# Equations of Motion; Surface Gravity Waves

• Joseph Pedlosky
Chapter

## Abstract

For a rotating stratified fluid, the general equations of motion can be written as:
1. 1.
Momentum equation:
$$\rho \left[ {\frac{{d\vec u}}{{dt}} + 2\Omega \times \vec u} \right] = - \nabla p + \mu {\nabla ^2}\vec u + k\nabla (\nabla \cdot \vec u)\;(if\;\mu \;cons\tan t,\;k\;is\sec ondvisosity)$$
(3.1)

2. 2.
Mass conservation:
$$\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot (\rho \vec u) = 0\;;and$$
(3.2)

3. 3.
Thermodynamic energy equation:
$$\frac{{ds}}{{dt}} = H$$
where s is specific entropy and H is the nonreversible heat addition. This can be rewritten, assuming that s is a thermodynamic function of p and ρ,
$${c_p}\frac{{dT}}{{dt}} - \frac{{\alpha T}}{{\rho t}}\frac{{dp}}{{dt}} = \Phi + \frac{k}{\rho }{\nabla ^2}T + Q + k{(\nabla \cdot \vec u)^2} \equiv H$$
(3.3)

where s is specific entropy and H is the nonreversible heat addition.

### Keywords

Entropy Beach Vorticity Compressibility Dition

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### References

1. Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, London, pp 615 (especially Chapter 1)Google Scholar
2. Kundu PK (1990) Fluid mechanics. Academic Press, San Diego, pp 638 (especially Chapter 5)Google Scholar
3. Lamb H (1945) Hydrodynamics, 6th edition. Dover Publications, New York, pp 738Google Scholar
4. Stoker JJ (1957) Water waves. Interscience, New York, pp 567Google Scholar