Richardson Number Simulator — Stability of Stratified Flow

Parameters

Temperature difference dT

Characteristic length L

Flow velocity V

m/s

Thermal-expansion coefficient B

x10^-3 /K

With the defaults (dT = 5 K, L = 10 m, V = 5 m/s, B = 3.4e-3 / K for air) Ri is about 0.067, the regime is Mixing occurs, V_crit is about 2.58 m/s, and N is about 0.129 rad/s. Lower V toward 2 m/s to push Ri above 0.25 (Neutral mixing), and below 1.6 m/s for Stable stratification. A negative dT inverts the stratification, and a non-positive Ri argument is reported as Convectively unstable.

Results

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Richardson number

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Stability

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Mixing-limit velocity

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Brunt-Vaisala frequency

Vertical stratification and horizontal flow

A vertical color gradient shows the temperature stratification (warm at the bottom, cool at the top when dT > 0). Horizontal arrows indicate the flow velocity V. With large Ri the horizontal layers are preserved (stable), and with Ri < 0.25 a Kelvin-Helmholtz wavy mixing develops at the interfaces. The wave amplitude grows as Ri decreases.

Ri stability chart (log axis)

Horizontal axis: Ri (log10, 0.01 to 100). Vertical axis: stability region. Red band: Ri < 0.25 (unstable, mixing). Yellow band: 0.25 to 1 (neutral mixing). Green band: Ri > 1 (stable stratification). Red vertical lines: Ri = 0.25 and Ri = 1 boundaries. Yellow marker: current Ri operating point. Slide V up and down to see the marker cross the bands.

Theory & Key Formulas

Richardson number: the dimensionless ratio of buoyancy (caused by density differences from temperature variation) to inertia in a stratified flow.

$$\mathrm{Ri} = \frac{g\,\beta\,\Delta T\,L}{V^2}$$

Brunt-Vaisala frequency: the natural angular frequency at which a vertically displaced fluid parcel oscillates under buoyancy in a stable stratification (rad/s):

$$N = \sqrt{\frac{g\,\beta\,\Delta T}{L}}$$

Mixing-limit velocity: the V that gives Ri = 0.25, the threshold above which Kelvin-Helmholtz mixing develops:

$$V_{\mathrm{crit}} = \sqrt{\frac{g\,\beta\,\Delta T\,L}{0.25}} = 2\sqrt{g\,\beta\,\Delta T\,L}$$

$g$ is gravity (9.81 m/s²), $\beta$ is the thermal-expansion coefficient (1/T for an ideal gas), $\Delta T$ is the vertical temperature difference across the layer, $L$ is the characteristic length (layer thickness), and $V$ is the flow velocity that crosses the layer. Ri < 0.25 is unstable (mixing), 0.25 ≤ Ri ≤ 1 is neutral mixing, and Ri > 1 is stable stratification (mixing suppressed).

What is the Richardson Number Simulator

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In atmospheric science class I learned that the wind drops at night, but does the Richardson number really explain that? I have never thought about Ri in daily life, what does it actually measure?

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A great question. The Richardson number Ri = g beta dT L / V^2 is the ratio of buoyancy to inertia in a stratified flow, and the night-time surface inversion is the textbook example. After sunset the ground cools by radiation; the air near the ground gets cold while the air aloft stays warmer, forming an inversion (warm above, cold below — a stable stratification). With a large dT and a small V, Ri exceeds 1 and the regime becomes Stable stratification, the wind (inertia) loses to the buoyancy, and turbulent mixing stops. That is why the wind goes calm at night. With the tool defaults (dT = 5 K, L = 10 m, V = 5 m/s, beta = 3.4e-3 / K) Ri is about 0.067 — in the Mixing occurs region — but lowering V to 1 m/s gives Ri about 1.7 and the label switches to Stable stratification.

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Why are the Ri = 0.25 and Ri = 1 boundaries the special numbers? 0.25 looks oddly specific.

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A fundamental question. Ri = 0.25 is the necessary condition that Miles and Howard proved in 1961 in the linear stability analysis: if Ri is below 0.25 then Kelvin-Helmholtz instability (wavy distortion of a shear layer) can grow. Conversely, if Ri is at or above 0.25 the linear theory guarantees stability. Ri = 1 is empirical, observed in experiments and field data as the threshold above which strong stable stratification suppresses turbulence dramatically. Slide V in this tool and you will first cross Ri = 0.25 and then Ri = 1, with a marked change in mixing character at each crossing. RANS turbulence closures in CFD also use Ri to correct the eddy viscosity in stably stratified flows.

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The Brunt-Vaisala frequency reads 0.129 rad/s. What is actually oscillating? The water in front of me does not look like it is shaking.

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Interesting point. N = sqrt(g beta dT / L) is the natural angular frequency at which a fluid parcel that has been displaced vertically inside a stable stratification oscillates back to its equilibrium position under buoyancy. Nothing macroscopic shakes; N is just the upper bound on internal-gravity-wave frequencies. Real-world examples include internal waves on a lake's thermocline (the surface is calm yet the temperature layer a few meters down undulates) and atmospheric mountain waves (gravity waves shed downwind of a mountain). In the ocean N is about 0.001 to 0.01 rad/s (10 to 100 minute periods); in the stratosphere N is about 0.02 rad/s (5 minute period). With the tool defaults T = 2 pi / N is about 48.7 seconds.

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The mixing-limit velocity V_crit = 2.58 m/s — how do I use that as a design number?

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Very practical question. V_crit is the velocity that gives Ri = 0.25, the threshold above which mixing develops and below which the stratification is preserved. With the tool defaults (dT = 5 K, L = 10 m, beta = 3.4e-3 / K) V_crit is about 2.58 m/s. For example, in a data center where cool air rises 10 m up a server rack with a 5 K vertical temperature difference, you can read it as: if the exhaust-driven air velocity goes above 2.58 m/s the carefully built cool layer breaks up and mixes. Likewise, in a stratified thermal-storage tank (hot water layered above cool water), keeping the circulation flow below V_crit preserves the stratification and improves the energy efficiency. Try a tank case (L = 2 m, dT = 20 K) in the tool to see V_crit directly.

Frequently Asked Questions

The Richardson number Ri = g beta dT L / V^2 is the dimensionless ratio of buoyancy (caused by the density difference from temperature variation) to inertia of the flow in a stratified fluid. Here g is gravity, beta is the thermal-expansion coefficient, dT is the vertical temperature difference, L is the layer thickness, and V is the velocity that crosses the layer. With the tool defaults (dT = 5 K, L = 10 m, V = 5 m/s, beta = 3.4e-3 / K, air) the result is Ri about 0.067, which lies inside the Kelvin-Helmholtz mixing region (Ri below 0.25). Ri is the basic parameter that decides whether turbulent mixing develops in atmospheric boundary layers, ocean thermoclines, or building ventilation.

Ri = 0.25 is the necessary condition from the Miles-Howard linear stability theorem: when Ri drops below 0.25, Kelvin-Helmholtz instability (interface waves that overturn into turbulence) can develop. Ri = 1 is an empirical strong-stratification threshold above which turbulence is greatly suppressed and the layered structure persists for a long time. This tool labels Ri below 0.25 as Mixing occurs, 0.25 to 1 as Neutral mixing, and above 1 as Stable stratification, and the marker on the Ri-stability chart moves between these bands as you slide V.

The Brunt-Vaisala frequency N = sqrt(g beta dT / L) is the natural angular frequency (rad/s) at which a fluid parcel that is displaced vertically inside a stable stratification oscillates back to its equilibrium position under buoyancy. A larger N means a stronger restoring force and gives the upper bound on the frequency of internal gravity waves. With the tool defaults N is about 0.129 rad/s, corresponding to a period T = 2 pi / N of about 48.7 seconds. In the ocean N is about 0.001 to 0.01 rad/s (10 to 100 minute periods); in the stratosphere N is about 0.02 rad/s (5 minute period).

Ri is used wherever the stability of stratified flow matters. Examples: (1) atmospheric stability assessment (radiative cooling at night creates a surface inversion, Ri grows large, and the wind stops), (2) mixing in ocean thermoclines (Ri below 0.25 causes internal-wave breaking), (3) buoyancy-driven natural ventilation in tall buildings, (4) fire safety (smoke layer integrity), (5) HVAC stratified cooling efficiency, and (6) the salt wedge in river estuaries. Increase V in this tool to push Ri below 0.25 and see the boundary at which mixing develops.

Real-World Applications

Night-time atmospheric stability: A weather observation with a 10 m wind speed V = 2 m/s and a 50 m vertical temperature difference dT = 3 K (warmer aloft, an inversion) gives Ri about 1.25 for air (beta = 3.4e-3 / K), which the tool labels Stable stratification. This is the condition under which pollutants and pollen accumulate near the surface until the morning wind picks up. Air-quality regulations use the duration of Ri above 0.25 to select dispersion stability classes (Pasquill-Gifford D to F).

Ocean thermocline and internal waves: In summer coastal waters a thermocline forms between a warm surface layer (20 C) and a cool deep layer (10 C), with a thickness L of about 10 m and dT of 10 K. With seawater beta about 0.21e-3 / K, N is about 0.0144 rad/s, period about 7.3 minutes. When the tidal current exceeds V_crit = sqrt(g beta dT L / 0.25) of about 0.575 m/s, Ri drops below 0.25 and internal-wave breaking mixes deep, nutrient-rich water into the surface layer, fueling phytoplankton blooms (a key process for coastal ecosystems). Try beta = 0.21, dT = 10 and L = 10 in this tool to reproduce the case.

Stratified ventilation in tall atria: Train stations, airports and large halls with tall atria (height L = 20 m, dT = 8 K) often develop stable thermal stratification. To avoid disrupting the cool floor layer, flow velocities V should stay below V_crit. The tool returns V_crit about 4.62 m/s for dT = 8, L = 20 and beta = 3.4e-3, so diffuser exit velocities below this preserve the stratification and only the occupied zone (1.8 m above the floor) is cooled — saving considerable HVAC energy.

Stratified thermal-storage tanks: Solar heating and district-heating storage tanks (height L = 4 m, dT = 30 K with hot water above) rely on keeping the inflow / outflow velocity below V_crit for thermal efficiency. With dT = 30, L = 4 and water (beta = 0.21e-3 / K), the tool returns V_crit about 0.314 m/s, so a low-velocity diffuser (large inlet, gentle flow) is needed. Exceeding V_crit triggers turbulent mixing that homogenizes the tank temperature and destroys the high-grade heat — storage efficiency depends critically on stratification preservation.

Common Pitfalls and Notes

The most common misconception is "mixing turns on instantly when Ri crosses 0.25". In reality Ri = 0.25 is only a necessary condition from linear stability, and around the threshold (0.1 to 0.5) the nonlinear growth of waves yields intermittent mixing rather than a sharp switch. Field observations in Ri = 0.2 to 0.3 show "turbulent intermittency" — bursts of turbulence — and the average mixing efficiency varies smoothly with Ri. This tool labels regions at Ri = 0.25 for convenience, but in design it is wise to treat Ri below 0.5 as "mixing possible" with a safety margin.

The next pitfall is "Ri can be represented by averaged values over the whole layer". This tool uses the bulk Richardson number with average dT and V over the layer thickness L. In reality temperature and velocity gradients vary vertically, and the local "gradient Richardson number" Ri_g = N^2 / (dU / dz)^2 predicts mixing onset more accurately. CFD evaluates Ri_g at each cell to determine the laminar / turbulent transition. Even when the bulk Ri is above 1, a thin shear layer with local Ri below 0.25 can trigger localized mixing.

The last pitfall is "Ri above 1 means a completely static fluid". Even within a stable stratification, internal gravity waves at frequencies up to N propagate over long distances (thousands of kilometers in the atmosphere, hundreds in the ocean) and transport energy to remote regions where they break. Mountain waves excited by topography reach the stratosphere and break, contributing to planetary-scale momentum transport. The N from this tool is the maximum allowed wave frequency inside the stratification; many lower-frequency internal waves coexist.

Richardson Number Simulator — Stability of Stratified Flow

What is the Richardson Number Simulator

Frequently Asked Questions

Real-World Applications

Common Pitfalls and Notes

Related Tools