Dynamical Criteria Determining Lee Cyclogenesis

The two-layer quasigeostrophic flow model is an intermediate system between the single-layer two-dimensional barotropic flow model and the continuously stratified three-dimensional baroclinic flow model. This model is widely used to investigate basic mechanisms in geophysical flows, such as baroclinic effects, the Gulf stream and subtropical gyres. We consider the two-layer quasigeostrophic flow model under stochastic wind forcing on the top layer. The fluctuating part of the wind forcing is modeled as the generalized time derivative of a Wiener process. We first transform this stochastic two-layer fluid system into a coupled system of random partial differential equations. Then we prove that the stochastic two-layer fluid system has finite sets of asymptotically determining functionals (such as determining modes and determining nodes) in probability. Furthermore, we show that the asymptotic probabilistic dynamics of this system depends only on the top fluid layer. Namely, in the probability sense and asymptotically, the dynamics of the two-layer quasigeostrophic fluid system is determined by the top layer. In other words, the bottom layer is slaved by the top layer. This conclusion is true provided that the Wiener process and the fluid parameters satisfy a certain condition. In particular, this latter condition is satisfied when the trace of the covariance operator of the Wiener process is small enough and the Ekman constant r is sufficiently large.

Local baroclinic instability is studied in a two-layer quasi-geostrophic model. Variable meridional bottom slope controls the local supercriticality of a uniform zonal flow. Solutions are found by matching pressure, velocity, and upper-layer vorticity across longitudes where the bottom slope changes abruptly so as to destabilize the flow in a central interval of limited zonal extent. In contrast to previous results from heuristic models, an infinite number of modes exist for arbitrarily short intervals. For long intervals, modal growth rates and frequencies approach the numerical and WKB results for the most unstable mode. For intervals of length comparable to and smaller than the wavelengths of unstable waves in the homogeneous problem, the WKB results lose accuracy. The modes retain large growth rates (about half maximum) for intervals as short as the internal deformation radius. Evidently, the deformation radius and not the homogeneous instability determines the fundamental scale for local instability. Maximum amplitudes occur near the downstream edge of the unstable interval. Lower-layer amplitudes decay downstream more rapidly than upper-layer amplitudes. For short intervals, the instability couples motions with widely disparate horizontal scales in the upper and lower layers. Heat flux is more strictly confined than amplitude. Growth rates increase linearly with weak supercriticality.

A data assimilation scheme for ocean acoustic tomography (OAT) data is developed and evaluated by an identical twin experiment to understand the dynamical processes and the evolution of flow pattern in the Kuroshio extension region. The scheme consists of three components, i.e., the two-layer quasi-geostrophic (QG) oceanic numerical model, the travel time data obtained from OAT, and the data assimilation technique which consists of the Kalman filter and observation equations relating the model variables (stream function and interface displacement) to the OAT-derived reciprocal travel time. The identical twin experiment using two different model fields made from the same model demonstrates both the stability and the availability of the scheme developed in the present study.

The quasigeostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras, we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Wherever possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.

A theory is presented for the propagation of wave packets in a slightly unstable baroclinic shear flow in a quasi-geostrophic two-layer model on the beta plane. The theory for inviscid motions is considered and packet solutions resembling solitary waves are found. It is shown that the propagation speed of the packet, which is a function of its amplitude, exceeds the most naturally defined group velocity. A physical explanation is presented and it is suggested that the enhancement of the signal velocity above the group velocity is a general property of systems possessing linear instability and nonlinear stability.

In this paper, the effects of bottom and interfacial friction on localized baroclinic instability are discussed in the weakly nonlinear, long-wave limit. Using a quasigeostrophic two-layer model in which the lower layer is assumed to be deep, we have derived a coupled evolution equation set that consists of a KdV-type equation for the upper layer and a linear long-wave equation for the lower layer. A perturbation theory reveals that there are multiple equilibria in this system, where baroclinic energy conversion and frictional dissipation are in balance; the flow is not forced externally, and multiplicity here refers to the presence or absence of solitary waves propagating steadily on a zonal flow. Further, direct numerical calculations show a rich variety of behavior of solitary waves, including steady, periodic, and complicated interacting evolutions. For a two-layer model to have multiple steady or oscillatory states, both bottom and interfacial friction should be included because if one of th...

The process of eddy detachment from a strong, eastward-flowing oceanic jet, which is modeled after the Gulf Stream, is studied using a two-layer quasi-geostrophic model. Numerical calculations are performed, in which an initial perturbation having a small amplitude is superimposed on a basic flow consisting of the eastward jet in the upper layer. The initial perturbation is located in the western part of the jet and consists of one wavelength of meanders. The initial perturbation propagates eastward and grows rapidly, for the most part because of a baroclinic instability. The extremes of large-amplitude meanders are forced to go along the edges of recirculating gyres which are generated on both sides of the jet in the western part, and as a result the meanders are cut off and form eddies. In some cases, meander interactions—by which an upstream meander overtakes the nearest downstream meander—precedes the detachment. The meander growth, which is the basic condition for the detachment, is highly dependent on the lower layer velocity of the basic jet, i.e., the fast eddy is detached later and farther downstream as the velocity increases from zero. The meander whose wavelength is ∼1.2 times as long as that of the fastest growing linear solution is the major source of the detached eddies, because nonlinearity has the stronger stabilizing effects on the shorter wavelength meanders and reduces group velocity of the longer wavelength meanders to a greater extent. A comparison of solutions on a β plane with ones on an f plane further reveals that the beta-effect assists detachment. The spatially growing solutions, which have larger amplitude in the eastern part, are more analogous to Gulf Stream meanders from which eddies are detached than are the spatially periodic, temporally growing solutions. The numerical results are applied to infrared images made from the NOAA-4 satellite of the Gulf Stream in which eddies are detached from meanders.

It is shown that the classical quasigeostrophic two-layer model of baroclinic instability possesses an optimal ratio of layer thicknesses that maximizes the growth rate, given the basic-state shear (thermal wind), beta, and the mean Rossby radius. This ratio is interpreted as the vertical structure of the most unstable mode. For positive shear and beta, the optimal thickness of the lower layer approaches the midheight of the model in the limit of strong criticality (shear/beta) but it is proportional to criticality in the opposite limit. For a set of parameters typical of the earth’s midlatitudes, the growth rate maximizes at a lower-layer thickness substantially less than the midheight and at a correspondingly larger zonal wavenumber. It is demonstrated that a turbulent baroclinic jet whose statistical steady state is marginally critical when run with equal layer thicknesses can remain highly supercritical when run with a nearly optimal thickness ratio.

The linear baroclinic and parametric instabilities of boundary currents with piecewise-constant potential vorticity are studied in a two-layer quasi-geostrophic model. The growth rates of both the exponential modes and of the optimal perturbations are calculated for the baroclinic instability of steady coastal currents. We show that the growth rates of the exponential modes are maximal for a vertically symmetric flow. Furthermore, the vertical asymmetries induced by different layer thicknesses, the presence of a barotropic potential vorticity or bottom topography, all act to dampen the growth rates and favor growth at shorter wavelengths. It is shown that this behavior can be predicted from the conditions for vertical resonance of Rossby waves on the two potential vorticity fronts. Also, the baroclinic instability of the optimal perturbations has larger growth rates at shorter wavelengths and shorter time scales. As well, the presence of a sloping bottom of moderate amplitude favors the growth of these optimal perturbations. Finally, we compute the growth rates of parametric instability of oscillatory coastal flows. We show that subharmonic resonance is the most unstable mode of growth. In addition, a second region of parametric instability is found (for the first time) away from marginality of exponential-mode baroclinic instability. It is shown that the functional dependency of the growth rates of parametric instability, for optimal excitation, are similar to that of the optimal perturbations of baroclinic instability. To explain this a mechanism for parametric instability, involving the rapid growth of short-wave optimal perturbations, is proposed.

We examine the potential vorticity (PV) flux produced by forced Rossby waves in a two-layer quasigeostrophic model, using a perturbation analysis. Rossby waves are excited by external forcing applied to the upper layer. The southward PV flux is produced in the lower layer by the higher-order Rossby waves that are excited by nonlinear wave–wave interactions, whereas the northward PV flux is produced in the upper layer. The direction of the PV flux is consistent with that obtained by an eddy-resolving model of the wind-driven circulation in previous studies. The southward PV flux is produced in a wide parameter range comparable to the eddy-resolving model. The basic features of the PV flux remain unchanged even in the limit of weak stratification. In this limit, stratification has nearly no effect on the flow, except that it isolates the lower layer from the direct effects of external forcing. The mechanism of the southward PV flux is explained using basic features of the barotropic Rossby waves and does not depend on details of the model. Furthermore, the resonant triad interaction of Rossby waves does not affect the PV flux. Stratification weakens or strengthens the PV flux depending on the horizontal scale of the external forcing.

We study the evolution of eastward propagating dipoles (modons) in a two-layer quasi-geostrophic β-plane model using high-resolution numerical simulations. Various combinations of background gradients of potential vorticity in the upper and lower layers (which may include sloping topography) shed light on the recently discovered breakdown mechanisms and rich dynamics of dipolar vortices. We consider dipoles with a circular separatrix in the upper layer and, owing to the β-effect in the same layer, the symmetry of the flow breaks due to an exponentially growing, rotating, asymmetric mode of linear instability associated with Rossby wave radiation. Further nonlinear transformation is found to consist of two phases: fast partner separation, resulting in a deceleration of the eastward drift, and subsequent slow separation with a saturated asymmetric mode accompanied by much weaker, shorter Rossby waves. This weakly radiating phase—with pulsating partners and homogenized potential vorticity between the partners and the separatrix—can be considered as a new type of long-lived dipole. Conversely, when no β-effect is present in the upper layer, the dipoles remain nearly symmetric, even when the topographic β-effect is present in the active lower layer. In this case, the development of a weak asymmetric mode is related to a small meridional shift of the dipole center on the numerical grid.

Motivated by the substantial sensitivity of eddies in two-layer quasi-geostrophic (QG) turbulence models to the strength of bottom drag, this study explores the sensitivity of eddies in more realistic ocean general circulation model (OGCM) simulations to bottom drag strength. The OGCM results are interpreted using previous results from horizontally homogeneous, two-layer, flat-bottom, f-plane, doubly periodic QG turbulence simulations and new results from two-layer β-plane QG turbulence simulations run in a basin geometry with both flat and rough bottoms. Baroclinicity in all of the simulations varies greatly with drag strength, with weak drag corresponding to more barotropic flow and strong drag corresponding to more baroclinic flow. The sensitivity of the baroclinicity in the QG basin simulations to bottom drag is considerably reduced, however, when rough topography is used in lieu of a flat bottom. Rough topography reduces the sensitivity of the eddy kinetic energy amplitude and horizontal length scales in the QG basin simulations to bottom drag to an even greater degree. The OGCM simulation behavior is qualitatively similar to that in the QG rough bottom basin simulations in that baroclinicity is more sensitive to bottom drag strength than are eddy amplitudes or horizontal length scales. Rough topography therefore appears to mediate the sensitivity of eddies in models to the strength of bottom drag. The sensitivity of eddies to parameterized topographic internal lee wave drag, which has recently been introduced into some OGCMs, is also briefly discussed. Wave drag acts like a strong bottom drag in that it increases the baroclinicity of the flow, without strongly affecting eddy horizontal length scales.

A quasi-linear two-layer quasigeostrophic β-plane model of the interaction between a baroclinic jet and a single zonal wavenumber perturbation is used to study the mechanics leading to a wave amplitude bifurcation—in particular, the role of the critical surfaces in the upper-tropospheric jet flanks. The jet is forced by Newtonian heating toward a radiative equilibrium state, and Ekman damping is applied at the surface. When the typical horizontal scale is approximately the Rossby radius of deformation, the waves equilibrate at a finite amplitude that is comparable to the mean flow. This state is obtained as a result of a wave-induced temporary destabilization of the mean flow, during which the waves grow to their finite-equilibrium amplitude. When the typical horizontal scale is wider, the model also supports a state in which the waves equilibrate at negligible amplitudes. The transition from small to finite-amplitude waves, which occurs at weak instabilities, is abrupt as the parameters of the system are gradually varied, and in a certain range of parameter values both equilibrated states are supported. The simple two-layer quasi-linear setting of the model allows a detailed examination of the temporary destabilization process inherent in the large-amplitude equilibration. As the waves grow they reduce the baroclinic growth by reducing the vertical shear of the mean flow, and reduce the barotropic decay by reducing the mean potential vorticity gradient at the inner sides of the upper-layer critical levels. Temporary destabilization occurs when the reduction in barotropic decay is larger than the reduction in baroclinic growth, leading to a larger total growth rate. Ekman friction and radiative damping are found to play a major role in sustaining the vertical shear of the mean flow and enabling the baroclinic growth to continue. By controlling the mean flow potential vorticity gradient near the critical level, the model evolution can be changed from one type of equilibration to the other.

: Research under this grant focused primarily on computations of unstable nonlinear periodic solutions, time-dependent normal modes (Floquet vectors), and singular vectors in a two-layer quasi-geostrophic channel model. The model was studied in weakly and strongly nonlinear regimes, in which small disturbances to an unstable, steady, zonal, baroclinic shear flow grow to finite amplitude and continue to vacillate irregularly for arbitrarily long times. The computation of time-dependent, normal-mode disturbances to unstable, nonlinear, time-periodic basic flows in a high-dimensional geophysical fluid model opens a new perspective on the analysis of disturbance growth in time-dependent flows, and on the closely related problem of error growth in predictive models of time- dependent flows.

This study describes the evolutionary mechanisms of nonlinear barotropic–baroclinic interactions, especially, on the excitations, propagations, and decreases of nonlinear coherent structures. Starting from the classical two-layer quasi-geostrophic potential vorticity conservation model equations, the barotropic and baroclinic model equations are derived from the classical work of Pedlosky and Thomson [J. Fluid Mech. 490, 189–215 (2003)]. By considering the effects of bottom topography and beta-plane approximation, the coupled nonlinear Korteweg–de Vries model equations for the evolutions of barotropic and baroclinic coherent structures are obtained by using the methods of multiple scales and perturbation expansions, respectively. Solitary wave solutions are given according to the method of elliptic function expansions, and the physical mechanisms for the evolutions of the nonlinear barotropic–baroclinic interactive coherent structures are analyzed based on the obtained solitary wave solutions. It will be potentially useful for further theoretical investigations on atmospheric blocking phenomena or wave–flow interactions.