Geophysical Fluid Dynamics

Pedlosky, Joseph

doi:10.1007/978-3-662-25730-2

Cited by 671 publications

(880 citation statements)

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“…These characteristics agree qualitatively with the linear solution (Gill 1976). The frequency σ of the oscillations, when they occur, is given to within ±20% by the frequency of the linear Poincaré waves (Pedlosky 1987) with lowest cross-channel mode, zero along-channel wavenumber and depth equal to the average of the initial levels on either side of the dam (1 + d 0 )/2,…”

Section: Mean Transportsupporting

confidence: 74%

Nonlinear Rossby adjustment in a channel

1999

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The Rossby adjustment problem for a homogeneous fluid in a channel is solved for large values of the initial depth discontinuity. We begin by analysing the classical dam break problem in which the depth on one side of the discontinuity is zero. An approximate solution for this case can be constructed by assuming semigeostrophic dynamics and using the method of characteristics. This theory is supplemented by numerical solutions to the full shallow water equations. The development of the flow and the final, equilibrium volume transport are governed by the ratio of the Rossby radius of deformation to the channel width, the only non-dimensional parameter. After the dam is destroyed the rotating fluid spills down the dry section of the channel forming a rarefying intrusion which, for northern hemisphere rotation, is banked against the right-hand wall (facing downstream). As the channel width is increased the speed of the leading edge (along the right-hand wall) exceeds the intrusion speed for the non-rotating case, reaching the limiting value of 3.80 times the linear Kelvin wave speed in the upstream basin. On the left side of the channel fluid separates from the sidewall at a point whose speed decreases to zero as the channel width approaches infinity. Numerical computations of the evolving flow show good agreement with the semigeostrophic theory for widths less than about a deformation radius. For larger widths cross-channel accelerations, absent in the semigeostrophic approximation, reduce the agreement. The final equilibrium transport down the channel is determined from the semigeostrophic theory and found to depart from the non-rotating result for channels widths greater than about one deformation radius. Rotation limits the transport to a constant maximum value for channel widths greater than about four deformation radii.The case in which the initial fluid depth downstream of the dam is non-zero is then examined numerically. The leading rarefying intrusion is now replaced by a Kelvin shock, or bore, whose speed is substantially less than the zero-depth intrusion speed. The shock is either straight across the channel or attached only to the righthand wall depending on the channel width and the additional parameter, the initial depth difference. The shock speeds and amplitudes on the right-hand wall, for fixed downstream depth, increase above the non-rotating values with increasing channel width. However, rotation reduces the speed of a shock of given amplitude below the non-rotating case. We also find evidence of resonant generation of Poincaré waves by the bore. Shock characteristics are compared to theories of rotating shocks and qualitative agreement is found except for the change in potential vorticity across the shock, which is very sensitive to the model dissipation. Behind the leading shock the flow evolves in much the same way as described by linear theory except for the generation of strongly nonlinear transverse oscillations and rapid advection down the 188 K. R. Helfrich, A. C. Kuo and L. J. Pratt rig...

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Section: Mean Transportsupporting

confidence: 74%

Nonlinear Rossby adjustment in a channel

1999

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“…We consider the stability of the zonal flow in the standard quasi-geostrophic two-layer model on the beta-plane (see for example Pedlosky 1987). The governing equations for the nonlinear evolution of the disturbances are, in non-dimensional form,…”

Section: Formulation Of the Linear Problemmentioning

confidence: 99%

“…The perturbation method required to achieve an amplitude equation governing the nonlinear evolution of the wave is standard and can be found described in many places, e.g. Pedlosky (1987) so that only the briefest outline of the development is given here. Naturally, a full understanding of the nonlinear problem must eventually move beyond weakly nonlinear theory but it seems to us useful to start with a dynamics in which the novel character of the timedependent problem is more easily clarified as in the weakly nonlinear perturbation problem.…”

Section: The Nonlinear Problemmentioning

confidence: 99%

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Baroclinic instability of time-dependent currents

Pedlosky

Thomson²

2003

J. Fluid Mech.

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The baroclinic instability of a zonal current on the beta-plane is studied in the context of the two-layer model when the shear of the basic current is a periodic function of time. The basic shear is contained in a zonal channel and is independent of the meridional direction. The instability properties are studied in the neighbourhood of the classical steady-shear threshold for marginal stability. It is shown that the linear problem shares common features with the behaviour of the well-known Mathieu equation. That is, the oscillatory nature of the shear tends to stabilize an otherwise unstable current while, on the contrary, the oscillation is able to destabilize a current whose time-averaged shear is stable. Indeed, this parametric instability can destabilize a flow that at every instant possesses a shear that is subcritical with respect to the standard stability threshold. This is a new source of growing disturbances. The nonlinear problem is studied in the same near neighbourhood of the marginal curve. When the time-averaged flow is unstable, the presence of the oscillation in the shear produces both periodic finite-amplitude motions and aperiodic behaviour. Generally speaking, the aperiodic behaviour appears when the amplitude of the oscillating shear exceeds a critical value depending on frequency and dissipation. When the time-averaged flow is stable, i.e. subcritical, finite-amplitude aperiodic motion occurs when the amplitude of the oscillating part of the shear is large enough to lift the flow into the unstable domain for at least part of the cycle of oscillation. A particularly interesting phenomenon occurs when the time-averaged flow is stable and the oscillating part is too small to ever render the flow unstable according to the standard criteria. Nevertheless, in this regime parametric instability occurs for ranges of frequency that expand as the amplitude of the oscillating shear increases. The amplitude of the resulting unstable wave is a function of frequency and the magnitude of the oscillating shear. For some ranges of shear amplitude and oscillation frequency there exist multiple solutions. It is suggested that the nature of the response of the finite-amplitude behaviour of the baroclinic waves in the presence of the oscillating mean flow may be indicative of the role of seasonal variability in shaping eddy activity in both the atmosphere and the ocean.

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“…Because the linear growth rates in this extended domain are small and the wave numbers are large, it is natural to wonder whether the effects of friction render these results of small importance in spite of their intrinsic interest. However, if the friction is primarily the result of a bottom Ekman layer type of interaction, the frictional effect relative to the inertial effects driving the instability actually diminishes as the wave number increases (Pedlosky 1987).…”

Section: Summary and Discussionmentioning

confidence: 99%

Baroclinic instability over topography: Unstable at any wave number

Pedlosky¹

2016

J Mar Res

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The instability of an inviscid, baroclinic vertically sheared current of uniform potential vorticity, flowing along a uniform topographic slope, becomes linearly unstable at all wave numbers if the flow is in the direction of propagation of topographic waves. The parameter region of instability in the plane of scaled topographic slope versus wave number then extends to arbitrarily large wave numbers at large slopes.The weakly nonlinear treatment of the problem reveals the existence of a nonlinear enhancement of the instability close to one of the two boundaries of this parametrically narrow unstable region. Because the domain of instability becomes exponentially narrow for large wave numbers, it is unclear how applicable the results of the asymptotic, weakly nonlinear theory are given that it must be limited to a region of small supercriticality.This question is pursued in that parameter domain through the use of a truncated model in which the approximations of weakly nonlinear theory are avoided. This more complex model demonstrates that the linearly most unstable wave in the narrow wedge in parameter space is nonlinearly stable and that the region of nonlinear destabilization is limited to a tiny region near one of the critical curves rendering both the linear and nonlinear growth essentially negligible.

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Geophysical Fluid Dynamics

Cited by 671 publications

References 0 publications

Nonlinear Rossby adjustment in a channel

Nonlinear Rossby adjustment in a channel

Baroclinic instability of time-dependent currents

Baroclinic instability over topography: Unstable at any wave number

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