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About 3,392 papers for “Primitive equations”

3. The Primitive Equations

Martin Ehrendorfer · Society for Industrial and Applied Mathematics eBooks · 2011

The angular-momentum and energy principles are fundamental in any treatment of the circulation. If in some approximate formulation of the equations they are not retained, the results are likely to be unrealistic. A spurious energy source may, for example, cause the wind to increase without limit.

Cited by 0

Primitive equations with continuous initial data

Igor Kukavica, Yuan Pei, Walter Rusin et al. · Nonlinearity · 2014

We address the well-posedness of the primitive equations of the ocean with continuous initial data. We show that the splitting of the initial data into a regular finite energy part and a small bounded part is preserved by the equations, thus leading to existence and uniqueness of solutions.

Cited by 53

GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS

Joseph Smagorinsky · Monthly Weather Review · 1963

An extended period numerical integration of a baroclinic primitive equation model has been made for the simulation and the study of the dynamics of the atmosphere's general circulation. The solution corresponding to external gravitational propagation is filtered by requiring the vertically integrated divergence to vanish identically. The vertical structure permits as dependent variables the horizontal wind at two internal levels and a single temperature, with the static stability entering as a parameter. The incoming radiation is a function of latitude only corresponding to the annual mean, and the outgoing radiation is taken to be a function of the local temperature. With the requirement for thermal equilibrium, the domain mean temperature is specified as a parameter. The role of condensation is taken into account only as it effectively reduces the static stability. All other external sources and sinks of heat are assumed to balance each other locally, and are thus omitted. The kinematics are that of a fluid on a sphere bounded by smooth zonal walls at the equator and at approximately 64° latitude. The dissipative sinks are provided by: (a) surface stresses proportional through a drag coefficient to the square of the surface wind which is suitably extrapolated from above, (b) internal convective stresses proportional to the vertical wind shear, and (c) lateral diffusion of momentum and heat through an exchange coefficient which depends on the local horizontal rate of strain—a horizontal length scale entering as the governing parameter. For a given specification of the parameters, an integration for 60 days has been made from initial conditions where random temperature disturbances have been superimposed on a zonally symmetric regime which is baroclinically unstable according to linear theory. This experiment not only displays the scale selective character of baroclinic instability, yielding zonal wave number 5 to 6, but also predicts an index or energy cycle. The period of this cycle is 11 to 12 days for the first 40 days of the experiment, then lengthening to 17 days while diminishing in amplitude during the latter part. The resulting mean zonal velocity profile is in good qualitative agreement with observation, but too intense, presumably because the effective static stability parameter is taken too large. Furthermore this profile is found to be no more than 5 percent super-geostrophic poleward of the angular momentum maximum and no more than 2 percent sub-geostrophic equatorward. The total zonal angular momentum remains constant to within 2 percent irrespective of the phase of the index cycle. This balance is controlled by the surface wind distribution which agrees quite well with observation. The poleward transport is mainly accomplished by the large-scale eddies, whereas the internal vertical flux is predominantly a transfer of the earth's angular momentum by the meridional circulation. The poleward heat transport is primarily accomplished by a Hadley circulation at low latitudes but by the large-scale horizontal eddies in mid-latitudes, where a Ferrel circulation tends to compensate through an equatorward flux. This compensation at mid-latitudes by an indirect meridional circulation is also quite evident, in the potential-kinetic energy transformations. Comparison of the momentum and heat transfer with observed data when available shows reasonably good quantitative agreement. The lateral transfer of momentum and heat by the non-linear diffusion, which parametrically is supposed to simulate the action of motions of sub-grid scale, accounts for a significant portion of the total eddy transfer. Although no direct comparison with the corresponding transfer in the real atmosphere is available, intuitively our small-scale diffusion appears to play too large a role. A diagnosis is made of the transformations among the baratropic and baroclinic parts of the kinetic energy as well as the zonal mean and zonal perturbation parts of the available potential and kinetic energy. This reveals the dominant paths that the energy passes through from source to ultimate sinks and the processes responsible for these transformations. It is found that the partitioning of dissipation by the energy components may differ considerably from estimates made from observation.

Cited by 13,701

Primitive equations with linearly growing initial data

Amru Hussein, Martin Saal, Okihiro Sawada · ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE · 2019

The primitive equations in a 3D infinite layer domain are considered with linearly growing initial data in the horizontal direction, which illustrates the global atmospheric rotating or straining flows. On the boundaries, Dirichlet, Neumann or mixed boundary conditions are imposed. The Ornstein-Uhlenbeck type operator appears in the linear parts, so the semigroup theory is established by Trotter's arguments due to decomposition of infinitesimal generators. To obtain smoothing properties of the semigroup, derivatives of the associated kernel are calculated. For proving time-local existence and uniqueness of mild solutions, the adapted Fujita-Kato scheme is used in certain Sobolev spaces.

Cited by 1Open access

Isentropic primitive equations for the moist troposphere

Shao‐Yi Lee, Tieh‐Yong Koh · Quarterly Journal of the Royal Meteorological Society · 2013

Despite the knowledge that the potential temperature of an air parcel has a dependence on its water vapour content, potential temperature is often still calculated as if the parcel were dry, assuming that this moisture dependence is negligible. We show that such a dry potential temperature approximation is not suitable for tropical regions. Moisture gradient terms are seen in the isentropic primitive equations when Exner and Montgomery functions are generalised with moist specific heat capacities, forming a contribution to the horizontal momentum tendency comparable to that by the Montgomery function. This reflects how local horizontal gradients in potential temperature created by inhomogeneous water vapour distribution are relatively significant compared to gradients created by inhomogeneous temperature, in a large‐scale background of weak horizontal temperature gradient. In such an environment, water plays an active role in tropical atmospheric dynamics without the uptake or release of latent heat during phase changes. Hence, we suggest that the tropical troposphere is a place where the atmosphere can behave dynamically as a binary‐component fluid at local and regional scales.

Cited by 1

The Navier-Stokes equations in primitive variables

F. Lam · arXiv (Cornell University) · 2019

The Navier-Stokes equations in the primitive formulation for incompressible flow describe the evolution of velocity and pressure, without recourse to vorticity. We show that, beyond the finite Leray-Hopf regularity interval, every postulated strong solution is accompanied by infinitely many diffusion-dominated percolations of arbitrary size, while the momentum deficit caused by the non-linearity is compensated by the pressure gradient. In the upper half space, we demonstrate how sequences of these collective companions can be re-scaled into an absurd singularity. Owning to the passive nature of the pressure, there exist no essential a priori bounds for establishing the uniqueness of primitive solutions. With the illustration of well-exploited examples of closed-form basic flows, we elucidate the reason why perturbations, infinitesimal or finite, instigate indeterminate states that render the concept of flow instability inadmissible. An effort has also been made to reappraise a number of important issues in fluid dynamics. Unfortunately, the primitive theory cannot serve as a reliable tool for prediction. Nevertheless, a dedicated effort has been made to elaborate a priori bounds for vorticity dynamics for ideal as well as real fluids. As a result, we are able to establish long-time regularity of the Cauchy problem for incompressible flows. A main conclusion is that no events of finite-time blow-up can ever occur in the Euler or Navier-Stokes equations for initial data of finite energy.

Cited by 1Open access

Boundary layers for the 2D linearized primitive equations

Makram Hamouda, Chang‐Yeol Jung, Roger Témam · Communications on Pure &amp Applied Analysis · 2008

In this article, we establish the asymptotic behavior, when theviscosity goes to zero, of the solutions of the Linearized PrimitiveEquations (LPEs) in space dimension $2$. More precisely, we provethat the LPEs solution behaves like the corresponding inviscidproblem solution inside the domain plus an explicit correctorfunction in the neighborhood of some parts of the boundary. Twocases are considered, the subcritical and supercritical modesdepending on the fact that the frequency mode is less or greaterthan the ratio between the reference stratified flow (around whichwe linearized) and the buoyancy frequency. The problem of boundarylayers for the LPEs is of a new type since the corresponding limitproblem displays a set of (unusual) nonlocal boundary conditions.

Cited by 15

Realizing Surface-Driven Flows in the Primitive Equations

Eric Bembenek, Francis Poulin, Michael L. Waite · Journal of Physical Oceanography · 2015

Abstract The surface quasigeostrophic (SQG) model describes flows with surface buoyancy perturbations with no interior quasigeostrophic potential vorticity at small Rossby number Ro and O (1) Burger number, where quasigeostrophic dynamics are expected to hold. Numerical simulations of SQG dynamics have shown that vortices are frequently generated at small scales, which may have O (1) Rossby numbers and therefore may be beyond the limits of SQG. This paper examines the dynamics of an initially geostrophically balanced elliptical surface buoyancy perturbation in both the SQG model and the nonhydrostatic Boussinesq primitive equations (PE). In the case of very small Rossby number, it is confirmed that both models agree, as expected. For larger Ro, non-SQG effects emerge and as a result the solution of the PE deviates significantly from that of SQG. In particular, an increase in the Rossby number has the following effects: (i) the buoyancy filaments at the surface are stabilized in that they generate fewer secondary vortices; (ii) the core of the vortex experiences inertial instability, which results in a uniform buoyancy profile in its interior; (iii) the divergent part of the energy spectrum increases in magnitude; (iv) the PE model has significantly more gravity waves that are radiated from the vortex; (v) the magnitude of the vertical velocity increases; and (vi) in the mature stages of evolution, there are gravitational instabilities that develop because of the complicated dynamics inside the vortex. It is demonstrated that significant non-SQG effects are evident when the large-scale Rossby number of the initial flow is about 0.05 and the local Rossby number is O (1).

Cited by 10Open access

Stable singularity formation for the inviscid primitive equations

Charles Collot, Slim Ibrahim, Quyuan Lin · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2023

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.

Cited by 6Open access