Dimensionless Numbers

Burger Number

• Definition: $S={\frac {c}{fr_{0}}}$, where $c={\sqrt {gH}}$ is the nonrotational baroclinic phase speed, $f$ is the inertial frequency under the f-plane approximation, and $r_{0}$ is the radius of basin. The Burger number is sometimes denotes as Bu.
• Interpretation: The ratio of stratification to rotation of a circular lake.
• Analysis: Small Burger numbers correspond to shallow water, wide lakes, or rotation dominated dynamics. Large Burger numbers correspond to deep water, narrow lakes, or stratification dominated dynamics.
• Example: Take a lake in the mid latitudes, then $f\approx 10^{-4}{\frac {1}{s}}$. If the lake has radius $r_{0}=1000m$ and a depth of $H=10m$, then $c={\sqrt {gH}}={\sqrt {9.81\cdot 10}}\approx 9.9{\frac {m}{s}}$ and so $S={\frac {c}{fr_{0}}}=99$, which means a stratification dominated lake.

Froude Number

• Definition: $Fr\propto \left[{\frac {\rho U^{2}/L}{\rho g}}\right]^{1/2}={\frac {U}{\sqrt {gL}}}$, where $g$ is the gravitational acceleration, and $U$ and $L$ are the characteristic velocity and length scales.
• Interpretation: The square root of the ratio of inertia force to gravity force.
• Analysis: The Froude number describes different regimes of open channel flow, including hydraulic jumps. $Fr=1$ describes critical flow; $Fr>1$ describes supercritical flow; $Fr<1$ describes subcritical flow. The Froude number is analogous to the Mach number in compressible flow.

Grashof Number

• Definition: $Gr={\frac {g\beta \Delta TV}{\nu ^{2}}}$, where $g$ is the gravitational acceleration, $\beta$ is the volumetric thermal expansion coefficient, $\Delta T$ is the temperature difference, $V$ is the volume, and $\nu$ is the kinematic viscosity.
• Interpretation: The ratio of buoyant forces to viscous forces.
• Analysis: When $Gr\gg 1$, the viscous force is negligible compared to the buoyancy and inertial forces, and the flow starts a transition to the turbulent regime.

Mach Number

• Definition: $M=U/c$, where $U$ is the characteristic velocity scale, and $c$ is the speed of sound.
• Interpretation: The ratio of inertia force to compressibility force.
• Analysis: Compressibility effects can be neglected if $M<0.3$.
• Example: The typical sound wave speed is $c\approx 340$ m/s in air. For wind with a speed $U=10$ m/s, the Mach number is $M\approx 0.03$, and hence the compressibility is negligible.

Prandtl Number

• Definition: $Pr=\nu /\kappa$, where $\nu$ is the viscosity, and $\kappa$ is the thermal diffusivity.
• Interpretation: The ratio of momentum diffusivity (i.e. kinematic viscosity) to heat diffusivity.
• Example: The typical Prandtl number for water is around 7 for water (At 20 degrees Celsius). See also Schmidt number.

Rayleigh Number

• Definition: $Ra={\frac {g\alpha \Gamma d^{4}}{\kappa \nu }}$, where $g$ is the acceleration due to gravity, $\alpha$ is the coefficient of thermal expansion, $\Gamma$ is the vertical temperature gradient of the background state, $d$ is the depth of the fluid layer, $\kappa$ is the thermal diffusivity, and $\nu$ is the kinematic viscosity.
• Interpretation: The ratio of the destabilizing effect of buoyancy force to the stabilizing effect of viscous force.

Reynolds Number

• Definition: $Re={\frac {UL}{\nu }}$, where $U$ and $L$ are the characteristic velocity and length scales, and $\nu$ is the kinematic viscosity.
• Interpretation: The ratio of inertia force to viscous force.
• Analysis: Small Reynolds numbers are often associated with viscous flows, whereas large Reynolds numbers are typically found in turbulent flows.
• Example: Take water with a $\nu =10^{-6}{\frac {m^{2}}{s}}$, and imagine a cube with side length 10 meters submerged in this water. The Reynolds number in this case is $Re={\frac {UL}{\nu }}={\frac {10^{2}}{10^{-6}}}=10^{8}$, and we expect a turbulent flow.

Richardson Number (flux)

• Definition: $Rf={\frac {-g\alpha {\overline {wT'}}}{-{\overline {uw}}U_{z}}}$, where the overlines denote the ensemble averages of variables.
• Interpretation: The ratio of the buoyant destruction of turbulent kinetic energy to the shear production.
• Analysis: The flux Richardson number is related to the gradient Richardson number by $Ri={\frac {\nu _{T}}{\kappa _{T}}}Rf$, where the ratio $\nu _{T}/\kappa _{T}$ is the turbulent Prandtl number.

Richardson Number (gradient)

• Definition: $Ri=N^{2}/U_{z}^{2}$, where $N$ is the buoyancy frequency, and $U$ is the background horizontal velocity.
• Interpretation: The ratio between the strength of stratification and velocity shear.
• Analysis: The gradient Richardson number is used for determining the linear stability of an inviscid stratified flow. A sufficient condition for the flow to be linearly stable is that the local Richardson number exceed 0.25 throughout the flow. However, $Ri<0.25$ does not mean the flow is necessarily unstable (the criterion is not sufficient). When the flow is not a parallel shear flow, the meaning of $Ri$ is not clear.

Rossby Number

• Definition: $Ro={\frac {U}{fL}}={\frac {1/f}{L/U}}$, where $U$ is the characteristic velocity, and $f$ is the Coriolis frequency, and $L$ is the characteristic length scale.
• Interpretation: The ratio of the advective forces to the Coriolis pseudo-forces, or the ratio of the planetary time scale to the time scale of fluid motion.
• Analysis: When $Ro\ll 1$, Coriolis effects dominate and the system approaches geostrophic balance.
• Example: The Coriolis frequency of Earth is $f=1\mathrm {day} ^{-1}$ or $f\approx 10^{-4}\mathrm {s} ^{-1}$ (for mid-latitudes). Assuming the characteristic speed of fluid flow is $U=0.1\mathrm {m} /\mathrm {s}$, then a flow in a laboratory tank of length $L=1\mathrm {m}$ has a Rossby number $Ro\approx 10^{3}$ and is not affected by the Coriolis force, whereas a flow on the ocean scale with $L=10\mathrm {km}$ has a Rossby number $Ro\approx 0.1$ and will be affected by Coriolis effects.

Schmidt Number

• Definition: $Sc=\nu /\kappa$, where $\nu$ is the viscosity, and $\kappa$ is the mass diffusivity.
• Interpretation: The ratio of momentum diffusivity (i.e. kinematic viscosity) to mass diffusivity.
• Example: The typical Schmidt number for water is around 500, depending on the temperature and salinity. For direct numerical simulations, $Sc=1$ is commonly used in the literature. See also Prandtl number.

Wedderburn Number

• Definition: $W={\frac {gH^{2}}{Fr_{0}}}$, where $g$ is the gravitational acceleration, $H$ is the depth, $F$ is the wind forcing, and $r_{0}$ is the radius of basin.
• Interpretation: The ratio of stratification to wind forcing of a circular lake.
• Analysis: Small Wedderburn numbers correspond to large wind stress, wide lakes, and shallow water. Large Wedderburn numbers correspond to low wind stress, narrow lakes, and deep water.