# Dimensionless Numbers

**Burger Number**

- Definition: , where is the nonrotational baroclinic phase speed, is the inertial frequency under the f-plane approximation, and 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 . If the lake has radius and a depth of , then and so , which means a stratification dominated lake.

**Froude Number**

- Definition: , where is the gravitational acceleration, and and 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. describes critical flow; describes supercritical flow; describes subcritical flow. The Froude number is analogous to the Mach number in compressible flow.

**Grashof Number**

- Definition: , where is the gravitational acceleration, is the volumetric thermal expansion coefficient, is the temperature difference, is the volume, and is the kinematic viscosity.
- Interpretation: The ratio of buoyant forces to viscous forces.
- Analysis: When , 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: , where is the characteristic velocity scale, and is the speed of sound.
- Interpretation: The ratio of inertia force to compressibility force.
- Analysis: Compressibility effects can be neglected if .
- Example: The typical sound wave speed is m/s in air. For wind with a speed m/s, the Mach number is , and hence the compressibility is negligible.

**Prandtl Number**

- Definition: , where is the kinematic viscosity, and 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: , where is the acceleration due to gravity, is the coefficient of thermal expansion, is the vertical temperature gradient of the background state, is the depth of the fluid layer, is the thermal diffusivity, and is the kinematic viscosity.
- Interpretation: The ratio of the destabilizing effect of buoyancy force to the stabilizing effect of viscous force.

**Reynolds Number**

- Definition: , where and are the characteristic velocity and length scales, and 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 , and imagine a cube with side length 10 meters submerged in this water. The Reynolds number in this case is , and we expect a turbulent flow.

**Richardson Number (flux)**

- Definition: , 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 , where the ratio is the
*turbulent Prandtl number*.

**Richardson Number (gradient)**

- Definition: , where is the buoyancy frequency, and 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, 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 is not clear.

**Rossby Number**

- Definition: , where is the characteristic velocity, and is the Coriolis frequency, and 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 , Coriolis effects dominate and the system approaches geostrophic balance.
- Example: The Coriolis frequency of Earth is or (for mid-latitudes). Assuming the characteristic speed of fluid flow is , then a flow in a laboratory tank of length has a Rossby number and is not affected by the Coriolis force, whereas a flow on the ocean scale with has a Rossby number and will be affected by Coriolis effects.

**Schmidt Number**

- Definition: , where is the kinematic viscosity, and 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, is commonly used in the literature. See also Prandtl number.

**Wedderburn Number**

- Definition: , where is the gravitational acceleration, is the depth, is the wind forcing, and 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.