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- Author or Editor: Melvin E. Stern x

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## Abstract

If an azimuthally symmetric barotropic eddy on the *f* plane is subject to a relatively small amplitude disturbance of unit azimuthal wavenumber (*m* = 1), it can propagate very many diameters away from its origin, as shown by a weak nonlinear theory for a piecewise uniform vorticity eddy, and also for one with continuous vorticity inside a finite area. In the former case an initial value contour dynamical calculation shows that the analytical solution is realizable over long distances; the same is true in the latter case, as shown by spectral calculations using the full two-dimensional vorticity equation (with small dissipation). The oceanographic significance of this effect lies in the ability of *almost* symmetric eddies to self-propagate over large distances and collide with other eddies, currents, and continents; this produces important mixing effects, as illustrated herein. It is also shown how the analysis and the effect is generalizeable to a 1½-layer density model on the *β* plane.

## Abstract

If an azimuthally symmetric barotropic eddy on the *f* plane is subject to a relatively small amplitude disturbance of unit azimuthal wavenumber (*m* = 1), it can propagate very many diameters away from its origin, as shown by a weak nonlinear theory for a piecewise uniform vorticity eddy, and also for one with continuous vorticity inside a finite area. In the former case an initial value contour dynamical calculation shows that the analytical solution is realizable over long distances; the same is true in the latter case, as shown by spectral calculations using the full two-dimensional vorticity equation (with small dissipation). The oceanographic significance of this effect lies in the ability of *almost* symmetric eddies to self-propagate over large distances and collide with other eddies, currents, and continents; this produces important mixing effects, as illustrated herein. It is also shown how the analysis and the effect is generalizeable to a 1½-layer density model on the *β* plane.

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## Abstract

This paper considers the equilibration of lateral intrusions in a doubly diffusive fluid with uniform unbounded basic-state gradients in temperature and salinity. These are density compensated in the horizontal direction and finger favorable in the vertical direction. Previous nonlinear studies of this effect have qualitative and quantitative limitations because of their fictitious parameterizations of the weak “turbulence” that arises. Here, two-dimensional direct numerical simulations (DNS) that resolve scales from the smallest to the intrusive are used to predict the equilibrium state. This is achieved by numerically tilting the *x*–*z* computational box so that the mean intrusion is represented by a mode with no lateral variation, but smaller-scale 2D eddies comparable to the intrusion thickness are resolved. The DNS show that the initial plane wave intrusion evolves to an equilibrium state containing both a salt finger interface and a diffusive interface, surrounded by well-mixed layers. The inversion of the horizontally averaged density in the mixed layer is negligibly small, but the salt finger buoyancy flux produces large transient density inversions that drive the mixed layer convection. For the considered values of horizontal/vertical gradients, the calculations yield small Cox numbers and buoyancy Reynolds numbers [comparable to those measured in staircases during the Caribbean-Sheets and Layers Transects (C-SALT) program]. An important testable result is the time-averaged maximum velocity of the fastest-growing intrusion *U*
_{max} = 18.0 (Σ*_{
z
}/Σ*_{
x
})^{+1/2}
*K _{T}
*(

*g*Θ*

_{ z }/

*νK*)

_{T}^{1/4}. Here Θ*

_{ z }is the undisturbed vertical temperature gradient in buoyancy units, Σ*

_{ z }and Σ*

_{ x }are the corresponding vertical and horizontal salinity gradients,

*g*is the gravity acceleration, and

*ν*and

*K*are the respective values of the molecular viscosity and heat diffusivity. The paradoxical inverse dependence on the horizontal gradient results from the assumption that the latter is unbounded.

_{T}## Abstract

This paper considers the equilibration of lateral intrusions in a doubly diffusive fluid with uniform unbounded basic-state gradients in temperature and salinity. These are density compensated in the horizontal direction and finger favorable in the vertical direction. Previous nonlinear studies of this effect have qualitative and quantitative limitations because of their fictitious parameterizations of the weak “turbulence” that arises. Here, two-dimensional direct numerical simulations (DNS) that resolve scales from the smallest to the intrusive are used to predict the equilibrium state. This is achieved by numerically tilting the *x*–*z* computational box so that the mean intrusion is represented by a mode with no lateral variation, but smaller-scale 2D eddies comparable to the intrusion thickness are resolved. The DNS show that the initial plane wave intrusion evolves to an equilibrium state containing both a salt finger interface and a diffusive interface, surrounded by well-mixed layers. The inversion of the horizontally averaged density in the mixed layer is negligibly small, but the salt finger buoyancy flux produces large transient density inversions that drive the mixed layer convection. For the considered values of horizontal/vertical gradients, the calculations yield small Cox numbers and buoyancy Reynolds numbers [comparable to those measured in staircases during the Caribbean-Sheets and Layers Transects (C-SALT) program]. An important testable result is the time-averaged maximum velocity of the fastest-growing intrusion *U*
_{max} = 18.0 (Σ*_{
z
}/Σ*_{
x
})^{+1/2}
*K _{T}
*(

*g*Θ*

_{ z }/

*νK*)

_{T}^{1/4}. Here Θ*

_{ z }is the undisturbed vertical temperature gradient in buoyancy units, Σ*

_{ z }and Σ*

_{ x }are the corresponding vertical and horizontal salinity gradients,

*g*is the gravity acceleration, and

*ν*and

*K*are the respective values of the molecular viscosity and heat diffusivity. The paradoxical inverse dependence on the horizontal gradient results from the assumption that the latter is unbounded.

_{T}^{ }

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## Abstract

The effect upon a stable atmosphere of a heat source (5–10 km width), which is a specified function of the coordinates, is investigated theoretically. A two-dimensional problem is chosen, with the *x*-coordinate in the direction of the mean undisturbed surface wind *U*, and the *z*-direction vertical. The heat source, a finite-width pulse in *x*, has maximum amplitude at the ground (*z* = 0) and decays with height. A steady state, in which the heat supplied by the source is continuously carried away downstream, is assumed, and the equations of motion are linearized by the method of perturbations. By Fourier analysis of the pulse function, the perturbation equations are made separable, and solutions for the streamline flow are obtained. It is shown that “lee waves,” *i.e.*, extended downstream oscillations in the streamlines, occur only if the undisturbed wind or stability undergoes a change in the vertical. The results of the analysis are compared with observations made over Nantucket Island, a flat sandy strip about 5 km in width.

## Abstract

The effect upon a stable atmosphere of a heat source (5–10 km width), which is a specified function of the coordinates, is investigated theoretically. A two-dimensional problem is chosen, with the *x*-coordinate in the direction of the mean undisturbed surface wind *U*, and the *z*-direction vertical. The heat source, a finite-width pulse in *x*, has maximum amplitude at the ground (*z* = 0) and decays with height. A steady state, in which the heat supplied by the source is continuously carried away downstream, is assumed, and the equations of motion are linearized by the method of perturbations. By Fourier analysis of the pulse function, the perturbation equations are made separable, and solutions for the streamline flow are obtained. It is shown that “lee waves,” *i.e.*, extended downstream oscillations in the streamlines, occur only if the undisturbed wind or stability undergoes a change in the vertical. The results of the analysis are compared with observations made over Nantucket Island, a flat sandy strip about 5 km in width.

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## Abstract

In Part I, the convective motions produced by the flow of a stable air stream over a small flat island were studied when the distribution of heating as a function of the coordinates was assumed. In Part II, the mechanism of the heating is examined. It is found that the heat source obeys an eddy-conduction equation and is established by turbulent eddying in the mixed ground layer. The streamline displacement may be divided into two components, one obeying the equation for air flow over a mountain ridge and the other obeying a heat-conduction equation, the latter component being important only in the near vicinity of the island. An “equivalent mountain” corresponding to the heated island may be specified analytically; it depends only upon the temperature distribution along the surface, the wind speed, the eddy conductivity in the ground layer, and the undisturbed stability. Its amplitude is related to the maximum streamline displacement. The “equivalent mountain” for Nantucket Island is calculated for two extreme observational cases. The complete streamline picture is constructed for several examples of an air stream whose properties remain unaltered to great heights. The basic current possessing a change in stability or wind speed at an upper level is also discussed, and the forecasting of lee waves is related to the development of the mixed ground layer *via* the height of the “equivalent mountain.” An expression for the sea breeze is also derived.

## Abstract

In Part I, the convective motions produced by the flow of a stable air stream over a small flat island were studied when the distribution of heating as a function of the coordinates was assumed. In Part II, the mechanism of the heating is examined. It is found that the heat source obeys an eddy-conduction equation and is established by turbulent eddying in the mixed ground layer. The streamline displacement may be divided into two components, one obeying the equation for air flow over a mountain ridge and the other obeying a heat-conduction equation, the latter component being important only in the near vicinity of the island. An “equivalent mountain” corresponding to the heated island may be specified analytically; it depends only upon the temperature distribution along the surface, the wind speed, the eddy conductivity in the ground layer, and the undisturbed stability. Its amplitude is related to the maximum streamline displacement. The “equivalent mountain” for Nantucket Island is calculated for two extreme observational cases. The complete streamline picture is constructed for several examples of an air stream whose properties remain unaltered to great heights. The basic current possessing a change in stability or wind speed at an upper level is also discussed, and the forecasting of lee waves is related to the development of the mixed ground layer *via* the height of the “equivalent mountain.” An expression for the sea breeze is also derived.

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## Abstract

At the northeast corner of Taiwan the direction of the continental slope isobaths changes rapidly relative to the oncoming Kuroshio, so that the inertia of a small inshore fraction of this current causes it to cross the slope, while the main branch follows the isobaths. It is suggested that the portion of the bifurcated current entering the shelf displaces ambient water of relatively high potential vorticity as a countercurrent, which flows across the slope. The vortex stretching and subsequent entrainment of this water into the main branch of the Kuroshio increases its maximum cyclonic vorticity and helps to maintain the inshore shear of the western boundary current. This Idea is supported by simple initial value and steady-state models, and also by dye observations of the flow from a source on the wall of a rotating tank.

## Abstract

At the northeast corner of Taiwan the direction of the continental slope isobaths changes rapidly relative to the oncoming Kuroshio, so that the inertia of a small inshore fraction of this current causes it to cross the slope, while the main branch follows the isobaths. It is suggested that the portion of the bifurcated current entering the shelf displaces ambient water of relatively high potential vorticity as a countercurrent, which flows across the slope. The vortex stretching and subsequent entrainment of this water into the main branch of the Kuroshio increases its maximum cyclonic vorticity and helps to maintain the inshore shear of the western boundary current. This Idea is supported by simple initial value and steady-state models, and also by dye observations of the flow from a source on the wall of a rotating tank.

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## Abstract

The formation and detachment of quasi-geostrophic eddies in a 1½ layer jet is studied using a piecewise uniform potential vorticity model. A vorticity front separates the two pieces, and thus the jet has cusplike character. The evolution of large amplitude initial disturbance (whose origin may be attributed to barotropic-baroclinic instability mechanisms not explicit in our model) is computed by the method of contour dynamics. Certain numerical results such as the steepening of the front prior to eddy detachment can be physically explained in terms of differential mean field advection and vortex induction. Computations are made for a variety of initial conditions and we indicate the amplitude/scale conditions necessary for the detachment of an eddy. The discussion is directed to the problem of the formation of warm/cold rings in the Gulf Stream. The effect of a coast on large perturbations of a jet is also briefly discussed.

## Abstract

The formation and detachment of quasi-geostrophic eddies in a 1½ layer jet is studied using a piecewise uniform potential vorticity model. A vorticity front separates the two pieces, and thus the jet has cusplike character. The evolution of large amplitude initial disturbance (whose origin may be attributed to barotropic-baroclinic instability mechanisms not explicit in our model) is computed by the method of contour dynamics. Certain numerical results such as the steepening of the front prior to eddy detachment can be physically explained in terms of differential mean field advection and vortex induction. Computations are made for a variety of initial conditions and we indicate the amplitude/scale conditions necessary for the detachment of an eddy. The discussion is directed to the problem of the formation of warm/cold rings in the Gulf Stream. The effect of a coast on large perturbations of a jet is also briefly discussed.

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## Abstract

The salt finger fluxes obtained in small-domain direct numerical simulations (DNSs) are used to parameterize the fluxes in a larger domain that resolves internal gravity waves. For the case in which the molecular diffusivity ratio *τ* = *K*
_{
S
}/*K*
_{
T
} < 1 is not excessively *small,* the wave amplification and overturning in a large-domain DNS that resolves *both* fingers and waves agrees with the result of a parameterized model. Since such a DNS is not feasible for heat–salt (*τ* = 1/80), the parametric model is used for 2D spectral calculations with periodic boundary conditions. For density ratios *R* = 1.25 and 1.5 an initialized low-frequency internal wave amplifies because of the wave strain; the irreversible (finger) fluxes and the wave potential energy increase until the isopycnals overturn. The maximum value of heat–salt flux produced by the wave is comparable to the finger fluxes in the absence of the wave. Even larger total fluxes should occur in the subsequent turbulent overturning stage (not computed), in which the viscous dissipation should be very much larger than in the pure finger stage. Although a finite-amplitude internal wave is generated for a larger *R* = 2.5, the overturning effect is very weak and on a relatively small scale; these “traumata” are attributed to wave–wave interactions.

## Abstract

The salt finger fluxes obtained in small-domain direct numerical simulations (DNSs) are used to parameterize the fluxes in a larger domain that resolves internal gravity waves. For the case in which the molecular diffusivity ratio *τ* = *K*
_{
S
}/*K*
_{
T
} < 1 is not excessively *small,* the wave amplification and overturning in a large-domain DNS that resolves *both* fingers and waves agrees with the result of a parameterized model. Since such a DNS is not feasible for heat–salt (*τ* = 1/80), the parametric model is used for 2D spectral calculations with periodic boundary conditions. For density ratios *R* = 1.25 and 1.5 an initialized low-frequency internal wave amplifies because of the wave strain; the irreversible (finger) fluxes and the wave potential energy increase until the isopycnals overturn. The maximum value of heat–salt flux produced by the wave is comparable to the finger fluxes in the absence of the wave. Even larger total fluxes should occur in the subsequent turbulent overturning stage (not computed), in which the viscous dissipation should be very much larger than in the pure finger stage. Although a finite-amplitude internal wave is generated for a larger *R* = 2.5, the overturning effect is very weak and on a relatively small scale; these “traumata” are attributed to wave–wave interactions.

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## Abstract

Two-dimensional direct numerical simulations (DNS) are used to investigate the growth and nonlinear equilibration of spatially periodic double-diffusive intrusion for negative vertical temperature *
T
*

_{ z }< 0 and salinity

S

_{ z }< 0 gradients, which are initially stable to small-scale double diffusion. The horizontal temperature

T

_{ x }and salinity

S

_{ x }gradients are assumed to be uniform, density compensated, and unbounded. The weakly sloping intrusion is represented as a mean lateral flow in a square computational box tilted with a slope equal to that of the fastest-growing linear theory mode; the vertical (

*η*) domain size of the box

*L**

_{ η }is a multiple of the fastest-growing wavelength. Solutions for the fastest-growing wavelength show that the intrusion growth is disrupted by salt fingers that develop when the rotation of the isotherms and isohalines by the intrusion shear results in temperature and salinity inversions; the thick inversion regions are separated by a thin interface supporting diffusive convection. These equilibrium solutions were always unstable to longer vertical wavelengths arising because of the merging of the inversion layers. The DNS predicts the following testable results for the maximum lateral velocity

*U**

_{ max}= 0.13

*N*

_{ S }

*L**

_{ η }, the lateral heat flux

*F** = 0.008

*ρ*

*C*

_{ P }(

S

_{ x }/

S

_{ z })

^{1/2}(

*N*

_{ S }/

*K*

_{ T })

^{1/4}

*N*

_{ S }

*L**

_{ η }

^{2.5}(

*β*

S

_{ z }/

*α*), and the interface thickness

*h*= 0.12

_{ρ}*L**

_{ η }, where

*N*

_{ S }=

*g*is the gravity acceleration,

*ρ*is the density,

*β*/

*α*is the haline contraction/heat expansion coefficient, and

*C*is the specific heat capacity. The results are compared with observations in the Arctic Ocean.

_{P}## Abstract

Two-dimensional direct numerical simulations (DNS) are used to investigate the growth and nonlinear equilibration of spatially periodic double-diffusive intrusion for negative vertical temperature *
T
*

_{ z }< 0 and salinity

S

_{ z }< 0 gradients, which are initially stable to small-scale double diffusion. The horizontal temperature

T

_{ x }and salinity

S

_{ x }gradients are assumed to be uniform, density compensated, and unbounded. The weakly sloping intrusion is represented as a mean lateral flow in a square computational box tilted with a slope equal to that of the fastest-growing linear theory mode; the vertical (

*η*) domain size of the box

*L**

_{ η }is a multiple of the fastest-growing wavelength. Solutions for the fastest-growing wavelength show that the intrusion growth is disrupted by salt fingers that develop when the rotation of the isotherms and isohalines by the intrusion shear results in temperature and salinity inversions; the thick inversion regions are separated by a thin interface supporting diffusive convection. These equilibrium solutions were always unstable to longer vertical wavelengths arising because of the merging of the inversion layers. The DNS predicts the following testable results for the maximum lateral velocity

*U**

_{ max}= 0.13

*N*

_{ S }

*L**

_{ η }, the lateral heat flux

*F** = 0.008

*ρ*

*C*

_{ P }(

S

_{ x }/

S

_{ z })

^{1/2}(

*N*

_{ S }/

*K*

_{ T })

^{1/4}

*N*

_{ S }

*L**

_{ η }

^{2.5}(

*β*

S

_{ z }/

*α*), and the interface thickness

*h*= 0.12

_{ρ}*L**

_{ η }, where

*N*

_{ S }=

*g*is the gravity acceleration,

*ρ*is the density,

*β*/

*α*is the haline contraction/heat expansion coefficient, and

*C*is the specific heat capacity. The results are compared with observations in the Arctic Ocean.

_{P}