Primitive Equation Research Articles

Numerical modelling of sea currents and tidalwaves

AbstractWe consider a mathematical model of sea currents and tidal waves based on the marine dynamics primitive equations. The equations are written in the orthogonal coordinate system on sphere with arbitrary position of the poles. It makes it possible to increase horizontal resolution due to placement of a pole into vicinity of the considered sub-area. Two problems are solved: (1) joint computation of wind-generated, baroclinic and tidal currents in the Black and Azov Seas; (2) simulating mesoscale variability of coastal currents in the Black Sea. The second problem is solved with increased horizontal resolution in the coastal zone of Gelendzhik.

First-Order Decoupled Finite Element Method of the three-Dimensional Primitive Equations of the Ocean

This paper is concerned with a first-order fully discrete decoupled method for solving the three-dimensional (3D) primitive equations of the ocean with the Dirichlet boundary conditions on the side, where a decoupled semi-implicit scheme is used for the time discretization, and the $P_1(P_1)-P_1-P_1(P_1)$ finite element for velocity, pressure, and density is used for the spatial discretization of these equations. The $H^1-L^2-H^1$ optimal error estimates for the numerical solution $(u_h^n,p_h^n,\theta_h^n)$ and the $L^2$ optimal error estimate for $(u^n_h,\theta_h^n)$ are established under the restriction of $0<h\le \beta_1$ and $0<\tau\le \beta_2$ for some positive constants $\beta_1$ and $\beta_2$. Moreover, numerical investigations are provided to show that the first-order decoupled method is of almost unconditional convergence with accuracy $\mathcal{O}(h+\tau)$ in the $H^1$-norm and $\mathcal{O}(h^2+\tau)$ in the $L^2$-norm for solving the 3D primitive equations of the ocean. Numerical results are gi...

Quasigeostrophic Diagnosis of Mixed Layer Dynamics Embedded in a Mesoscale Turbulent Field

AbstractA quasigeostrophic model is developed to diagnose the three-dimensional circulation, including the vertical velocity, in the upper ocean from high-resolution observations of sea surface height and buoyancy. The formulation for the adiabatic component departs from the classical surface quasigeostrophic framework considered before since it takes into account the stratification within the surface mixed layer that is usually much weaker than that in the ocean interior. To achieve this, the model approximates the ocean with two constant stratification layers: a finite-thickness surface layer (or the mixed layer) and an infinitely deep interior layer. It is shown that the leading-order adiabatic circulation is entirely determined if both the surface streamfunction and buoyancy anomalies are considered. The surface layer further includes a diabatic dynamical contribution. Parameterization of diabatic vertical velocities is based on their restoring impacts of the thermal wind balance that is perturbed by turbulent vertical mixing of momentum and buoyancy. The model skill in reproducing the three-dimensional circulation in the upper ocean from surface data is checked against the output of a high-resolution primitive equation numerical simulation.

Development of tropical cyclone wind field for simulation of storm surge/sea surface height using numerical ocean model

Coupled ocean-atmospheric phenomenon such as tropical cyclones (TC’s) is governed by geophysical fluid dynamics. TC associated strong wind stress transfer momentum energy to the ocean surface that acts as the prime mechanism in modulating the sea surface and in generating the storm surge. A primitive equation, Princeton ocean model (POM) with free surface, sigma (terrain-following) coordinates and realistic bottom topography is configured in Bay of Bengal (BoB) to simulate the storm surge/sea surface height (SSH) and surface currents during a super cyclone TC05B 1999. TC wind fields are developed by adopting a suitable formulation based on partial conservation of angular momentum. Modeled TC wind fields are superimposed with QuikSCAT Satellite/National Centre for Environmental Prediction (QSCAT/NCEP) blended ocean surface winds to drive the three-dimensional ocean model. Model simulated storm surge and SSH are compared with limited available surge estimates/observations and multi-satellite observed AVISO (Archive, Validation and Interpolation of Satellite Oceanography) SSH, respectively to evaluate its performance.

Forced Near-Inertial Motion and Dissipation of Low-Frequency Kinetic Energy in a Wind-Driven Channel Flow

AbstractUsing primitive equation simulations, a zonally periodic channel is considered. The channel flow is forced by a combination of steady and high-frequency winds. The high-frequency forcing excites near-inertial motion, and the focus is on how this influences the low-frequency, nearly geostrophic part of the flow. In particular, this study seeks to clarify how Reynolds stresses exerted by the near-inertial modes affect the low-frequency kinetic energy. In the system considered, the near-inertial Reynolds stresses (i) serve as a sink term in the low-frequency kinetic energy budget and (ii) transfer low-frequency kinetic energy downward from the mixed layer. Transfer spectra show the bulk of this sink to occur at relatively small horizontal wavenumber (i.e., in the mesoscale, not the submesoscale). The presence of near-inertial motion can also affect the kinetic-to-potential energy exchanges, especially within the low-frequency band.

Boundary layers for the 3D primitive equations in a cube: The supercritical modes

In this article we study the boundary layers for the viscous Linearized Primitive Equations (LPEs) when the viscosity is small. The LPEs are considered here in a cube. Besides the usual boundary layers that we analyze here too, corner layers due to the interaction between the different boundary layers are also studied.

Simulation of Turbulent Flows in River Confluences and Meandering Channels with a Cartesian 3D Free Surface Hydrodynamic Model

Three-dimensional primitive equations (3DPE) become a reasonable approach in hydrodynamics in terms of computational costs when the length of the computational domain and/or computational time scales increases. However, given the simplified set of equations used in the analysis, results with 3DPE-based models are expected to be approximate and before attempting to reproduce complex natural flows they first need to be validated against more simple flows observed in laboratory settings. Here, the validity of Cartesian free-surface hydrodynamic models to reproduce three turbulent flows characteristic of river environments is tested: (1) the development of shallow mixing layers, (2) flow pass a lateral cavity and (3) flow in open channel with mild curvature. Errors between measured and modeled values were generally less than 10%, proving their validity to reproduce such turbulent flows and their potential for simulations in more complex natural environments, such it is the case of the confluence between the Ebro and Segre rivers into Ribarroja reservoir.

Global observations and modeling of nonmigrating diurnal tides generated by tide‐planetary wave interactions

AbstractAdvective processes that couple planetary waves with tides have long been proposed as sources of nonmigrating diurnal tides. This paper reports observations of short‐term variability in global observations of nonmigrating tides predicted to arise from the interaction of the migrating diurnal tide (DW1) with a quasi‐stationary planetary wave number one (PW1). PW1 and tidal definitions are extracted from satellite temperatures and high‐altitude meteorological analyses. During winter months, the evolution of westward traveling diurnal tides with zonal wave number 2 (DW2) generally tracks that of strong‐amplitude stratospheric PW1. DW1 and PW1 spectra are used to compute nonlinear tidal forcing terms arising from advection. We then examine the response of a primitive equation model to the observation‐based nonlinear forcing. The model experiments indicate that meridional advection of PW1 zonal momentum by DW1 is a significant source of lower thermospheric DW2. Modeled DW2 amplitudes are very consistent with observed DW2 amplitudes when stratospheric PW1 penetrates to equatorial latitudes. The model experiments also indicate that the interaction can imprint short‐term variability associated with wintertime PW1 upon DW2 in the summer hemisphere and the lower thermosphere.

A New Development of Reduced Hamiltonian Equations for Ocean Surface Waves: an Extension From Small to Finite Amplitude

The reduced 3-wave and 4-wave Hamiltonian equations for ocean surface waves were widely used for the simplified structure with symmetric polynomial kernels and for the conservation of energy, etc. However, according to the related assumption for approximation in derivation, the range of applicability was limited to weakly nonlinear waves of small amplitude. Here the following issue was further studied: for nonlinear waves of finite amplitude within a certain range, was it also possible to obtain reduced Hamiltonian equations with symmetric polynomial kernels in a sense of sufficient accuracy? Because of complicated strongly nonlinear coupling, few development in this significant respect had been made as yet. A new approach was proposed based on the Chebyshev polynomials to best approximate the primitive water wave equations in the exact sense of strongly nonlinear coupling and derive new reduced Hamiltonian equations with symmetric polynomial kernels. The new results exhibit an extension from a weakly nonlinear case in which the product of the wave number and the wave steepness is small to a nonlinear case in which this product goes up to about 1.035. Moreover, within this range, the approximation errors are lower than 5%, and in particular, the new results prove exact whenever the said product lies close to 0.9.

Subgrid parameterisations for primitive equation atmospheric models

Dynamical and thermodynamical subgrid-scale parameterisations of eddy drain, net dissipation and stochastic backscatter are calculated for a multi-level primitive equation atmospheric general circulation model. The parameterisations have only moderate variability with height and a cusp behaviour with peaks near the largest retained wavenumber. Vertically integrated net dissipation functions for vorticity and temperature are very similar to corresponding results for barotropic simulations while the divergence dissipation is nearly four times stronger. Atmospheric general circulation model simulations with the new subgrid model improve kinetic energy spectra and zonal flows compared with control simulations. References J. S. Frederiksen. Self-energy closure for inhomogeneous turbulence and subgrid modelling. Entropy 14:769&ndash;799, 2012. doi:10.3390/e14040769 J. S. Frederiksen and A. G. Davies. Eddy viscosity and stochastic backscatter parameterizations on the sphere for atmospheric circulation models. J. Atmos. Sci 54:2475&ndash;2492, 1997. doi:10.1175/1520-0469(1997)054&lt;2475:EVASBP>2.0.CO;2 J. S. Frederiksen and S. M. Kepert. Dynamical subgrid-scale parameterizations from direct numerical simulations. J. Atmos. Sci. 63:3006&ndash;3019, 2006. doi:10.1175/JAS3795.1 J. S. Frederiksen, T. J. O'Kane and M. J. Zidikheri. Stochastic subgrid parameterizations for atmospheric and oceanic flows. Phys. Scripta 85:068202, 2012. doi:10.1088/0031-8949/85/06/068202 J. S. Frederiksen, M. R. Dix and S. M. Kepert. Systematic energy errors and the tendency toward canonical equilibrium in atmospheric circulation models. J. Atmos. Sci. 53:887&ndash;904, 1996. doi:10.1175/1520-0469(1996)053&lt;0887:SEEATT>2.0.CO;2 I. M. Held, and M. J. Suarez. A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Am. Met. Soc. 75:1825&ndash;1830, 1994. doi:10.1175/1520-0477(1994)075&lt;1825:APFTIO>2.0.CO;2 V. Kitsios, J. S. Frederiksen and M. J. Zidikheri. Scaling laws for parameterisations of subgrid eddy-eddy interactions in simulations of oceanic circulations. Ocean Model. 68:88&ndash;105, 2013. doi:10.1016/j.ocemod.2013.05.001 V. Kitsios, J. A. Sillero, J. S. Frederiksen and J. Soria. Proposed stochastic parameterisations of subgrid turbulence in large eddy simulations of turbulent channel flow. J. Turbulence 16:729&ndash;741, 2015. doi:10.1080/14685248.2015.1026970 J. L. McGregor, H. B. Gordon, I. G. Watterson, M. R. Dix and L. D. Rotstayn. The csiro 9-level atmospheric general circulation model. csiro Division of Atmospheric Research, Technical Report 26, 1993. https://publications.csiro.au/rpr/pub?list=BRO&pid=procite:d73112b4-4d15-481b-84c5-f488501e7e07 I. N. Smith, M. R. Dix and R. J. Allen. The effect of greenhouse SSTs on ENSO simulations with an agcm. J. Climate 10:342&ndash;352, 1997. doi:10.1175/1520-0442(1997)010&lt;0342:TEOGSO>2.0.CO;2

Sensitivity and variability redux in hot-Jupiter flow simulations

We revisit the issue of sensitivity to initial flow and intrinsic variability in hot-Jupiter atmospheric flow simulations, originally investigated by Cho et al. (2008) and Thrastarson & Cho (2010). The flow in the lower region (~1 to 20 MPa) `dragged' to immobility and uniform temperature on a very short timescale, as in Liu & Showman (2013), leads to effectively a complete cessation of variability as well as sensitivity in three-dimensional (3D) simulations with traditional primitive equations. Such momentum (Rayleigh) and thermal (Newtonian) drags are, however, ad hoc for 3D giant planet simulations. For 3D hot-Jupiter simulations, which typically already employ strong Newtonian drag in the upper region, sensitivity is not quenched if only the Newtonian drag is applied in the lower region, without the strong Rayleigh drag: in general, both sensitivity and variability persist if the two drags are not applied concurrently in the lower region. However, even when the drags are applied concurrently, vertically-propagating planetary waves give rise to significant variability in the ~0.05 to 0.5 MPa region, if the vertical resolution of the lower region is increased (e.g. here with 1000 layers for the entire domain). New observations on the effects of the physical setup and model convergence in ‘deep’ atmosphere simulations are also presented.

DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility

Abstract. The design of the icosahedral dynamical core DYNAMICO is presented. DYNAMICO solves the multi-layer rotating shallow-water equations, a compressible variant of the same equivalent to a discretization of the hydrostatic primitive equations in a Lagrangian vertical coordinate, and the primitive equations in a hybrid mass-based vertical coordinate. The common Hamiltonian structure of these sets of equations is exploited to formulate energy-conserving spatial discretizations in a unified way. The horizontal mesh is a quasi-uniform icosahedral C-grid obtained by subdivision of a regular icosahedron. Control volumes for mass, tracers and entropy/potential temperature are the hexagonal cells of the Voronoi mesh to avoid the fast numerical modes of the triangular C-grid. The horizontal discretization is that of Ringler et al. (2010), whose discrete quasi-Hamiltonian structure is identified. The prognostic variables are arranged vertically on a Lorenz grid with all thermodynamical variables collocated with mass. The vertical discretization is obtained from the three-dimensional Hamiltonian formulation. Tracers are transported using a second-order finite-volume scheme with slope limiting for positivity. Explicit Runge–Kutta time integration is used for dynamics, and forward-in-time integration with horizontal/vertical splitting is used for tracers. Most of the model code is common to the three sets of equations solved, making it easier to develop and validate each piece of the model separately. Representative three-dimensional test cases are run and analyzed, showing correctness of the model. The design permits to consider several extensions in the near future, from higher-order transport to more general dynamics, especially deep-atmosphere and non-hydrostatic equations.

Eddy‐topography interactions and the fate of the Persian Gulf Outflow

AbstractThe Persian Gulf feeds a warm and salty outflow in the Gulf of Oman (northern Arabian Sea). The salt climatological distribution is relatively smooth in the Gulf of Oman, and the signature of a slope current carrying salty waters is difficult to distinguish hundreds of kilometers past the Strait of Hormuz, in contrast to other outflows of the world ocean. This study focuses on the mechanisms involved in the spreading of Persian Gulf Water (PGW) in the Gulf of Oman, using a regional primitive equation numerical simulation. The authors show that the dispersion of PGW occurs through a regime that is distinct from, for example, the one responsible for the Mediterranean outflow dispersion. The background mesoscale eddy field is energetic and participates actively to the spreading of PGW. Remotely formed eddies propagate into the Gulf of Oman and interact with the topography, leading to submesoscales formation and PGW shedding. Eddy‐topography interactions are isolated in idealized simulations and reveal the formation of intense frictional boundary layers, generating submesoscale coherent vortices (SCVs). Interactions take place at depths encompassing the PGW depth, thus SCVs trap PGW and contribute to its redistribution from the boundaries to the interior of the Gulf of Oman. The overall efficiency of these processes is confirmed by a strong contribution of eddy salt fluxes in the interior of the basin, and is quantified using particle statistics. It is found to be a highly dispersive regime, with an approximated eddy diffusivity of .

Improve the Simulations of Near-Inertial Internal Waves in the Ocean General Circulation Models

AbstractThe near-inertial wind work and near-inertial internal waves (NIWs) in the ocean have been extensively studied using ocean general circulation models (OGCMs) forced by 6-hourly winds or wind stress obtained from atmospheric reanalysis data. However, the OGCMs interpolate the reanalysis winds or wind stress linearly onto each time step, which partially filters out the wind stress variance in the near-inertial band. In this study, the influence of the linear interpolation on the near-inertial wind work and NIWs is quantified using an eddy-resolving (°) primitive equation ocean model. In addition, a new interpolation method is proposed—the sinc-function interpolation—that overcomes the shortages of the linear interpolation.It is found that the linear interpolation of 6-hourly winds significantly underestimates the near-inertial wind work and NIWs at the midlatitudes. The underestimation of the near-inertial wind work and near-inertial kinetic energy is proportional to the loss of near-inertial wind stress variance due to the linear interpolation. This further weakens the diapycnal mixing in the ocean due to the reduced near-inertial shear variance. Compared to the linear interpolation, the sinc-function interpolation retains all the wind stress variance in the near-inertial band and yields correct magnitudes for the near-inertial wind work and NIWs at the midlatitudes.

Decaying capillary wave turbulence under broad-scale dissipation

We study the freely decaying weak turbulence of capillary waves by direct numerical solution of the primitive Euler equations. By introducing a small amount of wave dissipation, measured by the viscosity magnitude${\it\gamma}_{0}$, we are able to recover phenomena observed in experiments that are not described by weak-turbulence theory (WTT), including the exponential modal decay and time variation of the width and power-law spectral slope${\it\alpha}$of the inertial range. In contrast to WTT, this problem also involves non-constant inter-modal energy transfer across the inertial range, which imposes a difficulty in quantifying and measuring the energy flux$P$associated with a certain power-law spectrum. We propose an effective and novel way to evaluate$P$in such cases by physically considering the unsteady effects of the spectrum and variation of the inter-modal energy transfer. Our results show the fundamental difference between the energy flux$P$and the total energy dissipation rate${\it\Gamma}$, which is due to significant energy dissipation within the inertial range. This settles the previous debate on the measurement of$P$which assumes the equivalence of the two. Based on our numerical data, we obtain a general form of the time-evolving inertial-range spectrum, where the parameters involved are functions of${\it\gamma}_{0}$only. The value of the spectral slope${\it\alpha}$at each time moment in the decay, however, is found to be uniquely related to the spectral magnitude at that time and irrespective of${\it\gamma}_{0}$, in the range we consider. This physically reveals the dominant effect of nonlinear wave interaction in forming the power-law spectrum within the inertial range. The evolutions of the inertial-range energy are shown to be predicted by analytical integration of the evolving spectra for different values of${\it\gamma}_{0}$.

Mesoscale variability in the Arabian Sea from HYCOM model results and observations: impact on the Persian Gulf Water path

Abstract. The Arabian Sea and Sea of Oman circulation and water masses, subject to monsoon forcing, reveal a strong seasonal variability and intense mesoscale features. We describe and analyze this variability and these features, using both meteorological data (from ECMWF reanalyses), in situ observations (from the ARGO float program and the GDEM – Generalized Digital Environmental mode – climatology), satellite altimetry (from AVISO) and a regional simulation with a primitive equation model (HYCOM – the Hybrid Coordinate Ocean Model). The model and observations display comparable variability, and the model is then used to analyze the three-dimensional structure of eddies and water masses with higher temporal and spatial resolutions than the available observations. The mesoscale features are highly seasonal, with the formation of coastal currents, destabilizing into eddies, or the radiation of Rossby waves from the Indian coast. The mesoscale eddies have a deep dynamical influence and strongly drive the water masses at depth. In particular, in the Sea of Oman, the Persian Gulf Water presents several offshore ejection sites and a complex recirculation, depending on the mesoscale eddies. The associated mechanisms range from coastal ejection via dipoles, alongshore pulses due to a cyclonic eddy, to the formation of lee eddies downstream of Ra's Al Hamra. This water mass is also captured inside the eddies via several mechanisms, keeping high thermohaline characteristics in the Arabian Sea. The variations of the outflow characteristics near the Strait of Hormuz are compared with variations downstream.

Instability and Mixing of Zonal Jets along an Idealized Continental Shelf Break

AbstractThe interaction between an Antarctic Circumpolar Current–like channel flow and a continental shelf break is considered using eddy-permitting simulations of a quasigeostrophic and a primitive equation model. The experimental setup is motivated by the continental shelf of the West Antarctic Peninsula. Numerical experiments are performed to study how the width and slope of an idealized continental shelf topography affect the characteristics of the flow. The main focus is on the regime where the shelfbreak width is slightly greater than the eddy scale. In this regime, a strong baroclinic jet develops on the shelf break because of the locally stabilizing effect of the topographic slope. The velocity of this jet is set at first order by the gradient of the background barotropic geostrophic contours, which is dominated by the slope of the topography. At statistical equilibrium, an aperiodic cycle is observed. Initially, over a long stable period, an upper-layer jet develops over the shelf break. Once the vertical shear reaches the critical condition for baroclinic instability, the jet becomes unstable and drifts away from the shelf break. The cross-shelf mixing is intrinsically linked with the jet drifting, as most of the meridional flux occurs during this instability period. Investigation of the zonal momentum budget reveals that a strong Reynolds stress divergence inversion across the jet is associated with a drifting event, accelerating one flank of the jet and decelerating the other. The hypothesis that jet drifting may be due to one flank of the jet being more baroclinically unstable than the other is tested using topographic profiles with variable curvatures.

Cyclones and Anticyclones in Seismic Imaging

AbstractNearly all the subsurface eddies detected in seismic imaging of sections in the northeast Atlantic have been assumed to be anticyclones containing Mediterranean Water (MW). Fewer MW cyclones have been observed and studied. In this study, the work of previous numerical studies is extended to investigate some characteristics of layering surrounding MW cyclones, using a primitive equation model with equal diffusivities for salinity and temperature to suppress the effects of double diffusion. It is shown that, after a stable state is reached, both anticyclones and cyclones display similar patterns of layering: stacked thin layers of high acoustic reflectivity located above and below the core of each vortex, which do not match isopycnals. The authors conclude that it should not be possible to distinguish between MW cyclones and anticyclones based on their signature in seismic imaging alone. Complementary information is needed to determine the sense of rotation.

Energy‐conserving finite‐difference schemes for quasi‐hydrostatic equations

The pressure‐based hydrostatic primitive equations model LMD‐Z is extended to solve the quasi‐hydrostatic deep‐atmosphere as well as the non‐traditional shallow‐atmosphere equations (with a complete Coriolis force representation).The continuous equations are first derived in their curl form using Eulerian horizontal and non‐Eulerian vertical coordinates. The equations are then interpreted as a Hamiltonian system, as they are expressed in terms of functional derivatives of the Hamiltonian. Using a finite‐difference scheme on a longitude/latitude grid and based either on a Lagrangian or mass‐based vertical coordinate, the discrete scheme is obtained by imitating the Hamiltonian formulation at the discrete level. It is shown how this form leads straightforwardly to the conservation of discrete total energy. The relation between the discrete equations and the discrete antisymmetry property of the Poisson bracket is discussed.The computing infrastructure of the dynamical core is kept essentially unchanged but the modification of the hydrostatic balance requires a mass‐based vertical coordinate. Also, absolute angular momentum is used as a prognostic variable instead of relative velocity, which allows time‐dependent metric terms and the non‐traditional Coriolis force to be absorbed into it.The prototype implementation is applied to idealized circulations of an Earth‐like small planet and validates the stability and accuracy of the new dynamical core.

Assimilação de Dados Via Método 3D-Var em Dinâmica Caótica do Modelo de Lorenz

This paper explores some aspects of data assimilation by 3D-Var method implemented in the Lorenz model. This system was chosen because, although simple, has similar structure to the atmosphere under certain parameters. Conceptual models allow difficult judgments to be made in primitive equation models with their various nonlinear interactions at various scales. It is shown the need to apply data assimilation techniques in highly sensitive models to initial conditions. It was also shown that 3D-Var method is very dependent on the number of cycles during the process of minimizing of the cost function. In the experiment performed on this work was found that at least 1,000 interactions are necessary to solve this minimization problem. However the technique fail to 40% noise added to initial conditions, comparing to control. Additional experiments show the difficulty of performing high quality data assimilation in chaotic dynamics for underdetermined systems, such as the Earth's atmosphere.