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J. S. Allen

Abstract

The effects of alongshore variations in bottom topography and coastline on the wind-stress-forced barotropic motion over a continental shelf and slope are studied. Perturbation methods are used to obtain solutions for forced and free continental shelf waves on an idealized continental shelf and slope with small-amplitude alongshore variations in topography. The relevant alongshore scales, set by the wind stress and by the bottom and coastline topography, are assumed to be greater than the shelf-slope width. This enables the resulting motion to be treated in the long-wave nondispersive limit. As a result, the alongshore and time-dependent behavior of the perturbation flow is governed by a forced, first order wave equation, with terms from the interaction of the basic, lowest order flow with the bottom and coastline topography acting as the forcing function. To clarify the effects of topography alone, problems are considered where a uniform wind stress forces a basic unperturbed flow which is independent of the alongshore coordinate. In one example, a steady, alongshore-independent basic flow is established impulsively by a delta function application of the wind stress. The perturbation flow adjusts to the alongshore variations in topography through the propagation of disturbances as free continental shelf waves. There is an eventual establishment in the region of variable topography of a steady-state motion which follows contours of constant depth. Other problems in which single mode free shelf wave disturbances of limited alongshore extent propagate into regions of different topography are studied also. The basic disturbance is found to travel at the local wave speed with its cross shelf modal structure described by the local eigenfunctions. As a region of varying bottom topography is crossed, disturbances in other modes are generated. General features of this scattering process are examined for limiting cases where the alongshore scale of the disturbance is greater than, or less than, the scale of the topographic feature.

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S. J. M. ALLEN

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J. S. Allen

Abstract

An approximate model for small Rossby number ε that is close to the balance equations (BE) but that is based on approximate momentum equations is formulated for a rotating, continuously stratified fluid governed by the hydrostatic, Boussinesq, inviscid, adiabatic primitive equations with spatially variable Coriolis parameter. This model, referred to as BEM (balance equations based on momentum equations), conserves volume integrals of an appropriate energy density and also conserves potential vorticity on fluid particles and thus volume integrals of potential enstrophy density. The fact that, unlike the BE model which is derived from equations for the vertical component of vorticity and for the horizontal divergence, BEM is based on approximate momentum equations is important for two reasons. It allows the derivation of equations for the horizontal components of vorticity that are needed in the subsequent derivation of an equation for the potential vorticity and it allows the consistent formulation of boundary conditions at rigid surfaces. As is the case for BE, the BEM equations filter out high-frequency internal–gravity waves and remain valid for motion over O(1) variations in bottom topography and for flows with O(1) variations in the height of density surfaces. The governing equations for BEM may be conveniently expressed in a form similar to BE involving a vorticity and a divergence (balance) equation. In this formulation, the BE and BEM models involve identical equations for continuity, vorticity, and heat with differences represented only by the presence of additional higher order terms in the balance equation for BEM. Methods for the numerical solution of BEM and for the application of boundary conditions are presented.

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J. S. Allen

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J. S. Allen

Abstract

A two-layer model with an idealized continental shelf and slope bottom topography is utilized to study some properties of the response of stratified coastal regions to meteorological forcing with variations in the alongshore direction. The model is such that the coastline is straight, there are no alongshore variations in the bottom topography, and the parameter λ=δ′R/δ′B is small, i.e., λLt;1, where δ′R is the internal Rossby radius of deformation and δ′B is a scale length of the bottom topography. In that case, the inviscid response to forcing by an alongshore wind stress is composed of uncoupled baroclinic and barotropic components. The baroclinic component consists of forced internal Kelvin waves, with an offshore scale of the order of δ′R (∼15 km) and the barotropic component consists of forced continental shelf waves, with an offshore scale the order of the width L of the continental shelf and slope (L≈100 km). The alongshore scale of the forcing is assumed to be greater than L and the method of solution of Gill and Schumann and of Gill and Clarke is used. As a result, the alongshore and time-dependent behavior of the baroclinic and barotropic components is governed by forced, first-order wave equations. The response to an impulsively applied, upwelling-favorable wind stress with a specialized alongshore structure, i.e., a constant value for a distance of limited extent, is studied to give insight into the qualitative nature of the behavior of the forced time-dependent, baroclinic and barotropic components. The solutions show clearly how the region of forced upward motion of density surfaces may propagate alongshore to locations distant from that of the wind stress which causes the up-welling. They also illustrate how the barotropic onshore flow to the coast is influenced by the propagation of forced continental shelf waves such that the region of onshore flow from the interior to the slope and shelf may also propagate alongshore.

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J. S. Allen

Abstract

A two-layer model is used to study the properties of free coastal trapped waves which propagate over an idealized continental shelf and continental slope bottom topography. With both stratification and depth variations that are typical of continental shelf and slope regions, barotropic shelf waves and baroclinic internal Kelvin waves axe coupled. The internal Kelvin waves have an offshore scale given by the internal “Rossby radius of deformation” δ′R which is typically small (∼15 km) compared with the width L of the shelf and slope region (∼100 km), while the shelf waves have an offshore scale which, for the lower modes, is O(L). The strength of the coupling is represented by a parameter λ = δ′R/δ′B, where δ′B = H′/Hx is a characteristic length scale of the bottom topography and is essentially the distance over which the change in water depth H′, in the offshore direction x′, is the same magnitude as the depth itself. The nature of the interaction and modification of internal Kelvin waves and barotropic shelf waves is studied by a perturbation procedure for λ≪1. It is found that, for alongshore scales δ′y greater than L, O(1) internal Kelvin waves are accompanied by an O(λ) barotropic motion, which extends across the shelf and slope, and that O(1) shelf waves are accompanied by a weaker O(λ2) baroclinic motion. For short alongshore and onshore-offshore scales, i.e., for δ′y<O(L ½δ′R ½) and δ′x<O(L ½δ′R ½), the shelf wave solutions are coupled at the lowest order with baroclinic effects which alter the modal structure. For very short scales, δ′y, δ′x<δ′R, the shelf wave motion is primarily confined to the bottom layer and the waves are “bottom trapped.”

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J. S. Allen

Abstract

Previous studies of forced, long continental shelf waves on an f-plane have considered motion on the shelf and slope which is driven by an alongshore component of the wind stress, essentially through the suction of fluid into the surface layer at the coast. These studies have utilized a boundary condition, which arises consistently in the long-wave nondispersive limit for free shelf waves, that at the slope-interior junction the alongshore velocity component v≈0..This is an extremely useful condition for problems concerning forced motion on the shelf and slope, because it completely uncouples the motion in this region from that in the interior and it allows the shelf-slope problem to he solved independently of the interior problem. It is shown here, however, that this condition is not correct in general for wind-stress-forced f-plane motion. A proper formulation of the f-plane, forced shelf wave problem in the long wave limit is presented. The motion on the shelf and slope, in general, is coupled with and forced by the flow in the interior.

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J. S. Allen

Abstract

Intermediate models contain physics between that in the primitive equations and that in the quasigeostrophic equations and are capable of representing subinertial frequency motion over O(1) topographic variations typical of the continental slope while filtering out high-frequency gravity-inertial waves. We present here a formulation for stratified flow of a set of new intermediate models, termed iterated geostrophic (IG) models, derived under the assumption that the Rossby number ε is small. We consider the emotion of a rotating, continuously stratified fluid governed by the hydrostatic, Boussinesq, adiabatic primitive equations (PE) with a spatially variable Coriolis parameter and with weak biharmonic momentum diffusion. The IG models utilize the pressure field as the basic variable [as in the quasigeostrophic (QG) approximation], are capable of providing solutions of formally increasing accuracy in powers of ε in a systematic manner, and are straightforward to solve numerically. The 10 models are obtained by iteration, at a fixed time t = t 0, of the momentum and thermodynamic equations using the known pressure field ϕ(x, t). The iteration procedure products a sequence of estimates of increasing accuracy for the velocity components and for the time derivative of the pressure field ϕt(x, t 0). The formulation is asymptotic in the sense that, given the pressure field, at each iteration the velocity components and ϕt(x, t 0) are formally determined to a higher order of accuracy in powers of ε. The order of the IG model is specified by the predetermined fixed number of iterations N. Thus, a set of models is produced depending on the choice for N, and the different models are denoted by IGN. The value of ϕt(x, t 0) obtained from iteration N is used with a time difference scheme to advance the pressure field in time, and the process may be repeated. Energy and potential enstrophy conservation in the IG models are asymptotic. In the following companion paper (Allen and Newberger), the accuracies of several intermediate models, including IG2 and IG3, are investigated by a comparison of numerical finite-difference solutions to those of the primitive equations. For moderate Rossby number flows, it is found that IG2 gives approximate solutions of reasonable accuracy, with errors substantially smaller than those obtained from QG and several other intermediate models. The IG3 model is found to give extremely accurate approximate solutions for flows with Rossby numbers that range from moderately small to O(1).

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J. S. Allen

Abstract

The effect of bottom friction on the subinertial frequency motion of stratified shelf flow fields is studied in a two-layer f-plane model with idealized shelf and slope bottom topography. Coastal-trapped fire waves and motion forced by the alongshore component of the wind stress at the coast are considered. Vertical turbulent-diffusion effects are assumed to be present in thin surface and bottom-boundary layers, but not at the density interface. Simplifications are achieved by assuming that typical alongshore scales are larger than the offshore scales given by the internal Rossby radius of deformation δR and the shelf-slope width, that the upper-layer depth is small compared with the lower-layer depth, and that the topography of the continental margin may be represented by a linear bottom slope of small magnitude. Some results are not dependent on the presence of variable bottom topography; these are obtained first with a flat-bottom ocean adjacent to a vertical coast. A characteristic feature of free and forced motion with alongshore gradients is a decrease of lower-layer velocity and a resultant concentration of flow in the upper layer as the frequency approaches zero. For internal Kelvin waves of frequency ω, this change in velocity structure occurs for ω/α ≪ 1, where α−1 is barotropic spin-down time, and is accompanied by a decrease in frictional decay as ω/α → 0. As a result, coastal internal Kelvin waves may be able to participate with relatively small damping by bottom friction in low-frequency phenomena such as El Niño. For motion forced at frequency σ and alongshore wavenumber l, this change in structure occurs for σ/α ≪ l and σ(lδR)−1 ≪ 1. Concurrently, the magnitude of the barotropic, forced-shelf-wave component of the flow goes to zero as σ → 0. Thus, the “arrested topographic wave” is absent and plays no role in the steady solutions. Qualitatively similar behavior is found on the Oregon shelf in the summer where monthly mean alongshore current at midshelf have substantial vertical shear, but corresponding fluctuations on the several-day time scale are nearly depth-independent. Generalized first-order wave equations are derived to describe the alongshore (y) and time (t) dependence of the lowest-order baroclinic and barotropic components. The response to a wind strees with Heaviside-unit-function behavior in both y and t clearly illustrates how the effects of stratification liberate the “arrested topographic wave” and how a steady state is achieved where the current to the upper layer and to a region near the coast with offshore scale of O(δR).

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J. S. Allen

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A simple, linear, two-dimensional, f-plane model of coastal upwelling in a continuously stratified ocean is investigated. The transient response of the fluid to an impulsively applied alongshore wind stress, the nature of the approach to a steady state, and the final steady state flow are studied. Particular attention is given here to an examination, in the continuously stratified case, of the interrelation of upwelling and the phenomenon of a baroclinic coastal jet. The coastal jet is a feature which has appeared in studies of the transient response of two-layer fluid models near a coast. The stratification parameter S = (δN/f)2, where δ = II/L is the aspect ratio, H the depth, L a characteristic horizontal scale, N the Brunt-Väisälä frequency, and f the Coriolis parameter, is assumed to lie in the range O(Ev ½)<S≤O(1), where λ = Ev/EH and where EH and Ev are respectively the horizontal and vertical Ekman numbers. It is also assumed that L is large enough that S≪1 and λ≫1. These are the conditions for which, during the transient flow, a baroclinc coastal jet is present in an inviscid upwelling layer of width O(S ½). It also appears that these conditions are reasonable for oceanic upwelling regions. The jet development is initiated on a time scale of O(Ev −½). A steady state is reached in the O(S ½) upwelling layer on a longer diffusive time scale of O(SEH −1). For Sλ<O(1), the alongshore current outside the O(S ½) layer increases in magnitude with time, by diffusive spreading of the boundary effects, in an additional boundary layer of width O(λ−½). It reaches a steady state on a yet longer time scale of O(Ev −1). For Sλ = O(1), the two boundary layers merge and adjust on the same time scale. The final steady state is characterized by a coastal current which is confined to the O(S ½) and O(λ−½) boundary layers.

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