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Scale analysis (mathematics)

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Scale analysis is a powerful tool used in the mathematical sciences for the simplification of equations with many terms. First we must determine the approximate magnitude of individual terms. Then some negligibly small terms may be ignored. Consider for example the momentum equation in vertical coordinate in the atmosphere

${{\partial w }\over{\partial t }} + u {\frac{\partial w}{\partial x}} + v {\frac{\partial w}{\partial y}} + w {\frac{\partial w}{\partial z}} - {\frac{u^2 + v^2}{R}}= - { { \frac{1}{\varrho}}{\frac{\partial p}{\partial z}}} - g +2{\Omega u \cos \varphi} + \nu \left({\frac{\partial^2 w}{\partial x^2}}+{\frac{\partial^2 w}{\partial y^2}}+{\frac{\partial^2 w}{\partial z^2}}\right),\qquad(1)$

where R is Earth radius, Ω is frequency of rotation of the Earth, g is gravitational acceleration, φ is latitude ρ is density of air and ν is kinematic viscosity of air (we can neglect turbulence in free atmosphere).

In synoptic scale we can expect horizontal velocities about U = 10¹ m.s⁻¹ and vertical about W = 10⁻² m.s⁻¹. Horizontal scale is L = 10⁶ m and vertical scale is H = 10⁴ m. Typical time scale is T = L/U = 10⁵ s. Pressure differences in troposphere are dp = 10⁴ Pa and density of air ρ = 10⁰ kg·m⁻³. Other physical properties are approximately:

R = 6.378 × 10⁶ m;

Ω = 7.292 × 10⁵ rad·s^−1;

ν = 1.46 × 10⁻⁵ m·s^−1;

g = 9.81 m·s^−2.

Now we can introduce these values into the equation (1).

${\frac{10^{-2}}{10^5}}+10{\frac{10^{-2}}{10^6}} +10{\frac{10^{-2}}{10^6}} +10^{-2}{\frac{10^{-2}}{10^6}} -{\frac{10^2+10^2}{10^6}}$

$= - {{\frac{1}{1}} {\frac{10^4}{10^4}} } - 10 + 2 \times 10 \times 10^{-5} + 10^{-5} \left({\frac{10^{-2}}{10^{12}}} + {\frac{10^{-2}}{10^{12}}} + {\frac{10^{-2}}{10^{8}}} \right). \qquad (2)$

We can see that all terms on the right-hand side except the first and second are negligibly small. Thus we can simplify the vertical momentum equation to the hydrostatic equilibrium equation:

${ { \frac{1}{\varrho}}{\frac{\partial p}{\partial z}}} = - g. \qquad (3)$