Phenomena involved in convection 196x196px |A schematic representation of an open ocean deep convection chimney. The mixed patch (or chimney) created by convective plumes, the geostrophic eddies which contribute to the exchange of fluid and properties between the mixed patch and the surroundings and the peripheral boundary current (rim current) are shown. Deep convection is distinguished in small-scale and mesoscale processes. Plumes represent the smallest-scale process while chimneys (patch) and eddies represent the mesoscale. During the initial stage of the intensive deepening of the chimney, when the baroclinic instability effects are assumed to be unimportant, the depth can be found as a function of time using the buoyancy forcing. The buoyancy is defined as:Where is the acceleration due to gravity, the potential density and a constant reference value of density. The buoyancy equation for the mixed layer is:Where is the buoyancy and the buoyancy forcing. The buoyancy forcing is equal to where is the buoyancy loss. As a simplification, the assumption that the buoyancy loss is constant in time () is used. Neglecting horizontal advection and integrating the above equation over the mixed layer, we obtain:For a uniformly stratified fluid, the power of the buoyancy frequency is equal to:Therefore, the classical result for the nonpenetrative deepening of the upper mixed layer is: The chimney evolution equation 261x261px |A schematic view of a convective chimney in the stratified fluid As time progresses and the baroclinic instability effects are becoming important, the time evolution of the chimney cannot be described only by the buoyancy forcing. The maximum depth that a convection chimney reaches, must be found using the chimney evolution equation instead. Following Kovalevsky et al. (2020) and Visbeck et al. (1996), consider a chimney of radius and a time-dependent height . The driving force of the chimney deepening is the surface buoyancy loss which causes convective overturning leading to homogeneously mixed fluid in the interior of the chimney. Assuming that the density at the base of the chimney is continuous, the buoyancy anomaly of a particle which is displaced a distance Δz within the chimney is:According to Kovalevsky et al. (2020) the buoyancy budget equation is:The left-hand side represents the time evolution of the total buoyancy anomaly accumulated in the time-depended chimney volume . The first and second term on the right-hand side correspond to the total buoyancy loss from the sea-surface over the chimney and the buoyancy transfer between the interior of the chimney and the baroclinic eddies, respectively. Initially, the total buoyancy depends only on the total buoyancy loss through the sea-surface above the chimney. While time progresses, the buoyancy loss through the sea-surface above the chimney becomes partially equivalent with the lateral buoyancy exchange between the chimney and the baroclinic eddies, through the chimney’s side walls. The existence of the rim current plays an important role for the chimney’s collapse. At the center of the chimney, the mixed layer will deepen as , until the growing baroclinic instability begins to carry convected fluid outward while water from the exterior flows into the chimney. At this moment, the rim current around the cooling region becomes baroclinically unstable and the buoyancy laterally is transferred by the instability eddies. If the eddies are intense enough, deepening of the chimney will be limited. In this limit, when the lateral buoyancy flux completely balances the sea-surface buoyancy loss, a quasi-steady state can be established:By solving the above equation, the final depth of the convective chimney can be found to be: Consequently, the final mixing depth depends on the strength of the cooling, the radius of the cooling and the stratification. Therefore, the final mixing depth is not directly dependent of the rate of the rotation. However, baroclinic instability is a consequence of the thermal wind, which is crucially dependent on rotation. The length scale of baroclinic eddies, assumed to be set by the Rossby radius of deformation, scale as: 413x413px |The stages and properties of the convection process
Phenomena involved in convection 256x256px |A schematic representation of an open ocean deep convection chimney. The mixed patch (or chimney) created by convective plumes, the geostrophic eddies which contribute to the exchange of fluid and properties between the mixed patch and the surroundings and the peripheral boundary current (rim current) are shown. Deep convection is distinguished in small-scale and mesoscale processes. Plumes represent the smallest-scale process while chimneys (patch) and eddies represent the mesoscale. During the initial stage of the intensive deepening of the chimney, when the baroclinic instability effects are assumed to be unimportant, the depth can be found as a function of time using the buoyancy forcing. The buoyancy is defined as:Where is the acceleration due to gravity, the potential density and a constant reference value of density. The buoyancy equation for the mixed layer is:Where is the buoyancy and the buoyancy forcing. The buoyancy forcing is equal to where is the buoyancy loss. As a simplification, the assumption that the buoyancy loss is constant in time () is used. Neglecting horizontal advection and integrating the above equation over the mixed layer, we obtain:For a uniformly stratified fluid, the power of the buoyancy frequency is equal to:Therefore, the classical result for the nonpenetrative deepening of the upper mixed layer is: The chimney evolution equation 304x304px |A schematic view of a convective chimney in the stratified fluid As time progresses and the baroclinic instability effects are becoming important, the time evolution of the chimney cannot be described only by the buoyancy forcing. The maximum depth that a convection chimney reaches, must be found using the chimney evolution equation instead. Following Kovalevsky et al. (2020) and Visbeck et al. (1996), consider a chimney of radius and a time-dependent height . The driving force of the chimney deepening is the surface buoyancy loss which causes convective overturning leading to homogeneously mixed fluid in the interior of the chimney. Assuming that the density at the base of the chimney is continuous, the buoyancy anomaly of a particle which is displaced a distance Δz within the chimney is:According to Kovalevsky et al. (2020) the buoyancy budget equation is:The left-hand side represents the time evolution of the total buoyancy anomaly accumulated in the time-depended chimney volume . The first and second term on the right-hand side correspond to the total buoyancy loss from the sea-surface over the chimney and the buoyancy transfer between the interior of the chimney and the baroclinic eddies, respectively. Initially, the total buoyancy depends only on the total buoyancy loss through the sea-surface above the chimney. While time progresses, the buoyancy loss through the sea-surface above the chimney becomes partially equivalent with the lateral buoyancy exchange between the chimney and the baroclinic eddies, through the chimney’s side walls. The existence of the rim current plays an important role for the chimney’s collapse. At the center of the chimney, the mixed layer will deepen as , until the growing baroclinic instability begins to carry convected fluid outward while water from the exterior flows into the chimney. At this moment, the rim current around the cooling region becomes baroclinically unstable and the buoyancy laterally is transferred by the instability eddies. If the eddies are intense enough, deepening of the chimney will be limited. In this limit, when the lateral buoyancy flux completely balances the sea-surface buoyancy loss, a quasi-steady state can be established:By solving the above equation, the final depth of the convective chimney can be found to be: Consequently, the final mixing depth depends on the strength of the cooling, the radius of the cooling and the stratification. Therefore, the final mixing depth is not directly dependent of the rate of the rotation. However, baroclinic instability is a consequence of the thermal wind, which is crucially dependent on rotation. The length scale of baroclinic eddies, assumed to be set by the Rossby radius of deformation, scale as: 566x566px |The stages and properties of the convection process
R
196x196px
256x256px
R
261x261px
304x304px
R
413x413px
566x566px