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Doron Nof

Using hydrographic data and box models, it is shown that the presently discussed diversion of rivers such as the Yellow or the Yangtze for agricultural use is likely to cause the renewal of Bottom Water formation in the Japan/East Sea. Such formation was common (near the Siberian coast) in the 1930s, 1940s, and 1950s, but subsided since that time due to a warming trend (accompanied by a decreased salinity due to the melting of ice). Since a diversion of freshwater is analogous to evaporation, a (diversion induced) increase of salinity is expected and the increase is large enough to allow Bottom Water formation even at the present-day cooling rates. Even a modest diversion of “merely” 3000 m3 s−1 (which is 10% of the total freshwater flux) will probably cause Bottom Water formation at a rate of roughly 750 000 m3 s−1. This is the first study that predicts anthropogenic reversal of an existing vertical structure in a semienclosed sea.

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Doron Nof

Abstract

The behavior of an isolated pair of vortices consisting of two eddies situated on top of each other in a three-layer ocean is examined analytically. The amplitudes of both eddies are high and, consequently, the two eddies behave as one unit and migrate together in the ocean. For this reason, it is proposed to call the system joint vortices. The eddies are of equal or opposite sign; each vortex is situated in a different layer so that there are two active layers and one passive layer.

Attention is focused on the behavior of joint vortices on a sloping bottom in the deep ocean and on a β plane in the upper ocean. That is, we consider deep joint eddies situated on an inclined floor in the lowest two layers of a three-layer ocean and upper joint eddies in the upper two layers. Special attention is given to the cases where one of the vortices is a lens-like eddy. Approximate solutions for slope (or β) induced drifts in the east-west direction are obtained.

It is found that because of the high amplitudes and the resulting nonlinear coupling, the joint eddies have a mutual drift which is very different from the drift that each individual vortex would have. For example, while each individual vortex translates to the west in the absence of a conjugate vortex, the combined vortices may drift steadily to the east. This bizarre behavior stems from the presence of a “planetary lift” which is the oceanic equivalent of the side pressure force associated with the so-called Magnus effect. It is directed at 90° to the left of the drifting eddies.

Other results of interest are: (i) Under some conditions, the westward drift of joint eddies consisting of two cyclonic vortices it much faster than the long-wave speed. Such fag drift contradict previously held contentions that the speed of cyclonic eddies cannot exceed the long wave speed. (ii) As it translates westward, an anticyclonic lens-like eddy can carry a Taylor column on top of it.

Possible application of this theory to various eddies in the ocean is discussed.

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Doron Nof

Abstract

A simplified two-layer analytical model describing the interaction between a longshore current and a thin lenslike eddy is considered. The eddy is situated near a vertical wall and is embedded in a frictional boundary current which is flowing from one latitude to another. Attention is focused on the conditions under which the boundary current compensates for the tendency of the eddy is drift due to β so that the eddy is stationary. The model incorporates movements resulting from the circulation within the eddy, the longshore flow and β. Both the upper and lower layer are taken to be active; diffusion is neglected but bottom friction is included. Although our model is simplified, the movements within the eddy are not constrained to be quasi-geostrophic, in the sense that the Rossby number can be relatively large and the interface surfaces at a finite distance from the center. The desired solutions are constructed analytically.

It is found that a thin lenslike eddy adjacent to a western boundary can remain in a fixed position if the current in which it is embedded is flowing from low to high latitudes at a (“critical”) speed which depends on β, the inclination of the coastline, the frictional coefficient along the bottom of the ocean and the eddy's size, intensity and volume. Presumably, a northward flowing current whose speed is ten than “critical” will allow the eddy to drift upstream (southward), whereas a current whose speed is stronger than the critical will sweep the current downstream (northward).

In contrast to western boundaries, thin eddies embedded in eastern longshore flows can never be stationary regardless of the current's characteristics. This difference between western and eastern boundaries exists because as the current flows, it exert two forces on the eddy. One is parallel to the coastline (and can compensate for the eddy's β-induced force) and the other is perpendicular to the wall. In the western boundary case, the cross-stream force pushes the eddy toward the boundary causing it to lean against the wall. In the eastern boundary case, on the other hand, the force pushes the eddy away from the wall causing it to accelerate toward the open ocean.

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Doron Nof

Abstract

Shock waves are discontinuities (in the physical properties of a fluid) which behave in an organized manner. The possibility that such waves may occur in oceanic boundary currents is examined with a nonlinear two-layer analytical model. Attention is focused on separated boundary currents (i.e., light currents whose lower interface strikes the free surface or heavy currents whose upper interface intersects the floor) with zero potential vorticity. The shocks result from an increase in the upstream transport; they correspond to abrupt and violent changes in depth and velocity accompanied by a local energy loss. Nonlinear solutions for steadily translating shocks are constructed analytically by connecting the upstream and downstream fields without solving for the complicated region in the immediate vicinity of the shock.

It is found that, while stationary shocks are impossible, steadily propagating shocks can always occur. There are no special requirements on the boundary currents in question and the only necessary condition for steadily advancing shocks to occur is that the upstream depth is increased. Once formed the shocks propagate downstream at a speed greater than that of a Kelvin wave associated with the increased up-stream flow.

Possible application of this theory to the Mediterranean outflow is discussed. For this purpose, the results of the two-layer model are extended to a three-layer model corresponding to a wedge-like boundary current “sandwiched” between two infinitely deep layers. With the aid of this model it is suggested that the abrupt changes in temperature and depth observed in the Mediterranean outflow are a result of a shock wave advancing downstream. The observed changes in this region are so abrupt and violent that no other known kind of wave ran explain them.

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Doron Nof

Abstract

In this paper an analytical method is proposed for calculating the nonlinear β-induced translation of isolated baroclinic eddies. The study focuses on frictionless anticyclonic eddies with a uniform anomalous density and a lens-like cross section which translates steadily in a resting Ocean. The depth of these eddies vanishes along the outer edge so that as they translate westward their entire mass anomaly is caused along with them.

The proposed method for calculating the translation speed incorporates the nonlinear equations of motion in an integrated form and a simple perturbation scheme. It relates the translation of the eddy to its intensity, size and volume, but requires only an approximate knowledge of the corresponding numerical values.

The power and usefulness of the proposed method is demonstrated by its application to a class of simply-structured eddies whose swirl velocity increases monotonically with the distance from the center. It is found that the translation of these eddies is considerably smaller than that of a simple Rossby wave. A small Rossby number eddy whose swirl velocity increases monotonically with the distance from the center translates westward at approximately ⅓βR d 2 (where R d , is the deformation radius), whereas the most nonlinear eddy (whose negative relative vorticity approaches the vorticity of the earth) translates at ⅔βR d 2 .

The proposed method is tested by its application to more complicated anticyclonic eddies representing those shed by the Loop Current in the Gulf of Mexico. For these eddies, the predicted westward translation speed is 0.32βR d 2 which agrees very well with both numerical experiments and field observations.

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Doron Nof

Abstract

The interaction of two isolated lens-like eddies is examined with the aid of an inviscid nonlinear model. The barotropic layer in which the lenses are embedded is infinitely deep so that there is no interaction between the eddies unless their edges touch each other. It is assumed that the latter is brought about by a mean flow which relaxes after pushing the eddies against each other and forming a “figure 8” structure.

Using qualitative arguments (based on continuity and conservation of energy along the eddies’ edge) it is shown that, once a “figure 8” shape is established, intrusions along the eddies’ peripheries are generated. These intrusions resemble “arms” or “tentacles” and their structure gives the impression that one vortex is “hugging” the other. As time goes on the tentacles become longer and longer and, ultimately, the eddies are entirely converted into very long spiral-like tentacles. These spiraled tentacles are adjacent to each other so that the final result is a single vortex containing the fluid of the two parent eddies. It is speculated that the above process leads to the actual merging of lens-like eddies in the ocean.

Because of the inherent nonlinearity and the fact that the problem is three-dimensional (x, y, t), the complete details of the above process cannot be described analytically. Therefore, one cannot prove in a rigorous manner that the above process is the only possible merging mechanism. It is, however, possible to rigorously show analytically and experimentally that the intrusions and tentacles are inevitable. For this purpose, one of the interacting eddies is conceptually replaced by a solid cylinder. Initially, the cylinder drifts toward the eddy; subsequently, it is pushed slightly into the eddy and is then held fixed. The subsequent events are examined in a rigorous mathematical and experimental manner.

It is found that as the cylinder is forced into the eddy, a band of eddy water starts enveloping the cylinder in the clockwise direction. This tentacle continues to intrude along the cylinder parameter until it ultimately reattaches itself to the eddy, forming a “padlock” flow. Simple laboratory experiments on a rotating table clearly demonstrate that a “padlock” flow is indeed established when a lens is interacting with a solid cylinder. Using the details of this process it is argued that, in the actual eddy–eddy interaction case, intrusions must be established and that, consequently, merging of the two eddies is inevitable.

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Doron Nof

Abstract

The behavior of outflows resulting from channels cutting through broad continents and emptying into wedgelike oceans, or channels cutting in wedgelike continents and emptying into broad oceans, is examined analytically. The model is nonlinear and inviscid, and the vertical structure is approximated by two layers; the upper layer is active and the lower is passive.

Examination of the governing equations shows that, since outflows are externally driven (by gravity and mass flux), there exists an “outflow length scale” in the open ocean. This length scale (l) is given by [gHb/fU 0]½, where b is half the emptying channel width, g′ the “reduced gravity,” H the channel depth, f the Coriolis parameter, and U 0 the flow speed within the channel. Solutions are constructed using this new length scale and a power series expansion.

It is found that, due to the earth's rotation, an outflow can be deflected toward one of the coasts or bifurcate into two branches, depending on the basin geometry. When the outflow results from a channel cutting through a broad continent and emptying into a wedgelike ocean, there are two possibilities. If the wedge opening is less than 90°, the outflow deflects to the right (looking downstream); if the wedge opening is larger than 90°, the outflow deflects to the left. In contrast, when the channel is cutting through a deltalike continent and emptying into a broad ocean, the outflow bifurcates. If the angle between the two walls bounding the ocean is less than 270°, the outflow splits into a narrow band that flows to the right and a broad current that veers to the left and penetrates into the ocean interior as an isolated ocean. A mirrored picture is established when the angle between the walls is large than 270°.

Possible application of this theory to the two outflow modes observed near the Tsugaru Strait is mentioned.

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Doron Nof

Abstract

A nonlinear one-layer model is considered in order to describe the way that water with a relative vorticity intrudes into an otherwise stagnant channel. The channel has a uniform depth (D) and width (L) and the fluid is taken to be inviscid. The intruding fluid is separated from the (initially stagnant) water in the channel by a free dividing streamline that corresponds to a “vorticity front.” This front intersects the channel wall (at the head of the intrusion) and extends backwards upstream. As the fluid with relative vorticity is intruding into the channel, the fluid with no relative vorticity (i.e., the fluid present in the channel prior to the intrusion) escapes in the opposite direction. This flow compensates for the fluid displaced by the advancing intrusion. Solutions for steadily propagation intrusions are obtained analytically by equating the flow-force ahead of and behind the for steadily propagating intrusions are obtained analytically by equating the flow-force ahead of and behind the intrusion. Namely, steady state solutions correspond to a balance between the forward momentum flux and the form drag exerted on the intrusion by the escaping fluid. The nature of the intersection of the front with the wall is analyzed by methods similar to those employed by Stokes for analyzing the maximum steepness of surface gravity waves.

It is found that the vorticity in the intruding fluid “controls” the amount of fluid that flows through the channel. When the vorticity (ζ) of the intruding fluid is uniform, the width of the intrusion is always 2/3 of the channel width and the net volume flux of the intruding fluid is (2/27)ζDL 2. In the presence of weak dissipation, the channel can can transfer an amount less than (2/27)ζDL 2, but, under no circumstances can the channel the so-called hydraulic control {∼O[(gD)½ DL]}, which corresponds to the flux of an intrusion without any relative vorticity. When ζ∼O(f), the ratio between the maximum flux allowed by the vorticity control to the flux allowed by the hydraulic control is equivalent to about 1/10 of the ratio between the channel width and the barotropic deformation radius. Hence, for midlatitude channels, the vorticity control may limit the flux to a few percent of that associated with the hydraulic control.

Possible application of this theory to various oceanic situations is mentioned.

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Doron Nof

Abstract

Organized depth discontinuities involving a balance between steepening and dissipation are usually referred to as shock waves. An analytical “educed gravity” model is used to examine a special kind of shock wave. The wave under study is a depth discontinuity associated with a transition between a supercritical and subcritical flow in a channel. Even though the wave itself is highly nonlinear, the adjacent upstream and downstream fields are exactly geostrophic in the cross-stream direction. For this reason we term the wave a geostrophic shock wave. We focus on a stationary shock wave whose horizontal projection is a straight line perpendicular to the side walls. Solutions for the entire field are constructed analytically using power series expansions and shock conditions equivalent to the so-called Rankine-Hugoniot constraints.

It is found that, for particular upstream conditions, a geostrophic shock wave can be formed if the particle speed exceeds the surface gravity wave speed (i.e., the flow is “supercritical”). Specifically, in addition to supercriticality, a stationary geostrophic wave requires the upstream velocity to have a particular structure which depends on the strength of the shock and the channel width. When the latter condition is not met, a shock wave is still possible, but its adjacent fields will not be geostrophic and its shape will correspond to an “S” rather than a straight line.

Being the only known analytical solution for the entire field of shock waves on a rotating earth, the geostrophic shock provides useful information on the wave structure. For instance, it is shown that even though momentum is conserved across the shocks, relatively large changes in potential vorticity take place. For depth discontinuity of O(I) (i.e. high “amplitudes”), there is a generation of potential vorticity that is also of O(I). Such a phenomenon does not occur on a nonrotating plane where the (zero) potential vorticity may be altered through the action of shock waves in channels and passages. Possible application of this theory to various oceanic situations is mentioned.

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Doron Nof

Abstract

In this brief note it is demonstrated that the question of what is the mechanism(s) responsible for the southern migration of the Gulf Stream front during wanter–is still open.

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