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1. Introduction Advection is a fundamental physical process associated with many fluid applications, such as mass, energy, and tracer transportation in atmospheric and oceanic flows. The conservation of a fluid mass ψ with velocity v is represented by the advection equation: where t represents the time and the space gradient operator is over all space dimensions. Translation of the advection equation into an accurate discrete form is, however, not straightforward though modelers have been
1. Introduction Advection is a fundamental physical process associated with many fluid applications, such as mass, energy, and tracer transportation in atmospheric and oceanic flows. The conservation of a fluid mass ψ with velocity v is represented by the advection equation: where t represents the time and the space gradient operator is over all space dimensions. Translation of the advection equation into an accurate discrete form is, however, not straightforward though modelers have been
(exponential in time) separation between neighboring fluid particle trajectories. 1 As in the case of the polar vortex, the generation of small scales is controlled by long-lived, coherent structures, and the stirring process is therefore referred to as chaotic advection. By “long lived,” one generally assumes that the coherent structures remain intact over a time period long in comparison with the typical parcel winding time. It is not uncommon for flows undergoing chaotic advection to contain “barriers
(exponential in time) separation between neighboring fluid particle trajectories. 1 As in the case of the polar vortex, the generation of small scales is controlled by long-lived, coherent structures, and the stirring process is therefore referred to as chaotic advection. By “long lived,” one generally assumes that the coherent structures remain intact over a time period long in comparison with the typical parcel winding time. It is not uncommon for flows undergoing chaotic advection to contain “barriers
basic hydrodynamics ( Lerczak and Geyer 2004 ). The lateral structure of tidally averaged lateral advection ( υu y + wu z , where u , υ , and w are velocity components in along-estuary, cross-estuary, and vertical directions, respectively, and the subindex indicates partial differentiation) could be of similar shape to the pressure gradient and could help reinforce the two-layer estuarine circulation. In an estuary with asymmetric transverse variability of bathymetry, the tidally averaged
basic hydrodynamics ( Lerczak and Geyer 2004 ). The lateral structure of tidally averaged lateral advection ( υu y + wu z , where u , υ , and w are velocity components in along-estuary, cross-estuary, and vertical directions, respectively, and the subindex indicates partial differentiation) could be of similar shape to the pressure gradient and could help reinforce the two-layer estuarine circulation. In an estuary with asymmetric transverse variability of bathymetry, the tidally averaged
virtually all other global models defined on the QU grids, applies a vector-invariant (“weak conservative”) formulation to describe the momentum advection. On the other hand, consistently conservative dynamics, with the highlight in an Arakawa-type nonlinear momentum advection scheme written in an advective (“strong conservative”) form, is a common feature of both the regional Eta Model and the Nonhydrostatic Mesoscale Model (NMM) Weather Research and Forecasting (WRF), the NCEP’s new regional model
virtually all other global models defined on the QU grids, applies a vector-invariant (“weak conservative”) formulation to describe the momentum advection. On the other hand, consistently conservative dynamics, with the highlight in an Arakawa-type nonlinear momentum advection scheme written in an advective (“strong conservative”) form, is a common feature of both the regional Eta Model and the Nonhydrostatic Mesoscale Model (NMM) Weather Research and Forecasting (WRF), the NCEP’s new regional model
can be computed from a stable numerical advection scheme in a uniform flow, guaranteeing the initial shape of the tracer field is preserved in the absence of numerical diffusion ( Rasch and Williamson 1989 ). Indeed, numerical diffusion is introduced via interpolation errors when a field variable is approximated at intermediate positions in a given domain; hence, accuracy of spatial estimation is limited by the order of interpolation. Computationally efficient and accurate advection schemes have
can be computed from a stable numerical advection scheme in a uniform flow, guaranteeing the initial shape of the tracer field is preserved in the absence of numerical diffusion ( Rasch and Williamson 1989 ). Indeed, numerical diffusion is introduced via interpolation errors when a field variable is approximated at intermediate positions in a given domain; hence, accuracy of spatial estimation is limited by the order of interpolation. Computationally efficient and accurate advection schemes have
through latent heating and advection, and aerosols affect atmospheric radiation and cloud formation. Conservation is one requirement in the advection of such tracers, and a second requirement is to eliminate physically impossible negative values. Conservation of tracers can be achieved by using the finite-volume (FV) method to derive the discrete form of operators. Second-order-centered schemes have been commonly used (e.g., Tomita et al. 2001 ). However, second-order-centered schemes are influenced
through latent heating and advection, and aerosols affect atmospheric radiation and cloud formation. Conservation is one requirement in the advection of such tracers, and a second requirement is to eliminate physically impossible negative values. Conservation of tracers can be achieved by using the finite-volume (FV) method to derive the discrete form of operators. Second-order-centered schemes have been commonly used (e.g., Tomita et al. 2001 ). However, second-order-centered schemes are influenced
et al. 1986 ), multiple-Doppler thermodynamic retrievals (e.g., Gal-Chen 1978 ; Hane et al. 1981 ; Roux 1985 ; Parsons et al. 1987 ; Sun and Houze 1992 ; Crook 1994 ; Liou 2001 ), and multiple-radar merged reflectivity analyses ( Germann and Zawadzki 2002 ; Lakshmanan et al. 2006 ; Langston et al. 2007 ; Yang et al. 2009 ). Many of these algorithms are prone to substantial error if advection effects are not adequately accounted for or if the evolution time scale is comparable to (or
et al. 1986 ), multiple-Doppler thermodynamic retrievals (e.g., Gal-Chen 1978 ; Hane et al. 1981 ; Roux 1985 ; Parsons et al. 1987 ; Sun and Houze 1992 ; Crook 1994 ; Liou 2001 ), and multiple-radar merged reflectivity analyses ( Germann and Zawadzki 2002 ; Lakshmanan et al. 2006 ; Langston et al. 2007 ; Yang et al. 2009 ). Many of these algorithms are prone to substantial error if advection effects are not adequately accounted for or if the evolution time scale is comparable to (or
the density surfaces having a different slope than the local neutral tangent plane. These slope errors cause a fictitious diapycnal diffusivity ( Klocker et al. 2009a , b ) and an additional diapycnal advection ( Klocker and McDougall 2010 ), which have to be taken into account when using the density conservation equation on a continuous density surface. Recently, an algorithm has been introduced that significantly minimizes the slope errors between neutral tangent planes and density surfaces
the density surfaces having a different slope than the local neutral tangent plane. These slope errors cause a fictitious diapycnal diffusivity ( Klocker et al. 2009a , b ) and an additional diapycnal advection ( Klocker and McDougall 2010 ), which have to be taken into account when using the density conservation equation on a continuous density surface. Recently, an algorithm has been introduced that significantly minimizes the slope errors between neutral tangent planes and density surfaces
1. Introduction Advection or transport of trace substances and conserved properties by fluid is one of the most important processes to be represented in spatially gridded atmospheric models of many scales from boundary layer models to global climate models. Many numerical schemes are available (see, e.g., the review by Rood 1987 ). With models striving to maintain sharp spatial gradients on a grid whose resolution is severely limited by computational constraints, first-order schemes are
1. Introduction Advection or transport of trace substances and conserved properties by fluid is one of the most important processes to be represented in spatially gridded atmospheric models of many scales from boundary layer models to global climate models. Many numerical schemes are available (see, e.g., the review by Rood 1987 ). With models striving to maintain sharp spatial gradients on a grid whose resolution is severely limited by computational constraints, first-order schemes are
improve the advection calculation on the hexagonal geodesic grid ( Lipscomb and Ringler 2005 ; Miura 2007 ; Skamarock and Menchaca 2010 ). To our knowledge, the existing finite-volume schemes of flux form that make use of the cell-integrated average as the computational variable possess second-order accuracy at most. Our previous studies show that locally increasing the degrees of freedom (DOFs) is a practical and efficient alternative to construct high-order schemes ( Ii et al. 2005 ; Ii and Xiao
improve the advection calculation on the hexagonal geodesic grid ( Lipscomb and Ringler 2005 ; Miura 2007 ; Skamarock and Menchaca 2010 ). To our knowledge, the existing finite-volume schemes of flux form that make use of the cell-integrated average as the computational variable possess second-order accuracy at most. Our previous studies show that locally increasing the degrees of freedom (DOFs) is a practical and efficient alternative to construct high-order schemes ( Ii et al. 2005 ; Ii and Xiao
