Non-dimensionalisation for mortals: waves, nonlinearity and dispersion (part 1)
When you open a fluid mechanics or wave theory book, the first thing you meet are these monster PDEs: \[ \begin{aligned} & u_t + u u_x + w u_z = -\frac{1}{\rho_0} P_x, \\ & w_t + u w_x + w w_z = -g - \frac{1}{\rho_0} P_z, \\ & u_x + w_z = 0. \end{aligned} \]
This is just momentum conservation (horizontal and vertical, without viscosity) and mass conservation (incompressibility). But in this dimensional form, the equations don’t tell us when waves will behave linearly, when nonlinearity will dominate, or when dispersion matters. That’s the whole point of non-dimensionalisation.
What is non-dimensionalisation?
Think of it as putting the equations on a new measuring scale. Instead of dragging metres, seconds and Pascals everywhere, we rescale everything with characteristic values (length, depth, speed…). What’s left are pure numbers that show us which physical processes are important.
In wave theory, the two heroes are:
- Nonlinearity (\(\varepsilon\)): amplitude compared to depth. \[ \varepsilon = \frac{a_0}{d_0} \]
- Dispersion (\(\mu\)): depth compared to wavelength. \[ \mu = \left(\frac{d_0}{L_0}\right)^2 \]
Step 1: where do the scales come from? (not a cookbook!)
At first glance, the scaling rules can look like they were pulled straight from a recipe book. But each choice has a physical intuition:
- Horizontal scale: the wavelength is the natural measure, \(x = L_0 x'\).
- Vertical scale: the depth is the obvious candidate, \(z = d_0 z'\).
- Time scale: in shallow water, waves move with speed \(c_0 = \sqrt{g d_0}\). If it takes that speed to cover one wavelength, \[ T_0 = \frac{L_0}{c_0}, \quad t = \frac{L_0}{c_0} t'. \]
- Surface elevation: amplitude \(a_0\), written as \(a_0 = \varepsilon d_0\). Then \(\zeta = \varepsilon d_0 \zeta'\).
- Velocities:
- Horizontal: \(u \sim \varepsilon c_0\).
- Vertical: smaller, compressed by the aspect ratio: \[ w \sim \varepsilon \frac{L_0}{d_0} c_0. \] That factor \(L_0/d_0\) is chosen so that the continuity equation looks nice.
- Pressure: hydrostatic variation, \(P \sim \rho g d_0\).
Step 2: put it all together
So our non-dimensional variables are:
\[ x=L_0 x', \quad z=d_0 z', \quad t=\frac{L_0}{c_0} t', \] \[ u=\varepsilon c_0 u', \quad w=\varepsilon \frac{L_0}{d_0}c_0 w', \quad \zeta=\varepsilon d_0 \zeta', \quad P=\rho g d_0 P'. \]Notice: we deliberately inserted \(\varepsilon\) into \(u,w,\zeta\). This ensures that nonlinearity pops out automatically in the equations.
---Step 3: the equations with \(\varepsilon\) and \(\mu\)
Take the horizontal momentum equation:
\[ u_t + u u_x + w u_z = -\frac{1}{\rho_0} P_x. \]After substitution and simplification:
\[ \varepsilon u'_{t'} + \varepsilon^2 u' u'_{x'} + \frac{\varepsilon^2}{\mu} w' u'_{z'} = -P'_{x'}. \]Boom - now we see:
- \(\varepsilon\) tagging nonlinear terms,
- \(\mu\) tagging dispersive terms (from vertical derivatives).
Continuity becomes neat: \[ u'_{x'} + w'_{z'} = 0. \]
The vertical momentum equation gives terms with \(\varepsilon \sqrt{\mu}\) and \(\varepsilon^2/\sqrt{\mu}\), showing how vertical acceleration links nonlinearity and dispersion.
Step 4: boundary conditions
- Free surface: \[ \zeta'_{t'} + u' \zeta'_{x'} = w' \quad \text{at } z'=\varepsilon \zeta', \] \[ P'=0 \quad \text{at } z'=\varepsilon \zeta'. \]
- Bottom: \[ w'=0 \quad \text{at } z'=-1. \]
Why bother?
Because the result is a non-dimensional system where the key effects are clearly marked:
- Nonlinearity → controlled by \(\varepsilon\).
- Dispersion → controlled by \(\mu\).
Now we can choose our model like tuning an equaliser:
- \(\varepsilon \ll 1, \ \mu \ll 1\) → linear shallow water,
- \(\varepsilon = O(\mu)\) → Boussinesq,
- keep higher-order terms → Serre–Green–Naghdi.
Take-home message
Non-dimensionalisation is not a recipe from a textbook. It’s a way to reveal the underlying physics inside the equations.
- Geometry sets the scales (\(L_0, d_0\)).
- Hydrostatics sets the pressure.
- Wave motion sets time and velocity scales.
- Amplitude introduces \(\varepsilon\); depth-to-length introduces \(\mu\).
Coming up next…
In Part 2 we’ll take these non-dimensional equations and integrate them over the vertical. This will show us how to move from the full 3D Euler system to a 2D depth-averaged model, leading to a fully nonlinear hydrodynamic description of waves in shallow water. That’s where the famous Serre equations (also known as Serre–Green–Naghdi) come into play, letting us simulate nonlinear wave propagation and transformation on a beach. 🏖️
see ya later! Cesar
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