The pseudospectral approximation applied to the shallow water equations on a sphere

Abstract

The Radar data Analysis, Processing and Interactive Display (RAPID) system, developed by McGill University researchers, synthesizes spherical coordinate radar data onto Cartesian maps displaying Constant Altitude Plan Position Indicator (CAPPI) reflectivity, Vertically Integrated Liquid water content (VIL), and other radar‐based parameters. In this study, Carvel radar (53.34°N, 114.09°W) data from July 2000 were processed using McGill's RAPID software. Specifically, we compared observations of severe convection, as identified by selected radar‐based reflectivity parameters, with surface severe weather reports and atmospheric sounding data. July 2000 was characterized by frequent severe thunderstorm activity over central Alberta; there were seven days with golfball‐sized hail, and two days with confirmed tornadoes. The VIL, upper level VIL (UVIL), and the maximum reflectivity at 7 km (Z7) were employed to quantify the strength and frequency of storms within a 120‐km radial distance of Carvel. For each day, the intensity of the convection was quantified by counting the total number of 1‐km² pixels in the study area that exceeded the severe thresholds for VIL, UVIL and Z7. The severe thunderstorm algorithms were found to be very effective at correctly identifying the observed severe thunderstorm events. All three radar parameters indicated a diurnal cycle, with severe convection starting after noon and peaking between 16:00 and 18:00 Local Daylight Time. A positive correlation was evident between the observed storm severity and the daily UVIL and Z7 pixel counts. The daily UVIL and Z7 pixel counts were also positively correlated with the Convective Available Potential Energy (CAPE) calculated from proximity soundings.     

浅水波方程在全球上的变时间步长积分

本文讨论了浅水波方程在全球上的变时间步长(以下简称变步长)积分。并讨论了两步Lax-Wendroff的变时间步长(以下简称TLW变步长)积分,发现积分结果对平滑系数κ很敏感,必须选择合适。井将常数时间步长(以下简称定步长)的积分结果和两种变步长的积分结果进行比较。发现用实际资料作预报,变步长法的预报均方根误差比定步长法的小。还发现变步长的24小时预报对近极地系统移动偏慢的问题比定步长的有较大的改善。变步长积分的计算稳定性也优于定步长。     

An invariant theory of the linearized shallow water equations with rotation and its application to a sphere and a plane

The system of linearized shallow water equations is formulated in this paper on any rotating and smooth surface M in terms of differential geometry. The system decouples into two separate equations: a scalar one for the height deviation and a vector one for the velocity field. For low and high frequencies these equations yield asymptotic equations whose solutions are the generalizations of the Poincare and Rossby waves to smooth surface. The application of these equations to the β-plane yields both new and previously known equations for the height deviation and for the velocity components. The application of the equations to the rotating spherical Earth shows that the meridional amplitudes of Poincare and Rossby waves are both described by the prolate angular spheroidal wave functions. The asymptotic and the power series expansions of the eigenvalues of these functions yield new approximations for the dispersion relations of these waves on a sphere. The new dispersion relations are very accurate in the physically relevant range of the single nondimensional model parameter – the square of the nondimensional gravity waves’ phase speed. The invariant formulation can also be applied to other surfaces that are of geophysical interest such as an oblate ellipsoid of revolution.     

Revisiting well-posed boundary conditions for the shallow water equations

We derive a general form of well-posed open boundary conditions for the two-dimensional shallow water equations by using the energy method. Both the number and the type of boundary conditions are presented for subcritical and supercritical flows on a general domain. The boundary conditions are also discussed for a rectangular domain. We compare the results with a number of often used open boundary conditions and show that they are a subset of the derived general form.     

Integration of the shallow water equations in a plane geometry using semi-Lagrangian and Eulerian schemes

Summary Extended integrations of semi-Lagrangian and Eulerian shallow water primitive equation models are performed. The semi-Lagrangian model used the semi-implicit two-time-level scheme. The Eulerian model used a conserving nonlinear advection scheme.For low resolution and longer integrations, difficulties were encountered with the semi-Lagrangian model which were absent in the Eulerian model. These difficulties are discussed.With 14 Figures     

长江荆江分蓄洪区历史演变、前景和风险管理

用部分溃坝的波流与堰流相交法模拟淦河流域溃口点的流量。以溃口点的流量、DEM为基础数据, 利用GIS的暴雨洪涝淹没模型, 对2010年7月14日淦河流域由强降水引发的溃口式洪水淹没过程进行模拟, 并利用实际灾情对模拟结果进行检验分析。模拟结果表明:随着洪水演进, 淦河流域的淹没面积不断地增大, 其中0.5~1 m水深段的淹没面积增长最快; 14日18时, 洪水到达任窝村, 20时淹没至马桥镇, 22时淹没至严洲村。灾情调查检验结果显示, 对于洪水到达时间和地点, 淹没模型模拟值与实况值较为吻合, 表明该模型在溃口式洪水淹没过程方面具有较好的模拟效果。

     

On the drag coefficient over shallow water

It is shown that if the wind-wave spectrum in shallow water is approximately independent of wind speed due to the combined effects of white-capping and bottom friction, then the wave-induced drag coefficient has a maximum value when the wind speed is twice the maximum wave-speed; as the wind speed increases further, the drag coefficient slowly decreases. This result is consistent with the observations of Hicks et al. (1974).     

The role of shallow convection in the water and energy cycles of the atmosphere

The Canadian Centre for Climate Modelling and Analysis atmospheric general circulation model (AGCM4) is used to study the role of shallow convection in the hydrologic and energy cycles of the atmosphere. Sensitivity tests with AGCM4 show a marked effect of the parameterization of shallow convection in the model. In particular, including the parameterization of shallow convection produces considerably enhanced vertical mixing and decreased stratiform cloud amounts in the lower subtropical atmosphere over the oceans. The differences in simulated stratiform cloud amounts are associated with a change in the globally averaged outgoing shortwave radiative flux at the top of the atmosphere of about 11 W m⁻². Additionally, precipitation rates are considerably reduced for stratiform clouds and enhanced for convective clouds in the subtropics, if the parameterization of shallow convection is included in the model. Additional tests show that the simulated responses in cloud amounts and precipitation to the treatment of shallow convection are robust. Additional simulations with modified closures for deep convection and other changes to the treatment of convection in the model still lead to similar responses of the model results.     

Drag and bulk aerodynamic coefficients over shallow water

From direct measurements of the vertical turbulent fluxes of momentum, heat, and water vapor over shallow water, the drag and bulk aerodynamic coefficients are computed and compared with those obtained via the profile technique. The results show thatC _D =C _T =C _E =1.2×10^?3, with variability among their mean values being ? 20 %. All measurements were taken at 8 m above the water surface at Lake Hefner, Oklahoma. The measurement platform was a rectangular tower located within the lake and approximately 1 km from the nearest shore. The depth of the lake at the tower was 8 m.     

边界微分方程及其应用

本文讨论二维Helmholtz方程外边值问题的求解,以较为严格的方式建立了更精准的新的边界微分方程.在贴体坐标系下,Helmholtz方程可转化为非齐次Bessel方程.将Bessel方程的一般解代入Sommerfeld辐射条件可以得到等价于原Helmholtz方程的积分-微分方程,再利用分部积分消去其中积分,即可建立高频问题的边界微分方程.文中通过若干算例对新得到的边界微分方程进行了数值验证.     

A global spectral element model for poisson equations and advective flow over a sphere

A global spherical Fourier–Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre–Gauss–Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity,with only one element. Then, the partial differential equations are solved on the longitude–latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. Highprecision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.     

A regional spectral nested shallow water equation model

A method to expand meteorological elements in terms of finite double Fourier series in a limited-region and a spectral nested shallow water equation model based upon the method with conformal map projection in rectangular coordinates, have been proposed, and computational stability and efficiency of time integration have been discussed.     

Vortex erosion in a shallow water model of the polar vortex

The erosion of a model stratospheric polar vortex in response to bottom boundary forcing is investigated numerically. Stripping of filaments of air from the polar vortex has been implicated in the occurrence of stratospheric sudden warmings (SSWs) but it is not understood in detail what factors determine the rate and amount of stripping. Here a shallow water vortex forced by topography is used to investigate the factors initiating stripping and whether this leads the vortex to undergo an SSW. It is found that the amplitude of topographic forcing must exceed some threshold (of order 200–450 m) in order for significant stripping to occur. For larger forcing amplitudes significant stripping occurs, but not as an instantaneous response to the forcing; rather, the forcing appears to initiate a process that ultimately results in stripping several tens of days later. There appears to be no simple quantitative relationship between the amount of mass stripped and the topography amplitude. However, at least over the early stages of the experiments, there is a good correlation between the amount of mass stripped and the global integral of wave activity, which may be interpreted as a measure of the accumulated topographic forcing. Finally there does not appear to be a simple correspondence between amount of mass stripped and the occurrence of an SSW.     

Drag and bulk transfer coefficients associated with a shallow water surface

During September, 1971, measurements of Reynolds stress and sensible heat flux were made over a shallow coral reef off the coast of Papua. The water varied (with the tide) from 0 to 2.5 m in depth. Two independent measurements of the momentum flux confirmed that the surface deviated only slightly from aerodynamically smooth. The independent determinations of the sensible heat flux are consistent with this result, indicating a much lower Stanton number than would be expected over deeper water (viz., 1.1 × 10^–3 rather than about 1.5 × 10^–3).The near smoothness of the surface appears to be associated with either the attenuation of small surface waves resulting from the presence of a surface film or their inability to grow in the absence of longer wave components. Although it is not clear which of these effects was predominant, the data show that great care should be taken in extrapolating shallow water results to open ocean conditions, particularly when it is conceded that surface films may exist in either situation.Now with Argonne National Laboratory, Argonne, Illinois, U.S.A.Now with CSIRO Division of Mineral Physics, Sydney, N.S.W., Australia.     

Wind-Driven Ocean Circulation in Shallow Water Lattice Boltzmann Model

A lattice Boltzmann (LB) model with overall second-order accuracy is applied to the 1.5-layer shallow water equation for a wind-driven double-gyre ocean circulation. By introducing the second-order integral approximation for the collision operator, the model becomes fully explicit. In this case, any iterative technique is not needed. The Coriolis force and other external forces are included in the model with second-order accuracy, which is consistent with the discretized accuracy of the LB equation. The numerical results show correct physics of the ocean circulation driven by the double-gyre wind stress with different Reynolds numbers and different spatial resolutions. An intrinsic low-frequency variability of the shallow water model is also found. The wind-driven ocean circulation exhibits subannual and interannual oscillations, which are comparable to those of models in which the conventional numerical methods are used.     

Polar coordinates shallow water storm surge model for the coast of Bangladesh

A coastal shallow water model generally involves at least one coastal boundary which is curvilinear in nature. Representation of this curvilinear boundary needs special attention when a finite difference scheme is used. To incorporate the curvilinear coastal boundary and off-shore islands properly in the numerical scheme, we need fine resolution which may not be necessary away from the coast. In order to tackle this problem, a nested numerical scheme may be used where fine resolution for the coastal boundary region is nested into a course resolution scheme covering the whole analysis area. But this involves comparatively more memory and CPU time in the solution process. The present study aims at developing a storm surge model that may subsequently be modified for operational forecasting purpose for the coast of Bangladesh. A vertically integrated model is developed in a cylindrical polar coordinate system capable of incorporating bending of the coastline and off-shore islands with considerable accuracy without using any nesting. A comparison of CPU time and memory in the numerical computations between two types of modelling is described and it is found that the cylindrical polar coordinates model is economic. The model is applied for estimating the water levels at different coastal and island stations associated with a few storms that hit the coast of Bangladesh. It is found that the computed and observed water levels are in good agreement. Considering the computing time and other factors, it is found that the Polar Coordinates model is more suitable for operational forecasting purpose along the coast of Bangladesh.     

The solitary wave of barotropic atmosphere on a sphere

In this paper, we have investigated large scale disturbance in barotropic atmosphere on a sphere. It demonstrates that: considering nonlinear effects of interaction between waves in barotropic vorticity equation, the wave packet of the disturbance is governed by the famous equation-nonlinear Schrodinger equation. For the solitary wave, two fac-tors are very important: one is spherical effect of the disturbance and the other is meridional shear of blocking high. In comparison with the results of local Cartesian coordinates, the former factor is the individuality of spherical soliton.     

Probable effects of CO2-induced climatic warming on the thermal environment of ponded shallow water

A physical model was developed for describing the thermal environment of ponded shallow water as a model for rice fields in relation to climatic conditions. The model was used to assess probable effects of CO₂-induced warming on the thermal conditions of ponded shallow water. It was assumed that an altered equilibrium climate was produced by atmospheric CO₂ which was twice that of present levels. The 1951–80 climatic means of Japan were used as baseline data. Water temperature and energy balance characteristics predicted from the model were compared between both climates. The most notable results were that water temperature under CO₂ doubling rose 2 to 4 °C. These increases in temperature would induce a remarkable northward shift of the 15 °C isotherm which characterizes the isochrone of safe transplanting dates for rice seedlings. CO₂-warming would have a considerable influence on the energy balance characteristics, intensifying the evaporation rate from the water surface. Changes in thermal conditions of rice fields due to CO₂-induced climatic warming are, therefore, expected to bring about significant effects on aquatic environments and the life forms they support.