Potential Vorticity and the Conservation Laws of Energy and Momentum for a Stratified Incompressible Fluid

  • Felix V. Dolzhansky
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 103)


Note, however, an important feature of the motion of a stratified fluid: it is fibered into surfaces of constant density (isopycnic or iso-density surfaces) and remains such in the process of evolution: every fluid particle belonging to such a surface at the initial moments remains on the same surface due to the Lagrangian invariance of density. (This is why stratified fluid is also often called fibered.) In turn, the motion along any iso-density surface ρ(t,x)=ρ 0=const is the motion of a homogeneous incompressible fluid, for which the Kelvin theorem holds. In particular,
$$ K_{0} \doteq \oint\limits _{C_{0}} \mathbf{u} \delta\mathbf{l}= \boldsymbol {\Omega} d\boldsymbol {\sigma}_{0} $$
is a Lagrangian invariant (dK 0/dt=0), where C 0 is an infinitesimal closed contour on the iso-density surface, while d σ 0 is an element of this surface bounded by the contour C 0.


Incompressible Fluid Potential Vorticity Fluid Particle Fluid Domain Stream Line 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Felix V. Dolzhansky
    • 1
  1. 1.

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