Equations Of Atmospheric Dynamics Research Articles

Rigorous justification of the hydrostatic approximation limit of viscous compressible flows

This paper considers the asymptotic limit of small aspect ratio between vertical and horizontal spatial scales for viscous isothermal compressible flows. In particular, it is observed that fast vertical acoustic waves arise and induce an averaging mechanism of the density in the vertical variable, which at the limit leads to the hydrostatic approximation of compressible flows, i.e., the compressible primitive equations of atmospheric dynamics. We justify the hydrostatic approximation for general as well as “well-prepared” initial data. The initial data is called well-prepared when it is close to the hydrostatic balance in a strong topology. Moreover, the convergence rate is calculated in the well-prepared initial data case in terms of the aspect ratio, as the latter goes to zero.

Global Existence of Weak Solutions to 3D Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosity

In this work, we show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosity density-dependent for large initial data. With a pressure law of the form ρ2, we represent the vertical velocity as a function of the density and the horizontal one which will be important in using Faedo-Galerkin method to obtain the global existence of the approximate solutions. In analogy with the cases in [28] and [26, 27], we prove that the weak solutions satisfy the basic energy inequality and the Bresch-Desjardins entropy inequality. Based on these estimates and using compactness arguments, we prove the global existence of weak solutions of (1.1) by vanishing the parameters in our approximate system step by step.

Local well-posedness of strong solution to a climate dynamic model with phase transformation of water vapor

The primitive three-dimensional viscous equations for atmospheric dynamics with the phase transformation of water vapor are studied. According to the actual physical process, we give the heating rate, mass of water, and precipitation rate, which are related to temperature and pressure. In fact, this system strictly obeys the conservation of energy and is used to make better climate predictions. Providing H2 initial data and boundary conditions with physical significance, we prove the local well-posedness of a unique strong solution to the moist atmospheric equations by the contractive mapping principle and the energy method in the H2 framework.

Local Well-Posedness of Strong Solutions to the Three-Dimensional Compressible Primitive Equations

This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity.

100 Years of Progress on Mountain Meteorology Research

ABSTRACTMountains significantly influence weather and climate on Earth, including disturbed surface winds; altered distribution of precipitation; gravity waves reaching the upper atmosphere; and modified global patterns of storms, fronts, jet streams, and climate. All of these impacts arise because Earth’s mountains penetrate deeply into the atmosphere. This penetration can be quantified by comparing mountain heights to several atmospheric reference heights such as density scale height, water vapor scale height, airflow blocking height, and the height of natural atmospheric layers. The geometry of Earth’s terrain can be analyzed quantitatively using statistical, matrix, and spectral methods. In this review, we summarize how our understanding of orographic effects has progressed over 100 years using the equations for atmospheric dynamics and thermodynamics, numerical modeling, and many clever in situ and remote sensing methods. We explore how mountains disturb the surface winds on our planet, including mountaintop winds, severe downslope winds, barrier jets, gap jets, wakes, thermally generated winds, and cold pools. We consider the variety of physical mechanisms by which mountains modify precipitation patterns in different climate zones. We discuss the vertical propagation of mountain waves through the troposphere into the stratosphere, mesosphere, and thermosphere. Finally, we look at how mountains distort the global-scale westerly winds that circle the poles and how varying ice sheets and mountain uplift and erosion over geologic time may have contributed to climate change.

Global Existence of Weak Solutions to the Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosities

We show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosities. In analogy with the case of the compressible Navier-Stokes equations, the weak solutions satisfy the basic energy inequality, the Bresh-Desjardins entropy inequality and the Mellet-Vasseur estimate. These estimates play an important role in establishing the compactness of the vertical velocity of the approximating solutions, and therefore are essential to recover the vertical velocity in the weak solutions.

Three-pattern decomposition of global atmospheric circulation: part II—dynamical equations of horizontal, meridional and zonal circulations

The three-pattern decomposition of global atmospheric circulation (TPDGAC) partitions three-dimensional (3D) atmospheric circulation into horizontal, meridional and zonal components to study the 3D structures of global atmospheric circulation. This paper incorporates the three-pattern decomposition model (TPDM) into primitive equations of atmospheric dynamics and establishes a new set of dynamical equations of the horizontal, meridional and zonal circulations in which the operator properties are studied and energy conservation laws are preserved, as in the primitive equations. The physical significance of the newly established equations is demonstrated. Our findings reveal that the new equations are essentially the 3D vorticity equations of atmosphere and that the time evolution rules of the horizontal, meridional and zonal circulations can be described from the perspective of 3D vorticity evolution. The new set of dynamical equations includes decomposed expressions that can be used to explore the source terms of large-scale atmospheric circulation variations. A simplified model is presented to demonstrate the potential applications of the new equations for studying the dynamics of the Rossby, Hadley and Walker circulations. The model shows that the horizontal air temperature anomaly gradient (ATAG) induces changes in meridional and zonal circulations and promotes the baroclinic evolution of the horizontal circulation. The simplified model also indicates that the absolute vorticity of the horizontal circulation is not conserved, and its changes can be described by changes in the vertical vorticities of the meridional and zonal circulations. Moreover, the thermodynamic equation shows that the induced meridional and zonal circulations and advection transport by the horizontal circulation in turn cause a redistribution of the air temperature. The simplified model reveals the fundamental rules between the evolution of the air temperature and the horizontal, meridional and zonal components of global atmospheric circulation.

Numerical weather prediction revisions using the locally trained differential polynomial network

Meso-scale forecasts result from global numerical weather prediction models, which are based upon the differential equations for atmospheric dynamics that do not perfectly determine weather conditions near the ground. Statistical corrections can combine complex numerical models, based on the physics of the atmosphere to forecast the large-scale weather patterns, and regression in post-processing to clarify surface weather details according to local observations and climatological conditions. Neural networks trained with local relevant weather observations of fluctuant data relations in current conditions, entered by numerical model outcomes of the same data types, may revise its one target short-term prognosis (e.g. relative humidity or temperature) to stand for these methods. Polynomial neural networks can compose general partial differential equations, which allow model more complicated real system functions from discrete time-series observations than using standard soft-computing methods. This new neural network technique generates convergent series of substitution relative derivative terms, which combination sum can define and solve an unknown general partial differential equation, able to describe dynamic processes of the weather system in a local area, analogous to the differential equation systems of numerical models. The trained network model revises hourly-series of numerical prognosis of one target variable in sequence, applying the general differential equation solution of the correction multi-variable function to corresponding output variables of the 24-hour numerical forecast. The experimental results proved this polynomial network type can successfully revise some numerical weather prognoses after this manner.

On ellipticity of balance equations for atmospheric dynamics

Initial data for atmospheric multi-scale models need to be adjusted in order to ensure small amplitudes of high-frequency oscillations. Different adjustment methods lead to balance conditions in the form of time-independent partial differential systems with appropriate boundary conditions. One of the issues of such systems is a violation of the ellipticity conditions in a part of the problem domain. In this study we present the ellipticity conditions for balance equations based on diagnostic divergence relation with different levels of complexity and explore the existence of non-elliptic regions in the gridded fields of the atmospheric analysis data. It is shown that more physically justifiable balance equations are associated with much sparser and less intensive non-elliptic regions. The obtained results confirm Kasahara’s assumption that ellipticity conditions are violated in the actual atmospheric fields essentially due to approximations made under deriving balance equations.

Similarity parameters and a centrifugal convective instability of horizontally inhomogeneous circulations of the Hadley type

The stability of the zonal axisymmetric quasi-geostrophic hydrostatic solution to the equations of atmospheric dynamics that is determined by the horizontal temperature gradient is studied. Time-dependent regions of unstable solutions specified by the Rayleigh number describe ordinary convective (baroclinic) processes and the long-term weak growth of disturbances under the action of the centrifugal forces arising from the Earth’s rotation. Comparison with a centrifugal hydrodynamic instability is made. The spatiotemporal structure of the corresponding geophysical fields is described.

A Characterization of Tropical Transient Activity in the CAM3 Atmospheric Hydrologic Cycle

Abstract The Community Atmosphere Model version 3 (CAM3) is the latest generation of a long lineage of general circulation models produced by a collaboration between the National Center for Atmospheric Research (NCAR) and the scientific research community. Many aspects of the hydrological cycle have been changed relative to earlier versions of the model. It is the goal of this paper to document some aspects of the tropical variability of clouds and the hydrologic cycle in CAM3 on time scales shorter than 30 days and to discuss the differences compared to the observed atmosphere and earlier model versions, with a focus on cloud-top brightness temperature, precipitation, and cloud liquid water path. The transient behavior of the model in response to changes in resolution to various numerical methods used to solve the equations for atmospheric dynamics and transport and to the underlying lower boundary condition of sea surface temperature and surface fluxes has been explored. The ratio of stratiform to convective rainfall is much too low in CAM3, compared to observational estimates. It is much higher in CAM3 (10%) than the Community Climate Model version 3 (CCM3; order 1%–2%) but is still a factor of 4–5 too low compared to observational estimates. Some aspects of the model transients are sensitive to resolution. Higher-resolution versions of CAM3 show too much variability (both in amplitude and spatial extent) in brightness temperature on time scales of 2–10 days compared to observational estimates. Precipitation variance is underestimated on time scales from a few hours to 10 days, compared to observations over ocean, although again the biases are reduced compared to previous generations of the model. The diurnal cycle over tropical landmasses is somewhat too large, and there is not enough precipitation during evening hours. The model tends to produce maxima in precipitation and liquid water path that are a few hours earlier than that seen in the observations over both oceans and land.

Least action principle for a general, non‐hydrostatic, compressible, acoustically non‐filtered pressure‐coordinate model

AbstractThe least action principle is described for general (non‐hydrostatic, compressible, acoustically non‐filtered) pressure‐coordinate equations of atmospheric dynamics (Rõõm 1989), which represents a generalization of the Least Action Principle, first developed for the pressure‐coordinate representation of atmospheric dynamics by Salmon and Smith (1994).

A semi‐implicit scheme for the barotropic primitive equations of atmospheric dynamics

The primitive equations of a barotropic atmosphere in isobaric co‐ordinates are reformulated, in order to express the geopotential gradient as a function of the pressure at the Earth’s surface. Furthermore, the free surface equation is written in conservative form. A finite difference, semi‐implicit, semi‐Lagrangian scheme in isobaric co‐ordinates is developed. The numerical scheme is mass conservative, is proven to be stable and requires the solution of a single five‐diagonal system. Numerical simulations show that the model captures the main dynamical features of large scale atmospheric motion.

New formulations of the primitive equations of atmosphere and applications

The primitive equations are the fundamental equations of atmospheric dynamics. With the purpose of understanding the mechanism of long-term weather prediction and climate changes, the authors study as a first step towards this long-range project what is widely considered as the basic equations of atmospheric dynamics in meteorology, namely the primitive equations of the atmosphere. The mathematical formulation and attractors of the primitive equations, with or without vertical viscosity, are studied. First of all, by integrating the diagnostic equations they present a mathematical setting, and obtain the existence and time analyticity of solutions to the equations. They then establish some physically relevant estimates for the Hausdorff and fractal dimensions of the attractors of the problems.

A factored implicit scheme for numerical weather prediction

AbstractThe nonlinear partial differential equations of atmospheric dynamics govern motion on two time scales, a fast one and a slow one. Only the slow‐scale motions are relevant in predicting the evolution of large weather patterns. Implicit numerical methods are therefore attractive for weather prediction, since they permit a large time step chosen to resolve only the slow motions.To develop an implicit method which is efficient for problems in more than one spatial dimension, one must approximate the problem by smaller, usually one‐dimensional problems. A popular way to do so is to approximately factor the multidimensional implicit operator into one‐dimensional operators. The factorization error incurred in such methods, however, is often unacceptably large for problems with multiple time scales.We propose a new factorization method for numerical weather prediction which is based on factoring separately the fast and slow parts of the implicit operator. We show analytically that the new method has small factorization error, which is comparable to other discretization errors of the overall scheme. The analysis is based on properties of the shallow water equations, a simple two‐dimensional version of the fully three‐dimensional equations of atmospheric dynamics.

Existence and uniqueness of strong solutions in equations of global atmospheric convection

Much effort is currently devoted to the numerical solution of systems of equations of atmospheric dynamics for the puposes of both weather forecasting and theoretical studies in meteorology...