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NASA NTRS1989

An analysis of the vertical structure equation for arbitrary thermal profiles

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.

Cohn, Stephen E.↗

NASA NTRS1993

Analyzing Martian winds and tracer concentrations using Mars Observer data

During the courses of a day, the Mars Observer spacecraft will acquire globally distributed profiles of the martian atmosphere. It is highly desirable that this data be assembled into synoptic weather maps (complete specifications of the atmospheric pressure, temperature, and winds at a given time), which can in turn be used as starting points in the study of many meteorological phenomena. Unfortunately, the special nature of the Mars Observer data presents several challenges above and beyond the usual difficult problem of data initialization. Mars Observer atmospheric data will consist almost exclusively of asynoptic vertical profiles of temperatures (or radiances) and pressures, whereas winds are generally in balance with horizontal gradients of these quantities (which will not be observed). It will therefore be necessary to resort to dynamical models to analyze the wind fields. As a rule, data assimilation into atmospheric models can result in the generation of spurious gravity waves, so special steps must be taken to suppress these. In addition, the asynoptic nature of the data will require a four-dimensional (space and time) data assimilation scheme. The problem is to find a full set of meteorological fields (winds and temperatures) such that, when marched forward in time in the model, they achieve a best fit (in the weighted least-squares sense) to the data. The proposed solution is to develop a model especially for the Mars Observer data assimilation problem. Gravity waves are filtered from the model by eliminating all divergence terms from the prognostic divergence equation. This leaves a diagnostic gradient wind relation between the rotational wind and the temperature field. The divergent wind is diagnosed as the wind required to maintain the gradient wind balance in the presence of the diabatic heating. The primitive equations of atmospheric dynamics (with three principal dependent variables) are thus reduced to a simpler system with a single prognostic equation for temperature - the variable that will be best observed. (This balance system was apparently first derived by Charney as a first-order Rossby number expansion of the equations of motion). Experience with a full primitive equation model of the martian atmosphere indicates that a further simplification is possible: at least for short-term integrations, the model can be linearized about the zonally symmetric basic state.

Houben, Howard C.↗

NASA NTRS1984

Research Relative to High Spatial Resolution Passive Microwave Sounding Systems

Methods to obtain high resolution passive microwave weather observations, and understanding of their probable impact on numerical weather prediction accuracy were investigated. The development of synthetic aperture concepts for geosynchronous passive microwave sounders were studied. The effects of clouds, precipitation, surface phenomena, and atmospheric thermal fine structure on a scale of several kilometers were examined. High resolution passive microwave sounders (e.g., AMSU) with an increased number of channels will produce initialization data for numerical weather prediction (NWP) models with both increased spatial resolution and coverage. The development of statistical models for error growth in high resolution primitive equation NWP models which permit the consequences of various observing system alternatives, including sensors and assimilation times and procedures is discussed. A high resolution three dimensional primitive equation NWP model to determine parameters in an error growth model similar to that formulated by Lorenz, but with more degrees of freedom is utilized.

Staelin, D. H.↗

NASA NTRS1988

Balanced and unbalanced upper-level frontogenesis

The impacts of vertical shear in the cross-front component of the geostrophic wind, and of its implied along-front temperature gradient, on frontogenesis at upper levels were investigated using a hierarchy of three dynamical models based on the Keyser and Pecnick (1985) primitive-equation model. The three models are (in order of decreasing sophistication) the anelastic (AN), the geostrophic momentum (GM), and the quasi-geostsrophic (QG) approximations to the primitive equations. Intercomparison of the broad-scale structure of the upper-level jet-fronts as described by the AN and GM models showed close agreement. In contrast, the QG model exhibited marked differences with its AN counterpart, which became more pronounced with increased geostrophic wind.

Reeder, Michael J.↗

NASA NTRS1986

Vertical structure of the wind field during the Special Observing Period I of the Global Weather Experiment

The vertical structure of the global atmosphere is analyzed for selected periods of the Special Observing Period I (SOP-I) for the Global Weather Experiment (GWE). The analysis consists of projection of the stream-function and velocity potential at 200 and 850 mb on spherical harmonics and of the wind and height fields on the normal modes of a linearized form of the primitive equations for a basic state at rest. The kinematic vertical structure is discussed in terms of correlation coefficients of the 200 mb and 850 mb winds and analysis of the internal and external normal modes of the primitive equations. The reliability of the results is checked by applying the same analysis methods to data sets obtained from three different institutions: Geophysical Fluid Dynamics Laboratory (GFDL), European Center for Medium Range Weather Forecasting (ECMWF), and Goddard Laboratory for the Atmospheres (GLA). It is found that, on a global basis, vertically reversing circulations are as important as the equivalent barotropic structures. For the verticaly reversing components, the gravity and mixed Rossby-gravity modes have contributions of the same order of magnitude as those of the Rossby modes in tropical latitudes.

Paegle, J. N.↗

NASA NTRS1983

A linear analysis of the transition curve for the baroclinic annulus

A primitive equation linear model is shown to be capable of quantitatively predicting the neutral curve for a baroclinic annulus. Flows unstable to small nonaxisymmetric perturbations are assumed to lie on the nonaxisymmetric side of a transition curve, while the stable flows are on the axisymmetric side. The stability in the presence of perturbations is tested with linearized hydrostatic primitive equations. Attention is given to the fastest growing eigenmodes, demonstrating that the linear and nonlinear wave have similar structures at a particular point in the nonaxisymmetric regime. All the waves are similar to the Eddy wave except at the extreme to the transition curve, where the waves maintain little structure with height. The eddy kinetic energy is baroclinic with a potential energy suppresed by the large static stability of the basic state. Eddy dissipation is noted to be significant near all boundaries.

Miller, T. L.↗

NASA NTRS2003

Collaborative Research: Atlantic Ocean Tropical/Subtropical Processes from Seasonal to Decadal Time Scales: Model/Data, Model/Model Comparison and Model/Data Synthesis Through Assimilation

The effort of this first year of research has been focused on the assimilation of TOPEX/Poseidon altimetric data into a primitive equation model of the Atlantic tropical/subtropical circulation. A reduced-rank, stationary Kalman filter has been constructed to assimilate the altimetric sea surface height anomaly (SHA) into the model. The goal is to assess how the inter-hemispheric transports between the Atlantic subtropics and tropics are affected by the assimilation and how the subsurface thermocline structure , and its variability ,is dynamically constrained by the SHA. The model is a reduced-gravity primitive equation GCM of the upper Atlantic Ocean between 30 S and 30 N. The assimilation scheme is an approximation to the extended Kalman filter in which the error covariances of the state estimates are calculated only in a reduced- dimension subspace. The subspace is defined by the leading empirical orthogonal functions calculated from an unconstrained model calculation. Both an identical twin experiment using simulated SHA observations and assimilation of the real TOPEX data were performed. Results from the twin experiments demonstrate the ability of the method to constrain the ocean circulation and the subsurface temperature structure. The impact on the subsurface temperature structure of TOPEX assimilation was assessed using data from expandable bathythermographs. This showed a substantial improvement in the estimated temperature variability only within 13 degrees in latitude around the equator. The impact of TOPEX SHA assimilation on zonally integrated meridional transport across different latitudes was also estimated. Again within 13 degrees from the equator both the mean amplitude and interannual variability of the surface and subsurface transports were significantly enhanced, while the transports were insensitive to the assimilation in the subtropics.

Malannotte-Rizzoli, Paola↗

NASA NTRS1998

EOS Interdisciplinary Investigation: Observational and Modeling Studies of Radiative, Chemical, and Dynamical Interactions in the Earth's Atmosphere

A 3-dimensional model of atmospheric dynamics and photochemistry has been developed from the primitive equations in isentropic coordinates. The model extends from the upper troposphere through the middle atmosphere and is driven by observed dynamical and chemical structure in the troposphere. The model's formulation is entirely spectral: Horizontal structure is represented in terms of vector Hough functions, which explicitly partition the motion into its rotational and divergent components. This formulation enables the model's computational performance to be increased dramatically by filtering high-frequency gravity waves, which do not affect PV conservation. Vertical structure is represented in terms of eigenfunctions that follow from the primitive equations in isentropic coordinates. The model's fully spectral formulation enables scale-selective dissipation, necessary for numerical stability, to be applied at 6th order-in all 3 coordinate directions. This feature leaves all but the shortest vertical scales undamped and, consequently, allows potential vorticity and chemical tracers to be conserved quite accurately, distinctly better than in the model's layered counterpart. These dynamical features are complemented by a basic but fairly complete treatment of gas phase photochemistry, which accounts for some 48 chemical species, diurnally-varying SW absorption by O2 and O3, and LW cooling calculated via a detailed band calculation of radiative transfer. Driven by observed tropospheric structure, the 3D model reproduces observed structure in the middle atmosphere, inclusive of transport by the Brewer-Dobson circulation and accompanying chemical variations. Calculated residual motion is consistent with diabatic cooling rates, poleward transport, and descent in the polar night derived from satellite measurements of chemical tracers made from LIMS, SAGE, and UARS. The model also reproduces the observed structure of chemical species like O3, HNO3, and ClO. At the same time, it provides a respectable 3D simulation of the wintertime increase of midlatitude total ozone, as well as its geographical structure observed by Nimbus-7 TOMS. The 3D model has been used to explore the origin of diabatic upwelling in the tropical lower stratosphere, which has been related to planetary wave absorption in the winter- time stratosphere, as well as to influences from tropospheric convection at the tropical tropopause.

Salby, Murry L.↗

NASA NTRS1993

Multi-grid domain decomposition approach for solution of Navier-Stokes equations in primitive variable form

The new multigrid (or adaptive) pseudospectral element method was carried out for the solution of incompressible flow in terms of primitive variable formulation. The desired features of the proposed method include the following: (1) the ability to treat complex geometry; (2) high resolution adapted in the interesting areas; (3) requires minimal working space; and (4) effective in a multiprocessing environment. The approach for flow problems, complex geometry or not, is to first divide the computational domain into a number of fine-grid and coarse-grid subdomains with the inter-overlapping area. Next, it is necessary to implement the Schwarz alternating procedure (SAP) to exchange the data among subdomains, where the coarse-grid correction is used to remove the high frequency error that occurs when the data interpolation from the fine-grid subdomain to the coarse-grid subdomain is conducted. The strategy behind the coarse-grid correction is to adopt the operator of the divergence of the velocity field, which intrinsically links the pressure equation, into this process. The solution of each subdomain can be efficiently solved by the direct (or iterative) eigenfunction expansion technique with the least storage requirement, i.e. O(N(exp 3)) in 3-D and O(N(exp 2)) in 2-D. Numerical results of both driven cavity and jet flow will be presented in the paper to account for the versatility of the proposed method.

Ku, Hwar-Ching↗

NASA NTRS1981

An explicit mixed numerical method for mesoscale model

A mixed numerical method has been developed for mesoscale models. The technique consists of a forward difference scheme for time tendency terms, an upstream scheme for advective terms, and a central scheme for the other terms in a physical system. It is shown that the mixed method is conditionally stable and highly accurate for approximating the system of either shallow-water equations in one dimension or primitive equations in three dimensions. Since the technique is explicit and two time level, it conserves computer and programming resources.

Hsu, H.-M.↗

NASA NTRS1985

Marching iterative methods for the parabolized and thin layer Navier-Stokes equations

Downstream marching iterative schemes for the solution of the Parabolized or Thin Layer (PNS or TL) Navier-Stokes equations are described. Modifications of the primitive equation global relaxation sweep procedure result in efficient second-order marching schemes. These schemes take full account of the reduced order of the approximate equations as they behave like the SLOR for a single elliptic equation. The improved smoothing properties permit the introduction of Multi-Grid acceleration. The proposed algorithm is essentially Reynolds number independent and therefore can be applied to the solution of the subsonic Euler equations. The convergence rates are similar to those obtained by the Multi-Grid solution of a single elliptic equation; the storage is also comparable as only the pressure has to be stored on all levels. Extensions to three-dimensional and compressible subsonic flows are discussed. Numerical results are presented.

Israeli, M.↗

NASA NTRS1985

Marching iterative methods for the Parabolized and Thin Layer Navier-Stokes equations

Israeli, M.↗

NASA NTRS1985

A numerical study of global axisymmetric states and their stability to quasi-geostrophic disturbances

A steady, axisymmetric model of the general circulation is developed as a basis for climate stability studies. The model includes the effects of heating, rotation, and internal friction, but neglects topography. It is assumed that the axisymmetric flow may be modeled by making the Boussinesq and deep convection approximations. The hydrostatic assumption is not made, thus permitting the advective terms to be included in the vertical equation of motion. The initial set of five primitive equations is reduced to three equations in terms of the zonal velocity, meridional streamfunction, and the potential temperature perturbation. The application of the Boussinesq, deep convection, and quasi-geostrophic assumptions limits the ranges of the heating and rotation rates. For values not too far from typical atmospheric values, the model produces a stability boundary separating Hadley from Rossby flow. The boundary is characterized by a particular value of vertical wind shear, which suggests that baroclinic instability is the primary mechanism for the loss of stability. The initial growth rates are largest for longitudinal waves 4-7, also in agreement with studies of baroclinic instability.

Henderson, H. W.↗

NASA NTRS1977

A balanced diagnostic system compatible with a barotropic prognostic model

A system of diagnostic equations for the velocity field, or wind laws, for a barotropic primitive-equation model of large-scale atmospheric flow is derived. Attention is given to the classical balance equation and its ellipticity condition. Numerical solutions of the diagnostic system are presented, including examples of cases of the mixed elliptic-hyperbolic type and cases with non-zero divergence. Procedures for implementing such a system are outlined, along with a review of factors in using the technique for operational numerical weather prediction.

Ghil, M.↗

NASA NTRS1992

Quasigeostrophic vertical motions diagnosed from along- and cross-isentrope components of the Q vector

In a recent paper on the kinematics of frontogenesis, Keyser et al. (1988) conjectured that partitioning the Q vector into along- and cross-isentrope components yields vertical-motion patterns that are respectively cellular and banded: the former on the scale of the baroclinic disturbance, and the latter on the scale of the embedded frontal zones. This conjecture is examined diagnostically through solution of the quasi-geostrophic omega equation, using the output from a nearly adiabatic and frictionless f-plane primitive equation channel model of the evolution of a baroclinic disturbance to finite amplitude. The results of the present study support the proposed conjecture, suggesting the following interpretation of the characteristic comma structure of the vertical-motion field in midlatitude baroclinic disturbances: the dipole is associated with the along-isentrope component of the Q vector, reflecting the wavelike pattern in the potential temperature field within the baroclinic disturbance; the asymmetries are associated with the cross-isentrope component of the Q vector, reflecting the presence of frontal zones within the baroclinic disturbance.

Keyser, Daniel↗

NASA NTRS1979

Effect of subjectively enhancing the initial vorticity field in a short-range numerical forecast

The paper tests a simple method to incorporate subsynoptic-scale information into the initial conditions of a forecast model. The amplitude, orientation and vertical structure of the vorticity field associated with the subsynoptic-scale disturbance are estimated subjectively; this is done using a combination of climatological and physical models together with all available data. The initial values of geopotential are then obtained in the vicinity of the subsynoptic-scale disturbance, as over the rest of the domain, through the balance equation. Two sets of initial conditions are tested in 12-hr forecasts using a six-layer version of a primitive equation forecast model (Anthes and Warner, 1978). The analysis schemes consisted of (1) a careful subjective analysis of radiosonde data with a high-resolution objective interpolation, and (2) the same preceding analysis but including data derived from the enhanced vorticity fields in the vicinity of the subsynoptic-scale disturbance. The results are discussed in some detail.

Fried, G. D.↗

NASA NTRS1992

Quasigeostrophic diagnosis of three-dimensional ageostrophic circulations in an idealized baroclinic disturbance

Keyser et al.'s (1989) kinematic technique for the representation of 3D vertical circulations in baroclinic disturbances in terms of a vector eigenfunction, the 'psi vector', is presently projected onto the framework of quasi-geostrophic (QG) theory. The projection of the psi-vector equation onto the cross-front vertical plane leads to a generalization of the QG form of the Sawyer-Eliassen equation that is applicable to 3D flows. The diagnostic methodologies for the total ageostrophic flow and for the generalized Sawyer-Eliassen equation are illustrated by applications to upper-level and surface frontal zones that are simulated in an f-plane primitive equation channel model of a finite-amplitude baroclinic wave.

Keyser, Daniel↗

NASA NTRS1991

A Variational Formalism for the Radiative Transfer Equation and a Geostrophic, Hydrostatic Atmosphere: Prelude to Model 3

The second step in development of MODEL III is summarized. It combines the four radiative transfer equations of the first step with the equations for a geostrophic and hydrostatic atmosphere. This step is intended to bring radiance into a three dimensional balance with wind, height, and temperature. The use of the geostrophic approximation in place of the full set of primitive equations allows for an easier evaluation of how the inclusion of the radiative transfer equation increases the complexity of the variational equations. Seven different variational formulations were developed for geostrophic, hydrostatic, and radiative transfer equations. The first derivation was too complex to yield solutions that were physically meaningful. For the remaining six derivations, the variational method gave the same physical interpretation (the observed brightness temperatures could provide no meaningful input to a geostrophic, hydrostatic balance) at least through the problem solving methodology used in these studies. The variational method is presented and the Euler-Lagrange equations rederived for the geostrophic, hydrostatic, and radiative transfer equations.

Achtemeier, Gary L.↗