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- Author or Editor: Melvin E. Stern x

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## Abstract

When air flows over terrain where the surface temperature varies with position, non-adiabatic heat will be added or subtracted. We consider the time-dependent disturbances induced in a uniform basic current as the result of differential heating on a flat rotating earth.

The problem of obtaining a plausible mathematical-physical description of the heating function is then discussed. A method which was proposed in a previous paper, Stern and Malkus (1953), is re-examined and restated, to clarify the underlying assumptions and the point of departure from the classical theory of the eddy conduction of heat.

With this description of the heating function, it is shown that the mean motions may be specified in terms of an “equivalent mountain.” This depends, in general, on the Coriolis parameter as well as the surface temperature, undisturbed wind speed, and the eddy conductivity of the heated region. The theory is applied to the small-scale sea-breeze problem, and it is shown that, by retaining the linearized advective term in the momentum equation, it is possible to explain the frequently observed phase relation between the diurnal temperature wave and the sea breeze without introducing friction. Additional derived relations for the hodograph suggest tests by means of future observations.

## Abstract

When air flows over terrain where the surface temperature varies with position, non-adiabatic heat will be added or subtracted. We consider the time-dependent disturbances induced in a uniform basic current as the result of differential heating on a flat rotating earth.

The problem of obtaining a plausible mathematical-physical description of the heating function is then discussed. A method which was proposed in a previous paper, Stern and Malkus (1953), is re-examined and restated, to clarify the underlying assumptions and the point of departure from the classical theory of the eddy conduction of heat.

With this description of the heating function, it is shown that the mean motions may be specified in terms of an “equivalent mountain.” This depends, in general, on the Coriolis parameter as well as the surface temperature, undisturbed wind speed, and the eddy conductivity of the heated region. The theory is applied to the small-scale sea-breeze problem, and it is shown that, by retaining the linearized advective term in the momentum equation, it is possible to explain the frequently observed phase relation between the diurnal temperature wave and the sea breeze without introducing friction. Additional derived relations for the hodograph suggest tests by means of future observations.

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## Abstract

The evolution of large-amplitude disturbances at the outer edge of a quasi-geostrophic shear layer depends on the sign of the outward gradient of potential vorticity. Entrainment of ambient water can occur when the gradient of relative vorticity dominates in the potential vorticity, and detrainment from the current can occur when gradient of isopycnal thickness dominates. In the latter case long, thin filaments of finite area are “pinched off” into the surrounding water mass. This is verified using a quasi-geostrophic model having piecewise uniform potential vorticity. Contour dynamical calculations for many initial conditions allow us to define and tabulate an entrainment/detrainment velocity. This is used for an order of magnitude estimate of the flux of heat or salt on an isopycnal surface in a warm core ring.

## Abstract

The evolution of large-amplitude disturbances at the outer edge of a quasi-geostrophic shear layer depends on the sign of the outward gradient of potential vorticity. Entrainment of ambient water can occur when the gradient of relative vorticity dominates in the potential vorticity, and detrainment from the current can occur when gradient of isopycnal thickness dominates. In the latter case long, thin filaments of finite area are “pinched off” into the surrounding water mass. This is verified using a quasi-geostrophic model having piecewise uniform potential vorticity. Contour dynamical calculations for many initial conditions allow us to define and tabulate an entrainment/detrainment velocity. This is used for an order of magnitude estimate of the flux of heat or salt on an isopycnal surface in a warm core ring.

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## Abstract

We compute the evolution of disturbances on a circularly symmetric eddy having uniform vorticity in a central core, in a surrounding annulus, and in the irrotational exterior water mass. This vortex is known to be (Kelvin-Helmholtz) unstable when its annular width is less than the core radius. Our calculations for the nonlinear regime show that amplification of azimuthal wavenumber *n* = 2 causes the vortex to split into two dipoles, in agreement with previous numerical calculations for a smoothed version of our vorticity field. This paper concentrates on the evolution of large-amplitude disturbances on the outer edge of a stable and robust eddy. We show that lateral wave breaking of vorticity isopleths causes intrusions of the (irrotational) exterior water mass into the central core of the vortex, a physical process which is relevant to lateral diffusion and isopycnal mixing in baroclinic ocean eddies. Similar intrusive features occur for an *n* = 1 disturbance, which also causes a “self-propagation” of the entire eddy. The large-amplitude disturbances on the eddy can be initiated by the action of external eddies or currents. A simple model for this case exhibits filaments detraining from the eddy, as well as intrusive features.

## Abstract

We compute the evolution of disturbances on a circularly symmetric eddy having uniform vorticity in a central core, in a surrounding annulus, and in the irrotational exterior water mass. This vortex is known to be (Kelvin-Helmholtz) unstable when its annular width is less than the core radius. Our calculations for the nonlinear regime show that amplification of azimuthal wavenumber *n* = 2 causes the vortex to split into two dipoles, in agreement with previous numerical calculations for a smoothed version of our vorticity field. This paper concentrates on the evolution of large-amplitude disturbances on the outer edge of a stable and robust eddy. We show that lateral wave breaking of vorticity isopleths causes intrusions of the (irrotational) exterior water mass into the central core of the vortex, a physical process which is relevant to lateral diffusion and isopycnal mixing in baroclinic ocean eddies. Similar intrusive features occur for an *n* = 1 disturbance, which also causes a “self-propagation” of the entire eddy. The large-amplitude disturbances on the eddy can be initiated by the action of external eddies or currents. A simple model for this case exhibits filaments detraining from the eddy, as well as intrusive features.

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## Abstract

Separation from the continental slope of stratified jets like the Gulf Stream involves the sliding of successive isopycnal layers from a nearly horizontal bottom to the adjacent offshore isopycnal in the deep ocean. One mechanism for producing such an effect is due to a downstream convergence of slope isobaths, as shown herein for a 1-layer density model. Upstream of the convergence, a geostrophically balanced jet is assumed with an inshore region of cyclonic vorticity resting on the continental slope and an offshore anticyclonic region resting on the isopycnal interface above heavier water. For *O*(1) Rossby number and cross-stream topographic variation, the steady transverse current displacements forced by slowly varying downstream topography are computed. For “supercritical” upstream flow (i.e., fast compared to free topographic waves) *off*slope displacements are produced by converging isobaths; extrapolation of the small amplitude result suggests that the mechanism is quantitatively important for the explanation of the full separation of the Gulf Stream from the *bottom* of the continental slope. The kinematics involved in this process should apply to a continuously stratified jet, as well as to other forcing mechanisms known to be of importance in continental boundary current separation.

## Abstract

Separation from the continental slope of stratified jets like the Gulf Stream involves the sliding of successive isopycnal layers from a nearly horizontal bottom to the adjacent offshore isopycnal in the deep ocean. One mechanism for producing such an effect is due to a downstream convergence of slope isobaths, as shown herein for a 1-layer density model. Upstream of the convergence, a geostrophically balanced jet is assumed with an inshore region of cyclonic vorticity resting on the continental slope and an offshore anticyclonic region resting on the isopycnal interface above heavier water. For *O*(1) Rossby number and cross-stream topographic variation, the steady transverse current displacements forced by slowly varying downstream topography are computed. For “supercritical” upstream flow (i.e., fast compared to free topographic waves) *off*slope displacements are produced by converging isobaths; extrapolation of the small amplitude result suggests that the mechanism is quantitatively important for the explanation of the full separation of the Gulf Stream from the *bottom* of the continental slope. The kinematics involved in this process should apply to a continuously stratified jet, as well as to other forcing mechanisms known to be of importance in continental boundary current separation.

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## Abstract

The temporal evolution of large amplitude quasi-geostrophic disturbances in a piecewise uniform potential vorticity flow is elucidated by numerical solutions of the “contour dynamica” equations. Lateral wavebreaking occurs when the initial disturbance amplitude exceeds a certain value, and at later times tongues of the 1ower vorticity fluid are engulfed or entrained into the higher vorticity shear flow. The effect appears to be important for the evolution of “shingles” observed between the coastal water and the cyclonic side of the Gulf Stream. The effect may also be in important phase in initiating the mixing process at the perimeter of an eddy embedded in another water mass.

## Abstract

The temporal evolution of large amplitude quasi-geostrophic disturbances in a piecewise uniform potential vorticity flow is elucidated by numerical solutions of the “contour dynamica” equations. Lateral wavebreaking occurs when the initial disturbance amplitude exceeds a certain value, and at later times tongues of the 1ower vorticity fluid are engulfed or entrained into the higher vorticity shear flow. The effect appears to be important for the evolution of “shingles” observed between the coastal water and the cyclonic side of the Gulf Stream. The effect may also be in important phase in initiating the mixing process at the perimeter of an eddy embedded in another water mass.

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## Abstract

When a low Rossby number barotropic flow accelerates in the laterally converging half of a strait, the local propagation speed of long topographic waves can be reduced to zero, thereby blocking or preventing the formation of a steady flow downstream from the strait. An inviscid longwave theory is presented for the new steady upstream and downstream states that evolve from the blocking wave. The enhanced inshore cyclonic vorticity extending far downstream suggests that topographic jetogenesis, rather than lateral eddy diffusion, in major ocean straits (e.g., Yucatan and Florida) may be important in generating or reforming boundary currents.

## Abstract

When a low Rossby number barotropic flow accelerates in the laterally converging half of a strait, the local propagation speed of long topographic waves can be reduced to zero, thereby blocking or preventing the formation of a steady flow downstream from the strait. An inviscid longwave theory is presented for the new steady upstream and downstream states that evolve from the blocking wave. The enhanced inshore cyclonic vorticity extending far downstream suggests that topographic jetogenesis, rather than lateral eddy diffusion, in major ocean straits (e.g., Yucatan and Florida) may be important in generating or reforming boundary currents.

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## Abstract

It is suggested that the inshore shear of continental boundary flows like the Florida Current can be accounted for by a countergradient vorticity flux, rather than by lateral diffusion to the shore. Two simple barotropic models with cross-stream and downstream topographic variations illustrate the point. In the first case, a broad jet with piecewise uniform vorticity accelerates through a slowly converging strait, eventually becoming locally critical (in the hydraulic sense) and then undergoing a transition to a different downstream state in which the maximum inshore vorticity is increased. The topographic conditions for this to occur are determined by a nonlinear long-wave theory. In the second model, the computed flow around an idealized cape illustrates the role of lee waves in generating mean downstream vorticity. For large-amplitude capes a nonlinear long-wave theory shows that a downstream transition (similar to the strait problem) can occur as well as upstream “blocking.”

## Abstract

It is suggested that the inshore shear of continental boundary flows like the Florida Current can be accounted for by a countergradient vorticity flux, rather than by lateral diffusion to the shore. Two simple barotropic models with cross-stream and downstream topographic variations illustrate the point. In the first case, a broad jet with piecewise uniform vorticity accelerates through a slowly converging strait, eventually becoming locally critical (in the hydraulic sense) and then undergoing a transition to a different downstream state in which the maximum inshore vorticity is increased. The topographic conditions for this to occur are determined by a nonlinear long-wave theory. In the second model, the computed flow around an idealized cape illustrates the role of lee waves in generating mean downstream vorticity. For large-amplitude capes a nonlinear long-wave theory shows that a downstream transition (similar to the strait problem) can occur as well as upstream “blocking.”

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## Abstract

The free discharge of a layer of bottom water in a wide strait (e.g., the Denmark Strait) differs from the classical control problem because of the strong geostrophic turbulence. As a consequence, the cross-stream (*x*) variation of the time-averaged downstream velocity *υ*(*x*) is “underdetermined” and depends on more parameters than available conditions. To resolve this “degeneracy” the classical control condition is generalized; the result requires the discharge *Q* to be extremized with respect to the degeneracy parameters and with respect to the constraints. One of these constraints is that a branch point or a local stationary wave can be supported at some section of the long channel. By either maximizing *Q* or by requiring a stationary wave, useful approximations are obtained. Future work should consider the joint variational problem. Specific calculations are made for nonuniform potential vorticity in a rectangular channel and also for variable cross-stream bottom topography. In the latter case, the current has a free streamline whose point of intersection with the bottom is computed. It is suggested that a necessary condition for hydraulic control at any section is that the flow does not separate from either side of the channel. The mean isopycnals in the Denmark Strait satisfy this condition, thereby suggesting that it does in fact (topographically) control the upstream state in the Greenland Sea.

## Abstract

The free discharge of a layer of bottom water in a wide strait (e.g., the Denmark Strait) differs from the classical control problem because of the strong geostrophic turbulence. As a consequence, the cross-stream (*x*) variation of the time-averaged downstream velocity *υ*(*x*) is “underdetermined” and depends on more parameters than available conditions. To resolve this “degeneracy” the classical control condition is generalized; the result requires the discharge *Q* to be extremized with respect to the degeneracy parameters and with respect to the constraints. One of these constraints is that a branch point or a local stationary wave can be supported at some section of the long channel. By either maximizing *Q* or by requiring a stationary wave, useful approximations are obtained. Future work should consider the joint variational problem. Specific calculations are made for nonuniform potential vorticity in a rectangular channel and also for variable cross-stream bottom topography. In the latter case, the current has a free streamline whose point of intersection with the bottom is computed. It is suggested that a necessary condition for hydraulic control at any section is that the flow does not separate from either side of the channel. The mean isopycnals in the Denmark Strait satisfy this condition, thereby suggesting that it does in fact (topographically) control the upstream state in the Greenland Sea.

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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.