Primitive Equation Research Articles

Local Existence and Uniqueness of Strong Solution to the Inhomogeneous Primitive Equations with Vacuum

Local Existence and Uniqueness of Strong Solution to the Inhomogeneous Primitive Equations with Vacuum

A Liouville-type theorem for the 3D primitive equations

The 3D primitive equations are used in most geophysical fluid models to approximate the large scale oceanic and atmospheric dynamics. We prove that there do not exist smooth stationary solutions to the 3D primitive equations with compact support, independently of the presence of the Coriolis rotation term or the viscosity. This result is in strong contrast with the recently established existence of compactly supported smooth solutions to the incompressible 3D Euler equations.

Amplitude and oscillating assessment of thermal and magnetic boundary layer flow across circular heated cylinder with heat source/sink

The effects of heat source/sink and magnetohydrodynamics on the oscillatory and periodic quantities of heat transfer and current density characteristics of viscous fluid along the magnetized and heated circular cylinder has been investigated. The governing mathematical model in terms of partial differential equations is converted into dimensionless form. The dimensionless model is again converted into steady, real and imaginary part for oscillating results. The steady, real and imaginary part is transformed into primitive form for smooth algorithm with the help of Primitive Variable Formulation (PVF). The primitive equations are reduced into system of algebraic equations by using Finite Difference Method (FDM). The heat source/sink parameter δ, the magnetic force number ξ, the buoyancy parameter λ, magneto-Prandtl factor γ, and remaining secured factors are utilized to determine computational findings of unknown quantities. Graphs are displayed for velocity, temperature distribution and magnetic profile for all pertinent parameters by FORTRAN and Teplot 360 software. The main novelty of current work is to evaluate oscillatory and periodic quantities of heat transfer and current density by using the steady solutions. It was found that prominent results in temperature distribution are deduced at each angle with the heat source effects. It was expected physically because the heat source is used as a sporting agent to compute heat transfer performance in electrically conducting fluid. The significant amplitude of oscillation in heat transfer and current density is evaluated around every position of circular cylinder. The numerical results of skin friction are compared with existing literature with excellent agreement.

Blow-up criterion of solutions of the horizontal viscous primitive equations with horizontal eddy diffusivity

In this paper, we consider the blow-up criterion of the H1 norm of solutions to the primitive equations with only horizontal viscosity and horizontal thermal diffusivity. The maximum norm of the gradient of the vorticity is showed to control the break down of the H1 norm of solutions when there is no additional symmetry assumption for the data. The temperature irrelevant to the break down is also showed.

Anisotropic diffusion filters for flow-dependent variational data assimilation of sea surface temperature

It is well known that the background error covariance matrix in the data assimilation is anisotropic and varied with the flow field, that is, flow-dependent. However, it is generally assumed to be isotropic and static for ease of implementation. Therefore, to enhance the quality of the background error covariance, two easy-to-implement, flow-dependent three-dimensional variational analysis (3DVar) schemes, the advection–diffusion filter (ADF) and the Smagorinsky diffusion filter (SDF), are tentatively proposed. These serve as alternative schemes for the traditional methods that are based on the isotropic assumption. First, a correctness test of the gradient for the ADF is performed to verify the validity of the adjoint code. Then, the response analysis for the 1D advection–diffusion​ equation is investigated to validate the anisotropic characteristic of the ADF with various flow velocities. Second, the two flow-dependent 3DVar schemes are verified via single-observation assimilation experiments. Compared to the existing diffusion filter with a static diffusion coefficient, both new schemes show a flow-dependent capability in the assimilation experiment because they successfully spread the observational information anisotropically. Finally, a retroactive real-time forecast experiment framework based on a 3D baroclinic primitive equation ocean model is used to investigate the effects of the ADF/SDF on the analysis and forecast accuracy of sea surface temperature (SST). While the traditional variational assimilation method with an isotropic diffusion filter generally reduces initial field errors, both the ADF and SDF can further enhance the accuracy of the initial fields in strong current areas. In addition, better forecast skills can be obtained with the two flow-dependent schemes in the East China Sea.

Stable singularity formation for the inviscid primitive equations

The primitive equations (PEs) model large-scale dynamics of the oceans and the atmosphere. While it is by now well known that the three-dimensional viscous PEs are globally well posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blow-up mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic-in-time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameter family of nonsmooth profiles, and is stable only by smoother perturbations.

Global well‐posedness of the 3D primitive equations with only horizontal eddy diffusivity and delays in both convective and heat source terms

In ocean and atmospheric dynamics, the strong horizontal turbulent mixing tends to produce horizontal eddy viscosity, so the horizontal scale is much larger than the vertical scale. At the same time, as a fluid, the motion of the atmosphere or the ocean has a delayed property to some extent. In this paper, we consider 3D viscous primitive equations with only horizontal eddy diffusivity and delays in both convective and heat source terms. With some suitable assumptions on delay term, we prove the existence and uniqueness of strong solution with initial data and obtain the existence and uniqueness of the stationary solution. Moreover, we prove that the strong solution converges exponentially to the stationary solution under some suitable assumptions.

Global well-posedness of the primitive equations with the magnetic field with only horizontal viscosity

Global well-posedness of the primitive equations with the magnetic field with only horizontal viscosity

On the Rigorous Mathematical Derivation for the Viscous Primitive Equations with Density Stratification

In this paper, we rigorously derive the governing equations describing the motion of a stable stratified fluid, from the mathematical point of view. In particular, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ratio goes to zero, and the rate of convergence is of the same order as the aspect ratio. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.

Turbulence modeling to aid tidal energy resource characterization in the Western Passage, Maine, USA

Numerical models combined with field measurements are regularly used to characterize tidal energy resources at potential energetic sites. However, most existing works only focus on the tidal hydrodynamic characteristics, and turbulence parameters are often not reported because of the lack of high-quality turbulence measurements and the limitations of numerical models in resolving turbulent eddies. In this study, we used FVCOM – a hydrostatic primitive equation (HPE) model – to characterize the tidal energy resource in the Western Passage, Maine, USA, by taking care of the essential macro-scale turbulence properties. We observed an excellent model performance using the Mellor–Yamada Level 2.5 Turbulence Model; estimating the spatial and vertical distribution of the turbulent kinetic energy and intensity added a new perspective to the site ranking for tidal energy converter (TEC) deployments. In addition, we also examined the role of channel geometry and bathymetry, such as headlands and underwater sills, in enhancing turbulent eddies around potential TEC siting locations. Ultimately, the detailed analysis of the turbulent flow characteristics has changed the site-ranking results and demonstrated that the regional-scale HPE models could be used for the relative understanding of more or less turbulent sites for a refined resource assessment.

Mixed convection Casson polymeric flow from a nonlinear stretching surface with radiative flux and non‐Fourier thermal relaxation effects: Computation with CSNIS

AbstractThermal non‐Newtonian polymer coating flows is growing as a major area in materials processing. Inspired by new developments in this field which require more sophisticated mathematical models, the current investigation examines the laminar viscoplastic boundary layer flow and mixed convective heat transfer over a power‐law nonlinear stretching surface. To simulate thermal relaxation effects the hyperbolic Cattaneo‐Christov heat flux model is deployed. The non‐Newtonian polymer characteristics are described by employing the Casson flow model. High temperature conditions invoke thermal radiation flux which is analyzed with an algebraic flux model. Via robust similarity transformations, the primitive partial differential conservation equations for momentum and energy equations are rendered into a system of coupled non‐linear ordinary differential equations with associated wall and free stream boundary conditions. The emerging boundary value problem is solved numerically with an efficient Chebyshev Spectral Newton Iterative scheme (CSNIS), in the MATLAB platform. The resulting solutions are discussed for different emerging parameters using graphs and tables. Validation is included with special cases from the literature. With increasing power law stretching index increases, the flow is decelerated, and temperatures are reduced. Increment in mixed convection parameter boosts the velocity but suppresses temperature and thermal boundary layer thickness. Increasing non‐Fourier Deborah number, temperatures are depleted whereas with increasing radiative flux parameter they are increased. With elevation in Casson non‐Newtonian parameter, velocity is decreased whereas temperature is enhanced and Nusselt number is suppressed.

A Simple Family of Tropical Cyclone Models

This review discusses a simple family of models capable of simulating tropical cyclone life cycles, including intensification, the formation of the axisymmetric version of boundary layer shocks, and the development of an eyewall. Four models are discussed, all of which are axisymmetric, f-plane, three-layer models. All four models have the same parameterizations of convective mass flux and air–sea interaction, but differ in their formulations of the radial and tangential equations of motion, i.e., they have different dry dynamical cores. The most complete model is the primitive equation (PE) model, which uses the unapproximated momentum equations for each of the three layers. The simplest is the gradient balanced (GB) model, which replaces the three radial momentum equations with gradient balance relations and replaces the boundary layer tangential wind equation with a diagnostic equation that is essentially a high Rossby number version of the local Ekman balance. Numerical integrations of the boundary layer equations confirm that the PE model can produce boundary layer shocks, while the GB model cannot. To better understand these differences in GB and PE dynamics, we also consider two hybrid balanced models (HB1 and HB2), which differ from GB only in their treatment of the boundary layer momentum equations. Because their boundary layer dynamics is more accurate than GB, both HB1 and HB2 can produce results more similar to the PE model, if they are solved in an appropriate manner.

Wavenumber-Frequency Spectra of Normal Mode Function Decomposed Atmospheric Data: Departures from the Dry Linear Theory

The study of tropical tropospheric disturbances has led to important challenges from both observational and theoretical points of view. In particular, the observed wavenumber-frequency spectrum of tropical oscillations has helped bridge the gap between observations and the linear theory of equatorial waves. In this study, we obtained a similar wavenumber-frequency spectrum for each equatorial wave type by performing a normal mode function (NMF) decomposition of global Era–Interim reanalysis data. The NMF basis used here is provided by the eigensolutions of the primitive equations in spherical coordinates as linearized around a resting background state. In this methodology, the global multi-level horizontal velocity and geopotential height fields are projected onto the normal mode functions, characterized by a vertical mode, a zonal wavenumber, a meridional quantum index, and a mode type, namely, Rossby, Kelvin, mixed Rossby-gravity, and westward/eastward propagating inertio-gravity modes. The horizontal velocity and geopotential height fields associated with each mode type are then reconstructed in the physical space, as well as their corresponding filtered versions defined according to the vertical mode classes that exhibit barotropic and baroclinic structures within the troposphere. The results reveal expected structures, such as the dominant global-scale Rossby and Kelvin waves constituting the intraseasonal frequency associated with the Madden–Julian Oscillation. On the other hand, a number of unexpected features, such as eastward propagating westward inertio-gravity waves, are revealed by our observed 200 hPa zonal wind spectrum. Among all possible nonlinear processes, we focus on the analysis of the interaction between Kelvin and westward inertio-gravity waves, providing evidence for their coupling. Apart from the nonlinearity, we discuss the potential roles of a vertically/meridionally varying background state as well as the coupling with moist convection in explaining the departures of the observed spectra from the corresponding linear equatorial wave theory.

Seasonal Variability of the Ocean Circulation in Queen Charlotte Strait, British Columbia

ABSTRACT Queen Charlotte Strait (QCST) is a large marine area separating northern Vancouver Island from the British Columbia mainland. Although it is an important waterway connecting Queen Charlotte Sound with Johnstone Strait and eventually the Strait of Georgia, complex dynamics in QCST, including tides, surface winds, river runoffs, and coastal currents, are yet understudied. In this numerical study, a high-resolution (up to approximately 10 m in the horizontal) model was developed for QCST and adjacent areas to investigate the circulation in the strait, using the unstructured grid, Finite-Volume, primitive equation Community Ocean Model (FVCOM). Two operational larger-scale models force the FVCOM model: the Coastal Ice Ocean Prediction System for the West coast (CIOPS-W) at the open ocean (one-way nesting), and the High-Resolution Deterministic Prediction System (HRDPS) at surface boundaries. Prevailing winds over QCST are southeasterly in winter and northwesterly in summer and the model demonstrated the capacity to reproduce the ocean circulation regimes in both seasons of 2019. Complex empirical orthogonal function (CEOF) and correlation analyses determined that surface winds play a dominant role on the variability of residual flow in QCST in the winter, while river runoffs strongly influence the variability during the summer. The analyses highlight that QCST is a complex estuarine system with significant seasonal variability rather than just a simple conduit between Queen Charlotte Sound and the Strait of Georgia.

Rigorous derivation of the full primitive equations by the scaled Boussinesq equations with rotation

The primitive equations of large-scale oceanic dynamics form a fundamental model in geophysical flows. It is well-known that the primitive equations can be formally derived by the hydrostatic approximation. In this paper, we rigorously derive the full primitive equations from the mathematical point of view. More precisely, we prove that the scaled Boussinesq equations with rotation converge to the full primitive equations in a strong sense, globally in time, with the convergence rate $$O(\varepsilon )$$ , as the aspect ratio $$\varepsilon $$ goes to zero.

Climate‐change modelling at reduced floating‐point precision with stochastic rounding

AbstractReduced‐precision floating‐point arithmetic is now deployed routinely in numerical weather forecasting over short timescales. However, the applicability of these reduced‐precision techniques to longer‐timescale climate simulations—especially those that seek to describe a dynamical, changing climate—remains unclear. We investigate this question by deploying a global atmospheric, coarse‐resolution model known as Simplified Parameterizations PrimitivE Equation DYnamics (SPEEDY) to simulate a changing climate system subject to increased concentrations, over a 100‐year timescale. Whilst double precision is typically the operational standard for climate modelling, we find that reduced‐precision solutions are sufficiently accurate. Rounding the floating‐point numbers stochastically, rather than using the more common “round‐to‐nearest” technique, improves the performance of the reduced‐precision solutions notably. Over 100 years, the mean bias error (MBE) in the global mean surface temperature (precipitation) relative to the double‐precision solution is +1.8 K ( mm(6 hr)) when integrating numerically at half precision (10 significant bits) with stochastic rounding. By examining the resultant climatic distributions that arise after 100 years, the difference in the expected value of the global surface temperature relative to the double‐precision solution is K and that for precipitation is mm(6 hr). Whilst further research is necessary to extended these results to more complex and higher‐resolution models, they indicate that reduced‐precision techniques and stochastic rounding could be suitable for the next generation of climate models and motivate the use of low‐precision hardware to this end.

Spectral Budget of Rotational and Divergent Kinetic Energy in Global Analyses

Abstract To study the multiscale interactions between rotational and divergent components of atmospheric motion, a new formulation of spectral budget of rotational kinetic energy (RKE) and divergent kinetic energy (DKE) based on the primitive equations in the pressure coordinate is derived, with four main characteristics: 1) horizontal kinetic energy (HKE) spectral transfer is exactly divided into spectral transfer of RKE and DKE, 2) the exact spectral conversion term between DKE and RKE is constructed, 3) the Coriolis term is considered, and 4) both the baroclinic conversion from available potential energy (APE) and the vertical flux of HKE act only on DKE. With this new formulation, outputs from ERA5 global reanalysis are investigated. At planetary scales, HKE spectral transfer, mainly attributed to β effect, is dominated by downscale DKE transfer. At synoptic scales, it is dominated by an upscale transfer of RKE energized by conversion of DKE mainly due to the Coriolis effect. The ultimate source of DKE in the upper troposphere is conversion of APE, while in the stratosphere it is the vertical flux. At mesoscales, the spectral transfers of RKE and DKE are both downscale, and conversion from RKE to DKE exists at sub-800-km scales in the upper troposphere, which is mainly attributed to the contribution from relative vorticity. At different heights, the intersection scales of RKE and DKE spectra are affected by the scales of positive peaks of the local spectral conversion from DKE to RKE around total wavenumber 10. Significance Statement The purpose of this study is to explore more physical insights on the dynamics underlying the atmospheric energy spectra from the perspective of rotational and divergent components of motion. We derive a new formulation of the spectral rotational and divergent kinetic energy budget in the pressure coordinate for the global atmosphere, with application to ERA5 global reanalysis. Our results reveal the differences of spectral energy budget between rotational and divergent motions at different heights and scales. This new formulation provides a good tool for revealing the multiscale cascade and interaction between atmospheric rotational and divergent motions. Future work should investigate these dynamical processes with higher-resolution simulations and datasets.

Small-Time Solvability of the Primitive Equations for the Coupled Atmosphere and Ocean Model

Small-Time Solvability of the Primitive Equations for the Coupled Atmosphere and Ocean Model

The planetary boundary layer physical processes, the secondary thermal baroclinic circulation and inertial oscillation contribution to diurnal variation of the Etesian winds over the Aegean Sea

Etesian winds constitute an important climatological phenomenon, which does not only moderate the heat during the summer in the Aegean Sea, but provides a source of clean renewable energy as well. Even though several papers have attempted to explain their dynamical and physical characteristics, the respective processes that drive the diurnal variation of the wind speed are not fully understood. The objective of this paper is to identify the processes responsible for diurnal variation with observed maximum wind speed around noon and minimum around midnight. Analytical solutions of a primitive equation set in Eulerian form, after introducing suitable conditions and approximations, reveal an inertial oscillation over the Aegean Sea. Data based on direct observations, ECMWF IFS high resolution analyses and high-resolution simulations with the Weather Research and Forecasting (WRF) model are utilized to find out the type and structure of the planetary boundary layer (PBL) over the Aegean Sea. This PBL appears to be of a marine character and turbulent mostly during the day but less during the night. The direct impact of local and regional thermally-driven circulations is found to be the main cause of the diurnal variation of the observed wind and partly the inertial oscillation. Results from numerical simulations certify these findings. Furthermore, the momentum and Newtonian heating exchanges by the physical processes inside the PBL, where the gradient wind together with smaller scales of atmospheric motions exist, are also necessary for explaining the variability of the Etesian winds.

A Comparison of Stochastic and Deterministic Downscaling in Eddy Resolving Ocean Modelling: The Lakshadweep Sea Case Study

This study compares the skills of two numerical models at the same horizontal resolution but based on different principles in representing meso- and sub-mesoscale ocean features. The first model, titled LD20-NEMO, was based on solving primitive equations of ocean dynamics. The second model, titled LD20-SDD, used a newer stochastic–deterministic downscaling (SDD) method. Both models had 1/20° resolution, the same meteo forcing, and used outputs from a data assimilating global model at 1/12° resolution available from Copernicus Marine Service (CMEMS). The LD20 models did not assimilate observational data but were physically aware of observations via the parent model. The LD20-NEMO only used a 2D set of data from CMEMS as the lateral boundary conditions. The LD20-SDD consumed the full 3D set of data from CMEMS and exploited the stochastic properties of these data to generate the downscaled field variables at higher resolution than the parent model. The skills of the three models were assessed against remotely sensed and in situ observations for the four-year period 2015–2018. The models showed similar skills in reproducing temperature and salinity, however the SDD version performed slightly better than the NEMO, and was more computationally efficient by a large margin.