12.3 Stratified flows

Characteristics of Stratified Flows

Stratified flows occur when density varies through a fluid, typically because of temperature or salinity gradients. The result is a layered structure where denser fluid sits beneath lighter fluid. This layering has major consequences: it suppresses vertical mixing, reshapes how waves propagate, and fundamentally alters turbulence behavior. Understanding stratification is essential for analyzing everything from ocean circulation to atmospheric weather patterns.

In the atmosphere, temperature gradients are the primary driver of density variation, with cooler, denser air near the surface and warmer, lighter air above. In the ocean, both temperature and salinity matter. Colder, saltier water is denser than warmer, fresher water, and the interplay between these two properties creates complex density profiles.

Stability of Stratified Flows

The stability of a stratified flow depends on how density changes with depth. This determines whether vertical disturbances get damped out, amplified, or ignored entirely.

Stable Stratification

Stable stratification exists when density increases with depth, so lighter fluid sits on top of denser fluid. If a parcel of fluid gets displaced vertically, buoyancy forces push it back toward its original position. This restoring force suppresses vertical mixing and turbulence.

• The ocean's thermocline (the zone of rapid temperature change between warm surface water and cold deep water) is a classic example
• The troposphere under typical conditions also exhibits stable stratification above the boundary layer

Unstable Stratification

Unstable stratification occurs when denser fluid sits above lighter fluid. Any small vertical perturbation gets amplified by buoyancy, triggering convective overturning.

• The convective boundary layer in the atmosphere forms on sunny days when the ground heats the air above it, creating a bottom-heavy density profile that drives vigorous convection
• The ocean's mixed layer can become unstable when surface cooling makes the top water denser than the water below

Neutral Stratification

In neutral stratification, density is uniform with depth. Buoyancy forces play no role, and the flow behaves like homogeneous turbulence. This occurs in well-mixed regions, such as the ocean surface mixed layer under strong winds or the atmospheric boundary layer during overcast, windy conditions.

Internal Waves

Characteristics of Internal Waves

Internal waves are gravity waves that propagate within a stratified fluid rather than along the free surface. They're driven by buoyancy forces acting on displaced fluid parcels. Compared to surface waves, internal waves can have much larger amplitudes (tens of meters in the ocean) and much slower propagation speeds.

Their frequency, wavenumber, and vertical structure all depend on the local density stratification and background flow. A key constraint is that internal wave frequencies must fall between zero and the local buoyancy frequency $N$.

Generation of Internal Waves

Internal waves arise from several mechanisms:

• Flow over topography: When a stratified current encounters an obstacle like a seamount or ridge, the flow deflects vertically and generates internal waves that radiate away from the topography
• Wind stress: Surface winds can force vertical motions in the upper ocean, exciting internal waves through direct forcing or resonant interactions with surface waves
• Tidal forcing: Barotropic tides flowing over rough bottom topography convert energy into internal (baroclinic) tides, which are a major source of internal wave energy in the deep ocean

Propagation of Internal Waves

Internal waves propagate along surfaces of constant density called isopycnals. Their propagation direction and speed depend on frequency, wavenumber, and the stratification profile.

One distinctive feature: the group velocity (energy propagation direction) and phase velocity of internal waves are perpendicular to each other in the vertical-horizontal plane. As internal waves propagate, they can interact with other waves, transfer energy across scales, break, and drive mixing.

Mixing in Stratified Flows

Turbulent Mixing

Turbulent mixing in stratified flows is primarily driven by shear instabilities and breaking internal waves.

• Kelvin-Helmholtz instability develops when the velocity shear across a density interface exceeds a critical threshold. The criterion for this is the Richardson number $Ri = N^2 / (dU/dz)^2$. When $Ri < 0.25$, the flow is susceptible to shear instability, producing characteristic rolling billows that break down into turbulence and mix the fluid.
• Breaking internal waves generate turbulence particularly near critical layers (where the wave's phase speed matches the background flow speed) or in regions of high wave energy.

Molecular Diffusion

Molecular diffusion transports heat, salt, and other scalars through random molecular motion. It's typically far weaker than turbulent mixing, but it becomes important in regions of weak turbulence or very strong gradients.

Double-diffusive convection is a notable special case. Heat diffuses molecularly about 100 times faster than salt. This mismatch can drive instabilities even in otherwise stable stratification, producing distinctive staircase-like density profiles. Two forms exist: salt fingers (warm, salty water over cool, fresh water) and diffusive convection (cool, fresh water over warm, salty water).

Entrainment and Detrainment

Entrainment is the incorporation of fluid from a less turbulent layer into a more turbulent one. Detrainment is the reverse: fluid expelled from a turbulent region into a calmer one. Both processes transfer mass, momentum, and scalars across density interfaces and are important for understanding how stratified layers evolve over time.

Stratified Flow Regimes

Layered Flows

Layered flows contain distinct layers separated by sharp density interfaces. They arise from the interaction of different water masses (as in estuaries, where fresh river water meets salty ocean water) or from surface heating and cooling that creates a well-defined thermocline.

These flows tend to show reduced vertical mixing, enhanced horizontal dispersion, and the formation of internal waves or hydraulic jumps at the interfaces between layers.

Continuous Stratification

Continuously stratified flows have a smooth, gradual density profile with no sharp interfaces. This is common in the ocean interior and the upper atmosphere. Continuous stratification supports internal wave propagation across a range of frequencies and can produce thin layers of enhanced mixing or concentrated biological activity.

Froude Number in Stratified Flows

The Froude number is a dimensionless parameter that compares inertial forces to buoyancy forces:

$Fr = \frac{U}{NH}$

where $U$ is a characteristic velocity, $N$ is the buoyancy frequency, and $H$ is a characteristic length scale.

• $Fr \ll 1$: Stratification dominates. The flow is strongly constrained by buoyancy, and internal waves propagate freely. Vertical motions are heavily suppressed.
• $Fr \gg 1$: Inertial forces dominate. Stratification has little influence, and the flow behaves more like an unstratified fluid.
• $Fr \approx 1$: Inertial and buoyancy forces are comparable, often producing the most complex dynamics, including lee waves and upstream blocking.

Modeling Stratified Flows

Analytical Models

Analytical models use simplified equations to capture the essential physics of stratified flows. They typically assume idealized geometries, linear density profiles, and small perturbations to the background state.

• The Taylor-Goldstein equation describes linear internal wave propagation and stability in sheared, stratified flow
• The Korteweg-de Vries (KdV) equation captures nonlinear internal solitary waves, which maintain their shape as they propagate

Numerical Models

Numerical models simulate stratified flows in realistic geometries with complex forcing and boundary conditions. They solve the Navier-Stokes equations coupled with an equation of state relating density to temperature and salinity.

These models capture internal waves, turbulence, mixing, and their interactions. Widely used examples include the Regional Ocean Modeling System (ROMS) and the MIT General Circulation Model (MITgcm).

Applications of Stratified Flows

Atmospheric Stratification

Atmospheric stratification shapes weather and climate. The stability of the boundary layer controls cloud formation, pollutant dispersion, and turbulence intensity. Atmospheric gravity waves transport momentum and energy over large distances and contribute to forcing the global circulation.

Oceanic Stratification

Ocean stratification governs the storage and transport of heat, carbon, and nutrients. The density structure controls water mass formation, circulation patterns, and internal wave propagation. Mixing in the stratified ocean interior is critical for sustaining the global overturning circulation (the large-scale conveyor belt that redistributes heat and tracers around the planet).

Environmental Fluid Dynamics

Stratification affects a wide range of environmental problems:

• Air quality: Stable atmospheric stratification traps pollutants near the surface, reducing dispersion and increasing ground-level concentrations
• Oil spills: Ocean stratification influences how spilled oil spreads and at what depth it accumulates
• Biological productivity: Stratification controls nutrient supply to the sunlit surface layer, directly affecting phytoplankton growth

Experimental Techniques for Stratified Flows

Density Measurements

• CTD sensors (conductivity-temperature-depth) measure electrical conductivity and temperature to infer density profiles. These are the workhorse instruments for ocean stratification measurements.
• Density floats drift along surfaces of constant density (isopycnals), providing Lagrangian information about the density field
• Optical techniques like schlieren imaging detect refractive index variations caused by density changes

Velocity Measurements

• Acoustic Doppler Current Profilers (ADCPs) use the Doppler shift of sound waves scattered by particles to measure velocity profiles through the water column
• Particle Image Velocimetry (PIV) tracks tracer particles in a laser-illuminated plane to produce two-dimensional velocity fields
• Laser Doppler Velocimetry (LDV) measures point velocities from the Doppler shift of laser light scattered by particles

Flow Visualization

Dye tracers injected into the fluid highlight fluid parcel motion, internal wave structures, and turbulent features. Shadowgraph and schlieren imaging reveal density variations through refractive index changes, making internal waves and mixing events visible without disturbing the flow.

Mathematical Description of Stratified Flows

Governing Equations

The governing equations are the Navier-Stokes equations for conservation of mass, momentum, and energy, coupled with an equation of state relating density to temperature and salinity. A common linear form of the equation of state is:

$\rho = \rho_0 \left(1 - \alpha(T - T_0) + \beta(S - S_0)\right)$

where $\rho_0$ is a reference density, $\alpha$ is the thermal expansion coefficient, $\beta$ is the haline contraction coefficient, $T$ is temperature, and $S$ is salinity.

The Boussinesq approximation is widely used in stratified flow analysis. It assumes density variations are small relative to the reference density and only matter in the buoyancy term of the momentum equation. This simplification is valid when $\Delta\rho / \rho_0 \ll 1$, which holds for most oceanic and atmospheric flows.

Boundary Conditions

Boundary conditions depend on the specific problem:

• Free surface: Wind stress drives momentum exchange; heat and freshwater fluxes set the scalar boundary conditions
• Solid boundaries (e.g., the ocean floor): No-slip conditions for velocity and no-flux conditions for scalars are standard, though parameterized slip or flux conditions are sometimes used

Initial Conditions

Initial conditions specify the density and velocity fields at the start of a simulation or experiment. They can come from observations, analytical solutions, or prior simulations. The choice of initial conditions strongly affects the flow's subsequent evolution, especially for problems involving instabilities or transient dynamics.