Non-Hydrostatic Models

The motion of fluids, from the air we breathe to the oceans that cover our planet, is governed by fundamental physical laws. For decades, scientists have relied on a powerful simplification known as the hydrostatic approximation to model these vast systems. This assumption of a perfect balance between gravity and pressure works remarkably well for large-scale, slowly evolving flows, forming the bedrock of traditional weather and climate models. However, nature is often far from balanced; it is filled with violent updrafts, crashing waves, and turbulent eddies where vertical acceleration is not just present but paramount. This creates a critical knowledge gap: how do we accurately simulate these dynamic, smaller-scale phenomena?

This article delves into the world of non-hydrostatic models, the tools designed to fill this gap by embracing the full complexity of fluid motion. By retaining the vertical acceleration term in the governing equations, these models unlock a more truthful and detailed view of our planet's fluid dynamics. The following chapters will guide you through this complex and fascinating topic. First, "Principles and Mechanisms" will uncover the fundamental physics that distinguish non-hydrostatic from hydrostatic models, exploring the key criteria for their use and the profound computational consequences of this choice. Following that, "Applications and Interdisciplinary Connections" will showcase these models in action, revealing how they are revolutionizing our ability to understand and predict everything from individual thunderstorms and mountain waves to the intensity of hurricanes.

To understand the world of non-hydrostatic models, we must first journey to the heart of how we describe a fluid in motion. At its core, the motion of a fluid, whether it's the air in our atmosphere or the water in our oceans, is governed by a principle we all learn in our first physics class: Newton's second law, $F=ma$. For a small parcel of fluid, this law tells us that its acceleration is the result of the net force acting upon it. In the vertical direction, the story is dominated by a grand tug-of-war between two main forces: gravity, pulling the parcel down, and the pressure-gradient force, arising from differences in pressure above and below the parcel, which can push it either up or down.

Imagine the vast, tranquil expanse of the Pacific Ocean. Over thousands of kilometers, the water moves in slow, majestic gyres. If you were to follow a single parcel of water, you would find its vertical acceleration to be astonishingly small, practically zero. In such large-scale, slowly evolving systems, physicists and oceanographers made a brilliant simplification. They asked: what if we just assume vertical acceleration is exactly zero? If $a=0$, then $F=0$. The tug-of-war becomes a perfect, static balance. The upward push from the pressure gradient exactly cancels the downward pull of gravity. This beautiful equilibrium is known as ​​hydrostatic balance​​.

Under this ​​hydrostatic approximation​​, pressure becomes a wonderfully simple quantity. The pressure at any given depth is determined solely by the weight of the column of fluid sitting above it. This means if you know the density of the fluid everywhere, you can calculate the pressure by simply integrating downwards from the surface. This simplification is the cornerstone of ​​hydrostatic models​​, which have been the workhorses of large-scale climate and weather prediction for decades. They are computationally efficient and remarkably accurate for phenomena whose horizontal scale is much, much larger than their vertical scale [@4072640].

But nature is not always so calm and expansive. Think of a towering cumulonimbus cloud—a thunderstorm—punching through the atmosphere. Here, parcels of warm, moist air are not in a gentle balance; they are rocketing upwards with violent acceleration. Consider a wave rearing up to crash on a beach, or a dense current of cold, salty water spilling over an underwater mountain and plunging into the abyss. In these cases, vertical acceleration is not just a minor detail; it is the story. To ignore it would be to miss the entire point.

This is the domain of ​​non-hydrostatic models​​. They make no such simplifying assumption. They embrace the full dynamism of the vertical momentum equation, treating it as a true prognostic equation for vertical velocity. In this world, acceleration is not zero, and the balance of forces is a dynamic, ever-changing drama. The reward for this complexity is the ability to simulate the world as it truly is, in all its turbulent, convective, and wave-like glory [@3802927].

How do we know when we can get away with the simple hydrostatic world and when we must venture into the more complex non-hydrostatic realm? The answer, as is so often the case in physics, lies in comparing scales.

The most intuitive measure is the ​​aspect ratio​​ of a phenomenon: its height divided by its width. For a vast ocean basin that is thousands of kilometers wide but only a few kilometers deep, the aspect ratio $\delta = H/L$ is tiny. It's like a very, very thin sheet. Motions are overwhelmingly horizontal, and the hydrostatic approximation holds beautifully. But for a thunderstorm that is perhaps ten kilometers wide and ten kilometers tall, the aspect ratio is close to one. It is as tall as it is wide. It is no surprise that vertical motions are just as important as horizontal ones here [@4072640].

We can be more precise by using a dimensionless number called the ​​internal Froude number​​, $Fr$. You can think of $Fr$ as a ratio: the characteristic speed of the fluid flow ($W$) to the natural speed of internal gravity waves ($NH$), which are the waves that propagate on the density layers within a stratified fluid like the ocean or atmosphere. So, $Fr = W/(NH)$ [@4053471].

• When $Fr \ll 1$, the flow is slow and subcritical. The stratification is strong enough to keep vertical motions in check, and the hydrostatic approximation is generally valid. This is the case for near-inertial internal waves or basin-scale tides in the open ocean, which are very hydrostatic [@3802927].

• When $Fr \gtrsim 1$, the flow is fast and supercritical. The inertia of the vertical motion is strong enough to overcome the restoring force of stratification. Vertical accelerations become significant, and the hydrostatic balance breaks down.

Let's look at a concrete example. In the core of a deep convective updraft, characteristic vertical velocities can be $W = 20 \, \mathrm{m\,s^{-1}}$ over a depth of $H = 5 \, \mathrm{km}$. In a typical atmosphere, this might correspond to an internal Froude number of $Fr \approx 1$ [@4053471]. This calculation tells us, from first principles, that the vertical acceleration is of the same order of magnitude as the buoyancy force driving it. A hydrostatic model, which assumes these accelerations are zero, would be fundamentally blind to the core physics of the thunderstorm. The same principle applies to dense overflows down steep underwater sills, where the flow can become supercritical and form hydraulic jumps, or to surface waves shoaling on a beach, where the aspect ratio ($kH$) becomes order one [@3802927].

The most profound consequence of abandoning the hydrostatic approximation is not just that we get a dynamic equation for vertical velocity. It is that the very nature of pressure is transformed. In the non-hydrostatic world, pressure takes on a new and crucial role: it becomes the great enforcer of the conservation of mass.

In most oceanic and many atmospheric scenarios, we can treat the fluid as essentially ​​incompressible​​. This is a very good approximation for low-speed flows (low Mach number) and is formalized in the ​​Boussinesq approximation​​ for the ocean or the ​​anelastic approximation​​ for the atmosphere [@3802879]. The incompressibility constraint is elegantly simple: $\nabla \cdot \mathbf{u} = 0$. It states that the divergence of the velocity field is zero everywhere. In simple terms, you can't have more fluid flowing into a tiny box than is flowing out. Fluid cannot be magically created or destroyed at a point.

In a hydrostatic model, this constraint is used diagnostically to figure out the vertical velocity $w$ after the horizontal velocities have been determined [@3815102]. But in a non-hydrostatic model, how is this rigid mathematical constraint satisfied at every single point, at every single instant? The answer is pressure.

Imagine the model at one time step. It calculates all the forces—advection, rotation, buoyancy—and determines a "provisional" velocity field. This provisional field is what the fluid wants to do. But this motion might not satisfy $\nabla \cdot \mathbf{u} = 0$; it might have regions where fluid is piling up or thinning out. To correct this, the model must find a pressure field whose gradients will provide exactly the right pushes and pulls to adjust the velocity field and make it divergence-free.

This leads to one of the most important equations in computational fluid dynamics: the ​​pressure Poisson equation​​. It takes the form $\nabla^2 p' = \rho_0 \nabla \cdot \mathbf{R}$, where $\mathbf{R}$ represents all the non-pressure forces [@3813973]. This equation is a mathematical marvel. The term on the right is the divergence of the provisional velocity field—it's a measure of how much the "provisional" flow violates incompressibility. The equation dictates that the curvature of the pressure field ($\nabla^2 p'$) must respond to this violation. By solving this equation, we find the unique pressure field $p'$ that restores mass conservation.

Think about what this means. Hydrostatic pressure is a local property, determined only by the weight of the fluid directly above. Non-hydrostatic pressure is a global property. The pressure at one point in the domain is instantaneously influenced by what the flow is doing everywhere else, because the constraint $\nabla \cdot \mathbf{u} = 0$ must hold globally. Pressure acts as an unseen hand, providing instantaneous communication across the entire fluid to ensure its coherence. This is a profound shift from the simple, diagnostic role of pressure in a hydrostatic world.

This more truthful, beautiful physical picture comes at a cost. The power of non-hydrostatic models requires immense computational resources.

Solving the 3D pressure Poisson equation is the single most expensive part of a non-hydrostatic simulation. It's a massive system of linear equations that couples every single grid point in the domain to every other. This "global" nature makes it a notorious bottleneck for parallel computing [@3813973].

Furthermore, by resolving finer-scale physics, we become slaves to their faster time scales. The stability of explicit numerical schemes is governed by the ​​Courant-Friedrichs-Lewy (CFL) condition​​, which intuitively states that information cannot be allowed to travel more than one grid cell per time step. Non-hydrostatic models must respect the CFL limits imposed not just by advection (the speed of the flow itself), but also by the fastest waves they resolve. These include internal gravity waves, whose speeds can be quite high, forcing the model to take very small time steps [@3813973]. In a developing convective plume, for instance, buoyancy can cause vertical velocities to accelerate rapidly, which in turn tightens the vertical advective CFL limit, often requiring an ​​adaptive time step​​ that shrinks as the updraft strengthens [@3922276].

If the model is fully ​​compressible​​, it must also resolve sound waves. Since the speed of sound in air or water is extremely high (hundreds to over a thousand meters per second), a fully explicit compressible model would be forced to take impractically tiny time steps [@4068863]. This is why many non-hydrostatic models use "sound-proof" approximations. The ​​Boussinesq approximation​​, which assumes density is constant everywhere except in the buoyancy term, is perfect for the ocean where density variations are small. It simplifies the continuity equation to $\nabla \cdot \mathbf{u} = 0$. The ​​anelastic approximation​​, common in atmospheric science, is slightly less restrictive, allowing for a background density that changes with height, yielding a continuity equation of $\nabla \cdot (\rho_0 \mathbf{u}) = 0$ [@3802879]. Both elegantly filter out sound waves while retaining the essential non-hydrostatic dynamics. To deal with the remaining fast waves, modelers have developed clever numerical schemes—like ​​split-explicit​​ or ​​semi-implicit​​ methods—that treat the fast-wave terms differently from the slow advective terms, allowing for a much larger and more efficient overall time step [@4068863].

Why do we go to all this trouble? Because the reward is a model that captures the true physics of phenomena that are invisible to their hydrostatic counterparts. This has profound consequences for understanding how energy is moved and transformed in the ocean and atmosphere.

​​Internal Waves:​​ These waves travel along density surfaces within the fluid and are a crucial mechanism for transporting energy. A non-hydrostatic model correctly predicts their ​​dispersion relation​​, which governs how their speed depends on their wavelength. A hydrostatic model, by contrast, gets this relationship fundamentally wrong, especially for waves with a large vertical extent compared to their horizontal wavelength. The hydrostatic model predicts that such waves travel with an artificially high group velocity. This means a hydrostatic model will incorrectly simulate the propagation of energy by a large fraction of the internal wave spectrum, a critical flaw for understanding ocean mixing [@3925171].

​​Turbulence:​​ Turbulence is the chaotic, three-dimensional churning that mixes heat, salt, and momentum. A key process in stratified fluids is the production of ​​Turbulent Kinetic Energy (TKE)​​ by buoyancy: warm, light fluid parcels rise ($w' > 0, b' > 0$) and cold, dense parcels sink ($w' 0, b' 0$), converting potential energy to kinetic energy. This process is represented by the buoyancy flux term, $\overline{b'w'}$. In a non-hydrostatic model, where vertical velocity $w'$ is a fully dynamic, prognostic variable, this energy conversion is a natural part of the TKE budget. However, in a hydrostatic model, $w'$ is merely a diagnostic slave to the horizontal flow; there is no prognostic equation for vertical kinetic energy. The physical pathway for buoyancy to generate vertical turbulence is broken. Hydrostatic models are therefore constitutionally incapable of representing this fundamental aspect of geophysical turbulence [@3802868].

In the end, the choice between hydrostatic and non-hydrostatic models is a choice about which universe we wish to simulate. The hydrostatic world is a simplified, elegant approximation, powerful for the grand, slow dance of the planet's circulation. The non-hydrostatic world is the richer, more complex, and more truthful reality, a place of crashing waves, soaring thermals, and turbulent eddies—a world where, by embracing the simple principle of $F=ma$ in its full vertical glory, we can begin to see nature in its truest form.

Having journeyed through the principles that separate the hydrostatic and non-hydrostatic worlds, we now arrive at the most exciting part: seeing these models in action. Where does this seemingly subtle difference—keeping or discarding the vertical acceleration term—truly matter? The answer is that it unlocks a zoo of phenomena, from the whisper of wind over a mountain to the roaring eyewall of a hurricane. It allows us to build more faithful "digital twins" of our planet's fluid systems.

To truly appreciate the difference, imagine listening to an orchestra. A hydrostatic model is like listening with earmuffs that filter out all the high-pitched notes—the piccolos, the violins, the cymbals. You get the fundamental rhythm from the cellos and basses, but you miss the texture, the detail, and the sudden, sharp bursts of energy. A non-hydrostatic model takes off the earmuffs. Suddenly, you hear the full symphony. The tool we use to see this is the kinetic energy spectrum, which tells us how much energy exists at different spatial scales, from vast weather systems down to individual clouds. Hydrostatic models capture the energy of large-scale flows but show a steep, unrealistic drop-off at the scales of a few dozen kilometers. Non-hydrostatic models, by contrast, reveal a world teeming with energy at these smaller "mesoscales," often following a slope closer to $k^{-5/3}$, characteristic of turbulent motion. This is the energy of storms, waves, and circulations that the hydrostatic assumption renders invisible. Let's explore some of these newly audible notes.

The atmosphere is a canvas for the interplay of buoyancy, inertia, and pressure. Non-hydrostatic models are our finest brushes for painting its most intricate details.

The Atmosphere's Silent Wake: Mountain Waves

Imagine a steady wind flowing over a mountain range, like water flowing over a rock in a stream. If the mountain is very broad and gentle, like a slow swell in the ocean, the air has plenty of time to adjust vertically. The flow is smooth and largely hydrostatic.

But what if the mountain is sharp and narrow? The air is forced violently upward. It can't remain in hydrostatic balance; it accelerates vertically. This disturbance creates waves that ripple downstream and, more remarkably, propagate vertically, far up into the stratosphere. These are "mountain waves" or "lee waves." To simulate them correctly, a model must be non-hydrostatic. The key lies in comparing the wave's intrinsic frequency, set by the wind speed $U$ and the mountain's width (related to a wavenumber $k_h$), with the atmosphere's natural frequency of vertical oscillation, the Brunt–Väisälä frequency $N$. When the intrinsic frequency $\omega_{int} = U k_h$ is not much smaller than $N$, non-hydrostatic effects are dominant.

These waves are not just beautiful, lens-shaped clouds for pilots to observe. They carry a tremendous amount of momentum. As they propagate upward, they can break, just like ocean waves on a beach. When they break, they deposit their momentum, creating a powerful "drag" force on the atmosphere that can steer jet streams and influence global weather patterns. A non-hydrostatic model can explicitly calculate this "resolved mountain-wave drag" by simulating the waves and their momentum transport, a process described by the covariance of velocity perturbations, $\overline{u' w'}$. For climate and weather models that cannot resolve every mountain, this effect must be crudely estimated through "subgrid orographic drag parameterizations." As our models become more non-hydrostatic and resolution increases, they can directly see a larger fraction of this crucial drag, making our global forecasts more accurate.

Of course, this creates a new technical problem: what happens when these vertically propagating waves reach the top of the model, the artificial "lid"? A simple, rigid lid would act like a mirror, reflecting the wave energy back down and contaminating the simulation. To prevent this, modelers place a "sponge layer" near the top. This is a region where a frictional term, known as Rayleigh damping, is added to the equations. This damping term, $-\mu(z)\phi$, absorbs the energy of the upward-moving waves, mimicking the way they would naturally propagate out into space and dissipate in the real, unbounded atmosphere.

Convective Storms: The Engines of Weather

Let’s turn from the steady flow over a mountain to something far more violent: a thunderstorm. The heart of a thunderstorm is a powerful updraft, a plume of warm, moist air rising due to its buoyancy. At its simplest, we can model this as a "rising thermal bubble." A non-hydrostatic model captures the essential physics: the initial vertical acceleration driven by buoyancy, and the complex pressure field that develops around the bubble to push the surrounding air out of the way.

A mature thunderstorm is not just an updraft. It also produces powerful downdrafts of cold, rain-chilled air. When this dense air hits the ground, it spreads out horizontally like spilled paint. This advancing front of cold air is a "gust front," a miniature weather front that you can feel as a sudden, cool blast of wind just before a storm arrives. This phenomenon is a perfect example of a ​​density current​​. Its speed is not set by large-scale winds, but by a delicate balance between its own density (or "reduced gravity," $g' = g \Delta\rho/\rho_0$) and its depth, $H$. Non-hydrostatic models show that the propagation speed $U$ scales as $\sqrt{g'H}$, a relationship characterized by a Froude number, $Fr = U/\sqrt{g'H}$, of order one. The churning, turbulent head of the gust front, which lifts the warm air ahead of it and can trigger new storms, is a quintessentially non-hydrostatic feature.

Tropical Cyclones: The Pinnacle of Atmospheric Power

Nowhere is non-hydrostatic dynamics more critical than in the heart of a hurricane. The terrifying winds of a hurricane's eyewall are part of a vast rotating system, but the storm's engine is the intense convection within that eyewall. Updrafts here can scream upwards at $15\,\mathrm{m\,s^{-1}}$ or more. Is this motion hydrostatic?

We can ask the same question as we did for mountain waves, but now for vertical motion. We can define a vertical Froude number, $Fr_v = W/(NH)$, which compares the characteristic vertical velocity $W$ to the speed of gravity waves $NH$ over a vertical scale $H$. For a typical strong updraft in an eyewall, this number is around $0.25$. This is far from the $Fr_v \ll 1$ condition required for the hydrostatic approximation to hold. The vertical accelerations are significant, a vital part of the storm's dynamics.

This means that to accurately simulate the structure and, crucially, the intensity of a tropical cyclone, a model must be non-hydrostatic. A hydrostatic model simply cannot produce the strong, narrow updrafts that define the eyewall. It smears them out, weakening the storm's secondary circulation—the great "flywheel" that draws moisture in at the bottom and expels air at the top—and leading to a poor forecast of the storm's peak winds. The ability to run non-hydrostatic models at kilometer-scale resolution has been one of the great breakthroughs in modern hurricane forecasting.

The ocean, like the atmosphere, has its own complex vertical motions, but they are often hidden from our view. Here too, the choice between hydrostatic and non-hydrostatic models is a matter of scale.

Consider internal waves, which are waves that propagate not on the ocean's surface, but on the interfaces between layers of different density (pycnoclines) deep within the ocean. When the tide sloshes water back and forth over a continental shelf, it can generate vast ​​internal tides​​ with wavelengths of hundreds of kilometers. These are enormous, slow, gentle motions. A scale analysis confirms that their frequency $\omega$ is much, much smaller than the buoyancy frequency $N$. For these behemoths, the hydrostatic approximation is perfectly adequate. Likewise, for a storm surge—the large-scale rise in sea level during a hurricane—the horizontal scales are vast ($L \sim 100\,\mathrm{km}$) compared to the vertical depth ($H \sim 20\,\mathrm{m}$). Again, the flow is hydrostatic, and a hydrostatic model is the right tool for the job.

But if we zoom in on the flow over a sharp, steep feature on the seafloor, like an underwater mountain or the edge of the continental shelf, the story changes. Just like in the atmosphere, the flow can generate sharp, non-hydrostatic ​​lee waves​​. These nonlinear internal waves can be incredibly steep and energetic. To capture their dynamics, a non-hydrostatic ocean model is essential.

This non-hydrostatic capability comes at a significant computational cost. In a hydrostatic model, pressure is found by simply integrating the weight of the water above. In a non-hydrostatic model, the pressure must be calculated by solving a 3D elliptic (Poisson) equation that ensures the flow remains incompressible. This creates a computational headache, especially when using Adaptive Mesh Refinement (AMR) to zoom in on interesting features. The elliptic nature of the pressure equation means that information has to be communicated across all grid levels simultaneously, a much harder problem than the simple column physics of a hydrostatic model.

For decades, global climate and weather models were exclusively hydrostatic. Their grid cells were hundreds of kilometers wide, far too coarse to see individual thunderstorms. The collective effects of these unresolved storms had to be approximated using "convective parameterization" schemes, like the well-known Kain-Fritsch scheme. These schemes acted as a stand-in, attempting to deduce when and where convection would occur based on large-scale conditions like the Convective Available Potential Energy (CAPE), and then adding the resulting heating and moistening back to the coarse grid cells.

Today, we are on the cusp of a revolution: ​​Global Cloud-Resolving Modeling (GCRM)​​. Thanks to massive increases in computing power, we can now run non-hydrostatic global models with grid spacings of just a few kilometers ($1\text{–}5\,\mathrm{km}$). You might think this means we are "resolving" clouds, and the problem is solved. But nature is subtle.

The "effective resolution" of a model—the smallest feature it can simulate faithfully—is typically 6 to 10 times its grid spacing ($\lambda_e \approx 6\text{–}10\,\Delta x$). So, a model with a $1\,\mathrm{km}$ grid can only truly capture phenomena with scales of $6\text{–}10\,\mathrm{km}$ or larger. A typical deep convective updraft core, however, is only $1\text{–}5\,\mathrm{km}$ wide. A shallow cumulus cloud might be only a few hundred meters across. This means that even our best global models operate in a "convective gray zone." They are too fine for traditional parameterizations to work correctly, but still too coarse to fully resolve the inner life of a cloud. While they can now explicitly represent the grand organization of storms into mesoscale systems, they still struggle with the fundamental plume dynamics.

This is the frontier. The challenge now is to develop "scale-aware" parameterizations that can intelligently turn themselves down as the model grid becomes fine enough to see more and more of the convective process for itself. By embracing the full, non-hydrostatic equations, we have not solved all our problems, but we have been able to ask much deeper, more interesting questions about the multiscale nature of our world's weather and climate. We have taken off the earmuffs and are finally beginning to hear the full, complex, and beautiful symphony of the planet.