# 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 $r_0 = 1000 m$ and a depth of $H=10 m$, then $c = \sqrt{g H} = \sqrt{9.81 \cdot 10} \approx 9.9 \frac{m}{s}$ 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 kinematic 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.
• 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 kinematic 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 700, 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.