📄 earth_science

Simulating deep convective plumes in the framework of a Quasi-Non-Hydrostatic modelling: a heuristic approach

This paper proposes a heuristic "Quasi-Non-Hydrostatic" (QNH) modeling framework that preserves horizontal diffusion of vertical velocity to realistically simulate deep convective plumes in regional ocean models, offering a computationally efficient compromise between standard hydrostatic approximations and full non-hydrostatic systems.

Original authors: Pierre Garreau

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pierre Garreau

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the ocean as a giant, invisible bathtub. Usually, the water in this tub is calm and layered, like a parfait with distinct flavors of cold and warm. But sometimes, especially in winter, the air above gets so cold and dry that it steals heat from the surface. This makes the top layer of water heavy and dense, like a rock dropping into a pool. In the real world, this heavy water doesn't just sink straight down; it forms twisting, swirling columns called convective plumes. These plumes are like underwater tornadoes, diving deep while pushing lighter, warmer water up around them. They are the ocean's way of ventilating its deep corners, a process crucial for the planet's climate.

The problem is that computer models used to predict the ocean are often like low-resolution photos. They see the ocean in big, blocky pixels (usually 5 to 10 kilometers wide). When the heavy water tries to sink, these big blocks can't see the tiny, twisting plumes. Instead of forming a beautiful, organized swirl, the computer gets confused. It starts generating "ghost" movements—spurious, chaotic vertical speeds that shoot up to over 1 m/s (meters per second). That's like a waterfall appearing out of nowhere in a calm lake! The model essentially breaks, creating a noisy, checkerboard mess of fake physics that no one can trust.

For years, scientists tried to fix this by forcing the water to mix instantly whenever it got unstable, or by turning up the "vertical diffusion" (a fancy way of saying "let's stir it really hard"). But these methods are like using a sledgehammer to fix a watch; they get the result (mixed water) but miss the actual dance of the plumes. Others suggested using Non-Hydrostatic (NH) models, which solve the full, incredibly complex equations of fluid motion. While these are the most accurate, they are also so computationally expensive that they are like trying to run a supercomputer simulation on a toaster. They are too slow for the big, realistic maps scientists need.

Enter Pierre Garreau's Quasi-Non-Hydrostatic (QNH) approach. Think of this as a clever hack, a "heuristic" shortcut. Garreau suggests that the main reason the computer gets confused isn't just because it's ignoring the vertical acceleration (the "push" of the water), but because it's forgetting to let the sinking plumes "breathe" sideways.

In the real world, as a dense plume sinks, it doesn't stay a skinny, perfect cylinder. It widens, trading properties with the water around it, slowing down as it goes. Garreau's model adds a specific ingredient: horizontal diffusion of the vertical velocity. Imagine the sinking water column as a group of people running down a hallway. In a standard model, they run in a tight, rigid line and crash into the walls. In Garreau's QNH model, the runners are allowed to spread out, bump into their neighbors, and slow down a bit as they widen. This "spreading out" is controlled by a mathematical rule inspired by Large Eddy Simulation (LES), which acts like a smart filter.

When Garreau ran this new setup in a high-resolution simulation (using a grid size of 100 m), the magic happened. The chaotic, fake vertical speeds vanished. Instead, the model produced realistic plumes with vertical speeds between 0.10 m/s and 0.20 m/s (and peaking around 0.25 m/s), which matches what scientists actually observe in the ocean. The plumes organized themselves into beautiful, hexagonal or polygonal patterns at the surface, just like the ones seen in satellite radar images of the Greenland Sea.

The paper explicitly argues against the idea that we need the full, heavy machinery of Non-Hydrostatic modeling (including the local vertical acceleration term, $\partial w/\partial t$ ) to get these results. In fact, the simulations suggest that term plays only a secondary role for established deep convection. By removing that heavy term and focusing on the horizontal diffusion, the QNH model runs much faster—almost as fast as the old, broken hydrostatic models—but produces results that look almost identical to the expensive, full Non-Hydrostatic simulations.

The study confirms that the size of these plumes depends on the depth of the ocean. In a 2000 m deep basin, the plumes organize into structures roughly 1 km wide. If the ocean were 3000 m deep, the plumes would be even wider. The model suggests that to see these plumes clearly, you need a grid resolution of about 100 m, which is becoming possible with modern supercomputers.

So, what is the verdict? The paper doesn't claim to have solved the ocean's deepest mysteries forever, nor does it say this is a "perfect" solution for every single scenario. Instead, it suggests a very promising compromise. It shows that by tweaking how we let the water spread sideways (horizontal diffusion), we can simulate realistic, deep-diving plumes without needing a supercomputer the size of a city. It's a way to get the best of both worlds: the speed of the old models and the realistic, swirling beauty of the new ones, all while keeping the "ghost" vertical velocities in check. This approach could even help us understand smaller, local events, like how wastewater plumes spread near the coast, without needing complex, empirical workarounds.

Technical Summary: Simulating Deep Convective Plumes in a Quasi-Non-Hydrostatic Framework

Problem Statement
Modeling deep, partial, or diurnal ocean convection within Regional or Coastal Circulation Models remains a significant challenge. Deep convection, driven by negative buoyancy fluxes at the air-sea interface, generates convective plumes with non-negligible vertical velocities (order of 10 cm/s) and horizontal scales of hundreds of meters to kilometers. Standard Ocean General Circulation Models (OGCMs) and Regional Circulation Models (RCMs) typically rely on the Hydrostatic Primitive Equation (HPE) framework. In this framework, vertical velocity is diagnostically calculated rather than prognostically solved, rendering the models incapable of realistically simulating the intense vertical dynamics of convection.

To compensate, models often employ "convective adjustment" algorithms or artificially enhance vertical turbulent diffusion coefficients (e.g., to $10 \, m^2s^{-1}$ ) to restore stability. These methods are physically artificial; they homogenize the water column without representing the dynamical updrafts and downdrafts. Conversely, while full Non-Hydrostatic (NH) models solve the complete vertical momentum equation, they are computationally expensive and often require resolutions that are not yet feasible for realistic regional simulations. Furthermore, high-resolution HPE simulations often suffer from "grid instability," producing spurious, unrealistic vertical velocities (exceeding $1 \, ms^{-1}$ ) that exhibit a checkerboard distribution, particularly when horizontal diffusion is minimized to resolve sub-mesoscale features.

Methodology
The paper revisits the idealized deep convection test case originally proposed by Jones and Marshall (1993) using the MARS3D hydrostatic code. The study introduces a heuristic approach termed Quasi-Non-Hydrostatic (QNH) modelling. The methodology involves three key steps:

Idealized Setup: A 2,500 km² square domain with a 2,000 m deep non-stratified ocean is subjected to a localized surface heat loss (400 $Wm^{-2}$ ) over a 15 km radius disk. The experiment utilizes a high horizontal resolution (100 m) and 60 vertical levels.
Addressing Grid Instability: The study identifies that the lack of physical control over the horizontal scale of convection in HPE models leads to grid-scale instabilities. It proposes that horizontal diffusion of momentum and tracers is the missing physical process required to organize convection.
The QNH Formulation: The core innovation is the addition of the horizontal diffusion terms of the vertical velocity ( $w$ $w$ ) into the model, effectively introducing a non-hydrostatic perturbation of buoyancy ( $b_{nh}$ $b_{nh}$ ).
- The full non-hydrostatic vertical momentum equation includes local acceleration ( $\partial w/\partial t$ ), advection, Coriolis, and diffusion terms.
- The QNH approach retains the diffusion terms (specifically horizontal diffusion of $w$ ) but removes the local acceleration term ( $\partial w/\partial t$ ).
- This allows $w$ to remain a diagnostic variable (avoiding the need to solve a costly Poisson equation for pressure) while accounting for the "detrainment" of plumes—the physical process where a sinking plume widens and slows as it exchanges momentum with the surrounding fluid via turbulent diffusion.
- The horizontal diffusion coefficient is parameterized using a Large Eddy Simulation (LES) approach (Smagorinsky formulation) that includes the horizontal shear of the vertical velocity, a term often neglected in standard ocean models.

Key Contributions

Identification of Horizontal Diffusion: The paper demonstrates that horizontal diffusion is the critical mechanism for controlling the size and velocity of convective plumes in numerical models, preventing grid-scale noise.
The QNH Framework: It introduces a computationally efficient modification to standard hydrostatic codes. By retaining only the diffusion terms of the non-hydrostatic buoyancy perturbation, the model captures the physics of plume detrainment without the heavy computational cost of solving the full non-hydrostatic system.
LES Parameterization: The study applies a specific LES-type parameterization for horizontal diffusion that accounts for the shear of vertical velocity, which is essential for simulating the widening and deceleration of plumes.

Results
The QNH model was tested against the idealized case and compared with full Non-Hydrostatic (NH) simulations (using the CROCO-NBQ code) and observational data:

Suppression of Spurious Velocities: The QNH approach completely eliminates the unrealistic vertical velocity peaks ( $>1 \, ms^{-1}$ ) seen in high-resolution HPE models. Vertical velocities are organized into realistic plumes with magnitudes between 0.10 and 0.25 $ms^{-1}$ .
Plume Structure: The model successfully simulates the formation of a cold, dense surface lens that collapses into organized convection. The resulting plumes exhibit realistic horizontal scales (dependent on the depth of the convective layer) and vertical structures where downdrafts are stronger than updrafts.
Surface Patterns: The simulation reproduces the polygonal patterns (3 to 6 sides) of surface temperature and sea surface elevation observed in Synthetic Aperture Radar (SAR) images and full NH models. These patterns result from the convergence of cooled surface water into descending plumes and the divergence of warmer upwelling water.
Computational Efficiency: The QNH results are shown to be very close to those of the full NH model (CROCO-NBQ) but with significantly lower computational cost. The local acceleration term ( $\partial w/\partial t$ ) is found to play only a secondary role in established deep convection at this spatial resolution.
Scaling: The horizontal scale of the convective cells is shown to be proportional to the total depth of the convection, consistent with theoretical expectations.

Significance and Claims
The paper claims that the QNH approach offers a practical compromise for the ocean modeling community. It allows for the simulation of realistic convective plumes and vertical fluxes without the prohibitive computational time required for full non-hydrostatic simulations or the physical inaccuracies of convective adjustment schemes.

The authors assert that the horizontal diffusion of vertical velocity is the key process in controlling convection in the "gray zone" of resolution (where hydrostatic assumptions begin to fail but full turbulence resolution is not yet possible). By incorporating this heuristic, models can better represent the transfer of buoyancy from the surface to the deep ocean via realistic downdrafts and updrafts, rather than instantaneous mixing. The paper concludes that this method is a viable alternative for regional models and could also be applied to other problems involving buoyant releases, such as wastewater discharge, by managing both the local buoyant jet and far-field advection within a single framework.

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