12.3 Stratified flows

Stratified flows are characterized by density variations in fluids, often due to temperature or salinity gradients. These flows exhibit layered structures, with denser fluid underlying lighter fluid, leading to stable configurations. Stratification significantly influences fluid dynamics, suppressing vertical mixing and altering wave and turbulence propagation.

Density variations in stratified flows can arise from temperature differences or solute concentration variations. In the atmosphere, temperature gradients primarily cause density variations, while in the ocean, both temperature and salinity influence density. Understanding these variations is crucial for analyzing fluid behavior in various environmental systems.

Characteristics of stratified flows

• Stratified flows are characterized by the presence of density variations in the fluid, often due to temperature or salinity gradients
• These flows exhibit a layered structure, with denser fluid underlying lighter fluid, leading to a stable configuration
• Stratification can significantly influence the dynamics of fluid motion, suppressing vertical mixing and altering the propagation of waves and turbulence

Density variations in stratified flows

• Density variations in stratified flows can arise from temperature differences (thermal stratification) or variations in solute concentration (saline stratification)
• In the atmosphere, density variations are primarily caused by temperature gradients, with cooler, denser air near the Earth's surface and warmer, lighter air aloft
• In the ocean, density variations are influenced by both temperature and salinity, with colder, saltier water being denser than warmer, fresher water

Stability of stratified flows

Stable stratification

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• Stable stratification occurs when the density of the fluid increases with depth, leading to a configuration where lighter fluid overlies denser fluid
• In this case, the buoyancy force acts to restore any vertical displacements, suppressing vertical mixing and turbulence
• Examples of stable stratification include the thermocline in the ocean and the troposphere in the atmosphere

Unstable stratification

• Unstable stratification arises when the density of the fluid decreases with depth, resulting in a situation where denser fluid overlies lighter fluid
• This configuration is prone to convective instabilities, as any vertical perturbations will be amplified by the buoyancy force
• Examples of unstable stratification include the convective boundary layer in the atmosphere and the mixed layer in the ocean

Neutral stratification

• Neutral stratification refers to a situation where the density of the fluid remains constant with depth
• In this case, the buoyancy force does not influence vertical motions, and the flow behaves similarly to homogeneous turbulence
• Neutral stratification can occur in well-mixed regions, such as the surface mixed layer in the ocean or the atmospheric boundary layer under strong wind conditions

Internal waves

Characteristics of internal waves

• Internal waves are gravity waves that propagate within the interior of a stratified fluid, rather than on the free surface
• These waves are driven by buoyancy forces and can have much larger amplitudes and slower propagation speeds compared to surface waves
• Internal waves are characterized by their frequency, wavenumber, and vertical structure, which depend on the density stratification and background flow conditions

Generation of internal waves

• Internal waves can be generated by various mechanisms, including flow over topography, wind stress on the ocean surface, and tidal forcing
• When a stratified fluid encounters an obstacle (seamount), the flow is deflected vertically, generating internal waves that propagate away from the topography
• Wind stress can generate internal waves through resonant interactions with surface waves or by directly forcing vertical motions in the upper ocean

Propagation of internal waves

• Internal waves propagate along surfaces of constant density (isopycnals) in a stratified fluid
• The propagation direction and speed of internal waves depend on the frequency, wavenumber, and stratification profile
• As internal waves propagate, they can interact with other waves, leading to energy transfers, wave breaking, and mixing

Mixing in stratified flows

Turbulent mixing

• Turbulent mixing in stratified flows is driven by shear instabilities and breaking internal waves
• When the shear in the flow exceeds a critical threshold (Richardson number), Kelvin-Helmholtz instabilities can develop, leading to the formation of turbulent billows and enhanced mixing
• Breaking internal waves can also generate turbulence and mixing, particularly in regions of high wave activity or near critical layers

Molecular diffusion

• Molecular diffusion is the transport of heat, salt, or other scalars due to random molecular motions
• In stratified flows, molecular diffusion is often much weaker than turbulent mixing, but it can be important in regions of weak turbulence or strong gradients
• Double diffusive convection is a special case where the different molecular diffusivities of heat and salt can lead to enhanced mixing and the formation of staircase-like density profiles

Entrainment and detrainment

• Entrainment refers to the incorporation of fluid from one layer into another, typically from a less turbulent to a more turbulent region
• Detrainment is the opposite process, where fluid is expelled from a turbulent region into a less turbulent one
• In stratified flows, entrainment and detrainment can occur across density interfaces, leading to the exchange of mass, momentum, and scalars between layers

Stratified flow regimes

Layered flows

• Layered flows are characterized by the presence of distinct layers with sharp density interfaces separating them
• These flows can arise from the interaction of different water masses (estuaries), or from the effects of surface heating or cooling (thermocline)
• Layered flows often exhibit reduced vertical mixing and enhanced horizontal dispersion, as well as the formation of internal waves and hydraulic jumps at the interfaces

Continuous stratification

• Continuously stratified flows have a smooth density profile, with no distinct layers or interfaces
• These flows are common in the ocean interior and the upper atmosphere, where density varies gradually with depth
• Continuous stratification supports the propagation of internal waves and can lead to the formation of thin layers of enhanced mixing or biological activity

Froude number in stratified flows

• The Froude number is a dimensionless parameter that compares the inertial forces to the buoyancy forces in a stratified flow
• It is defined as $Fr = U / (N H)$, where $U$ is a characteristic velocity, $N$ is the buoyancy frequency, and $H$ is a characteristic length scale
• The Froude number is used to characterize the flow regime and the importance of stratification effects
  • For $Fr << 1$, the flow is strongly influenced by stratification, and internal waves can propagate freely
  • For $Fr >> 1$, the flow is dominated by inertial effects, and stratification has a weak influence on the dynamics

Modeling stratified flows

Analytical models

• Analytical models of stratified flows are based on simplified equations that capture the essential physics of the problem
• These models often assume idealized geometries, linear density profiles, and small perturbations to the background flow
• Examples of analytical models include the Taylor-Goldstein equation for linear internal waves and the Korteweg-de Vries equation for nonlinear internal solitary waves

Numerical models

• Numerical models are used to simulate stratified flows in more complex geometries and with realistic forcing and boundary conditions
• These models solve the Navier-Stokes equations with an equation of state that relates density to temperature and salinity
• Numerical models can capture a wide range of stratified flow phenomena, including internal waves, turbulence, and mixing
• Examples of numerical models for stratified flows include the Regional Ocean Modeling System (ROMS) and the MIT General Circulation Model (MITgcm)

Applications of stratified flows

Atmospheric stratification

• Atmospheric stratification plays a crucial role in weather and climate dynamics
• The stability of the atmospheric boundary layer influences the formation of clouds, the dispersion of pollutants, and the intensity of turbulence
• Internal waves in the atmosphere (gravity waves) can transport momentum and energy over large distances and contribute to the forcing of the global circulation

Oceanic stratification

• Oceanic stratification is important for the global climate system, as it controls the storage and transport of heat, carbon, and nutrients
• The ocean's density structure influences the formation and circulation of water masses, as well as the propagation of internal waves and tides
• Mixing in the stratified ocean interior is crucial for maintaining the global overturning circulation and the distribution of biogeochemical tracers

Environmental fluid dynamics

• Stratified flows are relevant to a wide range of environmental fluid dynamics problems, such as the dispersion of pollutants in the atmosphere and ocean
• The stability of the atmospheric boundary layer affects the transport and mixing of air pollutants, with stable conditions leading to reduced dispersion and higher concentrations
• In the ocean, stratification can influence the fate and transport of oil spills, as well as the dispersion of nutrients and biological productivity

Experimental techniques for stratified flows

Density measurements

• Density measurements in stratified flows can be performed using various techniques, such as conductivity-temperature-depth (CTD) sensors, which measure the electrical conductivity and temperature of the fluid to infer density
• Other methods include the use of density floats (drift along surfaces of constant density) or the analysis of refractive index variations using optical techniques (schlieren)

Velocity measurements

• Velocity measurements in stratified flows can be obtained using acoustic Doppler current profilers (ADCPs), which use the Doppler shift of sound waves to measure the velocity profile
• Other techniques include particle image velocimetry (PIV), which tracks the motion of tracer particles in the fluid, and laser Doppler velocimetry (LDV), which measures the velocity at a point based on the Doppler shift of scattered laser light

Flow visualization

• Flow visualization techniques are used to qualitatively observe the structure and dynamics of stratified flows
• Dye tracers can be injected into the fluid to highlight the motion of fluid parcels and the formation of internal waves or turbulent structures
• Shadowgraph and schlieren imaging techniques can be used to visualize density variations in the fluid based on changes in the refractive index

Mathematical description of stratified flows

Governing equations

• The governing equations for stratified flows are the Navier-Stokes equations, which describe the conservation of mass, momentum, and energy in the fluid
• The equations are coupled with an equation of state that relates density to temperature and salinity, such as the linear equation of state: $\rho = \rho_0 (1 - \alpha (T - T_0) + \beta (S - S_0))$
• The Boussinesq approximation is often used in stratified flows, which assumes that density variations are small and only affect the buoyancy term in the momentum equation

Boundary conditions

• Boundary conditions for stratified flows depend on the specific problem and geometry considered
• At the free surface, the boundary conditions typically include a wind stress condition for the momentum equation and a heat and salt flux condition for the scalar equations
• At solid boundaries (bottom), no-slip and no-flux conditions are often imposed, although slip or flux conditions can be used in some cases

Initial conditions

• Initial conditions for stratified flows specify the density and velocity fields at the start of the simulation or experiment
• These conditions can be based on observations, analytical solutions, or previous numerical simulations
• The choice of initial conditions can strongly influence the subsequent evolution of the flow, particularly in the case of instabilities or transient phenomena