Lord Kelvin

We return now to the concept of an ideal fluid. Let's recall that this type of fluid exhibits only a pressure force that acts orthogonal to the boundary, which neglects any tangential stresses (or viscous forces). Furthermore, we assume the fluid is incompressible. This gives us the following equations governing its motion \[ \begin{cases} \partial_t \vec{u}+\vec{u}\cdot\nabla \vec{u}=-\vec{\nabla p} \\ \nabla \cdot \vec{u}=0, \end{cases} \] which are known to us as the Euler equations.

In fluid mechanics, Kelvin's circulation theorem states that in a barotropic ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time. So sayeth our Lord Kelvin in 1869. Today's discussion will center around his theorem and hopefully provide us with more insight as to the formation of particular physical phenomena. So sit back, relax, pour yourself a glass of sherry and enjoy.

Before we start with the proof of Kelvin's theorem, let's recall an important fundamental concept in fluid mechanics: our particle trajectories. They are defined in the following manner (keeping in mind that we are in three dimensions)

\[ \begin{cases} \partial_t\vec{\varphi}(\vec{x},t)=\vec{u}(\vec{\varphi}(\vec{x},t),t) \\ \vec{\varphi}(\vec{x},0)=\vec{x}. \end{cases} \] Let $C$ be a simple closed contour in the fluid at $t=0$ and let $C_t=\vec{\varphi}_t(C)$ be the contour carried along by the flow. Then we have the circulation around $C_t$ to be defined as the line integral \[ \Gamma_{C_t}=\oint_{C_t}\vec{u}\cdot d\vec{s}. \] Now we are ready to prove the circulation theorem.

It suffices to show that \[ \frac{d}{dt}\Gamma_{C_t}=0. \] Let $\vec{x}(s)$ be a parametrization of $C$, $s \in [0,1]$. Then a parametrization of $C_t$ is $\vec{\varphi}(\vec{x}(s),t)$, $s\in [0,1]$. Thus, we have that % \begin{align*} \frac{d}{dt}\oint_{C_t}\vec{u}\cdot d\vec{s}&=\frac{d}{dt}\int_0^1\vec{u}(\vec{\varphi}(\vec{x}(s),t),t)\cdot\frac{\partial}{\partial s}\vec{\varphi}(\vec{x}(s),t)ds \\ &=\int_0^1\frac{D\vec{u}}{Dt}(\vec{\varphi}(\vec{x}(s),t))\cdot \frac{\partial}{\partial s}\vec{\varphi}(\vec{x}(s),t)ds + \int_0^1\vec{u}(\vec{\varphi}(\vec{x}(s),t),t)\cdot \frac{\partial}{\partial t}\frac{\partial}{\partial s}\vec{\varphi}(\vec{x}(s),t)ds \\ &=\int_0^1\frac{D\vec{u}}{Dt}(\vec{\varphi}(\vec{x}(s),t))\cdot \frac{\partial}{\partial s}\vec{\varphi}(\vec{x}(s),t)ds + \int_0^1\vec{u}(\vec{\varphi}(\vec{x}(s),t),t)\cdot \frac{\partial}{\partial s}\vec{u}(\vec{\varphi}(\vec{x}(s),t),t)ds \\ &=\int_0^1\frac{D\vec{u}}{Dt}(\vec{\varphi}(\vec{x}(s),t))\cdot \frac{\partial}{\partial s}\vec{\varphi}(\vec{x}(s),t)ds \\ &=\oint_{C_t}\frac{D\vec{u}}{Dt}\cdot d\vec{s} \\ &=-\oint_{C_t}\nabla p\cdot d\vec{s}=0. \ \ \ \Box \end{align*} Here we had $\frac{D}{Dt}=\partial_t + \vec{u} \cdot \nabla$ as our material derivative.

If we now take note of Stokes' Theorem, then we see that \[ \frac{d}{dt}\oint_{C_t}\vec{u}\cdot d\vec{s}=\frac{d}{dt}\int\int_{\Sigma_t}\vec{\omega}\cdot d\vec{A}=0, \] where $\vec{\omega}=\nabla \times \vec{u}$ is our vorticity. This tells us that not only is circulation conserved, but also the flux of vorticity. Now before proceeding, we define vortex lines and vortex tubes.

"A vortex line or vorticity line is a line which is everywhere tangent to the local vorticity vector. A vortex tube is the surface in the continuum formed by all vortex-lines passing through a given closed curve in the continuum."

Furthermore, the strength of a vortex tube is flux of vorticity across a cross-section of the tube.

It was in 1858 that Hermann von Helmholtz published his result on vortex tubes stating that the strength of a vortex tube for inviscid fluid motion remains constant with time, which can certainly be validated now by the work of Lord Kelvin (the flux of vorticity is conserved). From here, we observe that if a vortex tube is stretched and its cross-sectional area decreases, then the magnitude of our vorticity vector $\vec{\omega}$ must increase. To see this, consider the time evolution of a vortex tube. According to Helmholtz' theorem, tubes move with the fluid. Moreover, their strength does not change with time. Therefore, if the area of a cross-section of a tube should become very small, vorticity would have to amplify proportionally. Since the fluid is incompressible, however, the volume between two sections of the tube remains constant. Therefore, any shrinking of the cross-sectional area must be accompanied by a longitudinal stretching. We conclude that the local stretching of a vortex tube gives rise to a proportional amplification of the magnitude of vorticity. This is the so-called "vortex stretching mechanism."

We are now able to understand the basics of tornadogenesis. Here one envisions a circular shear flow parallel to the earth's surface. Any circular cylinder centered at the center of flow will have the flow field tangent to its boundary, and the vortex lines are straight up. Such a cylinder is therefore a vortex tube.

Now suppose there is a vertical density gradient caused by heating of the earth's surface and a resulting updraft caused by the buoyancy effect. In this, the air moves rapidly upward in localized regions, and to satisfy conservation of mass, other air has to move rapidly in those regions. The density variation is actually minor although the convective winds resulting from it are not minor. So the usual assumption of incompressibility is okay.

This movement is superimposed as a perturbation of the basic circular shear flow mentioned above. The vortices are now the prefered regions of updraft. So we have a vortex together with an updraft in the same direction. The updraft has the effect of stretching the vortex tube. In stretching the vortex tube, cross-sectional areas contract, which results in an intensification and localization of the vorticity, hence a strengthening of the associated swirling winds. A tornado is now born.

For further reading, I recommend

A. Chorin, J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, (1979).

P. Fife, A Gentle Introduction to the Physics and Mathematics of Incompressible Flow, Course Notes, 2000.