# Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics

@article{Cao2005GlobalWO, title={Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics}, author={Chongsheng Cao and Edriss S. Titi}, journal={Annals of Mathematics}, year={2005}, volume={166}, pages={245-267} }

In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics. 1. Introduction Large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations, with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification…

## 220 Citations

### Global Well–Posedness of the 3D Primitive Equations with Partial Vertical Turbulence Mixing Heat Diffusion

- Mathematics
- 2010

The three–dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly…

### Coupling of the ocean and atmosphere dynamics with sea ice

- Mathematics, Environmental Science
- 2022

. This article establishes local strong well-posedness of a model coupling the primitive equa- tions of the ocean and the atmosphere with a regularized version of Hibler’s viscous-plastic sea ice…

### SMALL-TIME SOLVABILITY OF PRIMITIVE EQUATIONS FOR THE OCEAN WITH SPATIALLY-VARYING VERTICAL MIXING

- Mathematics
- 2015

The small-time existence of a strong solution to the free surface problem of primitive equations for the ocean with variable turbulent viscosity terms is shown in this paper. In this model, the…

### Global well-posedness of large scale moist atmosphere system with only horizontal viscosity in the dynamic equation

- Mathematics
- 2022

In order to ﬁnd a better physical model to describe the large-scale cloud-water transforma-tion and rainfall, we consider a moist atmosphere model consisting of the primitive equations with only…

### The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

- MathematicsJournal of Differential Equations
- 2022

### Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere

- Mathematics
- 2011

### Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise

- Mathematics
- 2012

The primitive equations (PEs) are a basic model in the study of large scale oceanic and atmospheric dynamics. These systems form the analytical core of the most advanced general circulation models.…

### Rigorous derivation of the primitive equations with full viscosity and full diffusion by scaled Boussinesq equations

- Mathematics
- 2021

The primitive equations of large-scale ocean dynamics form the fundamental model in geophysical flows. It is well-known that the primitive equations can be formally derived by hydrostatic balance. On…

### Quasi-hydrostatic primitive equations for ocean global circulation models

- Mathematics, Environmental Science
- 2010

Global existence of weak and strong solutions to the quasi-hydrostatic primitive equations is studied in this paper. This model, that derives from the full non-hydrostatic model for geophysical fluid…

### Small-time existence of a strong solution of primitive equations for the ocean

- Mathematics
- 2012

Primitive equations derived originally by Richardson in 1920’s have been considered as the model equations describing the motion of atmosphere, ocean and coupled atmosphere and ocean. In this paper,…

## References

SHOWING 1-10 OF 41 REFERENCES

### Large-scale circulation with small diapycnal diffusion: The two-thermocline limit

- Environmental Science, Physics
- 1997

The structure and dynamics of the large-scale circulation of a single-hemisphere, closed-basin ocean with small diapycnal diffusion are studied by numerical and analytical methods. The investigation…

### On the equations of the large-scale ocean

- Environmental Science
- 1992

As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, the authors study the mathematical formulations…

### A Simple Friction and Diffusion Scheme for Planetary Geostrophic Basin Models

- Physics
- 1997

A simple friction and diffusion scheme is proposed for use with the time-dependent planetary geostrophic equations, which in their proper asymptotic form cannot be solved in a closed basin. The…

### Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation

- Physics, Environmental Science
- 1998

We study in this article the well-posdness of the planetary geostrophic equations of the gyre-scale midlatitude ocean, and address in particular the question of the existence and some properties of…

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

- Mathematics
- 2003

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 primitive equations on the large scale ocean under the small depth hypothesis

- Mathematics
- 2002

In this article we study the global existence
of strong solutions of the Primitive Equations (PEs)
for the large scale ocean under the small depth hypothesis.
The small depth hypothesis implies…

### The Use of the Primitive Equations of Motion in Numerical Prediction

- Environmental Science
- 1955

An obstacle to the use of the primitive hydrodynamical equations for numerical prediction is that the initial wind and pressure fields determined by conventional means give rise to spurious…

### FAST SINGULAR OSCILLATING LIMITS AND GLOBAL REGULARITY FOR THE 3D PRIMITIVE EQUATIONS OF GEOPHYSICS

- Mathematics
- 2000

Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D…

### Coastal Boundary Conditions and the Baroclinic Structure of Wind-Driven Continental Shelf Currents*

- Environmental Science
- 1997

Abstract The generation of continental shelf currents by wind forcing is investigated by analytical and numerical methods. The investigation is motivated by observations from the Coastal Ocean…