Open ocean convection

Plumes are the initial convectively driven vertical motions which are formed during the second phase of convection. They have horizontal scales between 100m and 1 km and their vertical scale is around 1–2 km with vertical velocities of up to 10 cm/s which are measured by acoustic doppler current profilers (ADCPs). Time scales associated with convective plumes are reported to be several hours to several days.[3][11][12]

The plumes act as “conduits” or as “mixing agents” in terms of their dynamic part. If they act as “conduits”, they transport cooled and dense surface water downward. This is the main mechanism of water transport toward lower depths and its renewal. However, plumes can act as “mixing agents” rather than as downward carriers of a flow. In this case, the convection cools and mixes a patch of water, creating a dense homogeneous cylinder, like a chimney, which ultimately collapses and adjusts under planetary rotation and gravity.[13]

Coriolis force and thermobaricity are important in deep convective plumes. Thermobaricity is the effect in which sinking cold saline water is formed under freezing conditions, resulting in downward acceleration. Additionally, many numerical and tank modeling experiments examine the role of rotation in the processes of the convection and in the morphology of the plumes. According to Paluszkiewicz et al. (1994), planetary rotation does not affect the individual plumes vertically, but do so horizontally. Under the influence of rotation, the diameter of the plumes becomes smaller compared to the diameter of the plumes in the absence of rotation. In contrast, chimneys and associated eddies are dominated by the effects of rotation due to thermal wind.[11]

The convective overturning of the water column occurs through a contribution of a high number of intense plumes, which strongly mix the column. The plumes can process large volumes of fluid to form what has become known as a “chimney” of homogenized fluid.[14] These vertically isolated homogenized water columns have a diameter of 10 to 50 km and they are 1–2 km deep. Surface waters growing denser and sinking drive the initial deepening stage, while the final deepening stage and the restratification phase are affected by a buoyancy transfer, through the lateral surface of the chimney, by baroclinic eddies.[2]

The chimneys of deep convection remain open for one to three months, during winter, in a quasi-stable state whereas they can collapse within a few weeks. The chimneys are destroyed in early spring when the sea-surface buoyancy flux weakens and reverses while the stratification of the water’s layers under the mixed layer starts becoming stable.[2]

Formation of the convection chimneys is preconditioned by two processes: strong heat fluxes from the sea-surface and cyclonic circulation. A chimney is formed when a relatively strong buoyancy flux from the ocean’s surface exists for at least 1 to 3 days. The time, depth and diameter development of the chimney clearly depends on the buoyancy flux and the stratification of the surrounding ocean.[2] As the surface water cools, it becomes denser and it overturns, forming a convectively modified layer of depth ${\displaystyle h}$. In the center of the chimney, the mixed layer deepens and the depth as a function of time is computed as described below.

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:

$${\displaystyle b=-g{\frac {\sigma }{\rho _{o}}}}$$

Where ${\displaystyle g}$ is the acceleration due to gravity, ${\displaystyle \sigma }$ the potential density and ${\displaystyle \rho _{o}}$a constant reference value of density. The buoyancy equation for the mixed layer is:

$${\displaystyle {Db \over Dt}=F_{B}}$$

Where ${\displaystyle b}$ is the buoyancy and ${\displaystyle F_{B}}$ the buoyancy forcing. The buoyancy forcing is equal to ${\displaystyle {\frac {\partial {B}}{\partial {z}}}}$where ${\displaystyle B}$ is the buoyancy loss. As a simplification, the assumption that the buoyancy loss is constant in time (${\displaystyle B_{o}}$) is used. Neglecting horizontal advection and integrating the above equation over the mixed layer, we obtain:

$${\displaystyle {\partial {b} \over \partial z}{\partial {h} \over \partial t}={\frac {B_{o}}{h}}}$$

For a uniformly stratified fluid, the power of the buoyancy frequency is equal to:

$${\displaystyle N^{2}={\frac {\partial {b}}{\partial {z}}}}$$

Therefore, the classical result for the nonpenetrative deepening of the upper mixed layer is:[14]

$${\displaystyle h={\frac {\sqrt {2B_{o}t}}{N}}}$$

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 ${\displaystyle r_{o}}$ and a time-dependent height ${\displaystyle h(t)}$. The driving force of the chimney deepening is the surface buoyancy loss ${\displaystyle B_{o}}$ 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 ${\displaystyle b'}$ of a particle which is displaced a distance Δz within the chimney is:

$${\displaystyle b'=N^{2}(z-h(t)),\;\;\;\;\;\;\;\;\;0<z<h(t)}$$

According to Kovalevsky et al. (2020) the buoyancy budget equation is:[2]

$${\displaystyle {d \over dt}\int _{V(t)}^{}b'dV=\int _{S_{top}}^{}B_{o}dS-\int _{S_{lat}}^{}{\overline {u'b'}}dS}$$

The left-hand side represents the time evolution of the total buoyancy anomaly accumulated in the time-depended chimney volume ${\displaystyle V(t)}$. 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.[2]

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.[2]

Visbeck et al. (1996), by using a suggestion by Green (1970) and Stone (1972), parameterized the eddy flux ${\displaystyle {\overline {u'b'}}}$as:

$${\displaystyle {\overline {u'b'}}=\alpha '{\frac {\overline {(b')^{2}}}{N}}}$$

Where ${\displaystyle \alpha '}$ is a constant of proportionality to be determined by observations and laboratory modeling. The variable ${\displaystyle u'}$represents the pulsations of the horizontal current velocity component perpendicular to the side-walls of the chimney while, by following Visbeck et al. (1996), ${\displaystyle {\overline {(b')^{2}}}}$ is equal to:[2]

$${\displaystyle {\overline {(b')^{2}}}=N^{4}h^{2}}$$

If the buoyancy loss is maintained for a sufficient time period, then the sea-surface cooling weakens and the restratification phase starts. At the surrounding of the convective regime, the stratification takes up an ambient value while in the center of the chimney the stratification is eroded away. As a result, around the periphery of the chimney, the isopycnal surfaces deviate from their resting level, tilting towards the ocean’s surface. Associated with the tilting isopycnal surfaces a thermal wind is set up generating the rim current around the edge of the convection regime. This current must be in thermal wind balance with the density gradient between the chimney’s interior and exterior. The width of the rim current’s region and its baroclinic zone will initially be of the order of the Rossby radius of deformation.[12][14]

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 ${\displaystyle {\sqrt {t}}}$, 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:[2][14]

$${\displaystyle \int _{S_{top}}^{}B_{o}dS=\int _{S_{lat}}^{}{\overline {u'b'}}dS}$$

By solving the above equation, the final depth of the convective chimney can be found to be:

$${\displaystyle h_{final}=\gamma {\frac {(B_{o}r_{o})^{1/3}}{N}},\;\;\;\;\;\;\;\;\;\gamma =\left({\frac {1}{2\alpha '}}\right)^{1/3}}$$

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.[14] The length scale of baroclinic eddies, assumed to be set by the Rossby radius of deformation, scale as:[14]

The stages and properties of the convection process

$${\displaystyle L_{eddy,final}={\frac {Nh_{final}}{f}}=\gamma {\frac {(B_{o}r_{o})^{1/3}}{f}}}$$

Which does depend on the rate of rotation f but is independent of the ambient stratification.

The least time that the chimney needs to reach the quasi-equilibrium state is equivalent with the time that it needs to reach the depth ${\displaystyle h_{final}}$ and it is equal to:[14]

$${\displaystyle t_{final}=\beta \left({\frac {r_{o}^{2}}{B_{o}}}\right)^{1/3},\;\;\;\;\;\;\;\;\;\beta ={\frac {\gamma ^{2}}{2}}}$$

The final timescale is independent of the rate of rotation, increases with the radius of the cooling region r and decreases with the surface buoyancy flux B_o. According to Visbeck at al. (1996), the constant of proportionality γ and β are found to be equal to 3.9 ± 0.9 and 12 ± 3 respectively, through laboratory experiments.[14]

Finally, the cooling of the surface as well as convective activity cease. Therefore, the chimney of homogenized cold water erodes into several small conical structures, named cones, which propagate outward. The cones travel outward, carrying cold water far from the area of cooling. As time progresses and the cones disperse, the magnitude of the rim current diminishes. The currents associated with the cones are intensified and cyclonic at the surface whereas they are weaker and anticyclonic at low depths.[15]