[Physics] Physical intuition for Lamb vector

aerodynamicsfluid dynamicsvortex

In many aerodynamical descriptions involving vortex structures (e.g. the Crocco's theorem) there is a useful physical quantity called Lamb vector:

\vec{\omega} \times \vec{v}

where $\vec{\omega}$ denotes local vorticity and $\vec{v}$ local velocity.

I lack a physical intuition for sch quantity, cause I have only see its derivation via "cold" vector calculus. It certainly has dimension of acceleration.

Could anybody provide a verbal explanation, what this quantity represents?

Best Answer

Consider the Navier-Stokes equations $$\frac{\partial \bf{u}}{\partial t}+\bf{u}\cdot\nabla \bf{u}=-\nabla p+\nu \nabla^2\bf{u}.$$

We can rewrite the inertial term as $$\bf{u}\cdot\nabla \bf{u}=\frac{1}{2}\bf{u}^2-\bf{u}\times\boldsymbol{\omega}.$$

Therefore, the Navier-Stokes equations can be rewritten as

$$\frac{\partial \bf{u}}{\partial t}=-\nabla\left( p+\frac{1}{2}\bf{u}^2\right)+\bf{u}\times \boldsymbol{\omega}+\nu \nabla^2\bf{u}.$$

We see that the Vortex force, i.e. $\bf{u}\times \boldsymbol{\omega}$, shows up on the RHS of our equations, and makes it clear that velocity-vorticity interactions can accelerate the flow. There are many examples where this perspective offers insight. I am going to give one below.

Let us consider Langmuir circulations. These were first reported by Nobel laureate Irving Langmuir$^*$ on a trip he was taking across the Atlantic. These circulations, basically counter-rotating rolls, are an extremely important process in upper ocean dynamics, as they mix the upper ocean, which then modulates air-sea interactions. A picture of these rolls is shown below. enter image description here

The derivation of the equations describing Langmuir circulation, known as the Craik-Leibovich equations, is pretty involved as one must grind through a great deal of algebra. Useful references are Craik & Leibovich (1976), and Leibovich's review in 1983.

Very heuristically, we consider a wave field, that contains Stokes drift $\bf{u}_s$. This drift is due to the slightly elliptical orbits that finite amplitude (Stokes) waves have. It is a flux of mass.

Now, if one properly sets up their scales, does the right averaging etc they end up with the equation

$$\frac{\partial \bf{v}}{\partial t}+\bf{v}\cdot \nabla \bf{v} = -\nabla p +\bf{F_{vor}}+\nu\nabla^2 \bf{v}$$

where $\bf{v}$ is the velocity of the (slow) mean flow, $p$ the pressure,$\nu$ the kinematic viscosity, and

$$\bf{F}_{vor}=\bf{u}_s\times \boldsymbol{\omega},$$

where $\boldsymbol{\omega} = \nabla \times \bf{v}$. Note this is the vortex force we discussed above.

So, in this context, the mean flow obeys something analogous to the Navier-Stokes equations, $\bf{PLUS}$ the vortex force due to the interaction of the Stokes drift and the mean flow.

Let's consider an example, shown in this figure enter image description here

The stokes drift is shown here as $U^s$. We consider an initial perturbation, $u'$, as shown by the arrows on the left side of the figure. If one works out the curl in $\bf{F}_{vor}$, they see that this would then lead to convergence of the flow, and then necessarily to downwelling. This then decreases with depth, forming the rolls known as Langmuir circulations.

$^*$ Langmuir worked with Kurt Vonnegut's brother (who was a climate scientist), and formed the basis for the character Felix Hoenikker in Cat's Cradle.

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