Reduced-gravity Shallow-water Model Research Articles

The fastest growing initial error in prediction of the Kuroshio Extension state transition processes and its growth

In this study, the predictability of the Kuroshio Extension (KE) transition processes is explored from an error growth perspective. The fastest growing initial errors (FGIEs) are obtained through the conditional nonlinear optimal perturbation approach within a reduced-gravity shallow-water model forced by steady winds, which provides a fairly realistic simulation of the KE low-frequency variability of intrinsic origin. The large amplitudes of the FGIEs for both the transitions from a typical low-energy state to a typical high-energy state (LH) and the opposite transition (HL), are found mainly in the Kuroshio large meander region south of Japan and in the KE region. The FGIE grows more rapidly for the HL process than for the LH process, implying that the HL transition process may be more difficult to predict. The evolution processes of the FGIEs and the related mechanisms are revealed by investigating the evolution of the potential vorticity anomalies caused by the FGIEs. The dominant physical processes governing the FGIE growth are found to be different for the LH and HL processes. For the LH process, the evolution is mainly governed by linear advection processes and interfacial friction, while for the HL process, in addition to these two processes, the nonlinear advection process also plays a vital role in the evolution. This indicates that nonlinear intrinsic oceanic processes affect considerably the error growth, especially in the HL transition process, suggesting that the intrinsic processes should be carefully considered when exploring the predictability and forecast of the KE low-frequency variability.

Generation of Rossby waves off the Cape Verde Peninsula: the role of the coastline

Abstract. In December 2002 and January 2003 satellite observations of chlorophyll showed a strong coastal signal along the west African coast between 10 and 22∘ N. In addition, a wavelike pattern with a wavelength of about 750 km was observed from 20 December 2002 and was detectable for 1 month in the open sea, south-west of the Cape Verde Peninsula. Such a pattern suggests the existence of a locally generated Rossby wave which slowly propagated westward during this period. This hypothesis was confirmed by analysing sea surface height provided by satellite altimeter during this period. To decipher the mechanisms at play, a numerical study based on a reduced-gravity shallow-water model has first been conducted. A wind burst, broadly extending over the region where the offshore oceanic signal is observed, is applied for 5 d. A Kelvin wave quickly develops along the northern edge of the cape, then propagates and leaves the area in a few days. Simultaneously, a Rossby wave whose characteristics seem similar to the observed pattern forms and slowly propagates westward. The existence of the peninsula limits the extent of the wave to the north. The spatial extent of the wind burst determines the extent of the response and correspondingly the timescale of the phenomenon (about 100 d in the present case). When the wind burst has a large zonal and small meridional extent, the behaviour of a wave to the north of the peninsula differs from that to the south. These results are corroborated and completed by an analytical study of a linear reduced-gravity model using a non-Cartesian coordinate system. This system is introduced to evaluate the potential impact of the coastline shape. The analytical computations confirm that a period of around 100 d can be associated with the observed wave considering the value of the wavelength; they also show that the role of the coastline remains moderate at such timescales. By contrast, when the period becomes shorter (smaller than 20–30 d), the behaviour of the waves is modified because of the shape of the coast. South of the peninsula, a narrow band of sea isolated from the rest of the ocean by two critical lines appears. Its meridional extent is about 100 km and Rossby waves could propagate there towards the coast.

Parameter sensitivity analysis of the short-range prediction of Kuroshio extension transition processes using an optimization approach

In this study, the optimization parameter sensitivity analysis (OPSA) method is applied to investigate the sensitivity of the short-range prediction of the Kuroshio extension (KE) transition process to parameter uncertainties as simulated with a reduced-gravity shallow water model. The results show that the uncertainties of the model parameters have important impacts on the short-range predictions of the KE transition processes: the predictions are found to be mainly affected by the uncertainties in the reduced-gravity and interfacial friction coefficients. On the contrary, the prediction is insensitive to the wind stress amplitude and the lateral eddy viscosity coefficient. Furthermore, if the sensitive parameters are more accurately estimated, the prediction will be significantly improved. It is therefore suggested that the OPSA method is a useful tool to assess parameter sensitivity for the prediction of the KE transition processes.

Effects of Singular-Vector-Type Initial Errors on the Short-Range Prediction of Kuroshio Extension Transition Processes

AbstractThe effects of optimal initial error on the short-range prediction of transition processes between the Kuroshio Extension (KE) bimodalities are analyzed using a reduced-gravity shallow-water model and the singular vector (SV) approach. Emphasis is placed on the spatial structures, growing processes, and effects of the SVs. The results show that the large values of the SVs are mainly located in the first crest region of the KE (around 35°N, 144°E) and in the Kuroshio large meander (KLM) region south of Japan (around 32°N, 139.5°E). The fast growths of the SVs have important impacts on the prediction of transition of the KE bimodality. The initial error with +SV pattern (with positive anomalies in the first crest region of the KE and negative anomalies in the KLM region) tends to strengthen the KE and shift it toward the high-energy state, while the error with −SV pattern is prone to weaken the KE and shift it toward the low-energy state. In addition, the SV-type initial errors grow more quickly in the transition phase of the KE from the high-energy to the low-energy state than in the opposite transition phase. A perturbation energy analysis illustrates that different physical processes are responsible for the error growth in the KE region for different transition phases of the KE; barotropic instability plays a dominant role in the error growth in the low-to-high (LH) energy phase, while the error evolution in the high-to-low (HL) energy phase is mainly caused by advection processes.

The stability of oceanic fronts in a shallow water model

We investigate mechanisms through which energy cascades from the mesoscale, $O(100~\text{km})$, to the submesoscale, $O(10~\text{km})$, for oceanic fronts in a reduced gravity shallow water model using two different profiles. The first idealization of an ocean front has an interfacial depth that is a smooth hyperbolic tangent profile and is an extension of the piecewise constant potential vorticity profile studied in Boss et al. (J. Fluid Mech., vol. 315, 1996, pp. 65–84). By considering a range of minimum depths, all of which have the same velocity profile, we are better able to isolate the effect of vanishing layer depths. We find that the most unstable mode exists in a one-layer model and does not need two layers, as previously speculated. Moreover, we find that even without a vanishing layer depth there are other modes that appear at both larger and smaller length scales that have a gravity wave structure. The second profile is the parabolic double front from Scherer & Zeitlin (J. Fluid Mech., vol. 613, 2008, pp. 309–327). We find more unstable modes than previously presented and also categorize them based on the mode number. In particular, we find there are pairs of unstable modes that have equal growth rates. We also study the nonlinear evolution of these oceanic fronts. It is determined that vanishing layer depths have significant effects on the unstable dynamics that arise. First, stronger gravity wave fields are generated. Second, cyclonic fluid that moves into the deeper waters is stretched preferentially more in comparison to the deep water scenario and destabilizes more easily. This results in smaller scale vortices, both cyclones and anticyclones, that have length scales in the submesoscale regime. Our results suggest that the nonlinear dynamics of a front can be very efficient at generating submesoscale motions.

Dynamics of turbulent western-boundary currents at low latitude in a shallow-water model

Abstract. The dynamics of low latitude turbulent western-boundary currents (WBCs) crossing the Equator are considered using numerical results from integrations of a reduced-gravity shallow-water model. For viscosity values of 1000 m2 s−1 and greater, the boundary layer dynamics compares well to the analytical Munk-layer solution. When the viscosity is reduced, the boundary layer becomes turbulent and coherent structures in the form of anticyclonic eddies, bursts (violent detachments of the viscous sub-layer, VSL) and dipoles appear. Three distinct boundary layers emerge, the VSL, the advective boundary layer and the extended boundary layer. The first is characterized by a dominant vorticity balance between the viscous transport and the advective transport of vorticity; the second by a balance between the advection of planetary vorticity and the advective transport of relative vorticity. The extended boundary layer is the area to which turbulent motion from the boundary extends. The scaling of the three boundary layer thicknesses with viscosity is evaluated. Characteristic scales of the dynamics and dissipation are determined. A pragmatic approach to determine the eddy viscosity diagnostically for coarse-resolution numerical models is proposed.

Measuring the Impact of Observations on the Predictability of the Kuroshio Extension in a Shallow-Water Model

Abstract In this paper, sequential importance sampling is used to assess the impact of observations on an ensemble prediction for the decadal path transitions of the Kuroshio Extension. This particle-filtering approach gives access to the probability density of the state vector, which allows the predictive power—an entropy-based measure—of the ensemble prediction to be determined. The proposed setup makes use of an ensemble that, at each time, samples the climatological probability distribution. Then, in a postprocessing step, the impact of different sets of observations is measured by the increase in predictive power of the ensemble over the climatological signal during one year. The method is applied in an identical-twin experiment for the Kuroshio Extension using a reduced-gravity shallow-water model. This study investigates the impact of assimilating velocity observations from different locations during the elongated and the contracted meandering states of the Kuroshio Extension. Optimal observation locations correspond to regions with strong potential vorticity gradients. For the elongated state the optimal location is in the first meander of the Kuroshio Extension. During the contracted state it is located south of Japan, where the Kuroshio separates from the coast.

The Evolution and Propagation of Quasigeostrophic Ocean Eddies*

Abstract The long-term evolution of initially Gaussian eddies is studied in a reduced-gravity shallow-water model using both linear and nonlinear quasigeostrophic theory in an attempt to understand westward-propagating mesoscale eddies observed and tracked by satellite altimetry. By examining both isolated eddies and a large basin seeded with eddies with statistical characteristics consistent with those of observed eddies, it is shown that long-term eddy coherence and the zonal wavenumber–frequency power spectral density are best matched by the nonlinear model. Individual characteristics of the eddies including amplitude decay, horizontal length scale decay, and zonal and meridional propagation speed of a previously unrecognized quasi-stable state are examined. The results show that the meridional deflections from purely westward flow (poleward for cyclones and equatorward for anticyclones) are consistent with satellite observations. Examination of the fluid transport properties of the eddies shows that an inner core of the eddy, defined by the zero relative vorticity contour, contains only fluid from the eddy origin, whereas a surrounding outer ring contains a mixture of ambient fluid from throughout the eddy’s lifetime.

Estimation of friction parameters in gravity currents by data assimilation in a model hierarchy

Abstract. This paper is the last in a series of three investigating the friction laws and their parametrisation in idealised gravity currents in a rotating frame. Results on the dynamics of a gravity current (Wirth, 2009) and on the estimation of friction laws by data assimilation (Wirth and Verron, 2008) are combined to estimate the friction parameters and discriminate between friction laws in non-hydrostatic numerical simulations of gravity current dynamics, using data assimilation and a reduced gravity shallow water model. I demonstrate, that friction parameters and laws in gravity currents can be estimated using data assimilation. The results clearly show that friction follows a linear Rayleigh law for small Reynolds numbers and the estimated value agrees well with the analytical value obtained for non-accelerating Ekman layers. A significant and sudden departure towards a quadratic drag law at an Ekman layer based Reynolds number of around 800 is shown, in agreement with classical laboratory experiments. The drag coefficient obtained compares well to friction values over smooth surfaces. I show that data assimilation can be used to determine friction parameters and discriminate between friction laws and that it is a powerful tool in systematically connecting models within a model hierarchy.

A Tropical Ocean Recharge Mechanism for Climate Variability. Part II: A Unified Theory for Decadal and ENSO Modes

Decadal to interdecadal timescale variability in the Pacific region, commonly referred to as the Pacific decadal oscillation (PDO), is studied in this research using analytical and numerical models. A coupled analytical model is formulated to analyze the physical mechanism of both the PDO and ENSO. It has the equatorial β-plane dynamics of a reduced-gravity model coupled with the wind stress of fixed spatial patterns. The amplitude of the latter is proportional to the sea surface temperature (SST) anomaly in the eastern equatorial Pacific. The SST anomaly is governed by a simple thermal dynamic equation used for ENSO modeling. It is found that when a warm SST is coupled with cyclonic wind stress patterns in the eastern subtropical Pacific, an oscillation with a timescale of around 10–15 yr could be generated. In contrast, when a warm SST is coupled with only a westerly wind stress in the central equatorial Pacific, an ENSO-like oscillation could be generated with a timescale of around 3–5 yr. Thus the present research is potentially relevant to aspects of the PDO and the mechanism of the PDO may be understood as a weakly coupled decadal recharge oscillator similar to the recharge oscillator dynamics of ENSO. The sensitivity of these two kinds of coupled modes to different parameters is tested. Numerical integrations with the reduced-gravity shallow-water model in a rectangular basin and a similar coupled framework confirm the results of the analytical model.

A Tropical Ocean Recharge Mechanism for Climate Variability. Part I: Equatorial Heat Content Changes Induced by the Off-Equatorial Wind

A reduced-gravity shallow-water model, an oceanic general circulation model for the Pacific region, and the analytical model of the equatorial b plane bounded in the zonal direction are used to investigate the equatorial thermocline response to tropical and subtropical wind stress forcing. The results show that the wind stress forcing in the tropical and subtropical region can generate a nearly zonal uniform thermocline depth change in the equatorial region. The response timescale is longer when the wind stress is placed farther away from the equator. There exist latitude bands around 108‐158N and 108‐158S where the forcing can cause a relatively large equatorial response. When the forcing is located in the eastern basin, the response timescale is longer and its magnitude is larger than the case when the forcing is located in the western basin. Thus the eastern tropical to subtropical region is a relatively effective area for off-equatorial wind stress to generate an equatorial thermocline response. When the wind stress forcing has a longer period, the response of the equatorial thermocline has a larger magnitude. The results from this study’s numerical experiments and the analytical solution are consistent. The present study has implications for the broad-scale ocean‐atmosphere interaction in the tropical region.

Small-amplitude coastally trapped disturbances and the reduced-gravity shallow-water approximation

Solutions are obtained for linear hydrostatic disturbances propagating parallel to the face of an uninterrupted topographic step in an infinitely deep, stably stratified fluid on an β-plane. These waves are vertically trapped because their frequencies are smaller than the Coriolis parameter and the height of the topographic step is finite. These waves are referred to as step-trapped Kelvin waves, because they are dynamically similar to internal Kelvin waves throughout the layer of fluid below the top of the topographic step. These waves appear to provide an idealized, semi-analytic model for the coastally trapped disturbances observed to propagate parallel to mountainous coastlines in several parts of the world. Computations are performed for a basic state with uniform static stability and for a three-layer basic state in which the two lowest layers represent the marine boundary layer and a strong capping inversion. One might suppose that the linear dynamics of hydrostatic disturbances in the three-layer basic state could be well approximated by a reduced-gravity shallow-water model, but this is not the case. In particular, the reduced-gravity shallow-water model does not provide reliable estimates for the phase speed of linear step-trapped Kelvin waves. This defect suggests that detailed quantitative comparisons between marine boundary-layer flows and the reduced-gravity shallow-water system may not have any intrinsic physical significance. Nevertheless, these results do not preclude the possibility of constructing useful qualitative analogies between marine boundary-layer flows and the reduced-gravity shallow-water model.

Decadal Variability in a Simplified Wind-Driven Ocean Model

The impact of an unsteady wind forcing on oceanic low-frequency variability is conceptually studied using a reduced-gravity shallow-water model. A time-averaged wind forcing and a simple ocean‐atmosphere coupling is completed by a stochastic component (spatially coherent white noise) representing the effect of atmospheric transient eddies. To account for the observed concentration of eddy activity along the North Atlantic and North Pacific storm tracks the variance of the stochastic forcing is chosen to be spatially inhomogeneous. Low-frequency variability of the basin-averaged energetics shows a dominant spectral peak with an amplitude depending on the inhomogeneity of the stochastic forcing and the time-averaged wind stress. The period of the variability is unexpected considering baroclinic Rossby waves forced by the ocean‐atmosphere coupling only. This variability can be explained by ‘‘spatial resonance’’ of the forced baroclinic Rossby wave and the Reynolds momentum flux induced by the spatially inhomogeneous white noise.

Imperfections of the North Atlantic wind-driven ocean circulation: Continental geometry and windstress shape

Multiple equilibria of the wind-driven gyres have been found in idealized quasi- geostrophic and shallow water models.In this paper we demonstrate that multiple equilibria persist within a reduced gravity shallow water model under quite realis- tic continental geometry and windstress orcing for the North-Atlantic.Multiple mean flow patterns of the Gulf Stream exist and differ with respect to their separation behavior along the North-American coast.The origin of these equilibria is investigated by determining the structure of steady solutions within a hierarchy of equivalent barotropic ocean models using continuation techniques.Within each model,the magnitude of lateral riction is used as a control parameter.It is shown that symmetry breaking, found in a quasi-geostrophic model for a rectan- gular ocean basin with idealized wind forcing is at the origin of two different mean states of the Gul Stream.The steady states ound become unstable only to a small number of oscillatory modes,which either have intermonthly or interannual periods.The modes of variability remain strongly related through the hierarchy of models indicating that their physics is not strongly dependent on the shape of the continents but is controlled by internal ocean dynamics.

The effects of a sloping bottom on the equatorial wave-induced deep residual currents

Based on an inverted one-and-one-half inviscid reduced gravity shallow water model with bottom topography representing an abyssal layer under a stagnant upper layer on the equatorial β-plane, a set of field equations governing the wave-induced Lagrangian residual currents is developed. The equations show that the wave-induced Lagrangian residual current satisfies generalized geostrophic dynamics. The relation of meridional residual current to vertical residual current resulted from the varied bottom is similar to the Sverdrup transport relation. The transport process of potential vorticity for zeroth order approximation is determined by the advection whose velocity is equal to that of the wave-induced Lagrangian residual current.

Tracking Nonlinear Solutions with Simulated Altimetric Data in a Shallow-Water Model*

Abstract Low-frequency variability of western boundary currents (WBCs) is pervasive in both observations and numerical models of the oceans. Because advection is of the essence in WBCs, nonlinearities are thought to be important in causing their variability. In numerical models, this variability can be distorted by our incomplete knowledge of the system’s dynamics, manifested in model errors. A reduced-gravity shallow-water model is used to study the interaction of model error with nonlinearity. Here our focus is on a purely periodic solution and a weakly aperiodic one. For the periodic case, the noise-corrupted system loses its periodicity due to nonlinear processes. For the aperiodic case, the intermittent occurrences of two relatively persistent states—a straight jet with high total energy and a meandering one with low total energy—in the perturbed model are almost out of phase with the unperturbed one. For both cases, the simulation errors are trapped in the WBC region, where the nonlinear dynamics is...

A wave-induced stirring mechanism in the mid-depth equatorial ocean

A wave-induced stirring and transport mechanism for the mid-depth equatorial ocean is proposed and examined using both analytic linear equatorial wave solutions and a fully nonlinear reduced-gravity model. The study of kinematic stirring using the linear solutions suggests that a superimposition of a few simple equatorial waves can lead to strong Lagrangian stirring and transport along the equator. In particular, a combination of an annual long Rossby wave and a high-frequency Yanai wave appears to be most effective in producing strong stirring in the interior equatorial region. Further investigations of stirring properties using an inverted, fully nonlinear reduced gravity shallow-water model support the results of the kinematic stirring study. By evaluating the finite-time estimates of Lyapunov exponents, we identified two regions where chaotic stirring is most active. One is the western boundary region where short Rossby waves likely play a dominant role in producing the strong chaotic stirring. The other is the equatorial waveguide where a low-frequency Rossby wave prescribes the pattern of the stirring geometry, and a high-frequency Yanai wave plays a role of stirring the fluid. The proposed stirring mechanism provides a plausible explanation of the observed chlorofluorocarbon distribution found in the mid-depth equatorial Atlantic Ocean.

Data Assimilation in a Simple Tropical Ocean Model with Wind Stress Errors

Abstract A major error source in the numerical simulation of tropical oceans is the uncertainty in wind stress forcing. A reduced-gravity shallow-water model has been used to test how assimilated ocean data correct simulation errors caused by erroneous wind stress in the tropics. The geometry of the basin is rectangular and symmetric about the equator, and the long-wave approximation is applied. All experiments are of the identical-twin type: the “observations” are generated by sampling the desired reference solution, and the data are assimilated by optimal interpolation into the same model, with wind stress forcing different from that in the reference case. In this paper, three types of wind stress errors are considered: errors of timing only, as well as persistent errors, systematic or stochastic. The relative usefulness of thermocline depth and current observations, and the effect of data distribution on state estimation are examined. The role of equatorial ocean waves in the process of data assimilati...

Paramater regimes for studying isolated eddies

Abstract The parameter ranges relevant to propagating oceanic eddies are explored with a view to simplifying the equation sets previously used to simulate such eddies. The reduced gravity shallow-water model is used as a starting point. We consider eddies whose width is of the same order as, or somewhat larger than, the Rossby deformation radius. Two model equations emerge from these scalings, both permitting modon solutions and easy numerical integration. Both, too, give fair simulations of the shallow-water model, but disagree in details. Using another approach, an ad hoc vortex model is used to predict the path of an eddy. This also gives a fair but not perfect simulation of the one-layer results. It appears that the only ideal way to reproduce primitive equation results accurately is to use the primitive equations themselves.