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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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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

The breakup of a long strip of dense fluid flowing over a sloping bottom is examined with the aid of a nonlinear two-layer analytical model. The inviscid strip is bounded by the sloping bottom from below and an interface (that intersects the bottom along the two edges) from the top. The infinitely deep upper layer in which the filament is embedded contains a uniform flow and is taken to be passive. Such flows represent an idealization of currants that result from various outflows and deep water spreading.

It is shown analytically that a dense filament can break up to a discrete set or closely packed anticyclonic eddies (lenses) propagating steadily along the isobaths. The lenses are arranged in a zig-zag manner with the e4cs of each tens touching its neighboring tens. Such a pattern results from the fact that the eddies are too large to fit into the area freed by the straight filament so that they push each other to the sides during the breakup. The solution for this pack of eddies is computed without solving for the detailed breakup process. As in other adjustment problems, the final and initial states are connected via known conservation properties even though the problem is highly nonlinear. Specifically, conservation of potential vorticity, integrated angular momentum and mass am applied. These conservation laws illustrate that about 10% of the initial energy is ,radiated away (via long gravity waves) during the breakup.

The theory suggest that some of the actual filaments in the ocean, such as the Mediterranean outflow, may not consist at a single continuous flow but rather of a stream of closely packed lenses translating steadily along the bottom.

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Doron Nof

Abstract

In this paper a mechanism is proposed which could be responsible for the formation of sharp horizontal density gradients such as those observed in shallow seas, from fluid which initially has weak horizontal density gradients. The sharp density gradients result from the mutual intrusion of several stratified bodies of water which were exposed to various degrees of vertical mixing for a limited amount of time. The dynamics of the intrusion are examined by a simplified nonrotating, frictionless multilayer model. The results are compared quantitatively to laboratory experiments and qualitatively to field observations.

The theoretical model contains an upper and lower portion, each of which consists of several bodies of fluids with different densities corresponding to various degrees of mixing. It predicts that in both the upper and the lower portions, fluids which were exposed to intermediate mixing sink rapidly from the surface, rise from the bottom, and after a finite amount of time concentrate in mid-depth. This results in a formation of density discontinuities (fronts) near the surface, bottom, and in the boundary between the upper and the lower portions.

Rotation is excluded from the simplified model, but it is expected that mutual intrusion will take place even if rotation is included, provided that the flow is not in an exact geostrophic balance. The theoretical predictions were tested in the laboratory in a tank which contained several bodies of water with different densities separated initially by a number of gates. The experimental results compare favorably with the theoretical predictions. Observations which suggest the existence of mutual intrusion in frontal zones are discussed.

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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

A frictionless nonlinear model with allowance for motions which are far from a state of geostrophic balance is considered in order to describe the dynamics of outflows consisting of two layers of fluids. The governing equations are solved by means of perturbation expansions, conformal mapping and Fourier series. The theory is compared with laboratory experiments.

The model predicts that an outflow from a channel with uniform velocity distribution deflects to the right in the Northern Hemisphere. The parameters of the problem are combined in such a way as to show that rotational effects are important whenever the ratio between the internal Froude number to the Rossby number is not negligible; the inverse of this ratio has a “critical” value, below which the flow separates from the left basin bank. The mathematical analysis shows that an outflow from a channel with initial negative relative vorticity approximately equal to the Coriolis parameter deflects to the left. As in the uniform flow case the flow separates from one of the banks under certain “critical” conditions.

Two experimental systems which included an abrupt cross-sectional variation in a rotating channel consisting of two layers were used. The experimental results compare favorably with the direction of deflection predicted by the mathematical model. Possible application of this study to the Straits of Gibraltar and other outflows are discussed.

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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

The author considers two oceanic basins separated by a meridional wall. The wall contains a gap that is initially blocked by a gate; westward winds are allowed to blow over the two-layered oceans creating western boundary currents and a sea level difference between the basins. The conceptual gate is then removed and the resulting nonlinear flow is computed.

The analytical calculations are based on a simple wind-driven general circulation model and a nonlinear integrated momentum constraint. Two classes of nonlinear solutions are constructed. One corresponds to a situation where the flow through the gap originates from the right-hand side (looking upstream) of the inner Pacific basin and the other to a situation where the flow originates from the left-hand side. It is suggested that the actual Indonesian Throughflow is composed of both of these classes of flows; that is, the throughflow corresponds to an exchange via two adjacent gaps.

Computations suggest that approximately 6 Sv (Sv ≡ 106 m3 s−1) enter the passages from the North Pacific and 1 Sv from the South Pacific giving a total of 7 Sv. This may resolve the apparent difficulty associated with existing linear theories (and nonlinear theories that neglect western boundary currents), which predict that without strong turbulent diffusion only South Pacific water can enter the passages.

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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

The analytical results for the splitting conditions of isolated barotropic eddies and the associated final equilibrium state are extended to: 1) nonlinear baroclinic eddies; 2) a group of four nonlinear closely packed eddies, two of which are cyclonic and two of which are anticyclonic (i.e., multiple eddies); and 3) joint nonlinear eddies (i.e., a system consisting of two eddies situated one above the other). The final equilibrium state associated with the group (of four) fission is related to a nonlinear version or geostrophic turbulence and, therefore, is referred to as ageostrophic turbulence.

Taking into account that inviscid fission may involve loss of energy via waves radiation, the breakup process is examined by conserving integrated angular momentum, potential vorticity, and mass. The analytical expressions for the conservation of these three properties provide a set of algebraic equations that are solved numerically.

For baroclinic eddies embedded in an infinitely deep lower layer, it is found that, as in the barotropic case, only intense cyclones can break up. This results from the fact that, despite the large amplitude of the nonlinear baroclinic eddies, the offspring are still forced a considerable distance away from their original prebirth center of rotation as is the case with the barotropic eddies. This causes a large gain in angular momentum implying that only eddies whose angular momentum is relatively large to begin with are capable of being potential parents. Again, as in the barotropic case, it turns out that only intense cyclones have large enough angular momentum to allow splitting (because the cyclonic orbital speed is in the same direction as the earth's rotation).

In ageostrophic turbulence, the cyclones break up and the anticyclones merge. Namely, the fission of the cyclones provides the energy necessary for the fusion of the anticyclones. Hence, the final result is a nonlinear system resembling a “Mickey Mouse”(with one large anticyclone and four small cyclones) whose total energy is identical to the total initial energy prior to the fission.

The impossibility of baroclinic anticyclones to break up appears initially to be in contradiction with classical laboratory experiments which show what seems to be an anticyclonic fission. The solution for joint eddies consisting of a cyclone situated above an anticyclone suggests, however, that what really really breaks up in the laboratory is the cyclone on top rather than the anticyclone underneath. (Recall that the generation of anticyclone in the laboratory is often accompanied by the formation of a cyclone on top due to convergence.)

It is suggested that the impossibility of baroclinic anticyclones to break up and the tendency of cyclones to split may provide an explanation for the relative abundance of anticyclones in the ocean.

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