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averaging in isopycnal coordinates. However, note that, following Nurser and Lee (2004) , the general case is discussed here, where isosurfaces of α define the material surfaces in Eq. (7) and where α is a general property of the fluid subject to the general conservation equation Dα / Dt = Q . The choice of α as semi-Lagrangian coordinate is motivated by the fact that α is conserved for vanishing forcing Q , such that particles stay on the material surfaces during their movements; only
averaging in isopycnal coordinates. However, note that, following Nurser and Lee (2004) , the general case is discussed here, where isosurfaces of α define the material surfaces in Eq. (7) and where α is a general property of the fluid subject to the general conservation equation Dα / Dt = Q . The choice of α as semi-Lagrangian coordinate is motivated by the fact that α is conserved for vanishing forcing Q , such that particles stay on the material surfaces during their movements; only
1. Introduction The problem of how best to construct a quasi-neutral density variable suitably corrected for pressure is a longstanding fundamental issue in oceanography whose answer is vital for many key applications ranging from the study of mixing to ocean climate studies. These include but are not limited to the separation of mixing into “isopycnal” and “diapycnal” components necessary for the construction of rotated diffusion tensors in numerical ocean models ( Redi 1982 ; Griffies 2004
1. Introduction The problem of how best to construct a quasi-neutral density variable suitably corrected for pressure is a longstanding fundamental issue in oceanography whose answer is vital for many key applications ranging from the study of mixing to ocean climate studies. These include but are not limited to the separation of mixing into “isopycnal” and “diapycnal” components necessary for the construction of rotated diffusion tensors in numerical ocean models ( Redi 1982 ; Griffies 2004
(2) D u D t + f × u = − ∇ ϕ , b = ∂ ϕ ∂ z , where b = − gδρ / ρ 0 is the buoyancy. The adiabatic thermodynamic equation is Db / Dt = 0. For adiabatic motions, the isopycnal coordinate is also the isentropic coordinate. Transforming the horizontal pressure gradient to the isentropic coordinates, we have (3) ( ∂ ϕ ∂ x ) z = ( ∂ ϕ ∂ x ) b − ( ∂ z ∂ x ) b ∂ ϕ ∂ z = ( ∂ ϕ ∂ x ) b − b ( ∂ z ∂ x ) b = ( ∂ M ∂ x ) b , where M = ϕ − zb , so ∂ M /∂ b = − z . Thus, the horizontal
(2) D u D t + f × u = − ∇ ϕ , b = ∂ ϕ ∂ z , where b = − gδρ / ρ 0 is the buoyancy. The adiabatic thermodynamic equation is Db / Dt = 0. For adiabatic motions, the isopycnal coordinate is also the isentropic coordinate. Transforming the horizontal pressure gradient to the isentropic coordinates, we have (3) ( ∂ ϕ ∂ x ) z = ( ∂ ϕ ∂ x ) b − ( ∂ z ∂ x ) b ∂ ϕ ∂ z = ( ∂ ϕ ∂ x ) b − b ( ∂ z ∂ x ) b = ( ∂ M ∂ x ) b , where M = ϕ − zb , so ∂ M /∂ b = − z . Thus, the horizontal
formulation, but the value of this form of potential energy has been largely overlooked. Indeed, Hughes et al. (2009) , von Storch et al. (2012) , Zemskova et al. (2015) , and others discuss the oceanic energy cycle in z or pressure coordinates. Isopycnal models [such as the Miami Isopycnal Coordinate Ocean Model (MICOM)], introduced in oceanography by Bleck and Boudra (1981) and much developed since then (see Bleck 2002 ; Hallberg 1995 ), calculate potential energy in density coordinates
formulation, but the value of this form of potential energy has been largely overlooked. Indeed, Hughes et al. (2009) , von Storch et al. (2012) , Zemskova et al. (2015) , and others discuss the oceanic energy cycle in z or pressure coordinates. Isopycnal models [such as the Miami Isopycnal Coordinate Ocean Model (MICOM)], introduced in oceanography by Bleck and Boudra (1981) and much developed since then (see Bleck 2002 ; Hallberg 1995 ), calculate potential energy in density coordinates
yellow shaded boxes alone) are highlighted (green shaded boxes). Note that the atmospheric chemistry and aerosol distributions in the versions described here are externally specified. These independent frameworks for describing the ocean allow us to explore the sensitivity of the ocean response under anthropogenic CO 2 forcing and climate change to ocean configuration. In traditional depth coordinates, the ocean is divided into boxes in which horizontally adjacent pressures interact, typically
yellow shaded boxes alone) are highlighted (green shaded boxes). Note that the atmospheric chemistry and aerosol distributions in the versions described here are externally specified. These independent frameworks for describing the ocean allow us to explore the sensitivity of the ocean response under anthropogenic CO 2 forcing and climate change to ocean configuration. In traditional depth coordinates, the ocean is divided into boxes in which horizontally adjacent pressures interact, typically
is based on the Generalized Ocean Layer Dynamics (GOLD) ocean code, which is presented in detail in Adcroft and Hallberg (2006) . It is configured using isopycnal coordinates, which eliminates the spurious diapycnal mixing produced by numerical ocean models that use a fixed vertical coordinate ( Ilicak et al. 2012 ). a. Model configuration We model a representative section of the Southern Ocean by a 9600 × 1600 km 2 channel on a β plane (with β = 1.5 × 10 −11 m −1 s −1 ), with a
is based on the Generalized Ocean Layer Dynamics (GOLD) ocean code, which is presented in detail in Adcroft and Hallberg (2006) . It is configured using isopycnal coordinates, which eliminates the spurious diapycnal mixing produced by numerical ocean models that use a fixed vertical coordinate ( Ilicak et al. 2012 ). a. Model configuration We model a representative section of the Southern Ocean by a 9600 × 1600 km 2 channel on a β plane (with β = 1.5 × 10 −11 m −1 s −1 ), with a
oceanic mesoscale eddies, it has become common to introduce a vertically Lagrangian (VL) coordinate system, using density or neutral density for the vertical coordinate (e.g., isopycnal or isoneutral coordinates), following the example from the atmospheric literature where potential temperature is typically used ( Andrews et al. 1987 ). In the horizontal the standard Eulerian coordinates are retained. Equations averaged in isopycnal coordinates have been widely used to develop parameterizations of the
oceanic mesoscale eddies, it has become common to introduce a vertically Lagrangian (VL) coordinate system, using density or neutral density for the vertical coordinate (e.g., isopycnal or isoneutral coordinates), following the example from the atmospheric literature where potential temperature is typically used ( Andrews et al. 1987 ). In the horizontal the standard Eulerian coordinates are retained. Equations averaged in isopycnal coordinates have been widely used to develop parameterizations of the
trunk in the three-model ensemble where an atmosphere swap, AM3 for AM2, leads to the CM3 branch, and an ocean swap, the Generalized Ocean Layer Dynamics (GOLD) isopycnal model for the depth-based Modular Ocean Model (MOM), leads to the ESM2G branch. This three-model ensemble is well suited to distinguish the influence of the atmosphere and ocean on the sensitivities. Table 2. Summary of GFDL climate model components. The development of an isopycnal ocean component for the climate model was
trunk in the three-model ensemble where an atmosphere swap, AM3 for AM2, leads to the CM3 branch, and an ocean swap, the Generalized Ocean Layer Dynamics (GOLD) isopycnal model for the depth-based Modular Ocean Model (MOM), leads to the ESM2G branch. This three-model ensemble is well suited to distinguish the influence of the atmosphere and ocean on the sensitivities. Table 2. Summary of GFDL climate model components. The development of an isopycnal ocean component for the climate model was
along streamlines, as in de Szoeke and Levine (1981) . The technical details of this coordinate transform are described in Viebahn and Eden (2010) , while its influence on the resulting overturning are discussed in Treguier et al. (2007) . For clarity, in the following discussion, we will use the notation s and η as zonal and meridional stream coordinates, in place of the usual x and y notation used in the majority of the literature (e.g., Marshall and Radko 2003 ). On an isopycnal
along streamlines, as in de Szoeke and Levine (1981) . The technical details of this coordinate transform are described in Viebahn and Eden (2010) , while its influence on the resulting overturning are discussed in Treguier et al. (2007) . For clarity, in the following discussion, we will use the notation s and η as zonal and meridional stream coordinates, in place of the usual x and y notation used in the majority of the literature (e.g., Marshall and Radko 2003 ). On an isopycnal
sigma coordinates are used. The horizontal resolution is horizontal triangular spectral truncation at total wavenumber 42 (T42) with 40 vertical layers. The physical parameterizations incorporate a higher-order turbulence closure, prognostic cloud scheme, cloud microphysics, and a prognostic scheme for number concentrations of cloud droplets and ice crystals [see Watanabe et al. (2010) for the individual schemes]. The convection scheme of Chikira and Sugiyama (2010) is newly developed to control
sigma coordinates are used. The horizontal resolution is horizontal triangular spectral truncation at total wavenumber 42 (T42) with 40 vertical layers. The physical parameterizations incorporate a higher-order turbulence closure, prognostic cloud scheme, cloud microphysics, and a prognostic scheme for number concentrations of cloud droplets and ice crystals [see Watanabe et al. (2010) for the individual schemes]. The convection scheme of Chikira and Sugiyama (2010) is newly developed to control
