Planetary Geostrophic Model Research Articles

Regularity “in Large” for the 3DSalmon’s Planetary Geostrophic Model of Ocean Dynamics

AbstractIt is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction/damping, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the lateral boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no-heat-flux physical boundary condition. Consequently, the second order parabolic heat equation is over-determined with two different boundary conditions. In a previous work we proposed one remedy to this problem by introducing a fourth-order artificial hyper-diffusion to the heat transport equation and proved global regularity for the proposed model. A shortcoming of this higher-oder diffusion is the loss of the maximum/minimum principle for the heat equation. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction/damping term for the vertical component of the velocity in the hydrostatic balance equation. In this paper we prove the global, for all time and all initial data, well-posedness of strong solutions to the three-dimensional Salmon’s planetary geostrophic model of ocean dynamics. That is, we show global existence, uniqueness and continuous dependence of the strong solutions on initial data for this model. Unlike the 3D viscous PG model, we are still unable to show the uniqueness of the weak solution. Notably, we also demonstrate in what sense the additional damping term, suggested by Salmon, annihilate the ill-posedness in the original system; consequently, it can be viewed as “regularizing” term that can possibly be used to regularize other related systems.

On the Charney Conjecture of Data Assimilation Employing Temperature Measurements Alone: The Paradigm of 3D Planetary Geostrophic Model

AbstractAnalyzing the validity and success of a data assimilation algorithmwhen some state variable observations are not available is an important problem in meteorology and engineering. We present an improved data assimilation algorithm for recovering the exact full reference solution (i.e. the velocity and temperature) of the 3D Planetary Geostrophic model, at an exponential rate in time, by employing coarse spatial mesh observations of the temperature alone. This provides, in the case of this paradigm, a rigorous justification to an earlier conjecture of Charney which states that temperature history of the atmosphere, for certain simple atmospheric models, determines all other state variables.

Water Mass Transformations Driven by Ekman Upwelling and Surface Warming in Subpolar Gyres

AbstractThe Sverdrup relationship when applied to the Southern Ocean suggests that some isopycnals that are deep in the eastern Pacific will shoal in the Atlantic. Cold waters surfacing in the South Atlantic at midlatitudes would be warmed by the atmosphere. The potential for water mass transformations in this region is studied by applying a three-layer planetary geostrophic model to a wide ocean basin driven by the Ekman upwelling typical of the Southern Ocean surface winds. The model uses a simple physically based parameterization of the entrainment of mass into the surface layer with zonally symmetric atmospheric surface fields to find steady-state subpolar gyre solutions. The solutions are found numerically by specifying suitable boundary conditions and integrating along characteristics. With reasonable parameter settings, transformations of more than 10 Sverdrups (Sv; 1 Sv ≡ 106 m3 s−1) of water between layers are obtained. The water mass transformations are sensitive to the strength of the wind stress curl and the width of the basin and relatively insensitive to the parameterization of the surface heat fluxes. On the western side of the basin where the cold waters are near the surface, there is a large region where there is a local balance between the Ekman pumping and the exchange of mass between layers. Simple formulas are derived for the water mass transformation rates in terms of the difference between the maximum and minimum northward Ekman transports integrated across the basin and the depths of the isopycnal layers on the eastern boundary. The relevance of the model to the Southern Ocean and the Atlantic meridional overturning circulation is briefly discussed.

Flow of grounded abyssal ocean currents along zonally-varying topography on a rotating sphere

A steady nonlinear planetary–geostrophic model in spherical coordinates is presented describing the hemispheric-scale meridional flow of grounded abyssal currents on a zonally-sloping bottom. The model, which corresponds mathematically to a quasi-linear hyperbolic partial differential equation, can be solved explicitly for a cross-slope isopycnal field that is grounded (i.e. intersects the bottom on the up slope and down slope sides). As a consequence of the conservation of potential vorticity, the abyssal currents possess decreasing thickness in the equatorward direction while maintaining constant meridional volume transport. There is a small westward zonal transport in the interior of these currents that results in westward intensification as they flow toward the equator. Conditions for the possible formation of a shock to develop on the up slope flank of the current are derived.

Impact of the winter North-Atlantic weather regimes on subtropical sea-surface height variability

Interannual variability of subtropical sea-surface-height (SSH) anomalies, estimated by satellite and tide-gauge data, is investigated in relation to wintertime daily North-Atlantic weather regimes. Sea-level anomalies can be viewed as proxies for the subtropical gyre intensity because of the intrinsic baroclinic structure of the circulation. Our results show that the strongest correlation between SSH and weather regimes is found with the so-called Atlantic-Ridge (AR) while no significant values are obtained for the other regimes, including those related to the North Atlantic Oscillation (NAO), known as the primary actor of the Atlantic dynamics. Wintertime AR events are characterized by anticyclonic wind anomalies off Europe leading to a northward shift of the climatological wind-stress curl. The latter affects subtropical SSH annual variability by altered Sverdrup balance and ocean Rossby wave dynamics propagating westward from the African coast towards the Caribbean. The use of a simple linear planetary geostrophic model allows to quantify those effects and confirms the primary importance of the winter season to explain the largest part of SSH interannual variability in the Atlantic subtropical gyre. Our results open new perspectives in the comprehension of North-Atlantic Ocean variability emphasizing the role of AR as a driver of interannual variability at least of comparable importance to NAO.

Averaging of the planetary 3D geostrophic equations with oscillating external forces

In this article, we consider a non-autonomous three-dimensional planetary geostrophic model of the ocean with a singularly oscillating external force depending on a small parameter ϵ. We prove the existence of the uniform global attractor A ϵ . Furthermore, using the method of [11] in the case of the two-dimensional Navier–Stokes systems, we study the convergence of A ϵ as ϵ goes to zero.

Non-autonomous planetary 3D geostrophic equations with oscillating external force and its global attractor

In this article, we consider a non-autonomous three-dimensional planetary geostrophic model of the ocean with a singulary oscillating external force depending on a small parameter ϵ . We prove the existence of the uniform global attractor A ϵ . Furthermore, using the method of Chepyzhov and Vishik (2007) [3] in the case of the two-dimensional Navier–Stokes systems, we study the convergence of A ϵ as ϵ goes to zero.

Smooth solutions and attractor dimension bounds for planetary geostrophic ocean models

Recent results on the existence and uniqueness of global strong solutions of the planetary geostrophic model of large-scale ocean circulation, with traditional parametrizations of horizontal and vertical diffusion of heat and horizontal momentum, are summarized. The new results are stronger than existing results for the hydrostatic primitive equations. A rigorous upper bound on attractor dimension for the planetary geostrophic model is presented, which scales as the cube of the product of the Reynolds number Re and the Peclet number Pe, or as Re 3/2 .Pe 9/4 if some additional assumptions are made. The dimension bound demonstrates that the model dynamics are controlled asymptotically by a finite number of effective degrees of freedom, and raises new physical questions regarding the associated estimate of this number. Available numerical evidence appears to indicate that, in contrast to analogous results for homogeneous turbulence described by the Navier-Stokes equations, substantial improvements to these bounds may be possible.

Nonnormal Multidecadal Response of the Thermohaline Circulation Induced by Optimal Surface Salinity Perturbations

Abstract Optimal perturbations of sea surface salinity are obtained for an idealized North Atlantic basin using a 3D planetary geostrophic model—optimality is defined with respect to the intensity of the meridional overturning circulation. Both optimal initial and stochastic perturbations are computed in two experiments corresponding to two different formulations of the surface boundary conditions: the first experiment uses mixed boundary conditions (i.e., restoring surface temperature and prescribed freshwater flux), whereas the second experiment uses flux boundary conditions for both temperature and salinity. The latter reveals greater responses to both initial and stochastic perturbations that are related to the existence of a weakly damped oscillatory eigenmode of the Jacobian matrix, the optimal perturbations being closely related to its biorthogonal. The optimal initial perturbation induces a transient modification of the circulation after 24 yr. The spectral response to the optimal stochastic perturbation reveals a strong peak at 35 yr, corresponding to the period of this oscillatory eigenmode. This study provides an upper bound of the meridional overturning response at multidecadal time scales to freshwater flux perturbation: for typical amplitudes of Great Salinity Anomalies, initial perturbations can alter the circulation by +2.25 Sv (1 Sv ≡ 106 m3 s−1; i.e., 12.5% of the mean circulation) at most; stochastic perturbations with amplitudes typical of the interannual variability of the freshwater flux in midlatitudes induce a circulation variability with a standard deviation of 1 Sv (i.e., 5.5% of the mean circulation) at most.

On strong solutions of a planetary geostrophic model of the ocean

We study, in this article, the well-posedness of a planetary geostrophic (PG) model of the ocean. This model is derived from the PG studied in [17] by adding a (non-linear) advection term in the horizontal momentum equation. We address in particular the question of existence and uniqueness of global (in time) weak solution to the model. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions. We prove that any weak solution to the model converges exponentially with time to a weak solution to the primitive equations of the ocean provided that the data are small.

A Comparison of Data Assimilation Methods Using a Planetary Geostrophic Model

Abstract Assimilating hydrographic observations into a planetary geostrophic model is posed as a problem in control theory. The cost functional is the sum of weighted model and data residuals. Model errors are assumed to be spatially correlated, and hydrographic station data are assimilated directly. Searches in state space and data space, for minimizing the cost functional, are compared to a direct matrix inversion algorithm in the data space. State-space methods seek the minimizer of the cost functional by performing a preconditioned search in an N-dimensional space of state or control variables, where N is approximately 650 000 in the present calculations. Data-space methods solve the Euler–Lagrange equations for the extremum of the cost functional by working in an M-dimensional dual space, where M is the number of measurements. The following four solvers are compared: (i) an iterative state-space solver, with a naive diagonal matrix preconditioner; (ii) an iterative state-space solver, with a sophisticated preconditioner based on the inverse of the model’s dynamical operators; (iii) an iterative data-space solver, with no preconditioning; and (iv) a direct, M × M matrix inversion, data-space solver. The best solver is the iterative data-space solver, (iii), which is approximately 10 times faster than the sophisticated preconditioned state-space solver, (ii), and 100 times faster than the direct data-space solver, (iv).

Buoyancy-forced circulations in shallow marginal seas

The properties of water mass transformation and the thermohaline circulation in shallow marginal seas with topography and subject to surface cooling are discussed in the context of an eddy-resolving primitive equation model and an analytic planetary geostrophic model. A unique and important aspect of the model configuration is that the geostrophic contours, or characteristics of the system, extend from a region where temperature is restored toward a uniform value, providing a source of heat, through the cooling region. This removes a degree of symmetry that has often been imposed in previous studies of deep convection. The heat loss within the marginal sea is balanced by lateral advection from the restoring region. The planetary geostrophic model shows that the basic temperature distribution can be well predicted by integrating along geostrophic contours from their entry into the marginal sea to their exit. Scaling estimates for the exchange rate and density of the waters formed within the marginal sea are derived and compare well with a series of numerical model calculations. In contrast to many previous buoyancy-forced deep convection problems, basin-scale cooling is balanced mainly by the mean flow, with mesoscale eddies serving primarily to restratify locally but not to provide a net heat flux to balance cooling. However, eddy fluxes and the mean flow are locally of comparable importance for cases with a localized patch of surface cooling.

A ‘horizontal’ hyper-diffusion three-dimensional thermocline planetary geostrophic model: well-posedness and long-time behaviour

In this paper, we study a three-dimensional thermocline planetary geostrophic ‘horizontal’ hyper-diffusion model of the gyre-scale midlatitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite-dimensional global attractor to this dissipative evolution system. Preliminary computational tests indicate that our hyper-diffusion model does not exhibit any of the non-physical instabilities near the lateral boundary which are observed numerically in other models.

Linear planetary wave dynamics in a 2.5-layer ventilated thermocline model

Linear planetary wave dynamics in a 2.5-layer ventilated thermocline model is investigated by a local eigenvalue analysis and simple numerical computations. It is known that there are two types of waves in this system; we refer to one of them as the N-mode (Non-Doppler shift mode), which propagates almost westward, and the other as the A-mode (Advection mode), which propagates almost along the second layer basic current. First, we study the local longwave dynamics, assuming that the wavelength is much longer than the deformation radius and shorter than the gyre scale. It is shown that the N-mode and the A-mode cannot neatly be separated in the ventilated zone, and even in the Rhines and Young pool, a compact A-mode disturbance cannot exist by itself. When anomalous Ekman pumping is applied in the ventilated zone, the N-mode is generated in the forcing region, and the A-mode is generated at the wave front of the N-mode. For diabatic forcing, a similar phenomenon occurs. It is also found that the shadow zone is unstable to longwave disturbances. Secondly, the wave behavior in the linear planetary geostrophic model is numerically investigated. The main features can be interpreted by the local wave dynamics, and the disturbances around the maximum amplitudes are dominated by the waves whose wavenumber vectors are perpendicular to the second layer potential vorticity contours. The amplitude changes during the wave propagation are also discussed. Finally, the effects of the finite wavelength are studied. The N-mode is strongly dispersive at the scale of the Rossby deformation radius, while the A-mode is weakly dispersive. It is also shown that the ventilated zone is unstable to shortwave disturbances, although it is stable to longwave disturbances.

Large-Scale Circulations Forced by Localized Mixing over a Sloping Bottom*

Abstract A simple, nonlinear, two-layer, planetary geostrophic model of the large-scale circulation forced by localized mixing over a sloping bottom is discussed. The model is forced by parameterized diapycnal mixing at the density interface and/or by a mass flux downward into (unresolved) deep topographic canyons. Two nondimensional parameters are identified: the ratio of the change in Coriolis parameter over the horizontal mixing length scale to the nominal Coriolis parameter and the ratio of the advective speed to the Rossby wave phase speed. The former controls the strength of horizontal recirculation gyres that are forced by spatially variable diapycnal mixing, while the latter is a measure of the importance of nonlinearity in the density equation. When bottom topography is introduced, bottom pressure torque becomes important and the traditional strong horizontal recirculation gyre found for mixing over a flat bottom (beta plume) is gradually replaced by a zonal flow into or out of the mixing region ...

Remarks on the planetary geostrophic model of gyre scale ocean circulation

We study in this article the mathematical formulation of the planetary geostrophic (PG) equations of large-scale ocean circulation, in the case where small-scale processes are parameterized by the traditional Laplacian eddy diffusion and eddy viscosity. We prove the existence and uniqueness of global in time strong solutions of these equations with either $L^\infty$ or $H^2$ initial data. Due essentially to the high nonlinearity (comparable to a squared gradient) of the equations, two problems remain open. First, the existence of more regular solutions with $L^\infty \cap H^1$ initial data is still unknown, although more regular solutions are obtained with $H^2$ initial data. Second, the existence of global attractor and its dimension estimates are open, and related to that are the time uniform boundedness of the norm in $H^2$ and higher order Sobolev spaces of the solutions.

Dynamics of the Mediterranean Salinity Tongue

Abstract A reduced-gravity planetary-geostrophic model of the North Atlantic consisting of two active layers overlying a motionless abyss is developed to investigate the effect of the wind field in shaping the dynamics of the Mediterranean salinity tongue. The model is driven by climatological winds and eastern boundary ventilation in a basin of realistic geometry and includes a parameterization of meddies. The upper-layer depth from the model shows a clear similarity to observations, both in terms of the location and intensity of the subtropical gyre and also the position of the outcropping line in the northern basin. Potential vorticity in layer two reproduces the sweep of potential-vorticity contours southwestward from the eastern boundary and extending westward into the interior, and provides the pathways along which Mediterranean Water spreads into the model interior. The authors solve for the steady salinity field in the second layer, including sources of Upper Labrador Sea Water and Antarctic Inter...

A simple model of the large-scale circulation of Mediterranean Water and Labrador Sea Water

A steady, 2 1 2 layer planetary geostrophic model is used to study the circulation and mixing of Mediterranean Water (MW) and Labrador Sea Water (LSW). The model includes parameterizions of salt fingering, mesoscale variability, and meddies. A close relationship between the vertical density ratio (and expected strength of salt fingering) and the potential vorticity anomaly of the upper salt tongue is identified and used to parameterize salt fingering as a spatially nonhomogeneous diapycnal mass flux. A balance between baroclinic Rossby wave propagation and salt fingering produces an asymmetric tongue of potential vorticity extending into the interior from the eastern boundary, consistent with the observed Mediterranean salt tongue. The resulting internal pressure gradients give rise to a large-scale anticyclonic circulation in the upper layer of the salt tongue with northward flow at mid-latitudes, consistent with geostrophic flow estimates based on hydrographic data. Salt fingering that attains maximum strength at mid-latitudes forces a large-scale circulation in the lower MW/LSW depth range that advects water from the western boundary into the eastern basin near 50°N, where it then flows to the south under the upper MW, and finally returns to the western boundary. This pattern is generally consistent with a variety of estimates for the circulation of LSW, and suggests that the eastward flow of LSW off the western boundary may be, at least in part, forced by salt fingering in the interior. The addition of mixing due to mesoscale eddies modifies the details of the potential vorticity distribution, but does not change the qualitative behavior. Representing meddies as a distributed source of mass to the upper layer results in a further westward extension of the salt tongue signature and enhances the strength of the deep recirculation.

Simple Model Prediction of Horizontal Temperature Fields in the Subtropical–Subpolar System Caused by Sudden Change in Wind Stress Curl

Abstract A four-level numerical model, which is driven by wind stress and surface heat flux, is used to study responses of horizontal temperature fields in the subtropical–subpolar system to a sudden change in the magnitude of the wind stress curl. Weakly nonlinear responses to O(1) change in the wind stress curl are examined, according to the effects on the Ekman pumping, convection, westward baroclinic wave, and advection. For this purpose, a quasi-analytical method, that is, characteristics associated with effects of both wave propagation and advection, is constructed based on a planetary geostrophic model with four-level geometry. Characteristics obtained for the first and second baroclinic modes are used to diagnose steady-state and time-dependent solutions. One feature of time-dependent motions predicted by the model is a westward propagation of the first baroclinic mode with a significantly higher speed than the combined speed of the nondispersive first-mode baroclinic Rossby wave and barotropic ci...

Seasonally ice free glacial nordic seas without deep water ventilation

ABSTRACTAt present the Nordic Seas are a key region of North Atlantic Deep Water (NADW) formation. Two alternative scenarios have been suggested by some authors for the Last Glacial Maximum: (i) the Nordic Seas were permanently covered by sea ice, preventing the formation of NADW, or (ii) that they were seasonally free of ice and that deep water formation did occur. A modified scenario is presented here based on parallel ocean circulation modelling results from the GFDL primitive equation model and a planetary geostrophic model. It is suggested that the glacial Nordic Seas were at least seasonally ice free, but it is observed that there was never deep water formation from the surface; rather it occurred only in the North Atlantic south of 40°–50°N. North of 40°N, the weaker LGM northward flowing thermohaline conveyor is subducted below a reverse conveyor which occurred to a depth of over 1000 m. Various modelling experiments presented here indicate that the reversed conveyor was primarily caused by the colder conditions of the glacial North Atlantic that led to far stronger zonality of glacial analogue of the North Atlantic Current.