The non-uniqueness of solution and compatibility between the coupled boundary conditions in computing velocity potential and streamfunction from horizontal velocity in a limited domain of arbitrary shape are revisited...The non-uniqueness of solution and compatibility between the coupled boundary conditions in computing velocity potential and streamfunction from horizontal velocity in a limited domain of arbitrary shape are revisited theoretically with rigorous mathematic treatments.Classic integral formulas and their variants are used to formulate solutions for the coupled problems.In the absence of data holes,the total solution is the sum of two integral solutions.One is the internally induced solution produced purely and uniquely by the domain internal divergence and vorticity,and its two components（velocity potential and streamfunction） can be constructed by applying Green＇s function for Poisson equation in unbounded domain to the divergence and vorticity inside the domain.The other is the externally induced solution produced purely but non-uniquely by the domain external divergence and vorticity,and the non-uniqueness is caused by the harmonic nature of the solution and the unknown divergence and vorticity distributions outside the domain.By setting either the velocity potential（or streamfunction） component to zero,the other component of the externally induced solution can be expressed by the imaginary（or real） part of the Cauchy integral constructed using the coupled boundary conditions and solvability conditions that exclude the internally induced solution.The streamfunction（or velocity potential） for the externally induced solution can also be expressed by the boundary integral of a double-layer（or singlelayer） density function.In the presence of data holes,the total solution includes a data-hole-induced solution in addition to the above internally and externally induced solutions.展开更多
Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary（around a data hole） inside the domain,...Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary（around a data hole） inside the domain,the total solution is the sum of the internally and externally induced parts.For the internally induced part,three numerical schemes（grid-staggering,local-nesting and piecewise continuous integration） are designed to deal with the singularity of the Green＇s function encountered in numerical calculations.For the externally induced part,by setting the velocity potential（or streamfunction） component to zero,the other component of the solution can be computed in two ways：（1） Solve for the density function from its boundary integral equation and then construct the solution from the boundary integral of the density function.（2） Use the Cauchy integral to construct the solution directly.The boundary integral can be discretized on a uniform grid along the boundary.By using local-nesting（or piecewise continuous integration）,the scheme is refined to enhance the discretization accuracy of the boundary integral around each corner point（or along the entire boundary）.When the domain is not free of data holes,the total solution contains a data-hole-induced part,and the Cauchy integral method is extended to construct the externally induced solution with irregular external and internal boundaries.An automated algorithm is designed to facilitate the integrations along the irregular external and internal boundaries.Numerical experiments are performed to evaluate the accuracy and efficiency of each scheme relative to others.展开更多
Three types of previously used numerical methods are revisited for computing the streamfunctionψand velocity potentialχfrom the horizontal velocity v in limited domains.The first type,called the SOR-based method,use...Three types of previously used numerical methods are revisited for computing the streamfunctionψand velocity potentialχfrom the horizontal velocity v in limited domains.The first type,called the SOR-based method,uses a classical successive over-relaxation(SOR)scheme to computeψ(orχ)first with an arbitrary boundary condition(BC)and thenχ(orψ)with the BC derived from v.The second type,called the spectral method,uses spectral formulations to construct the inner part of(ψ,χ)-the inversion of(vorticity,divergence)with a homogeneous BC,and then the remaining harmonic part of(ψ,χ)with BCs from v.The third type,called the integral method,uses integral formulas to compute the internally induced(ψ,χ)-the inversion of domain-internal(vorticity,divergence)using the free-space Greenꞌs function without BCs and then the remaining harmonicψ(orχ)with BCs from v minus the internally-induced part.Although these methods have previously been successfully applied to flows in large-scale and synoptic-scale domains,their accuracy is compromised when applied to complex flows over mesoscale domains,as shown in this paper.To resolve this problem,two hybrid approaches,the integral-SOR method and the integral-spectral method,are developed by combining the first step of the integral method with the second step adopted from the SOR-based and spectral methods,respectively.Upon testing these methods on real-case complex flows,the integral-SOR method is significantly more accurate than the integral-spectral method,noting that the latter is still generally more accurate than the three previously-used methods.The integral-SOR method is recommended for future applications and diagnostic studies of complex flows.展开更多
After first reviewing historical and current difficulties in solving streamfunction and velocity potential in a limited domain,and describing recent developments in obtaining accurate solutions in a limited domain wit...After first reviewing historical and current difficulties in solving streamfunction and velocity potential in a limited domain,and describing recent developments in obtaining accurate solutions in a limited domain with arbitrary shape,a newly proposed approach is introduced and its application to a torrential rain event is reported.The results show that the newly developed method has advantages in capturing mesoscale information,compared with horizontal winds.展开更多
A high resolution upwind compact streamfunction numerical algorithm for two-dimensional(2D)double-diffusive convection(DDC)is developed.The unsteady Navier-Stokes(N-S)equations in the streamfunction-velocity form and ...A high resolution upwind compact streamfunction numerical algorithm for two-dimensional(2D)double-diffusive convection(DDC)is developed.The unsteady Navier-Stokes(N-S)equations in the streamfunction-velocity form and the scalar temperature and concentration equations are used.An optimized third-order upwind compact(UCD3 opt)scheme with a low dispersion error for the first derivatives is utilized to approximate the third derivatives of the streamfunction in the advection terms of the N-S equations and the first derivatives in the advection terms of the scalar temperature and concentration equations.The remaining first derivatives of the streamfunction(velocity),temperature,and concentration variables used in the governing equations are discretized by the fourth-order compact Pade(SCD4)schemes.With the temperature and concentration variables and their approximate values of the first derivatives obtained by the SCD4 schemes,the explicit fourth-order compact schemes are suggested to approximate the second derivatives of temperature and concentration in the diffusion terms of the energy and concentration equations.The discretization of the temporal term is executed with the second-order Crank-Nicolson(C-N)scheme.To assess the spatial behavior capability of the established numerical algorithm and verify the developed computer code,the DDC flow is numerically solved.The obtained results agree well with the benchmark solutions and some accurate results available in the literature,verifying the accuracy,effectiveness,and robustness of the provided algorithm.Finally,a preliminary application of the proposed method to the DDC is carried out.展开更多
A streamfunction projection method called gravest empirical mode(GEM) is applied to the hydrographic section at 137°E to filter out eddy noises in the western North Pacific and derive quantitative ensemble-averag...A streamfunction projection method called gravest empirical mode(GEM) is applied to the hydrographic section at 137°E to filter out eddy noises in the western North Pacific and derive quantitative ensemble-average water mass properties in the North Equatorial Current region. The GEM fields capture more than 80% of total property variances in the thermocline layer. The core layer structures of key water masses, including the North Pacific Tropical Water(NPTW) and the North Pacific Intermediate Water(NPIW), are examined with a definition of water mass boundary based on property gradient. It shows that a tongue of maximal root-mean-square(RMS) residual exists in the upper half of NPIW for all water properties. These subsurface RMS tongues appear to be close to sharp property gradients. It is the first time a GEM diagnosis is applied to nutrient data, which reveals a drastic difference of N/P reaction rate ratio above and below the maximal-nutrient core at 1250 m. Additionally, a GEM velocity reconstruction successfully produces the North Equatorial Undercurrent(NEUC), demonstrating the stable thermal-wind nature of this newly-discovered current.展开更多
In this paper we consider a geometric inverse problem which requires detecting an unknown obstacle such as a submarine or an aquatic mine immersed in a Stokes slow viscous stationary flow of an incompressible fluid,fr...In this paper we consider a geometric inverse problem which requires detecting an unknown obstacle such as a submarine or an aquatic mine immersed in a Stokes slow viscous stationary flow of an incompressible fluid,from a single set of Cauchy(fluid velocity and stress force)boundary measurements.The numerical reconstruction is based on the method of fundamental solutions(MFS)for the pressure and streamfunction in two dimensions combined with regularization.Numerical results are presented and discussed.展开更多
The calculation of the meridional overturning streamfunction in the southern Indian Ocean is biased by the Indonesian Throughflow.Therefore,this study applies the vertical overturning streamfunction to diagnose the sh...The calculation of the meridional overturning streamfunction in the southern Indian Ocean is biased by the Indonesian Throughflow.Therefore,this study applies the vertical overturning streamfunction to diagnose the shallow overturning circulation in the Indian Ocean.Using the Ocean General Circulation Model for the Earth simulator output,improvements with the vertical overturning streamfunction compared with the meridional overturning streamfunction are explored.The results show that the vertical overturning streamfunction smoothly connects the shallow overturning circulations of the northern Indian Ocean and the southern Indian Ocean with the whole cycle of the subtropical cell and the cross-equatorial cell.The vertical overturning streamfunction shows a much cleaner shallow overturning circulation,which is underestimated by the meridional overturning streamfunction.It shows that the shallow overturning circulation has a magnitude of~13 Sv(1 Sv≡106 m 3 s−1),of which the subtropical cell accounts for~8 Sv.In addition,the vertical overturning streamfunction captures a clockwise overturning cell in the upper 600 m layer between 30°S and 34°S.This cell has a magnitude of about−5 Sv and probably corresponds to the wind-forced subtropical gyre.Therefore,the vertical overturning streamfunction provides a new approach for estimating the shallow overturning circulation in the Indian Ocean.展开更多
In this study, kinetic energy budget equations of rotational and divergent flow in pressure coordinates are derived on terrain-following coordinates. The new formulation explicitly shows the terrain effects and can be...In this study, kinetic energy budget equations of rotational and divergent flow in pressure coordinates are derived on terrain-following coordinates. The new formulation explicitly shows the terrain effects and can be applied directly to model-simulated dynamic and thermodynamic fields on the model's original vertical grid. Such application eliminates interpolation error and avoids errors in virtual weather systems in mountainous areas. These advantages and their significance are demonstrated by a numerical study in terrain-following coordinates of a developing vortex after it moves over the Tibetan Plateau in China.展开更多
The Island Rule, derived from the Sverdrup theory, is widely used to estimate and analyze water transport through a strait. Previous studies presented single-or multi-island rules with either lateral or bottom frictio...The Island Rule, derived from the Sverdrup theory, is widely used to estimate and analyze water transport through a strait. Previous studies presented single-or multi-island rules with either lateral or bottom friction. In this paper,an analytical model of wind-driven circulation is assumed based on linear dynamics. Considering both lateral and bottom friction, the analytic solutions of the transport streamfunction around the islands are derived and the volume transport through the channel is presented. The results are similar to those of Wajsowicz, but the frictional constants represent different values. The analytic solution shows that the relationship between the lateral frictional and bottom frictional dissipation is complex in terms of the frictional constants. To understand the interaction between the two friction types, lateral and bottom friction values were randomly chosen on a barotropic beta plane. The result shows an approximately linear relationship between the lateral and bottom friction in consisting of the combined frictional constants. We studied the effect of the channel width on the transport through the channel. The results show that the friction enhances the flow under some widths, which is similar to the flow behavior when only the lateral friction is considered. We also compared the transport through the channel at different depths and founded that the deeper the water, the smaller the transport reduction ratio when the horizontal eddy viscosity coefficient and the bottom drag coefficient remained constants. To further present the combined role of lateral frictional and bottom frictional dissipation, we compared our model with the model of Wajsowicz for two islands, where only the lateral or bottom friction were considered, with different channel widths. The results showed that the effect of the lateral friction is greater than the bottom friction when the channel is narrow, especially in the Munk boundary layer thickness. When the channel is much wider than the Munk boundary layer thickness, the role of the bottom friction is greater than that of the lateral friction. The model was applied to the Indonesian throughflow and yielded a reduction of approximately 20% in the transport.展开更多
基金supported by the Office of Naval Research (Grant No. N000141010778) to the University of Oklahomathe National Natural Sciences Foundation of China (Grant Nos. 40930950,41075043,and 4092116037) to the Institute of Atmospheric Physicsprovided by NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreement (No. NA17RJ1227),U.S. Department of Commerce
文摘The non-uniqueness of solution and compatibility between the coupled boundary conditions in computing velocity potential and streamfunction from horizontal velocity in a limited domain of arbitrary shape are revisited theoretically with rigorous mathematic treatments.Classic integral formulas and their variants are used to formulate solutions for the coupled problems.In the absence of data holes,the total solution is the sum of two integral solutions.One is the internally induced solution produced purely and uniquely by the domain internal divergence and vorticity,and its two components（velocity potential and streamfunction） can be constructed by applying Green＇s function for Poisson equation in unbounded domain to the divergence and vorticity inside the domain.The other is the externally induced solution produced purely but non-uniquely by the domain external divergence and vorticity,and the non-uniqueness is caused by the harmonic nature of the solution and the unknown divergence and vorticity distributions outside the domain.By setting either the velocity potential（or streamfunction） component to zero,the other component of the externally induced solution can be expressed by the imaginary（or real） part of the Cauchy integral constructed using the coupled boundary conditions and solvability conditions that exclude the internally induced solution.The streamfunction（or velocity potential） for the externally induced solution can also be expressed by the boundary integral of a double-layer（or singlelayer） density function.In the presence of data holes,the total solution includes a data-hole-induced solution in addition to the above internally and externally induced solutions.
基金supported by the Office of Naval Research (Grant No.N000141010778) to the University of Oklahomathe National Natural Sciences Foundation of China (Grant Nos. 40930950,41075043,and 4092116037) to the Institute of Atmospheric Physicsprovided by NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreement No. (NA17RJ1227),U.S. Department of Commerce
文摘Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary（around a data hole） inside the domain,the total solution is the sum of the internally and externally induced parts.For the internally induced part,three numerical schemes（grid-staggering,local-nesting and piecewise continuous integration） are designed to deal with the singularity of the Green＇s function encountered in numerical calculations.For the externally induced part,by setting the velocity potential（or streamfunction） component to zero,the other component of the solution can be computed in two ways：（1） Solve for the density function from its boundary integral equation and then construct the solution from the boundary integral of the density function.（2） Use the Cauchy integral to construct the solution directly.The boundary integral can be discretized on a uniform grid along the boundary.By using local-nesting（or piecewise continuous integration）,the scheme is refined to enhance the discretization accuracy of the boundary integral around each corner point（or along the entire boundary）.When the domain is not free of data holes,the total solution contains a data-hole-induced part,and the Cauchy integral method is extended to construct the externally induced solution with irregular external and internal boundaries.An automated algorithm is designed to facilitate the integrations along the irregular external and internal boundaries.Numerical experiments are performed to evaluate the accuracy and efficiency of each scheme relative to others.
基金supported by the National Natural Science Foundation of China under Grant Nos. 91937301, 41875074, and 41675060the Second Tibetan Plateau Comprehensive Scientific Expedition 2019QZKK0104+1 种基金the National Key Scientific and Technological Infrastructure Project “EarthLab”provided by NOAA/OAR under NOAA–OU Cooperative Agreement #NA16OAR4320072, U.S. Department of Commerce
文摘Three types of previously used numerical methods are revisited for computing the streamfunctionψand velocity potentialχfrom the horizontal velocity v in limited domains.The first type,called the SOR-based method,uses a classical successive over-relaxation(SOR)scheme to computeψ(orχ)first with an arbitrary boundary condition(BC)and thenχ(orψ)with the BC derived from v.The second type,called the spectral method,uses spectral formulations to construct the inner part of(ψ,χ)-the inversion of(vorticity,divergence)with a homogeneous BC,and then the remaining harmonic part of(ψ,χ)with BCs from v.The third type,called the integral method,uses integral formulas to compute the internally induced(ψ,χ)-the inversion of domain-internal(vorticity,divergence)using the free-space Greenꞌs function without BCs and then the remaining harmonicψ(orχ)with BCs from v minus the internally-induced part.Although these methods have previously been successfully applied to flows in large-scale and synoptic-scale domains,their accuracy is compromised when applied to complex flows over mesoscale domains,as shown in this paper.To resolve this problem,two hybrid approaches,the integral-SOR method and the integral-spectral method,are developed by combining the first step of the integral method with the second step adopted from the SOR-based and spectral methods,respectively.Upon testing these methods on real-case complex flows,the integral-SOR method is significantly more accurate than the integral-spectral method,noting that the latter is still generally more accurate than the three previously-used methods.The integral-SOR method is recommended for future applications and diagnostic studies of complex flows.
基金supported by the National Basic Research Program of China(Grant No.2012CB417201)the National Science and Technology Support Program(Grant No.GYHY201406001)+1 种基金the National Natural Science Foundation of China(Grant No.41205033)the Key Project of the Key Laboratory of Atmosphere and Environments on the Plateau in Sichuan Province(Grant No.PAEKL-2014-C1)
文摘After first reviewing historical and current difficulties in solving streamfunction and velocity potential in a limited domain,and describing recent developments in obtaining accurate solutions in a limited domain with arbitrary shape,a newly proposed approach is introduced and its application to a torrential rain event is reported.The results show that the newly developed method has advantages in capturing mesoscale information,compared with horizontal winds.
基金supported by the National Natural Science Foundation of China(Nos.11872151,11372075,and 91330112)。
文摘A high resolution upwind compact streamfunction numerical algorithm for two-dimensional(2D)double-diffusive convection(DDC)is developed.The unsteady Navier-Stokes(N-S)equations in the streamfunction-velocity form and the scalar temperature and concentration equations are used.An optimized third-order upwind compact(UCD3 opt)scheme with a low dispersion error for the first derivatives is utilized to approximate the third derivatives of the streamfunction in the advection terms of the N-S equations and the first derivatives in the advection terms of the scalar temperature and concentration equations.The remaining first derivatives of the streamfunction(velocity),temperature,and concentration variables used in the governing equations are discretized by the fourth-order compact Pade(SCD4)schemes.With the temperature and concentration variables and their approximate values of the first derivatives obtained by the SCD4 schemes,the explicit fourth-order compact schemes are suggested to approximate the second derivatives of temperature and concentration in the diffusion terms of the energy and concentration equations.The discretization of the temporal term is executed with the second-order Crank-Nicolson(C-N)scheme.To assess the spatial behavior capability of the established numerical algorithm and verify the developed computer code,the DDC flow is numerically solved.The obtained results agree well with the benchmark solutions and some accurate results available in the literature,verifying the accuracy,effectiveness,and robustness of the provided algorithm.Finally,a preliminary application of the proposed method to the DDC is carried out.
基金supported by the National Basic Research Program of China(Grant No.2012CB417400)the National Natural Science Foundation of China(Grant Nos.41421005,U1406401)
文摘A streamfunction projection method called gravest empirical mode(GEM) is applied to the hydrographic section at 137°E to filter out eddy noises in the western North Pacific and derive quantitative ensemble-average water mass properties in the North Equatorial Current region. The GEM fields capture more than 80% of total property variances in the thermocline layer. The core layer structures of key water masses, including the North Pacific Tropical Water(NPTW) and the North Pacific Intermediate Water(NPIW), are examined with a definition of water mass boundary based on property gradient. It shows that a tongue of maximal root-mean-square(RMS) residual exists in the upper half of NPIW for all water properties. These subsurface RMS tongues appear to be close to sharp property gradients. It is the first time a GEM diagnosis is applied to nutrient data, which reveals a drastic difference of N/P reaction rate ratio above and below the maximal-nutrient core at 1250 m. Additionally, a GEM velocity reconstruction successfully produces the North Equatorial Undercurrent(NEUC), demonstrating the stable thermal-wind nature of this newly-discovered current.
文摘In this paper we consider a geometric inverse problem which requires detecting an unknown obstacle such as a submarine or an aquatic mine immersed in a Stokes slow viscous stationary flow of an incompressible fluid,from a single set of Cauchy(fluid velocity and stress force)boundary measurements.The numerical reconstruction is based on the method of fundamental solutions(MFS)for the pressure and streamfunction in two dimensions combined with regularization.Numerical results are presented and discussed.
基金supported by the National Key Research and Development Program of China[grant number 2016YFC1401803]the National Natural Science Foundation of China[grant numbers 41976019 and 42076020]+3 种基金the Strategic Priority Research Program of the Chinese Academy of Sciences[grant number XDA20060502]the open project of the State Key Laboratory of Tropical Oceanography,South China Sea Institute of Oceanology,Chinese Academy of Sciences[grant number LTO1910]the Research Program of the Southern Marine Science and Engineering Guangdong Laboratory(Guangzhou)[grant number GML2019ZD0306]the Key Research Program of the Chinese Academy of Sciences[grant number ZDRW-XH-2019-2]。
文摘The calculation of the meridional overturning streamfunction in the southern Indian Ocean is biased by the Indonesian Throughflow.Therefore,this study applies the vertical overturning streamfunction to diagnose the shallow overturning circulation in the Indian Ocean.Using the Ocean General Circulation Model for the Earth simulator output,improvements with the vertical overturning streamfunction compared with the meridional overturning streamfunction are explored.The results show that the vertical overturning streamfunction smoothly connects the shallow overturning circulations of the northern Indian Ocean and the southern Indian Ocean with the whole cycle of the subtropical cell and the cross-equatorial cell.The vertical overturning streamfunction shows a much cleaner shallow overturning circulation,which is underestimated by the meridional overturning streamfunction.It shows that the shallow overturning circulation has a magnitude of~13 Sv(1 Sv≡106 m 3 s−1),of which the subtropical cell accounts for~8 Sv.In addition,the vertical overturning streamfunction captures a clockwise overturning cell in the upper 600 m layer between 30°S and 34°S.This cell has a magnitude of about−5 Sv and probably corresponds to the wind-forced subtropical gyre.Therefore,the vertical overturning streamfunction provides a new approach for estimating the shallow overturning circulation in the Indian Ocean.
基金supported by the Key Program of the Chinese Academy of Sciences(No.KZZD-EW-05-01)the Supporting Program for Science and Technological Research of China(No.2008BAC37B01)+1 种基金the National Basic Research Program of China(Nos.2012CB417201 and 2009CB421505)the National Natural Sciences Foundation of China(Nos.41205033 and 41175056)
文摘In this study, kinetic energy budget equations of rotational and divergent flow in pressure coordinates are derived on terrain-following coordinates. The new formulation explicitly shows the terrain effects and can be applied directly to model-simulated dynamic and thermodynamic fields on the model's original vertical grid. Such application eliminates interpolation error and avoids errors in virtual weather systems in mountainous areas. These advantages and their significance are demonstrated by a numerical study in terrain-following coordinates of a developing vortex after it moves over the Tibetan Plateau in China.
基金The National Key Research and Development Program of China under contract No.2016YFC0301103the Scientific and Technological Innovation Project financially supported by Qingdao National Laboratory for Marine Science and Technology under contract No.2015ASKJ01+2 种基金the SOA Program on Global Change and Air-Sea Interactions under contract Nos GASI-IPOVAI-03,GASI-IPOVAI-02 and GASI-IPOVAI-01-02the National Natural Science Foundation of China under contract Nos 40476025,41876027and 41506036the Office of Naval Research of United States under contract No.N00014-08-01-0618
文摘The Island Rule, derived from the Sverdrup theory, is widely used to estimate and analyze water transport through a strait. Previous studies presented single-or multi-island rules with either lateral or bottom friction. In this paper,an analytical model of wind-driven circulation is assumed based on linear dynamics. Considering both lateral and bottom friction, the analytic solutions of the transport streamfunction around the islands are derived and the volume transport through the channel is presented. The results are similar to those of Wajsowicz, but the frictional constants represent different values. The analytic solution shows that the relationship between the lateral frictional and bottom frictional dissipation is complex in terms of the frictional constants. To understand the interaction between the two friction types, lateral and bottom friction values were randomly chosen on a barotropic beta plane. The result shows an approximately linear relationship between the lateral and bottom friction in consisting of the combined frictional constants. We studied the effect of the channel width on the transport through the channel. The results show that the friction enhances the flow under some widths, which is similar to the flow behavior when only the lateral friction is considered. We also compared the transport through the channel at different depths and founded that the deeper the water, the smaller the transport reduction ratio when the horizontal eddy viscosity coefficient and the bottom drag coefficient remained constants. To further present the combined role of lateral frictional and bottom frictional dissipation, we compared our model with the model of Wajsowicz for two islands, where only the lateral or bottom friction were considered, with different channel widths. The results showed that the effect of the lateral friction is greater than the bottom friction when the channel is narrow, especially in the Munk boundary layer thickness. When the channel is much wider than the Munk boundary layer thickness, the role of the bottom friction is greater than that of the lateral friction. The model was applied to the Indonesian throughflow and yielded a reduction of approximately 20% in the transport.
