Dynamic effect of open latitude on wind-driven circulation in a zonal β-plane channel

A class of generalized Fofonoff modes, both barotropic and baroclinic, are found in a β-plane channel presumably to model the circumpolar ocean in the presence of a bottom ridge. These generalized Fofonoff modes share many dynamic features with those in closed basins. Conserved quantities are employed to show that these generalized Fofonoff modes, like those in closed basins, are stable with respect to finite amplitude perturbations.     

The exchange of fluid mass between quasi-geostrophic and ageostrophic motions during the reflection of Rossby waves from a coast. I. The case of an infinite rectilinear coast

When the problem of the reflection of spatially localized Rossby waves from a coast is treated using the quasigeostrophic (QG) approximation, the total fluid mass and the along-shore circulation calculated from the geostrophic height field are not conserved. To understand the correct mass balance and the degree to which the QG equations and boundary conditions may be in error, we analyze an initial-value problem for the Laplace tidal equations on a β-plane in the asymptotic limit 1, where is the ratio of the spatial scale of the motion to the Earth's radius.It is shown that there is a coupling between QG and O() fields. Physically, the coupling occurs by a peculiar adjustment process in the O() approximation in which fast gravity waves are permanently generated to build up a quasi-stationary edge Kelvin wave. Different temporal scales (large for O(1) Rossby waves and small for the O() gravity waves make comparable the contributions of the waves to the mass and circulation balance equations. However, QG analysis itself describes the reflection of Rossby waves correctly, but is incomplete, and for satisfactory balances one has to take into account the fields of both orders of the approximation.Applications of the results to closed basins, baroclinicity, and variable bottom topography are discussed. It is conjectured that an interaction of strong oceanic eddies with a coast (continental slope) may give rise to noticeable along-shore jet currents.     

Nonlinear interactions of spherical Rossby modes

Weakly nonlinear triad interactions between spherical Rossby harmonics are studied. Each of the harmonics has the form AP_n^m(sin θ)exp[i(mλ − σt)], where A is an amplitude and P_n^m is the associated Legendre function. Equations for the amplitudes are derived and resonance conditions are analysed. The resonance conditions differ substantially from the usual resonance conditions on a β-plane and include a Diophantine equation and a few inequalities for the integer wavenumbers n and m of the interacting modes. Particular analytical series of solutions to the resonance conditions are constructed, which show that modes with arbitrary large wavenumbers can participate in the interactions. A numerical study of the resonance conditions reveals that no more than 21% of the Rossby harmonics can participate in the triad interactions and that chains of the interacting triads soon break. Thus precise interactions (for which the resonance conditions hold exactly) do not result in any significant redistribution of energy over the spectrum. On the other hand, approximate interactions (for which the resonance conditions hold approximately) generate an intensive energy redistribution among short Rossby modes with typical scales much smaller than the Earth's radius. Thus the energy cascade is concentrated mainly in the short-wave part of the spectrum and is very weak in the large-scale domain. The efficiency of the triad interaction of Rossby modes with scales much smaller than the Earth's radius depends strongly on the existence of the so-called interaction latitude at which the local wave-vectors and frequencies of the interacting modes satisfy resonance conditions for plane Rossby waves on the β-plane approximating the neighbourhood of the latitude. If the interaction latitude exists, the interaction is intensive; in the opposite case, it is weak.     

Refraction of Rossby waves on a multiple β-plane

A multiple β-plane is introduced to explore the relation between plane and spherical Rossby waves. The fundamental problem, the refraction of a plane Rossby wave across a discontinuity in β, is solved. It is shown that refraction on the multiple β-plane agrees in the limit with refraction on the full sphere only if a suitable correction is made for the geometric distortion of the β-plane. The full spherical modes of Rossby waves trapped in a band about the equator (Longuet-Higgins, 1964) have their counterpart in a simple model consisting of an “equatorial” β-plane bounded above and below by “polar” β-planes.     

Nonlinear spatial baroclinic instability in slowly varying zonal flow

The linear and weakly nonlinear dynamics of long, low frequency, spatially growing baroclinic waves embedded in slowly varying zonal flow on a β-plane channel are examined in a continuous model of the atmosphere. For a basic state jet flow possessing a locally unstable region, the nonlinear solution yields a maximum amplitude that is located near the region of maximum baroclinicity and substantially upstream of the maximum amplitude obtained from linear theory. The difference between the linear and nonlinear solutions is due to the time-averaged wave fluxes becoming large enough in the nonlinear problem to stabilize the flow prior to reaching the location (jet center) where the basic state baroclinicity and locally computed linear growth rate are maximized.     

A laboratory study of open ocean barometric response

An attempt is made to explain the physical mechanism for the Mid-Ocean Dynamics Experiment (MODE) bottom pressure observations of Brown et al. (1975) by means of a joint laboratory and theoretical study. First, possible relevant effects in an f-plane model are considered, where pressure driving forms and transient effects can be evaluated. If atmospheric pressure change is local (of a standing wave nature), sea surface response increases with increasing subinertial frequency. If pressure changes are advective, sea surface response decreases with increasing speed. Overreaction of the free surface to forcing, causing overshoot, can occur when the free inertia-gravity modes of the basin are excited. Laboratory experiments are performed which agree with the theory of the local pressure change mechanism to within assinged experimental error.The study is extended to consider the response of an equivalent β-plane system in a more laboratory-oriented approach. It is found that the β-effect is important in the entire range of frequencies studied in the laboratory, with the response approaching that of an f-plane ocean at higher frequencies. The excitation of free planetary modes of the basin proves to be important in the barometric response, allowing for local oversoot and undershoot and for the generation of a substantial pressure signal at a distance from the driving, allowing a significant contribution to the incoherent bottom signal to be forced by distant atmospheric pressure disturbances in the ocean basin.     

The effect of barotropic shear on baroclinic instability Part I: Normal mode problem

The effect of barotropic shear in the basic flow on baroclinic instability is investigated using a linear multilevel quasi-geostrophic β-plane channel model and a nonlinear spherical primitive equation model. Barotropic shear has a profound effect on baroclinic instability. It reduces the growth rates of normal modes by severely restricting their structure, confirming earlier results with a two-layer model. Dissipation, in the form of Ekman pumping and Newtonian cooling, does not change the main characteristics of the effect of the shear on normal mode instability.Barotropic shear in the basic state, characterized by large shear vorticity with small horizontal curvature, also effects the nonlinear development of baroclinic waves. The shear limits the energy conversion from the zonal available potential energy to eddy energy, reducing the maximum eddy kinetic energy level reached by baroclinic waves. Barotropic shear, which controls the level of eddy activity, is a major factor which should be considered when parameterizing the eddy temperature and momentum fluxes induced by baroclinic waves in a climate model.     

A parcel theory approach to β-plane inertial stability

The problem of the orbit of a parcel from its initial location at which there might be a momentum anomaly is here considered for a β-plane. The solutions represent a generalisation of the recent work of Wan and Yang (1990, Adv. Atmos. Sci., 7:409–422). A physical interpretation of the orbits, in which they are related to flow stability, is given for both extratropics and tropics. The possible effect of dissipation on these results is also discussed.     

A homogeneous model of the Deacon cell in a circumpolar ocean

Dynamics of the three-dimensional structure of the wind-driven Deacon cell in a β-plane channel are discussed in a homogeneous model in the presence of a sufficiently high ridge. The emphasis is on the water mass balance: how the northward surface Ekman drift is returned. It is demonstrated that a sufficiently high ridge can break up the geostrophic constraint and a net geostrophic volume flux across the open latitude band is allowed. It is found that: (1) the Deacon cell is a fundamentally three-dimensional structure, (2) wind forcing can drive an inter-basin water mass exchange in the Southern Ocean, and (3) zonal through-channel transport in the circumpolar ocean varies at different longitudes.     

Laboratory modeling of topographic Rossby normal modes

The physical modeling of topographic Rossby normal modes carried out at the “Coriolis” Rotating Platform (Grenoble), is presented. The basic feature of the bottom topography is a linear slope of 4.3 m×2 m delimited by two lateral walls. Since the studied motions are essentially barotropic, homogeneous water was used. Unsheared currents were generated by a simple movement of a wavemaker located in front of the topographic barrier. The conservation of potential vorticity for the currents flowing onto the channel slope produced Rossby waves: reflections at the lateral boundaries then led to the formation of propagating barotropic Rossby normal modes, whose frequencies and spatial structures were selected by the physical system. The currents were measured through the correlation imaging velocimetry (CIV) method, which allowed an extremely detailed synoptic map of the horizontal velocities in an area (13 m²) including the slope to be obtained every 30 s.A variety of experiments were performed in order to provide a complete process study in which the effect of different channel lengths and rotation periods could be tested. Two different lengths of the linear slope, 4.3 and 3.3 m, and rotation periods ranging from 30 to 50 s were considered. The qualitative analysis of the 2D current patterns, and the good agreement found between the measured eigenperiods and the periods obtained by means of a simple analytical model, show that in all cases the first Rossby normal mode was generated. Moreover, numerical simulations based on the shallow-water equations, for a geometry and paddle movements that match closely the experimental setup, allow to calibrate the analytical model and provide useful information on a discrepancy found between experimental and analytical eigenperiods due to an oscillation of the normal mode trajectory.     

Characteristics of adjoint sensitivity to potential vorticity

Structures of adjoint sensitivities to potential vorticity for specific initial and final norm are investigated for a short-range cyclone forecast in a three-dimensional quasigeostrophic (QG) model. Moreover, adjoint sensitivities to potential vorticity are compared with nonlinear sensitivities calculated for the same cyclogenesis case in the QG model. The adjoint sensitivities using different initial and final norms (e.g., total QG disturbance energy and potential enstrophy) show approximately similar characteristics for the horizontal and vertical structures and evolutions. Consistent with previous studies, the horizontal structure of the adjoint sensitivity is smaller for the energy norm than for the potential enstrophy norm. The dynamical mechanism of cyclone development by adjoint sensitivity coincides with that of nonlinear sensitivity, with slight differences in the region of sensitivity maxima over the upstream (nascent) low for the adjoint (nonlinear) sensitivity. The adjoint sensitivities show different vertical distributions from the nonlinear sensitivities. Consistent with the sensitive regions denoted by singular vectors and error evolution in the QG model, maxima of the adjoint sensitivities are located at both the upper and lower boundaries, with prominent secondary peaks in the lower to mid-troposphere of the domain. The level of the secondary maxima changes depending on the initial and final norm used. The secondary peak is located in the lower to mid- (mid-) troposphere for the total QG disturbance energy (potential enstrophy) as the initial and final norm. Based on the correspondence in the level of the sensitivity maxima in the interior of the domain between the adjoint and nonlinear sensitivities, adjoint sensitivities may serve as an alternative to nonlinear sensitivities given the enormous computing expenses in nonlinear sensitivity calculation.     

The domain of validity of the KdV-type solitary rossby waves in the shallow water β-plane model

The Domain, where the necessary and sufficient conditions for the existence of the KdV-type solitary Rossby waves are satisfied is defined in the shallow water β-plane model. The KdV-type solitary Rossby waves are the Rossby waves whose time-longitude dependence is determined by the KdV equation. As far as an appropriate amplitude and an appropriate ratio of the scales of the east-west and north-south directions are given, the KdV-type solitary Rossby waves can exist for every basic zonal flow. This result suggests the large validity of the soliton model in geophysical fluid dynamics. The KdV-type solitary Rossby waves are classified into four categories: (1) shear solitons studied by Long, Larsen, Benny, Redekop, and Hukuda, (2) β-divergent solitons studied by Clarke, Yamagata, and Nogami, (3) β-solitons found in the case of the strong stratification, and (4) divergent solitons which exist in the planetary-geostrophic-scale zonal flow. A remarkable result is that, in addition to the conventional east-west elongated solitons, the north-south elongated solitons can also exist for the case of the divergent solitons.     

A STUDY ON THE EXCITATION, ESTABLISHMENT AND TRANSITION OF MULTIPLE EQUILIBRIUM STATES PRODUCED BY NEARLY RESONANT THERMAL FORCING---- PART I: ASYMPTOTIC SOLUTIONS OF MULTIPLE EQUILIBRIUM STATES

A two-layer quasi-geostrophic baroclinic model in a narrow, longitudinally periodic channel on a β-plane is used, which involves near-resonant thermal forcing, frictional dissipation and a uniformly sheared basic current. By means of the multi-scale technique, a system of simplified differential equations or disturbances due to the steady thermal-forcing waves projected on the X-T plane is derived, which contains the nonlinear interactions between forced waves and free waves. The asymptotic solutions of the equilibrium states of these equations are analytically obtained by a singular perturbation method. The results show that these mul-tiple equilibrium state (MES) solutions can exist in a wide range of the parameters used.     

Evaluation of statistical downscaling in short range precipitation forecasting

The objective of this study is to compare several statistical downscaling methods for the development of an operational short-term forecast of precipitation in the area of Bilbao (Spain). The ability of statistical downscaling methods nested inside numerical simulations run by both coarse and regional model simulations is tested with several selections of predictors and domain sizes. The selection of predictors is performed both in terms of sound physical mechanisms and also by means of “blind” criteria, such as “give the statistical downscaling methods all the information they can process”.Results show that the use of statistical downscaling methods improves the ability of the mesoscale and coarse resolution models to provide quantitative precipitation forecasts. The selection of predictors in terms of sound physical principles does not necessarily improve the ability of the statistical downscaling method to select the most relevant inputs to feed the precipitation forecasting model, due to the fact that the numerical models do not always fulfil conservation laws or because precipitation events do not reflect simple phenomenological laws. Coarse resolution models are able to provide information usable in combination with a statistical downscaling method to achieve a quantitative precipitation forecast skill comparable to that obtained by other systems currently in use.     

Vortex–ridge interaction in a rotating fluid

The evolution of barotropic vortices interacting with a topographic ridge on a f-plane is studied by means of laboratory experiments in a rotating tank and numerical simulations. The initial condition in all experiments is a cyclonic vortex created at a certain distance from the ridge. The results are presented in two main scenarios: (a) weak interactions, which occur at early stages of the experiments, when the vortex is far from the ridge, and thus weakly experiences the influence of the topography. In these situations, the vortex slowly drifts towards the ridge with a leftward inclination due to the ascending slope of the topography. Such a behaviour is similar to the “northwestern” motion of cyclones over a weak sloping bottom. The circular shape of the monopolar vortex is preserved. (b) Strong interactions, in which the vortex core reaches the ridge and presents a more complicated evolution. The cyclone “climbs” to the top of the topography and crosses to the other side. Once the vortex experiences the opposite slope, it moves backwards trying to return to the original side of the ridge. For strong enough vortices, this process may be repeated a number of times until the vortex is dissipated by viscous effects. During these interactions the shape of the vortex is strongly deformed and several filaments are produced. In some cases the vortex is cleaved in two parts when crossing the ridge, one at each side of it and moving in opposite directions.Weak and strong interactions are numerically simulated by using a quasi-two-dimensional model. The results confirm that the vortex behaviour is governed by stretching and squeezing effects associated with changes in depth over the ridge and, at latter stages, by Ekman damping due to the solid bottom. The main results observed during strong interactions on a f-plane are also found on preliminar topographic β-plane experiments.     

On the interactions between planetary geostrophy and mesoscale eddies

Multiscale asymptotics are used to derive three systems of equations connecting the planetary geostrophic (PG) equations for gyre-scale flow to a quasigeostrophic (QG) equation set for mesoscale eddies. Pedlosky (1984), following similar analysis, found eddy buoyancy fluxes to have only a small effect on the large-scale flow; however, numerical simulations disagree. While the impact of eddies is relatively small in most regions, in keeping with Pedlosky’s result, eddies have a significant effect on the mean flow in the vicinity of strong, narrow currents.First, the multiple-scales analysis of Pedlosky is reviewed and amplified. Novel results of this analysis include new multiple-scales models connecting large-scale PG equations to sets of QG eddy equations. However, only introducing anisotropic scaling of the large-scale coordinates allows us to derive a model with strong two-way coupling between the QG eddies and the PG mean flow. This finding reconciles the analysis with simulations, viz. that strong two-way coupling is observed in the vicinity of anisotropic features of the mean flow like boundary currents and jets. The relevant coupling terms are shown to be eddy buoyancy fluxes. Using the Gent-McWilliams parameterization to approximate these fluxes allows solution of the PG equations with closed tracer fluxes in a closed domain, which is not possible without mesoscale eddy (or other small-scale) effects. The boundary layer width is comparable to an eddy mixing length when the typical eddy velocity is taken to be the long Rossby wave phase speed, which is the same result found by Fox-Kemper and Ferrari (2009) in a reduced gravity layer.     

On deep mean flow generation mechanisms and the abyssal circulation of numerical model gyres

Mesoscale resolution ocean general circulation model (EGCM) experiments have been carried out under a variety of different model physical assumptions, and the different model systems often produce very different deep mean flow fields. The flat bottom, rectangular basin experiments exhibit two distinct types of deep mean flow, which are here called “corotating” and “counterrotating”. Counterrotating deep flow, in which two adjacent deep gyres, with circulation of opposite senses, underlie the upper ocean eastward jet and its recirculation, has been found only in models with adiabetic two-layer model physics. None of the more complex model systems exhibit counterrotating deep flows; this type of flow is apparently restricted to a particular range of forcing/dissipation parameter space and/or particular model physical assumptions.Since the deep flow in these EGCM systems is generally weak, geostrophic dynamics provides the basic deep flow interior balance and the mean vertical velocity field, through the lower layer vorticity equation, largely determines the deep interior flow. The dynamical constraints on the mean vertical velocity field introduced by different model physical equations are reviewed and the adiabatic quasi-geostrophic (QG) two-layer model system is shown to be strongly constrained in several respects. In particular, the idea that eddy and mean heat flux divergence (or “layer thickness flux divergence”) drive the mean vertical velocity does not generalize to more complicated dynamical systems in which there is the possibility of altering the mean vertical density profile and/or in which the horizontal flow can be divergent. As a consequence of the constraints, there can be no basin net vorticity input to the lower layer via vortex stretching in the QG system.Because of the adiabatic QG constraints and the particular parametric regime in which the published adiabatic QG EGCM experiments exist, a very plausible explanation can be found for the existence of the deep cyclonic circulation of the model subtropical gyre. It is this cyclonic circulation that causes these deep flows to differ so dramatically from those of the more physically complex model systems. Because all the published adiabatic QG experiments that have non-trivial deep flows exhibit the counterrotating behavior, and because available ocean data do not support the existence of such a gyre in the North Atlantic, it seems important to thoroughly understand the reasons for the existence or absence of the deep cyclonic circulations. If they are an invitable feature of adiabatic QG systems, these models may need to be treated with caution as tools for understanding the mean ocean circulation.     

Potential vorticity aspects of the MJO

Considering linearized motion about a resting basic state, we derive analytical solutions of the equatorial β-plane primitive equations under the assumption that the flow is steady in a reference frame moving eastward with a diabatic forcing resembling a Madden–Julian Oscillation (MJO) convective envelope. The solutions are analyzed in terms of potential vorticity (PV) dynamics. Because the diabatic source term for PV contains a factor βy, the diabatic heat source is ineffective at generating a PV anomaly at the equator but maximizes the PV response near the poleward edges of the heat source. In this way a moving heat source can produce two ribbons of lower tropospheric PV anomaly, a positive one off the equator in the northern hemisphere and a negative one off the equator in the southern hemisphere, with oppositely signed PV anomalies in the upper troposphere. Associated with these PV anomalies are geopotential anomalies that are shifted several hundred kilometers poleward. In the lower troposphere these zonally elongated geopotential anomalies resemble ITCZ trough zones, which demonstrates the close connection between the MJO wake dynamics and the formation of double ITCZs.To demonstrate that the MJO wake response can be described by simple PV dynamics, we propose an invertibility principle relating the PV to the streamfunction, which in turn is locally related to the geopotential. This equatorial invertibility principle accurately recovers the balanced wind and mass fields found in the MJO wake in the primitive equation model. However, while the invertibility principle highlights the ability of simple PV dynamics to accurately describe the flow in the wake of an MJO convective envelope, it also clearly illustrates the inability of such dynamics to describe the Kelvin-like flow pattern ahead of the convection.     

Propagation of barotropic dipoles over topography in a rotating tank

It is shown how symmetric dipolar vortices can be formed by the action of an impulsive jet in a homogeneous single layer of fluid in a rotating tank. These dipoles are allowed to interact with a constant topographic slope, which can model a β-plane or a continental shelf. A dipole's trajectory bends toward the right when climbing a slope and to the left when descending, as predicted by numerical simulations and analytical arguments. The maximum penetration of the dipoles over a slope, the adjustment to the slope, and formation of trailing lobes are compared with both numerical simulations and a two-point vortex model. The results suggest that Rossby wave radiation plays an important role in the interaction process.     

Very-Large-Scale Motions in the Atmospheric Boundary Layer Educed by Snapshot Proper Orthogonal Decomposition

Large-eddy simulations of the atmospheric boundary layer (ABL) under a wide range of stabilities are conducted to educe very-large-scale motions and then to study their dynamics and how they are influenced by buoyancy. Preliminary flow visualizations suggest that smaller-scale motions that resemble hairpins are embedded in much larger scale streamwise meandering rolls. Using simulations that represent more than 150 h of physical time, many snapshots in the \(xy\) -, \(yz\) - and \(xz\) -planes are then collected to perform snapshot proper orthogonal decomposition and further investigate the large structures. These analyses confirm that large streamwise rolls that share several features with the very-large-scale motions observed in laboratory studies arise as the dominant modes under most stabilities, but the effect of the surface kinematic buoyancy flux on the energy content of these dominant modes is very significant. The first two modes in the \(yz\) -plane in the neutral case contain up to 3 % of the total turbulent kinetic energy; they also have a vertical tilt angle in the \(yz\) -plane of about 0 to 30 \(^\circ \) due to the turning effect associated with the Coriolis force. Unstable cases also feature streamwise rolls, but in the convective ABL they are strengthened by rising plumes in between them, with two to four rolls spanning the whole domain in the first few modes; the Coriolis effect is much weaker in the unstable ABL. These rolls are no longer the dominant modes under stable conditions where the first mode is observed to contain sheet-like motions with high turbulent kinetic energy. Using these proper orthogonal decomposition modes, we are also able to extract the vertical velocity fields corresponding to individual modes and then to correlate them with the horizontal velocity or temperature fields to obtain the momentum and heat flux carried by individual modes. Structurally, the fluxes are explained by the topology of their corresponding modes. However, the fraction of the fluxes produced by the modes is invariably smaller than the fraction of energy they contain, particularly under stable conditions where the first modes are found to perform weak counter-gradient fluxes.