Lagrangian Coordinates in Fluid Flow

Question

I apologize if this is not the right place to ask this question:

I am currently reading a paper by Y. Brenier, where for the fluid flow he introduces a Lagrangian label $a$ instead of the vertical coordinate $z$, and defines a "Lagrangian foliation to be a family of sheets $z = Z(t,x,a)$, labelled by $a \in [0,1]$, where $Z$ is a smooth function such that: $$ 0\leq Z(t,x,a)\leq 1, \,\,\,Z(t,x,0) = 0, \,\,\,Z(t,x,1) = 1\\ \partial_a Z(t,x,a) > 0\\ \partial_t Z(t,x,a) + u(x,t,Z(t,x,a))\cdot \nabla_x Z(t,x,a) = w(t,x,Z(t,x,a))" $$ where $x \in \mathbb{R}^2$, $u = (u_1,u_2)$ is the horizontal flow velocity and $w$ is the vertical velocity and $\nabla_x = (\partial_{x_1},\partial_{x_2})$.

A possible initial choice is $Z(0,x,a) = a$.

My question is: What is the physical interpretation for this? It seems like instead of labeling each individual fluid parcel in the classical Lagrangian coordinates, we are only labeling horizontal planes instead, and then seeing how these sheets move and deform with the flow?

Sorry - "Homogeneous Hydrostatic Flows with Convex Velocity Profiles" by Yann Brenier. — Mark14, Mar 28 '13 at 00:37
Right, it's a foliation/coordinate that just happens to be more useful ("a clever substitution", although implicitly described) to study the particular problem. Why don't you consider the previous sentence to be a physical interpretation? — Luboš Motl, Mar 28 '13 at 06:25

score 1 · Answer 1 · answered Mar 28 '13 at 11:59

Your question 'What is the physical interpretation for this?' is not too clear. What is the physical meaning of which aspect of the Initial Boundary Value Problem (IBVP) that you describe?

Regardless of the above, from what you have written and the title and abstract of the paper we are dealing with a periodic solution in $x$ for a convex flow in $z$. From this if we take our velocity field parallel to the $z$ coordinate to be $u(t, x, z)$ then you can write

$$ \partial_{zz} u(t, x, z) > 0.$$

So using $z = Z(t, x, a)$ with $a \in [0, 1]$ (which specifies a 'layering' of the flow), the condition $\partial_{a} Z(t, z, a) > 0$ is directly linked to the stated convexity of the velocity profiles (given by the expression above) and given the 'layering' of the flow you can write

$$[\partial_{zz} u(t, x, Z(t, x, a))]^{-1} = \partial_{a} Z(t, z, a),$$

So $\partial_{a} Z(t, z, a) > 0$ is just a condition of the convexity of the flow. The other conditions are merely assumed boundary conditions. So the flow is convex in the z-direction, and the sheets are layered 'vertically' (if we take z as 'vertical'), not 'horizontal planes' (the $x_{1}, x_{2}$ plane) as you have put it.

To see this, from the convexity condition above we can write

$$\partial_{z} u(t, x, z) = \kappa,$$

where $\kappa \in \mathbb{R}$. Using the advection equation (shown for an arbitrary scalar field $\lambda$ with no source or sink terms)

$$\partial_{t}\lambda + u . \nabla \lambda = 0,$$

for this flow we can write

$$\partial_{t}(\partial_{z} u(t, x, z)) + u\partial_{x}(\partial_{z} u(t, x, z)) + w\partial_{z}(\partial_{z} u(t, x, z)) = 0,$$

as $\partial_{z} u(t, x, z) = \kappa,$ this shows that the quantity $\partial_{z} u(t, x, z)$ is advected with the flow.

I hope this helps.

Lagrangian Coordinates in Fluid Flow

1 Answers1