Gordon E. Swaters

On the baroclinic instability of cold-core coupled density fronts on a sloping continental shelf

A theory is presented to describe the linear baroclinic instability of coupled density fronts on a sloping continental shelf. The new baroclinic model equations used to study the instability process correspond to an ‘intermediate lengthscale ’ dynamical balance. Specifically, the frontal dynamics, while geostrophic, is not quasigeostrophic because frontal height deflections are not small in comparison with the frontal scale height. The evolution of the frontal height is strongly coupled to the geostrophic pressure in the surrounding slope water through the hydrostatic balance which expresses the continuity of the dynamic pressures across the frontal interface. The deeper surrounding slope water evolves quasi-geostrophically and is coupled to the front by baroclinic vortex-tube stretching/compression associated with the perturbed density front (allowing the release of mean frontal potential energy) and the topographic vorticity gradient associated with the sloping bottom. It is shown that the baroclinic stability characteristics are principally determined by a so-called non-dimensional interaction parameter (denoted p) which physically measures the ratio of the destabilizing baroclinic vortex-tube stretching/compression to the stabilizing topographic vorticity gradient. For a given along-front mode wavenumber it is shown that a minimum p is required for instability. Several other general stability results are presented : necessary conditions for instability, growth rate and phase speed bounds, the existence of a high wavenumber cutoff, and a semicircle theorem for the unstable modes. The linear stability equations are solved exactly for a parabolic coupled density front and a detailed description of the spatial and temporal characteristics of the instabilities is given. For physically realistic parameter values the instabilities are manifested as amplifying topographic Rossby waves in the slope water, and on the density front the unstable perturbations take the form of amplifying anticyclones which have maximum amplitude on the offshore side.

The source area influencing a measurement in the Planetary Boundary Layer: The “footprint” and the “distribution of contact distance”

This paper considers the ground area which affects the properties of fluid parcels observed at a given spot in the Planetary Boundary Layer (PBL). We examine two source-area functions; the “footprint,” giving the source area for a measurement of vertical flux: and the distribution of “contact distance”, the distance since a particle observed aloft last made contact with the surface. We explain why the distribution of contact distance extends vastly farther upwind than the footprint, and suggest for the extent of the footprint the inequalities: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9Lq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9W8qr0-vr% 0-viWZqaceaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyvam% aalaaabaGaamiAaaqaaiabeo8aZnaaBaaaleaacaWGxbaabeaakiaa% cIcacaWGObGaaiykaaaacqGH8aapcaWG4bGaeyipaWJaamyvaKazaa% iadaGabaqaamaaDaaajqwaacqaaiaadIgacaGGVaGabmOEayaacaGa% aiilaiaabccacaGGVbGaaiiDaiaacIgacaGGLbGaaiOCaiaacEhaca% GGPbGaai4CaiaacwgaaeaacaWGubWaaSbaaKazcaiabaGaamitaaqa% baqcKfaGaiaacIcacaWGObGaaiykaiaabYcacaqGGaGaaeiAaiaabc% cacaGGHbGaaiOyaiaac+gacaGG2bGaaiyzaiaabccacaGGZbGaaiyD% aiaackhacaGGMbGaaiyyaiaacogacaGGLbGaeyOeI0IaaiiBaiaacg% gacaGG5bGaaiyzaiaackhaaaaajqgaacGaay5EaaaakeaaaeaacaGG% 8bGaamyEaiaacYhacqGH8aapcqaHdpWCdaWgaaWcbaGaamODaaqaba% GccaGGOaGaamiAaiaacMcadaWcaaqaaiaadIhaaeaacaWGvbaaaaaa% aa!7877!\[\begin{array}{l} U\frac{h}{{\sigma _W (h)}} < x < U\left\{ {_{h/\dot z,{\rm{ }}otherwise}^{T_L (h){\rm{, h }}above{\rm{ }}surface - layer} } \right. \\ \\ |y| < \sigma _v (h)\frac{x}{U} \\ \end{array}\] where U is the mean streamwise (x) velocity, h is the observation height, ΤL is the Lagrangian timescale, Σv and Σw are the standard deviations of the cross-stream horizontal (y) and vertical (z) velocity fluctuations, and ż is the Lagrangian Similarity prediction for the rate of rise of the centre of gravity of a puff released at ground.Simple analytical solutions for the contact-time and the footprint are derived, by treating the PBL as consisting of two sub-layers. The contact-time solutions agree very well with the predictions of a Lagrangian stochastic model, which we adopt in the absence of measurements as our best estimate of reality, but the footprint solution offers no improvement over the above inequality.

Dynamics of ventilated coherent cold eddies on a sloping bottom

eddy is modelled with nonlinear ‘intermediate lengthscale ’ geostrophic dynamics which is coupled to the surrounding fluid. The process of ventilation is modelled with a simple cross-interfacial mass flux parameterization. The surrounding fluid is governed by nonlinear quasi-geostrophic dynamics including eddy-induced vortextube compression. Assuming a relatively weak ventilation rate, a multiple-scale asymptotic theory is constructed to describe the propagation of an initially isolated or coherent baroclinic eddy. Throughout the evolution the eddy is assumed to be interacting strongly with the surrounding fluid. To leading order, the eddy and surrounding fluid satisfy the Stern isolation constraint. The magnitude of the Eulerian velocity field in the surrounding fluid above the eddy is shown to be larger than the swirl velocities in the eddy interior as suggested by experimental data. Also, to leading order, the along-shelf translation speed is given by the Nof formula. The process of ventilation is shown to induce a slowly decaying upslope translation in the propagating eddy, and acts to stimulate a weak slowly decaying topographic Rossby wave field in the surrounding fluid. The important features of the theory are illustrated with a simple example calculation.

Topographically-Induced Baroclinic Eddies near a Coastline, with Application to he Northeast Pacific

Abstract A mathematical model is developed to describe the interaction between variable bottom topography and a steady, horizontally-sheared baroclinic coastal current. The topography modeled in this study consists of an offshore seamount and a seaward protrusion of the continental slope. The fluid motions are assumed to conserve potential vorticity on the f-plane, and expressions for the pressure, density, velocity and mass transport fields are obtained using a normal mode analysis and the appropriate Greens function for the horizontal problem. The theory is applied to the northeast Pacific Ocean in an attempt to model the anticyclonic eddy which has been observed by Tabata west of Sitka, Alaska. The numerical calculations of the model and the observed location, dimensions, velocities and transports of the Sitka eddy are in good agreement. The axial velocities and dimensions of the calculated eddy field are largest for upstream surface and bottom currents of approximately 5–7 and 1–2 cm s−1, respectivel...

Numerical simulations of the baroclinic dynamics of density‐driven coupled fronts and eddies on a sloping bottom

Numerical simulations of the baroclinic dynamics of density-driven coupled fronts and eddies are described. The simulations are based on a two-layer intermediate length scale model which filters out barotropic instability and focuses on the subinertial baroclinic evolution of density-driven flows within the context of allowing finite-amplitude height variations in the lower layer. The baroclinic destabilization of a bottom-trapped coupled front on a sloping bottom is described. In the overlying fluid the instability takes the form of amplifying topographic Rossby waves. In the coupled front the perturbations to the downslope incropping are preferentially amplified compared to those on the upslope incropping. The perturbations to the downslope incropping develop into downslope propagating plumes which eventually evolve into relatively coherent along-slope propagating domes. We discuss the propagation characteristics of these domes. We also simulate the evolution of density-driven eddies or domes. The first eddy simulation we describe is for an initial eddy configuration which satisfies a zero topographic Rossby wave condition in the upper layer. We show that these traveling solutions remain remarkably coherent over a period of about 40 eddy circulation times or about 250 days for typical continental slope values. For a sufficiently large initial eddy height, upper layer fluid parcels can be transported in the along-slope direction by the baroclinic eddy. We also simulate the evolution of an initial eddy configuration which does not satisfy a zero topographic Rossby wave condition in the upper layer. A relatively intense cyclonic circulation develops in the overlying fluid over the traveling dome as does a topographic Rossby wave tail. However, even these solutions remain surprisingly coherent over many eddy circulation times.

Stability conditions and a priori estimates for equivalent-barotropic modons

Linear Liapunov stability conditions are obtained for all classes of equivalent‐barotropic modons. The stability conditions are stated in terms of a parameter η called the ‘‘generalized disturbance wavenumber’’ related to the ratio of the initial values of disturbance enstrophy to energy. It is shown that c>0 modons (c is the drift speed, nondimensionalized by the long wave speed) are stable when η≤κ (κ is the modon wavenumber), and that c<−1 modons are stable when η≥κ. This dependency of the stability on the initial spectral structure of the disturbance is observed in numerical calculations. A priori L2‐type estimates bounding the growth of perturbations are derived. The instability mechanism is interpreted in terms of Fj≂rtoft’s energy cascade theorem [Tellus 5, 225 (1953)].

Barotropic modon propagation over slowly varying topography

Abstract A perturbation theory is developed to describe modon propagation over slowly varying topography. The theory is developed from the rigid-lid shallow-water equations on an infinite β-plane. Nonlinear hyperbolic equations are derived, based on the conservation of energy, enstrophy and vorticity, to describe the evolution of the slowly varying modon radius, translation speed and wavenumber for arbitrary finite-amplitude topography. To leading order, the modon is unaffected by meridional gradients in topography. Analytical perturbation solutions for the modon radius, translation speed and wavenumber are obtained for small-amplitude topography. The perturbations take the form of hyperbolic transients and a stationary component proportional to the topography. The solution predicts that as the modon moves into a region of shallower (deeper) fluid the modon radius increases (decreases), the translation speed decreases (increases) and the modon wavenumber decreases (increases). In addition, as the modon pr...

Nonlinear stability of intermediate baroclinic flow on a sloping bottom

The governing equations describing the dynamics of mesoscale gravity currents or coupled density fronts and steadily-travelling coherent cold eddies on a sloping bottom are shown to possess a non-canonical hamiltonian structure. We exploit the hamiltonian formalism to obtain a variational principle that describes arbitrary steady solutions in terms of a suitably constrained hamiltonian. Two Arnol’d-like stability theorems are obtained which can establish the linear stability in the sense of Liapunov of these steady solutions. Based on this analyses two a priori estimates are derived which bound the disturbance energy and the Liapunov norm with respect to the initial disturbance potential enstrophy and energy. In the limit of parallel shear flow solutions corresponding to a current flowing along isobaths, the first formal stability theorem reduces to a previously established normal-mode stability result. Based on the formal stability analysis, convexity conditions are given for the constrained hamiltonian that can rigorously establish nonlinear stability in the sense of Liapunov for the steady current solutions. A variational principle is also presented which can describe steadily-travelling isolated cold eddy solutions of the model. The principle is based on constraining the hamiltonian with appropriately chosen Casimir and momentum invariants. It is shown that a suitably extended form of Andrews’ theorem holds for our model equations. Therefore, the stability of the steadily travelling isolated eddy solutions cannot be established using the energy-Casimir analysis developed here.

Finite-amplitude baroclinic instability of a mesoscale gravity current in a channel

Abstract A finite amplitude theory is developed for the evolution of marginally unstable modes for a mesoscale gravity current on a sloping bottom. The theory is based on a nonquasigeostrophic, baroclinic model of the convective destabilization of gravity currents which allows for large amplitude isopycnal deflections while filtering out barotropic instabilities. Two calculations are presented. First, a purely temporal amplitude equation is derived for marginally unstable modes not located at the minimum of the marginal stability curve. These modes eventually equilibrate with a new finite amplitude periodic solution formed. Second, the evolution of a packet of marginally unstable modes located at the minimum of the marginal stability curve is presented. These two models are dramatically different due to fundamental physical differences. For marginally unstable modes not located at the minimum of the marginal stability curve, it is possible to determine the evolution of a single normal mode amplitude. For ...

Stability Characteristics of Deep-Water Replacement in the Strait of Georgia

Abstract It has been suggested that low-frequency current fluctuations in the southern Strait of Georgia are the result of baroclinic instability. However, data extracted from cyclesonde and fixed current meter moorings suggest that the conditions for baroclinic instability are highly variable in space and time. It has been recently discovered that there are summertime bottom-intensified gravity currents with fortnightly and monthly periods associated with the introduction of salty waters from the Juan de Fuca Strait during periods of neap tides. These currents are the dominant mechanism for deep-water renewal in the Strait of Georgia. It is argued that these currents are baroclinically unstable and that the stability characteristics are reasonably consistent with the observed structure of the low-frequency current fluctuations. The episodic nature of these unstable bottom flows may help to explain the spatial and temporal variability of the low-frequency current fluctuations observed in the Strait of Geo...