaped lines of flow in the pressure field around
the mounts’ tops and the vertical disturbances of the isopycnals over the tops, obtained from the
numerical experiment, should manifest themselves in the real ocean in the form of anomalies of the
water masses near the tops, compared to the background conditions of the North Pacific waters.
Micrography of the Mt. Milwaukee area found such anomalies in oceanographic characteristics.
A multinuclear vortical structure in the temperature field was observed to exist from the
surface down to the 1000 m horizon, with maximum intensity in the 100–800 m thick layer,
evidenced by increase of temperature horizontal gradients between the nuclei of the cold and warm
waters.
The vortical structure of the field of mass in the vicinity of the tops could also be traced in
the distribution of salinity between the surface and the last horizon observed. In the 100 –1000 m
layer, isolated areas with increased salinity values were observed. Positive anomalies increased from
0.4‰ at a depth of 1000 m (35.0‰–34.6‰), to 0.5‰ on the 300 m horizon (34.9–34.4‰), to 0.7‰
over the mounts’ tops (34.8‰ – 34.1‰). The anomalies did not retain their positions strictly above
the centers of the tops through all the layers, but gravitated toward the tops. Along the slopes, lowsalinity intermediate waters were noticed to move into the upper horizons. It is notable that to the
south of Japan, oceanologists observing captured vortices over mounts Daiiti and Daini–Kinan have
also observed anomalous ring-shaped areas of high salinity in the vortex centers over the tops
(Konaga and Nishiyma 1978).
Hydrochemical characteristics are much less dependent on stratification of waters, and can
often serve as good tracers of water dynamics. The distribution of the biogenic elements phosphates
and silicon was surveyed on a section across the southern top of Mt. Milwaukee (Figure
elements were distributed in the form of a dense column over the mount’s top, where their
6
). These
concentration exceeded background by several times. High contents of silicon were observed in the
100–900 m layer, but in the upper 100 m layer, there were much lower concentrations, approaching
analytical zero.
A dense column of phosphates was observed at depths between 200 and 700 m in the layer
encompassing the top and extending over it. In the upper 200 m layer phosphate concentration was
typical of the undisturbed ocean, monotonically increasing with depth and with isolines oriented
horizontally. It follows that the greater the topographic effect, the less traceable it is, which was also
corroborated by a numerical experiment.
The distribution of oxygen also testified to a blocking effect: in the 200–700 m layer the isolines
with the same percentage of oxygen registered a “splash” over the mount’s top. The dissolved
oxygen had closed lens-shaped nuclei oriented vertically over the mount’s top, which is not typical
of an undisturbed flow. The periodic descent of surface waters over the tops of the mounts is
evidenced by the nuclei of positive oxygen anomalies, which reached 1.5–4.5 ml/l, observed in the
100–800 m layer in the vicinity of the mountains. The distribution of the oxygen anomalies showed
evidence of small-scale turbulence and of biochemical processes, but topographic effects play the
dominant role in the immediate vicinity of the tops.
Under the influence of quasi-cylindrical currents, hydrobionts act as passive tracers and are
distributed vertically in the inner part of the vortex. Biological research showed that the largest
amount of boarfish roe (up to 500 grains per catch using fish roe net IKS-80) was observed near one
of Milwaukee’s tops. Grains of roe, which are passive and have a neutral buoyancy, can also serve
as tracers of dynamic processes; the effect of vortical currents on their distribution is indubitable,
and known to be common in other areas of the World Ocean.
The anomalies observed in the distribution of a number of oceanographic characteristics
have quasi-cylindrical or quasi-conic shapes located over the tops of mountains. Their presence is
not necessarily revealed on the surface, which is one reason why this effect was not noticed for a
long time. The anomalous properties of water masses and their localization in the vicinity of
seamounts are caused by the blocking effect of Taylor vortexes and Taylor-Hogg cones, which
appear from time to time over the tops of the mounts (Kozlov, 1983, Darnitski, 1987, 1991,
Zyryanov, 1985, 1995).
Hydroacoustic echolocation data also bear out the blocking activity of quasi-cylindrical
currents over the tops. During three crossings of one of Mt. Milwaukee’s tops in August 1980,
hydroacoustic equipment registered echoes concentrated only over the top and with a discontinuous
columnar form. Similar echoes of sound-diffusing layers (ESDL) had earlier been registered
7
hydroacoustically by researchers on the SFS Professor Nikolsy over one of Mt. Fiberling’s tops in
the eastern Pacific (Figure ). Echoes were not registered over deep waters near the mount (tacks 4
and 6), but after numerous crossings of the top, ESDL were registered in the 50–370 m layer (Figure
). The ESDL record made over Mt. Fiberling had a vertical orientation only above the mount’s
rocky slopes, and a horizontal one above the flat surface of the top.
Band-shaped, almost discontinuous records of ESDL of significant horizontal duration are
common for the ocean. The specific character of ESDL oriented vertically over a top is
characteristic of seamounts and is apparently caused by the blocking effect of quasi-cylindrical
currents. It is obvious that both hydrological observation and hydroacoustic monitoring of ESDL are
promising avenues for successful exploration of Taylor vortex dynamics.
References Cited
Darnitski. 1987.
Darnitski. 1991.
Description. 1989.
Eide. 1979.
Happert and Bryan. 1976.
Hide. 1961.
Hogg, N. 1973.
Huppert. 1975.
Ingersoll. 1969.
Johnson. 1977.
Johnson. 1978.
Konaga and Nishiyma. 1978.
Kozlov. 1981.
Kozlov, V. F. 1983.
Kozlov and Sokolovski. 1978.
Kozlov et al. 1979.
Kozlov et al. 1982.
McCartney. 1975.
Monin et al. 1974.
Proudman. 1916.
Taylor. 1923.
8
Zyryanov, V. N. 1985.
Zyryanov, V. N. 1995.
9