Integration of a semi‐implicit model with time‐dependent boundary conditions

A semi‐implicit barotropic primitive equations model is integrated over a limited area with time dependent boundary conditions using the standard mesh and a finer mesh. Following a theorem by Charney, a minimum number of variables are specified as boundary conditions for the limited area integrations in order to avoid mathematical over‐specification. The comparison of coarse mesh limited area forecasts with the corresponding forecasts made over a much larger domain demonstrates that the essential features, namely the Rossby type perturbations, are handled almost perfectly. The fine mesh forecasts over the same limited area are also very good. Finally, the effect of specifying inaccurate boundary conditions, in the form of twelve‐hour forecasts, is briefly illustrated.     

Errors near the poles generated by a semi‐Lagrangian integration scheme in a global spectral model

Abstract

When a semi‐Lagrangian integration scheme was introduced into a global spectral model by Ritchie (1988), large errors developed in the neighbourhood of the poles. It took approximately 6 months of effort to diagnose the problem and find an appropriate correction. The method that was used to diagnose the source of error was quite tedious, but it was successful. Furthermore, it seems that this method could be used to diagnose other sources of error that occasionally show up in numerical integrations. For this reason, it was felt that this method should be described in a separate article. This is the main objective of the presentation that follows.

An integration is carried out with the original version of the model and some results are presented in order to illustrate these errors. In order to identify their exact cause, the model is stripped down in two steps. At each step, some checks are made to ensure that the errors are stillpresent in the degraded version of the model. In the end, the remaining equations are sufficiently simple to ensure that the cause of the errors becomes obvious. The diagnosis immediately suggests some alternative computational designs. A modification that completely eliminates these errors is then proposed and tested. An integration with the modified spectral model is carried out and results are presented to show that the errors have disappeared.     

Sea‐ice dynamics and CO2 sensitivity in a global climate model

Abstract

Present‐day results and CO₂ sensitivity are described for two versions of a global climate model (genesis) with and without sea‐ice dynamics. Sea‐ice dynamics is modelled using the cavitating‐fluid method of Flato and Hibler (1990, 1992). The atmospheric general circulation model originated from the NCAR Community Climate Model version 1, but is heavily modified to include new treatments of clouds, penetrative convection, planetary boundary‐layer mixing, solar radiation, the diurnal cycle and the semi‐Lagrangian transport of water vapour. The surface models include an explicit model of vegetation (similar to BATS and SiB), multilayer models of soil, snow and sea ice, and a slab ocean mixed layer.

When sea‐ice dynamics is turned off, the CO₂‐induced warming increases drastically around ～60–80°S in winter and spring. This is due to the much greater (and unrealistic) compactness of the Antarctic ice cover without dynamics, which is reduced considerably when CO₂ is doubled and exposes more open ocean to the atmosphere. With dynamics, the winter ice is already quite dispersed for 1 × CO₂ so that its compactness does not decrease as much when CO₂ is doubled.     

Onset of convection in a rotating semi‐infinite fluid: Theory and experiment

Abstract

We look at the development of the first plumes that emerge from a convectively unstable boundary layer by modelling the process as the instability of a fluid with a time‐dependent mean density field. The fluid is semi‐infinite, rotating, dissipative ‐ characterized by the ratio of its viscosity to thermal diffusivity (Prandtl number Pr = ν/κ) ‐ and initially homogeneous. A constant destabilizing heat flux is applied at the boundary and the stability of the evolving density field is investigated both mathematically and in laboratory experiments.

Using a “natural convective” scaling, we show that the behaviour of the non‐dimensional governing equations depends on Pr and the parameter γ = f(ν/B)^1/2, where f is the Coriolis parameter, and B is the applied buoyancy flux. For the ocean, γ ≈ 0.1, whilst for the atmosphere γ ≈ 0.01. In the absence of rotation, the behaviour of the differential equations is independent of B, depending only on Pr. The boundary‐layer Rayleigh number (Ra_bl) is also independent of B. We show that Ra_bl, evaluated at the onset of rapid vertical motion, depends on the form of the perturbation.

Due to the time‐dependence of the mean density field, analytic instability analysis is difficult, so we use a numerical technique. The governing equations are transformed to a stretched vertical coordinate and their stability investigated for a particular form of perturbation function. The model predictions are, for the ocean: instability time ～2–4 h, density difference ～0.002–0.013 kg m^‐3, boundary‐layer thickness ～50–75 m and horizontal scale ～200–300 m; and for the atmosphere: instability time ～10 min, temperature difference ～2.0–3.0°C, boundary‐layer thickness ～400–500 m and horizontal scale ～1.5–2.0 km.

Laboratory experiments are performed to compare with the numerical predictions. The time development of the mean field closely matches the assumed analytic form. Furthermore, the model predictions of the instability timescale agree well with the laboratory measurements. This supports the other predictions of the model, such as the lengthscales and buoyancy anomaly.     

Non‐linear coupling between modes in a low‐dimensional model of ENSO

Abstract

An intermediate coupled model of the tropical Pacific ocean‐atmosphere system was reduced by projecting the non‐linear model onto a truncated basis set of its own empirical orthogonal functions (EOFs). For moderate coupling strengths, the simulated El Niño/Southern Oscillation (ENSO) variability consists of a dominant quasi‐quadrennial mode with a period of approximately four years and a smaller quasi‐biennial mode at a period of approximately two years. In the absence of a seasonal cycle, the leading two EOFs capture the dynamics of the leading interannual mode, with a further two EOFs being required to capture the secondary oscillation. The presence of seasonal forcing increases the EOF requirement by two, the leading pair of EOFs being dominated by the annual cycle. Normal mode analysis of the reduced models indicates that the quasi‐biennial mode manifests itself, even though it is linearly stable, by non‐linear coupling to the quasi‐quadrennial mode. The nonlinearity does not produce the quasi‐biennial signal unless the spatial degrees of freedom associated with the linear quasi‐biennial mode are present. Other linearly stable modes also couple non‐linearly to the leading interannual mode and to the seasonal cycle, but the quasi‐biennial mode is favoured over other, less‐damped linear modes because of its proximity to a multiple of the quasi‐quadrennial frequency.     

Grid‐size dependency of the meridional heat transport in a numerical model of an ocean

Abstract

The meridional heat transport across a latitude circle in a model ocean is calculated by using a general circulation model with a coarse grid, a medium grid and a fine grid capable of resolving the mesoscale eddies in order to show to what extent this transport depends on grid size. Although the grid size strikingly affects the current velocities, it has almost no effect upon the meridional heat transport.     

Calibration of a three‐component angular distribution model of sky radiance

Abstract

A three‐component (isotropic, circumsolar and horizon‐brightening) model of the angular distribution of sky (short‐wave) radiance has been tested and validated against a data base of measured sky radiance. The data base encompasses cloud cover 0.0 to 1.0 and solar zenith angles 30 to 80°. Empirical constants have been derived for the model enabling the prediction of sky radiances for any sky condition.     

The steady‐state structure of the natural stratosphere and ozone distribution in a 2‐D model incorporating radiation and O‐H‐N photochemistry and the effects of stratospheric pollutants

A mean meridional circulation model of the stratosphere, incorporating radiative heating and photochemistry of the oxygen‐hydrogen‐nitrogen atmosphere, is used to simulate the meridional distributions of O₃, HO_X, N₂O,NO_X, temperature and the three components of mean motion for the summer and winter seasons under steady‐state conditions. The results are generally in good agreement with the available observations in the normal stratosphere. The model has been applied to assess the effects of water vapour and nitrogen oxide perturbations resulting from aircraft emissions in the stratosphere. It is found that a fleet of 500 Boeing‐type sst's, flying at 20 km and 45°N in the summer hemisphere and inserting NO_x at a rate of 1.8 megatons per year, has the effect of reducing the global total ozone by 14.7%. Similar calculations for 342 Concorde/TU‐114's, cruising at 17 km and injecting NO_x at a rate of 0.35 megatons per year, show a global‐average total‐ozone reduction of 1.85%. Although water vapour is considered important, because of its ability to convert NO₂ into HNO₃, the direct effect on global‐average total‐ozone reduction resulting from the 100% increase in the stratospheric water content is less than 1%. The changes in the chemical structure (HO^NO^), temperature, and mean motions associated with the ozone reduction are also investigated in the case of the 1.8‐megaton‐per‐year NO_X perturbation. It is shown that the reduced meridional temperature gradient in the middle and upper stratosphere resulting from the NO_x perturbation leads to the weakening of the tropical easterly jet in the summer hemisphere and mid‐latitude westerlies in the winter season.

The sensitivity of the model solutions to an alternate choice of input parameters (diffusion coefficients and solar photodissociation data) is tested and the main deficiency of the model is pointed out.     

Some properties and comparative performance of the semi‐Lagrangian method of Robert in the solution of the advection‐diffusion equation

Abstract

A numerical method for solving the advection‐diffusion equation, based on the semi‐Lagrangian algorithm of Robert (1981, 1982) is described, analysed and evaluated in comparison with other methods through a series of test problems. It is found that this method is generally better than other semi‐Lagrangian methods, and is a viable alternative to existing methods for LRTAP and other meteorological modelling because of its flexibility in application, its computational stability and its accuracy.     

Application of a coupled ice‐ocean model to the Labrador Sea

Abstract

The steady, coupled ice‐ocean circulation model of Willmott and Mysak (1989) for a meridional channel is applied to the Labrador Sea for the winter season. The model consists of a thermodynamic reduced‐gravity ocean combined with a variable thickness ice cover that is in thermal equilibrium. Upon specifying the forcing fields of surface air temperature, wind stress and water temperature along the open southern boundary, the winter climatological ice‐edge position, ice thickness, ocean circulation and temperature fields are determined in the channel domain. The sensitivity of the results to the various model parameters is examined. In particular, the optimum heat exchange coefficients for the interfaces of air‐water, ice‐water and air‐ice are found.

The model ice‐edge position compares favourably with the 50% winter climatological ice concentration isoline obtained from an analysis of 32 years (1953–84) of sea‐ice concentration data. The simulations of the ocean temperature and ice thickness are also quite realistic according to the observed records available. The model is also applied to two specific winters (1981 and 1983) during which anomalous sea‐ice and weather conditions prevailed in the Labrador Sea.     

The climatology of the Canadian Climate Centre general circulation model as obtained from a five‐year simulation

Abstract

The climatology of the Canadian Climate Centre atmospheric general circulation model is presented and compared with the observed climatology of the atmosphere. The model climatology is obtained from a simulation over five annual cycles and the results are presented in terms of averages for the four seasons.

The climatology of the model is discussed in terms of zonally and time averaged values of the primary atmospheric variables as well as in terms of the spatial distributions of the important surface parameters and of the rotational and divergent components of the tropospheric flow. Some measure of model variability is also presented.

The model is generally quite successful in reproducing the mean observed climatology of the atmosphere.     

On the influence of stratospheric conditions on forced tropospheric waves in a steady‐state primitive equation model

Abstract

The structure of a forced planetary wave is computed by means of a linearized steady‐state primitive equation model on a hemisphere. The vertical velocity in pressure coordinates is specified at the lower boundary to simulate orographie forcing. The vertical finite differences are on equally spaced pressure levels with a moderately high vertical resolution. The upper boundary condition dp/dt = 0 is applied at p =0 in the model. Numerical experiments show that the tropospheric structure of forced planetary waves is sensitive to the stratospheric background conditions in the model.     

The application of non‐linear normal mode initialization to an operational forecast model

Abstract

The non‐linear normal mode initialization technique used in shallow water equation models by Baer (1977) and Machenhauer (1977) is now applied to a full baroclinic primitive equations forecast model. The initialization procedure is shown to be capable of completely removing high frequency oscillations from model integrations, even in the presence of topography. The procedure also produces a consistent and physically realistic initial vertical motion field.     

Surface‐absorbed and top‐of‐atmosphere radiation fluxes for the Mackenzie River basin from satellite observations and a regional climate model and an evaluation of the model

Abstract

Both the earth‐reflected shortwave and outgoing longwave radiation (OLR) fluxes at the top of the atmosphere (TOA) as well as surface‐absorbed solar fluxes from Canadian Regional Climate Model (CRCM) simulations of the Mackenzie River Basin for the period March 2000 to September 2003 are compared with the radiation fluxes deduced from satellite observations. The differences between the model and satellite solar fluxes at the TOA and at the surface, which are used in this paper to evaluate the CRCM performance, have opposite biases under clear skies and overcast conditions, suggesting that the surface albedo is underestimated while cloud albedo is overestimated. The slightly larger differences between the model and satellite fluxes at the surface compared to those at the TOA indicate the existence of a small positive atmospheric absorption bias in the model. The persistent overestimation of TOA reflected solar fluxes and underestimation of the surface‐absorbed solar fluxes by the CRCM under all sky conditions are consistent with the overestimation of cloud fraction by the CRCM. This results in a larger shortwave cloud radiative forcing (CRF) both at the TOA and at the surface in the CRCM simulation. The OLR from the CRCM agrees well with the satellite observations except for persistent negative biases during the winter months under all sky conditions. Under clear skies, the OLR is slightly underestimated by the CRCM during the winter months and overestimated in the other months. Under overcast conditions the OLR is underestimated by the CRCM, suggesting an underestimation of cloud‐top temperature by the CRCM. There is an improvement in differences between model and satellite fluxes compared to previously reported results largely because of changes to the treatment of the surface in the model.     

Graupel growth and trajectories in a shallow cb cloud determined by a forced 1‐D model

Abstract

A kinematic model is applied to graupel growth. The vertical velocity and ther‐modynamic field data are taken from the forced 1‐D time‐dependent model of Cb cloud developed by Curie and Jane (1988). The graupel embryo pocket was released at the height of the — 10°C isotherm. The influence of the forced lifting on further graupel growth and its trajectory is analysed by sensitivity experiments based on the amplitude of the forced lifting, and initial graupel radius, density and cloud droplet concentration for the forced lifting initiation time derived from the model and the forced lifting duration time that agreed with observations. In particular, the sensitivity analysis was carried out for the forced lifting initiation and duration times.

It is shown that for large values of the forced lifting amplitudes, the residence time of the graupel within cloud and the final graupel radius may be significantly larger compared with those in the non‐forced case. The residence time in a cloud can also be significantly larger for the smallest amplitude, whereas the final radius is rather insensitive owing to oscillations around the melting level. For some cases the forced lifting causes recycling inside the updraft, contrary to the results of previous non‐forced numerical models. The recycling mechanism is sensitive to the forced lifting duration time and the time interval between the graupel pocket injection in cloud and the initiation of the forced lifting. Initially the observed recycling mechanism is a consequence of the periodic forced lifting mechanism, but then combines with recycling of the Pflaum type (1980).     

Optimal estimation of Eddy viscosity and friction coefficients for a Quasi‐three‐dimensional numerical tidal model

Abstract

It is shown that the parameters in a quasi‐three‐dimensional numerical tidal model can be estimated accurately by assimilation of data from current meters and tide gauges. The tidal model considered is a semi‐linearized one in which advective nonlinearities are neglected but nonlinear bottom friction is included. The parameters estimated are the eddy viscosity, bottom friction coefficient, water depth and wind drag coefficient, the first three of which are allowed to be position‐dependent. The adjoint method is used to construct the gradient of a cost function defined as a certain norm of the difference between computed and observed current and surface elevations. On the basis of a number of tests, it is shown that very effective estimation of the nodal values of the parameters can be achieved using the current data either alone or in combination with elevation data. When random errors are introduced into the data, the estimated parameters are quite sensitive to the magnitude of the errors, and in particular the eddy viscosity is unstably sensitive. The sensitivity of the viscosity can be stabilized by incorporating an appropriate penalty term in the cost function or alternatively by reducing the number of estimated viscosity values via a finite element approximation. Once stabilized, the sensitivity of the estimates to data errors is significantly reduced by assimilating a longer data record.     

The energy of near‐surface internal waves in the strait of Georgia

An estimate of the energy content of near‐surface internal waves in the Strait of Georgia is obtained from a combination of aerial photographs and in‐situ measurements. The role of these waves in the tidal energy budget and in the mixing processes in the Strait is discussed.     

Short‐term forecasting with a multi‐level spectral primitive equation model part I ‐ model formulation

A global baroclinic primitive equation model using the spectral technique has been constructed for short‐ and medium‐range numerical weather prediction. The spectral technique, which is a special case of the Galerkin method, employs spherical harmonic basis functions in the evaluation of all horizontal derivatives. The use of a transform technique allows all the horizontal operations to be performed efficiently and allows physical processes to be evaluated in real space. The model employs a semi‐implicit algorithm for time integration and finite differencing in the vertical. Physical processes include orography, moist convection, large scale precipitation and boundary layer processes.     

A decision‐analytic study of the joint value of seasonal precipitation and temperature forecasts in a choice‐of‐crop problem

Abstract

A static decision‐analytic method is used to investigate the economic value of bivariate ‐ precipitation and temperature ‐ seasonal forecasts of the form currently issued by the U.S. National Weather Service. This method is applied to a corn versus spring wheat choice‐of‐crop decision‐making problem by considering a transect of four counties across the northwestern margin of the North American corn belt. Numerical results indicate that seasonal forecasts of current quality can be of appreciable value (≥$1/ha) for some locations when the optimal action chosen on the basis of climatological information is only marginally preferred to another action. Increases in forecast value follow from hypothetical increases in the quality of both the precipitation and temperature components of the forecasts in the spring wheat region, whereas forecast value increases primarily as a function of the quality of the precipitation forecasts alone in the corn belt region. The results are very sensitive to absolute and relative crop prices.