Hydrostatic Primitive Equation Model Research Articles

Elimination of spectral blocking by ensuring rotation‐free property of discretized pressure gradient within a spectral semi‐implicit semi‐Lagrangian global atmospheric model

AbstractThe widely‐adopted discretization of the horizontal pressure gradient term formulated by Simmons and Burridge for atmospheric models on σ‐p hybrid vertical coordinate is found to incur spectral blocking for rotational wind components at high vertical levels when used in a spectral semi‐Lagrangian model run on a linear grid. A remedy to this issue is proposed and tested using a spectral semi‐implicit semi‐Lagrangian hydrostatic primitive equations model. The proposed method removes aliasing errors at high wave‐numbers by ensuring that the rotation‐free property of the pressure gradient term on the isobaric surface, a feature possessed by the continuous system, is preserved in the discretized system, which highlights the significance of mimetic discretization within the context of numerical weather prediction models.

Energy‐conserving finite‐difference schemes for quasi‐hydrostatic equations

The pressure‐based hydrostatic primitive equations model LMD‐Z is extended to solve the quasi‐hydrostatic deep‐atmosphere as well as the non‐traditional shallow‐atmosphere equations (with a complete Coriolis force representation).The continuous equations are first derived in their curl form using Eulerian horizontal and non‐Eulerian vertical coordinates. The equations are then interpreted as a Hamiltonian system, as they are expressed in terms of functional derivatives of the Hamiltonian. Using a finite‐difference scheme on a longitude/latitude grid and based either on a Lagrangian or mass‐based vertical coordinate, the discrete scheme is obtained by imitating the Hamiltonian formulation at the discrete level. It is shown how this form leads straightforwardly to the conservation of discrete total energy. The relation between the discrete equations and the discrete antisymmetry property of the Poisson bracket is discussed.The computing infrastructure of the dynamical core is kept essentially unchanged but the modification of the hydrostatic balance requires a mass‐based vertical coordinate. Also, absolute angular momentum is used as a prognostic variable instead of relative velocity, which allows time‐dependent metric terms and the non‐traditional Coriolis force to be absorbed into it.The prototype implementation is applied to idealized circulations of an Earth‐like small planet and validates the stability and accuracy of the new dynamical core.

Numerical study of upper ocean response to a typhoon moving zonally across the Luzon Strait

The Luzon Strait (LS) is a wide channel between Taiwan and the Luzon islands. Eastward of the LS, the Kuroshio Current (KC) flows northward along the eastern coasts of Luzon and Taiwan. A typhoon is a strong and localized low-pressure weather system that occurs frequently in the vicinity of the Taiwan area. One typical typhoon track, passing through the seas surrounding Taiwan, is a zonal path across the LS. The satellite measured SST, corresponding to typhoons Pabuk (5 Aug. 2007–12 Aug. 2007) and Dujuan (29 Aug. 2003–5 Sep. 2003), which both moved along this path, demonstrated that a classic right bias cooling occurs to the north of the storm track, during the typhoon forced period. However, some cold anomaly water also present toward left (south) of the storm track east of the LS in the relaxation period. This paper adopted a three-dimensional hydrostatic primitive equation model to study the possible causes of this southward transport of cold waters. Both model results and the observed SST anomaly revealed that the strength of the upper ocean cooling depends on whether a resonant regime between the typhoon winds and the near-inertial currents can be excited. To the east of the LS, the convergence between the warm Kuroshio water and the cold wakes in the poststorm period will enhance the southward spreading of cold anomaly water. The enhanced vertical mixing, induced by the southward propagation of nearly inertial waves associated with the cold wakes, can also produce some cold anomaly to the south of a storm track in the poststorm period. Both mechanisms can contribute to the occurrence of some cold anomaly water to the south of the storm track east of the KC. To the west of the LS, the convergence between the warm Kuroshio water and the cold upwelling water from the northern South China Sea can further strengthen the Kuroshio front in the LS.

A semi-geostrophic model incorporating well-mixed boundary layers

Semi-geostrophic theory has proved a powerful framework for understanding the dynamics of mid-latitude weather systems. However, one limitation is the lack of a realistic boundary-layer representation. Semi-geostrophic theory can be modified to include an atmospheric boundary layer by replacing the geostrophic wind with the ‘geotriptic’ (or Ekman-balanced) value in the substantive derivative and appropriately approximating the momentum diffusion term– the so-called semi-geotriptic theory. However, until now, solutions of the semi-geotriptic equations using predictor– corrector methods have not been possible for the important case of well-mixed boundary layers. Existing predictor– corrector methods require a Brunt– Väisälä frequency greater than zero to be solvable. Here we describe a method of incorporating well-mixed boundary layers into semi-geotriptic theory. We modify the hydrostatic relationship by including a small horizontal diffusion of vertical velocity. This enables the formation of a well-posed predictor– corrector method. Given well-mixed boundary layers are a ubiquitous feature of the lower atmosphere, the modification increases the usability of the model. Calculations are also performed at much higher vertical resolution than before. The revised semi-geotriptic model is compared with a hydrostatic primitive-equation model for a test case of a two-dimensional idealized diurnal cycle of a sea breeze. The performance of the revised semi-geotriptic model in the growth phase of the sea breeze is improved, as a well-mixed boundary layer is now permitted. The additional vertical resolution captures the capping inversion and the sea-breeze circulation better. The hydrostatic primitive-equation model is shown to produce inertial oscillations that persist beyond the evening decay of the boundary layer until the following morning. In contrast, the semi-geotriptic model decays following the boundary-layer state in a more realistic way. The semi-geotriptic model thus demonstrates its usefulness as a critical tool in understanding boundary-layer dynamics coupling issues in operational models. Copyright © 2010 Royal Meteorological Society and Crown Copyright.

A Modeling Study of Coastal-Trapped Wave Propagation in the Gulf of California. Part II: Response to Idealized Forcing

The propagation of incident coastal-trapped waves in the Gulf of California is investigated using a hydrostatic primitive equation model. The behavior of idealized incident wave disturbances with different amplitudes and time scales is examined. The incident wave disturbances propagate northward up-gulf along the east side with no significant change. At the sill, which is 600 km north of the entrance, the wave splits and most of the energy is steered to the west side where it propagates southward down-gulf with decreased amplitude (50%). A small fraction (10%–20%) of the incident energy enters the north where it is dissipated. Sea level at the entrance of the gulf is well correlated with sea level everywhere inside the gulf. In contrast, correlations of depth-averaged velocity between Topolobampo (close to the entrance of the gulf) and locations around the gulf decrease along the propagation path of the wave. Most of the dissipation of wave energy in the gulf takes place through bottom friction in the vicinity of the sill. Incident waves with large, but realistic, sea level displacement magnitudes exhibit nonlinear properties. Phase speeds increase as the sea level displacements of the incident waves increase from −30 to +30 cm. Waves of sea level elevation steepen. On the east side, large-amplitude elevation waves produce a down-gulf current adjacent to the coast such that the up-gulf currents associated with the wave separate from the coast. The separation process seems to be connected with subsequent downslope propagation of energy. Energetic anticyclonic eddies with spatial scales of 50–80 km can be generated by long-time-scale or large-amplitude elevation waves.

A Modeling Study of Coastal-Trapped Wave Propagation in the Gulf of California. Part I: Response to Remote Forcing

The evolution of remotely forced coastal-trapped waves in the Gulf of California is studied using a hydrostatic primitive equation model. The sea level time variability at a remote station south of the gulf is assumed to propagate northward into the gulf as a mode-1 coastal-trapped wave (CTW). To validate this assumption, observations and model results are compared. In general, sea level fluctuations are reasonably well represented by the model, with model–data correlations decreasing from 0.76 at Topolobampo, close to the entrance of the gulf, to 0.52 at Santa Rosalia in the central gulf. Model–data correlations of velocity are lower (<0.6). In the gulf, CTWs propagate northward along the east side with no significant changes south of the sill, which is 600 km north of the entrance. When incident waves propagating northward in the gulf along the east side arrive at the sill, a small fraction of the wave energy enters the northern gulf and is dissipated. Most of the wave energy is steered at the sill to the west side of the gulf where it propagates southward with decreased sea level amplitude. The weakened waves leave the gulf at the southwest boundary approximately 6–7 days after entering. Some of the incident wave energy is lost into downslope propagating disturbances generated as the CTWs pass, resulting in relatively intense bottom currents. Wave disturbances exhibit nonlinear characteristics while propagating. For example, isopycnal displacements in the wave fronts steepen. This occurs primarily for waves of sea level elevation. The contribution of remotely forced CTWs in the Gulf of California to the total kinetic energy is comparable to that produced by the wind.

Stratosphere–Troposphere Coupling in a Relatively Simple AGCM: The Role of Eddies

The extratropical circulation response to cooling of the polar-winter stratosphere in a simple AGCM is investigated. The AGCM is a dry hydrostatic primitive equation model with zonally symmetric boundary conditions and analytically specified physics. It is found that, as the polar-winter stratosphere is cooled, the tropospheric jet shifts poleward. This response projects almost entirely and positively (by convention) onto the AGCM’s annular mode. At the same time, the vertical flux of wave activity from the troposphere to the stratosphere is reduced and the meridional flux of wave activity from high to low latitudes is increased. Thus, as the stratosphere is cooled, the stratospheric wave drag is reduced. In order to understand this response, the transient adjustment of the stratosphere‐troposphere system is investigated using an ensemble of ‘‘switch on’’ stratospheric cooling runs of the AGCM. The response to the switch-on stratospheric cooling descends from the upper stratosphere into the troposphere on a time scale that matches simple downward-control theory estimates. The downward-control analysis is pursued with a zonally symmetric model that uses as input the thermal and eddy-driving terms from the eddying AGCM. With this model, the contributions to the response from the thermal and eddy-driving perturbations can be investigated separately, in the absence of eddy feedbacks. It is found that the stratospheric thermal perturbation, in the absence of such feedbacks, induces a response that is confined to the stratosphere. The stratospheric eddy-driving perturbation, on the other hand, induces a response that penetrates into the midtroposphere but does not account, in the zonally symmetric model, for the tropospheric jet shift. Furthermore, the tropospheric eddy-driving perturbation, in the zonally symmetric model, induces a strong upward response in the stratospheric winds. From these experiments and from additional experiments with the eddying AGCM, it is concluded that the stratospheric eddy-driving response induces a tropospheric response, but that the full circulation response results from a two-way coupling between the stratosphere and the troposphere.

A Sensitivity Study Of The Subtropical Ocean Surface Energy Balance To The Parameterization Of Precipitation From Stratocumulus Clouds

In the `First Lagrangian' of the Atlantic Stratocumulus Experiment(ASTEX), a cloudy air mass was tracked as it was advected by thetrade winds toward higher sea surface temperatures. In this study,a full diurnal cycle observed during this experiment is simulated andthe impact of the precipitation parameterization is examined. The modelwe use is the one dimensional version of the hydrostatic primitiveequation model MAR (Modele Atmospherique Regional) developed at the Universite catholique de Louvain (UCL).It includes an E-∈ turbulence closure, a wide-band formulationof the radiative transfer, and a parameterized microphysical schemeallowing partial condensation. The model realistically reproducesthe diurnal clearing of the cloud layer as well as the formation ofcumulus clouds under the stratocumulus deck. Nevertheless, as thesurface warms and the boundary layer becomes more convective,the simulation progressively differs from the observed evolution.Further experiments are carried out with different precipitationparameterizations commonly used in mesoscale modelsand general circulation models (GCMs).A strong sensitivity of the simulated liquid water path evolution isfound. The impact on the surface energy flux and the solar fluxreflected by the cloud is also examined. For both fluxes averagedover 24 hours, differences as large as 20 W m-2 are obtainedbetween the various simulations. Low cloudiness covers large areasover the ocean and such errors on the reflected solar flux may stronglyaffect the Earth's radiative budget in GCM simulations. We estimatethat the impact on the globally averaged outgoing solar flux could beas large as 5 W m-2. Furthermore, when atmospheric models arecoupled to ocean models, errors in the surface energy exchanges mayinduce significant drift in the simulated climate.

Fractionally modified flow in a deep ocean channel: Application to the Vema Channel

The modification of the exchange flow in a deep southern hemisphere passage, resembling the Vema Channel, by frictionally induced secondary circulation is investigated numerically. The hydrostatic primitive equation model is a two‐dimensional version of the sigma‐coordinate Princeton Ocean Model. The time dependent response of a stratified along‐channel flow, forced by barotropic or baroclinic pressure gradients, is examined. Near the bottom, where the along‐channel flow is retarded, there is cross‐channel Ekman flux that is associated with downwelling on the eastern side and upwelling on the western side of the channel. In the presence of stratification the cross‐channel flow rearranges the density structure, which in turn acts on the along‐channel velocity via the thermal wind relation. Eventually the cross‐isobath Ekman flux is shut down. In the case of baroclinically driven flow of Antarctic Bottom Water through the Vema Channel the model reproduces the observed shape of the deep temperature profiles and their cross‐channel asymmetry. The model offers an explanation that is alternative or supplementary to inviscid multilayer hydraulic theory that was proposed in earlier studies. It explains the extremely thick bottom boundary layers in the center and on the western slope of the channel. The deep thermocline is spread out in the west and sharpened in the east, and the coldest water is found on the eastern side of the deep trough. The modified density field reduces the along‐channel flow near the bottom and focuses it into a narrow jet on the eastern side of the channel.

Responses of Different Nonhydrostatic, Pressure-Coordinate Models to Orographic Forcing

Abstract Estimation of accuracy of different high-resolution, sound-filtered, pressure-coordinate models is carried out by modeling their response to orographic forcing. Evaluated models are the elastic nonhydrostatic model (EFM), the anelastic nonhydrostatic model (AEM), and the hydrostatic primitive equation model (HSM). These models are compared to the exact, nonfiltered, nonhydrostatic, pressure-coordinate dynamics (ExM). All model equations are linearized, a wave equation for vertical displacements of air particles is derived, and exact analytical stationary solutions for each model are compared for uniform flow over given smooth orography (bell-shaped mountain). These linear solutions reveal that compressible (ExM, EFM) and incompressible (AEM, HSM) models are different at long horizontal scales l ∼ 1000 km. Differences are especially large in the vertical velocity field (up to 100%) at the medium and upper levels of the atmosphere, where incompressible models give systematic reductions of wave ampl...

Simulation of marine stratocumulus: effect of precipitation parameterization and sensitivity to droplet number concentration

Marine stratocumulus observations show a large variability in cloud droplet number concentration (CDNC) related to variability in aerosol concentration. Changes in CDNC modify the cloud reflectivity, but also affect cloud water content, cloud lifetime, and cloudiness, through changes in precipitation. In mesoscale models and general circulation models (GCMs), precipitation mechanisms are parameterized. Here we examine how the precipitation parameterization can affect the simulated cloud. Simulations are carried out with the one-dimensional version of the hydrostatic primitive equation model MAR (Modele Atmospherique Regional) developed at the Universite catholique de Louvain. It includes a E-epsilon turbulence closure, a wide-band formulation of the radiative transfer, and a parameterized microphysics including prognostic equations for water vapour, cloud droplets and rain drops concentrations. In a first step, the model is used to simulate a horizontally homogeneous stratocumulus deck observed during the Atlantic Stratocumulus Transition Experiment (ASTEX) on the night of 12-13 June 1992. The observations show that the model is able to realistically reproduce the vertical structure of the cloud-topped boundary layer. In a second step, several precipitation parameterizations commonly used in mesoscale models and GCMs are tested. It is found that most parameterizations tend to overestimate the precipitation, which results in an underestimation of the vertically integrated liquid water content. Afterwards, using those parameterizations that are sensitive to CDNC, several simulations are performed to estimate the effect of CDNC variations on the simulated cloud. Based upon the simulation results, we argue that currently used parameterizations do not enable assessment of such a sensitivity.

Air‐sea interactions over Terra Nova Bay during winter: Simulation with a coupled atmosphere‐polynya model

A preliminary simulation of the Terra Nova Bay polynya has been performed with a coupled atmosphere–polynya model. The atmospheric model is a hydrostatic primitive equations model that has been validated previously by a simulation of the strong katabatic winds observed in that area. The polynya model includes a representation of the free drift of frazil ice and simple sea‐ice dynamics and thermodynamics. Two and three‐dimensional experiments have been performed under polar night conditions. Two‐dimensional experiments show that an open (warm) water area influences significantly the atmospheric circulation in the antarctic coastal zone: an additional ice‐breeze effect is simulated and is responsible for the strenghthening of the katabatic winds near the coast. Because of the important temperature difference between the continental air and the ice‐free ocean (up to 40°C), strong surface heat fluxes are simulated over the polynya. Finally, a three‐dimensional experiment has been performed. The integration domain includes Terra Nova Bay. The polynya observed in that region is well simulated. It is found that heat losses from the polynya surface are stronger than previously thought but are probably constrained by the idealized representation of frazil ice, which is assumed to be uniform in each grid box. This stresses the need for having a better knowledge of frazil ice evolution in large polynyas.

Numerical Simulations of Maritime Frontogenesis

Abstract A hydrostatic primitive equation model initialized in a highly baroclinically unstable state was used to simulate maritime cyclogenesis and frontogenesis. In order to identify boundary layer physical processes important in maritime frontogenesis, several different simulations were performed. In an effort to isolate impacts due solely to the boundary layer, moist processes were not included. An adiabatic and inviscid simulation provided the control for these experiments. Two different boundary layer parameterizations were used: a K-theory parameterization featuring Richardson-number-dependent eddy diffusivity and a second-order closure scheme with prognostic equations for the turbulence quantities. Results indicated that strong warm and cold fronts formed in the adiabatic and inviscid case but that the vertical motion fields were weak. In the K-theory simulation, the results were somewhat more realistic with stronger vertical motion. In both the K-theory and second-order closure simulations, the b...

Three-Dimensional Mass-Conserving Semi-Lagrangian Scheme Employing Forward Trajectories

Abstract Through the use of the dimensional splitting “cascade” method of grid-to-grid interpolation, it is shown that consistently high-order-accurate semi-Lagrangian integration of a three-dimensional hydrostatic primitive equations model can be carried out using forward (downstream) trajectories instead of the backward (upstream) trajectory computations that are more commonly employed in semi-Lagrangian models. Apart from the efficiency resulting directly from the adoption of the cascade method, improved computational performance is achieved partly by the selective implicit treatment of only the deepest vertical gravity modes and partly by obviating the need to iterate the estimation of each trajectory's location. Perhaps the main distinction of our present semi-Lagrangian method is its inherent exact conservation of mass and passive tracers. This is achieved by adopting a simple variant of the cascade interpolation that incorporates mass (and tracer) conservation directly and at only a very modest add...

A Numerical Study of the Effects of Differential Cloud Cover on Cold Frontal Structure and Dynamics

Abstract The effects of sensible heating and momentum mixing on the low-level structure and dynamics of a two-dimensional cold front are studied with a hydrostatic primitive equation model. Effects of inhomogeneous heating arising from a contrast in low-level cloud cover across the front are emphasized. The relative importance of grid resolution and the choice of method for parameterizing planetary boundary layer (PBL) processes in the model are also examined. Frontal updraft dynamics are studied in terms of the following inquiries: (a) the relative importance of turbulent momentum transport, differential sensible heating, and the reduction in static stability in the heated region ahead of the front; (b) the nature of the interaction between the adiabatic, semigeostrophic frontal circulation and the thermally forced circulation; and (c) possible roles played by dry symmetric instability and density current dynamics. The terms in the frontogenesis and divergence budget equations are computed to determine t...

Preliminary simulations of the flow field over the Attic Peninsula

In the frame of the APSIS experiment, three mesoscale meteorological models have been used over the Attic Peninsula. TVM (IAG UCl) developed from the URBMET code (USA, 1985) is hydrostatic, boussinesq and written in vorticity. MAR (IAG-UCL) is a fully compressible hydrostatic primitive equations model, and RAMS (CSU, USA) is non hydrostatic and fully compressible. These three models have been tested on the Athens region where strong interactions exist between different sea-breeze cells and the complex topography. Suring unstable daytime conditions, all models are shown to produce similar wind fields and good agreement with measurements, while the stable night-time leads to leads to larger differences.

A Test of an Upstream Spline Interpolation Technique for the Advective Terms in a Numerical Mesoscale Model

Abstract Upstream interpolation with a cubic spline is used to integrate the advective terms in a two-dimensional hydrostatic primitive equation model. The model is applied to study the problems of air flow over a mountain, and sea and land breezes. Results are compared against existing numerical models and against an identical model which uses a simple upstream differencing scheme for the advective terms. It was found that under certain atmospheric conditions, where the atmosphere is being continuously forced, accurate results may be obtained near the surface even with the simple upstream scheme. Also it is shown that a sophisticated planetary boundary layer parameterisation is needed in order to obtain accurate results in both the lower and upper troposphere.

Release of Potential Instability: Part I. A Sequential Plume Model within a Hydrostatic Primitive Equation Model

Abstract The release of potential instability by large-scale lifting and the subsequent interaction of cumulus convection and the hydrostatic mesoscale flow in a most complex scale-interaction process. This process is an essential part of tropical weather but it is also important in extratropical cyclones through the formation of mesoscale rainbands that contribute much of the precipitation. The purpose of this paper is to qualitatively and quantitatively clarify the potential instability release process within a framework that will permit calculation of convective/mesoscale interactions. The approach is to use an extension of the Lagrangian form of the one-dimensional cumulus model to provide values of convective-scale changes to a hydrostatic primitive equation model. This cumulus sub-routine locates the base of the convection, computes the cumulus plume that will build, accounts for the environmental subsidence, and mixes the subsided environment with the cumulus plume after rainout. These plumes build...