Hydraulics and instabilities of quasi-geostrophic zonal flows (original) (raw)
Related papers
Journal of Fluid Mechanics, 2006
Unstable modes of a linear shear flow in shallow water on an equatorial β-plane are obtained over a wide range of values of a non-dimensional parameter and are interpreted in terms of resonance between neutral waves. The non-dimensional parameter in the system is E ≡ γ 4 /(gHβ 2), where γ , g, H and β are the meridional shear of basic zonal flow, gravitational constant, equivalent depth and the northsouth gradient of the Coriolis parameter, respectively. The value of E is varied within the range −2.50 6 log E 6 7.50. The problem is solved numerically in a channel of width 5γ /β. The structures of the most unstable modes, and the combinations of resonating neutral waves that cause the instability, change according to the value of E as follows. For log E < 2.00, the most unstable modes have zonally non-symmetric structures; the most unstable modes for log E < 1.00 are caused by resonance between equatorial Kelvin modes and continuous modes, and those for 1.00 6 log E < 2.00 are caused by resonance between equatorial Kelvin modes and westward mixed Rossby-gravity modes. The most unstable modes for log E > 2.00 have symmetric structures and are identical with inertially unstable modes. Examinations of dispersion curves suggest that nonsymmetric unstable modes for 1.00 6 log E < 2.00 and inertially unstable modes for log E > 2.00 are the same kind of instability.
1994
The thesis addresses the applicability of traditional hydraulic theory to an unstable, mid-latitude jet where the only wave present is the Rossby wave modified by shear. While others (Armi 1989, Pratt 1989, Haynes et aL1993 and Woods 1993) have exam-ined specific examples of shear flow "hydraulics", my goal was to find general criteria for the types of flows that may exhibit hydraulic behavior. In addition, a goal was to determine whether a hydraulic mechanism could be important if smaller scale shear instabilities were present. A flow may exhibit hydraulic behavior if there is an alternate steady state with the same functional relationship between potential vorticity and streamfunction. Us-ing theorems for uniqueness and existence of two point boundary value problems, a necessary condition for the existence of multiple states was established. Only certain flows with non-constant, negative dQ6P) have alternate states. Using a shooting method for a given transport and a giv...
Global instabilities in diverging channel flows
Theoretical and Computational Fluid Dynamics, 2011
A global stability study of a divergent channel flow reveals features not obtained hitherto by making either the parallel or the weakly non-parallel (WNP) flow assumption. A divergent channel flow is chosen for this study since it is the simplest spatially developing flow: the Reynolds number is constant downstream, and for a theoretical Jeffery-Hamel flow, the velocity profile obeys similarity. Even in this simple flow, the global modes are shown to be qualitatively different from the parallel or WNP. In particular, the disturbance modes are often not wave-like, and the local scale, estimated from a wavelet analysis, can be a function of both streamwise and normal coordinates. The streamwise variation of the scales is often very different from the expected linear variation. Given recent global stability studies on boundary layers, such spatially extended modes which are not wave-like are unexpected. A scaling argument for why the critical Reynolds number is so sensitive to divergence is offered.
Response to a Steady Poleward Outflow. Part I: The Linear, Quasigeostrophic Problem
Journal of Physical Oceanography, 2009
The response of a zonal channel to a uniform, switched-on but subsequently steady poleward outflow is presented. An eastward coastal current with a Kelvin wave's cross-shore structure is found to be generated instantly upon initiation of the outflow. The current is essentially in geostrophic balance everywhere except for the vicinity of the outflow channel mouth, where the streamlines must cross planetary vorticity contours to feed the current. The adjustment of this region generates a plume that propagates westward at Rossby wave speeds. The cross-shore structure of the plume varies with longitude, and at any given longitude it evolves with time. The authors show that the plume evolution can be understood both conceptually and quantitatively as the westward propagation of the Kelvin current's meridional spectrum, with each spectral element propagating at its own Rossby wave group velocity.
A simple model of Rossby-wave hydraulic behaviour
Journal of Fluid Mechanics, 1993
This paper considers hydraulic control and upstream influence in systems where the only wave propagation mechanism arises from the variation of vorticity or potential vorticity. These systems include two-dimensional shear flows as well as many simple paradigms for large-scale geophysical flows. The simplest is a flow in which the vorticity or potential vorticity is piecewise constant. We consider such a flow confined to a rotating channel and disturbed by a topographic perturbation. We analyse the behaviour of the system using steady nonlinear long-wave theory and demonstrate that it exhibits behaviour analogous to open-channel hydraulics, with the possibility of different upstream and downstream states. The manner by which the system achieves such states is examined using time-dependent long-wave theory via integration along characteristics and using full numerical solution via the contour-dynamics technique. The full integrations agree well with the hydraulic interpretation of the steady-state theory. One aspect of the behaviour of the system that is not seen in open-channel hydraulics is that for strong subcritical flows there is a critical topographic amplitude beyond which information from the control cannot propagate far upstream. Instead flow upstream of the topographic perturbation adjusts until the long-wave speed is zero, the control moves to the leading edge of the obstacle and flow downstream of the control is supercritical, with a transition from one supercritical branch to another on the downstream slope of the obstacle.
On the Baroclinic Instability of Nonplanar Flows
Nuovo Cimento Della Societa Italiana Di Fisica C-Physics and Astronomy, 1985
Different ocean models with one or two layers having constant static stability and supporting constant-shear flows, whose directions are allowed to change with depth, are examined in the framework of the linear nonzonal baroclinic stability theory and in the absence of the fl-effect. The analysis is reduced to solving a simple Sturm-Liouville boundary value problem in one dimension. A fairly general dispersion relation is found which correctly reproduces several special cases analysed by other authors. This relation shows a fair variety of possible behaviours for stability curves of two-layer models. The results show that the presence of a nonplanar shear-flow may have profound consequences on the stability character of the stationary geostrophic flow. In fact, it appears that the stability properties are strongly dependent on the propagation angle of the disturbance so that wave numbers which appear stable in the usual zonal theory may result unstable on such a nonzonal flow, and vice versa.
Instability of marginally stable streamwise varying shear flow to long rossby waves
Communications in Mathematical Sciences, 2007
The instability of streamwise varying shear flow that is marginally stable to long Rossby waves is examined. Both Hamiltonian and non-Hamiltonian flows are considered within the framework of coupled wave instability (CWI). The CWI is shown to be mediated by a 'physical' wave and a 'virtual' wave. The physical/virtual wave model concisely describes the differences between the instabilties that develop on locally supercritical flow and those that develop on globally subcritical flow. Globally subcritical Hamiltonian flows are proven to be stable. In contrast, non-Hamiltonian flows may be unstable in both locally supercritical and globally subcritical regimes. In locally supercritical flow, the CWI grows via wave-resonance between the physical/virtual wave pair and is highly localized to the supercritical region. In contrast, in globally subcritical flow, the CWI grows via pseudomomentum extraction from the background flow and either radiates away or remains trapped to the streamwise variation in the flow.
On the stability of ocean overflows
Journal of Fluid Mechanics, 2008
The stability of a hydraulically driven sill flow in a rotating channel with smoothly varying cross-section is considered. The smooth topography forces the thickness of the moving layer to vanish at its two edges. The basic flow is assumed to have zero potential vorticity, as is the case in elementary models of the hydraulic behaviour of deep ocean straits. Such flows are found to always satisfy Ripa's necessary condition for instability. Direct calculation of the linear growth rates and numerical simulation of finite-amplitude behaviour suggests that the flows are, in fact, always unstable. The growth rates and nonlinear evolution depend largely on the dimensionless channel curvature κ=2αg′/f2, where 2α is the dimensional curvature, g′ is the reduced gravity, and f is the Coriolis parameter. Very small positive (or negative) values of κ correspond to dynamically wide channels and are associated with strong instability and the breakup of the basic flow into a train of eddies. Fo...