Blog - fluid flows and infinite dimensional manifolds (part 4)

This page is a blog article in progress, written by Tim van Beek.

In part 2 we defined what an ideal incompressible fluid is. And how the equation of motion, Euler’s equation, can be seen as a geodesic equation on $\mathrm{SDiff}(M)$, the infinite dimensional manifold of volume preserving diffeomorphisms. Last time I have promised to talk about the pressure function. I have also mentioned that Arnold used the geometric setup to put a bound on weather forecasts. I will try to fulfill both promises in this blog post! So let us start with the pressure function.

Euler’s equation is a geodesic equation:

$\frac{d^2 \phi}{d t^2} = \partial_t u + \nabla_u u = \text{something
coming from curvature}$

The geodesic equation describes the acceleration that geodesics need to stay on the manifold. If we think about a surface $S$ in $\mathbb{R}^3$ for a moment, then we have at every tangent space $T_p S$ a orthonormal basis consisting of two tangent vectors $e_1, e_2$ and a normal vector $n$, with the orthogonal sum decomposition with respect to the induced scalar product:

$\mathbb{R}^3 = \span(e_1, e_2) \oplus \span(n) = T_p S \oplus \span(n)$

where $\span(e_i)$ denotes the linear span of the vector(s) $e_i$. Sidenote for math nerds: Maybe some of you already know that there is a way to use this as the defining property of the Ehresmann connection.

How about Euler’s equation? If we think of $\mathrm{SDiff}(M)$ as a submanifold of $\mathrm{Diff}(M)$ just as our surface $S$ is a submanifold of $\mathbb{R}^3$, we see that there is an analog: If we can use the Helmholtz decomposition, we see that we have an $(L^2-)$ orthogonal decomposition of an arbitrary vector field $w \in \mathrm{Vec}(M)$ into a divergence free vector field $v \in \mathrm{SVec}(M)$ and the gradient of a potential. So, we have an orthogonal decomposition

$\mathrm{Vec}(M) = \mathrm{SVec}(M) \oplus \{ \text{gradients} \}$

I won’t go into the question to which function spaces each player needs to belong .

But one of the references on Wikipedia has a thorough exposition of this:

• Vivette Girault and Pierre-Arnaud Raviart: *Finite element methods for Navier-Stokes equations. Theory and algorithms.* Springer Series in Computational Mathematics, Berlin, 1986.

Since the acceleration in the geodesic equation is orthogonal to the tangent space, we see that in the equation

$\frac{d^2 \phi}{d t^2} = \partial_t u + \nabla_u u = \text{something coming from curvature}$

the “something coming from curvature” needs to be the gradient of a scalar function:

$\frac{d^2 \phi}{d t^2} = \partial_t u + \nabla_u u = - \nabla p$

Note that if we want to keep the interpretation of p as pressure, we should put a minus sign before the gradient: The gradient points into the direction of steepest increase, but the force resulting from pressure is into the *opposite* direction. Some authors don’t put a minus sign there, so be aware!

So, using our Riemanninan geometry setup, we note that pressure can be seen as the source of acceleration that keeps the geodesics, the fluid flows, in the manifold of volume preserving diffeomorphisms.

Predicting the weather means that you try to calculate what will happen the next days based on how it looks today, which is an initial value problem. When you solve an initial value problem you have to take into account errors in the initial data. For the weather, initial data are clearly limited by the measurement devices that are available. For the moment, let us accept as a vastly simplified model for the weather on Earth the flow of the atmosphere as an ideal incompressible fluid.

Actually, the weather forecast for the next 24 hours pretty much depends on a prediction of what will be blown your way, so the model is not quite as bad as it may seem at first. (TODO: add reference).

In this model, predicting the weather means calculating a geodesic in $\mathrm{SDiff}(M)$ where we could set $M$ to the unit sphere, if we think in two dimensions. You could also extend $M$ to a three dimensional manifold by extending it horizontally. Anyway, errors in your initial data means that your geodesic will not start in the exact right point. So, the question is, what can one say about geodesics that start from nearby points? Do they stay close or do they deviate from each other?

Since we have Riemann geometry at our disposal, we can use the **geodesic deviation equation** to explore this question. Since not everybody will know about this equation, I will talk a little bit about it before we get to apply it to fluid flows. I will throw around some buzz words for those who would like to look them up in the literature.

For the moment and for simplicity, think about a smooth orientable surface $S$ embedded in $\mathbb{R}^3$. Let us assume that we can parametrize a family of geodesics $\gamma_{\alpha}$ in $S$ smoothly by a real parameter $\alpha$. We can pick two geodesics $\gamma_{\alpha = 0}$ and $\gamma_{\alpha}$ that start close to each other in the sense that

$d(\gamma_{\alpha = 0}(0), \gamma_{\alpha}(0)) \lt \epsilon$

for some small positive $\epsilon$. We can talk about the distance as a function of $\alpha$ and the affine parameter $t$:

$(t, \alpha) \mapsto d(\gamma_{0}(t), \gamma_{\alpha}(t))$

This could turn out to be a rather complicated function, but we can try to see what we can find out about the derivative in the $\alpha$ direction, thought of as a tangential vector of $S$:

$J \coloneqq \frac{\partial \gamma_{\alpha}(t)}{\partial \alpha} |_{\alpha = 0}$

It turns out that $J$ as defined above satisfies a differential equation in the parameter $t$. That is the geodesic deviation equation. It says that:

$\nabla^2 J + R(J, \dot{\gamma}) \dot{\gamma} = 0$

R is the Riemann curvature tensor. At this point, in the literature about Riemann geometry, people turn the definition around and define the term **Jacobi field** to mean any vector field that satisfies this differential equation. It is important to keep in mind that a Jacobi field describes only a local, linear approximation.

Remember that you can feed the Riemann curvature tensor two vector fields $X$ and $Y$ and get an automorphism on the tangential bundle:

$R(X, Y): TM \to TM$

$Z \mapsto R(X, Y) \; Z$

The operator that appears in the geodesic deviation equation takes one vector field $X$ and yields an automorphism on the tangential bundle this way:

$R(X, \cdot): TM \to TM$

$Z \mapsto R(X, Z) \; Z$

Mathematicians who don’t know general relativity call this the **directional curvature operator**. It would seem that the derived concept of sectional curvature is more widely known. This is for linearly independent vectors $X_p, Y_p \in T_p M$ defined to be this normalized linear form:

$K(X, Y) \coloneqq \frac{1}{\| X \wedge Y \|^2} \; \langle R(X, Y) Y, X \rangle$

Of course everybody who knows general relativity should call the directional curvature operator rather the **tidal force operator**! The reason for this is that in general relativity, free falling test masses follow geodesics in space time. This applies to you if you are in a geostationary satellite, for example. If you leave a screwdriver outside on the side of the Earth where gravitation is slightly stronger, you will notice that the screwdriver moves farther away from you and your satellite over time. And you will attribute this to gravity, of course. The acceleration of the screwdriver is approximately described by the tidal force operator via the geodesic deviation equation, hence the name.

In our case a geodesic describes approximately the fluid flow of air on the surface of the Earth. The geodesic deviation equation tells us how the curvature of the manifold we are talking about influences the stability of a “prediction” (a geodesic) against errors in the initial data (geodesics starting at close points of the prediction). This is what I have mentioned in the introduction of the first post of this series, and one of the highlights of Arnold’s work.

To get an idea what the Jacobi equation is about in the case of an orientable smooth surface $S$ embedded in $\mathbb{R}^3$, we can calculate the concrete form of the geodesic deviation equation in the following way:

Let $\gamma$ be a geodesic and $J$ be a Jacobi field along $\gamma$. Choose an orthonormal base consisting of $(\dot{\gamma}, v)$, that is the vectors $\dot{\gamma_p}$ and $v_p$ are an orthonormal base of $T_p S$ at every point $p \in S$. Then we can expand $J$ in this basis:

$J(t) = x(t) \; \dot{\gamma} + y(t) \; v$

The Riemann curvature tensor is

$R^1_{121} = \langle R(\dot{\gamma}, v) \dot{\gamma}, v \rangle = K$

where $K$ denotes the **Gaussian curvature** of our surface $S$. So the Jacobi equation in our case is

$\frac{d^2 y}{d t^2} + K y = 0$

If $K$ is positive, then we get an oscillating $y$, and when it is negative, $y$ will escape to infinity. Since all of this is only a *local* approximation, we cannot conclude what happens *globally* without assuming more about $S$. A sphere, for example, has a constant positive curvature. On a sphere, geodesics like meridians “oscillate”, which means, starting in a pole, they first deviate but then meet again in the opposite pole. This example is the starting point of a more general definition:

A Riemannian manifold $M$ is called a space of constant sectional curvature, or a **space form**, if the sectional curvature $K(X_p, Y_p)$ is constant for all linearly independent tangent vectors $X_p, Y_p$ and all $p \in M$. A space form is called **spherical, flat, hyperbolic** if $K \gt 0, = 0, \lt 0$.

Why these names? Because it is possible to prove (for finite dimensional manifolds, TODO: don’t know about infinite dimensional ones), that a flat space form ist locally isometric to Euclidean space.

Interestingly, the sectional curvature already determines the whole curvature tensor. We can make the influence of the sectional curvature on a Jacobi field $J$ more explicit. Let us fix a tangent space $T_p M$, a geodesic $\gamma$ with $\gamma(0) = p$ and $\cdot \gamma(0) = v$. Choose another vector $w \in T_p M$ that is orthogonal to $v$ and has lenght 1, that is $\| w \| = 1$. We can write down a Jacobi field $J(t)$ that is given by the exponential function:

$J(t) = (d \mathrm{exp}_p)_{t v} (t w)$

Then the norm of $J(t)$ has a Taylor expansion that involves the sectional curvature $K(v, w)$:

$\| J(t) \| = t - \frac{1}{6} K(v, w) t^3 + O(|t^3|)$

If you would like to look that up, you can find it here (corollary 2.10):

• Manfredo Perdigao do Carmo: *Riemannian Geometry*, (Birkhäuser, Boston, 1992)

So the sectional curvature turns up as the first term that describes a deviation from linear growth. So we will need to calculate or estimate the sectional curvature. We need to do this for a Riemannian manifold that is also a Lie group and has a right invariant metric. The latter part means that right translations map the group to itself isometrically, by definition. Therefore it would suffice to calculate or estimate the curvature of two dimensional planes in the tangent space at identiy, which is the Lie algebra.

As a simple example we will calculate the sectional curvature of the diffeormorphism group of the circle. I will repeat some of the formulas from the first blog post for convenience.

We need to calculate $\nabla_X Y$ for right invariant vector fields. I will write again $X = u(x) \partial_x$, $Y = v(x) \partial_x$ and $Z = w(x) \partial_x$ with smooth real valued functions $u, v, w$ with the boundary condition $u(x +1) = u(x)$.

For a right invariant metric we have

$\nabla_X Y = \frac{1}{2} (\mathrm{ad}_X Y - \mathrm{ad}^*_Y X - \mathrm{ad}^*_X Y)$

We can calculate the action of the coadjoint operator via

$\langle \mathrm{ad}^*_X Y, Z \rangle = \int_{S^1} v (u w_x - u_x w) \mathrm{d x} = - \int_{S^1} (u v_x + 2 u_x v) w \mathrm{d x}$

So we get for the coadjoint operator

$\mathrm{ad}^*_X Y = - (u v_x + 2 u_x v) \partial_x$

We get for the Riemannian connection:

$\nabla_X Y = (2 u v_x + v u_x) \partial_x$

The inertial operator is defined to be:

$R(X,Y) Y = \nabla_X (\nabla_Y Y) - \nabla_Y (\nabla_X Y) - \nabla_{[X, Y]}$

We can calculate each term involved:

$\nabla_X (\nabla_Y Y) = 6 u(v v_x)^' + 3 v v_x u_x$

$\nabla_Y (\nabla_X Y) = 2 v (2 u v_x + v u_x)' + (2 u v_x + v u_x) v_x$

$\nabla_{[X, Y]} Y = 2 (u v_x - v u_x) v + v (u v_x - v u_x)'$

Combining these, we get:

$R(X,Y) Y = 2 v_x (u v_x - v u_x) v + v (u v_x - v u_x)^'$

Surprisingly, this results in a very simple formula for the sectional curvature:

$K(X, Y) = \int_{S^1} (u v_x - v u_x)^2 d x$

The sectional curvature is non-negative! So geodesics on $Diff(S^1)$ should be stable. The explicit calculation of the sectional curvature will turn out to be unfeasable for more complicated diffeomorphism groups. We will see how one can get approximate results in one of the next blog posts.

I would like to mention some references for the mathematically inclined: It is not hard to do Riemannian geometry on Hilbert manifolds. Or course one needs to take care that one does not have local compactness. This means for example that the theorem of Hopf-Rinow fails. That is, infinite dimensional manifolds could turn out to be not geodesically complete.

A nice textbook that systematically handles Hilbert manifolds is this one:

• Wilhelm P.A. Klingenberg: *Riemannian Geometry*, 2nd edition, Walter de Gruyter, Berlin, 1995

But it seems to me to be more appropriate to start thinking about Fréchet manifolds, because for example diffeomorphism groups can be equipped with different scalar products on their tangent spaces. These in turn result in different partial differential equations as geodesic equations. A nice paper that explains this in more detail is this one:

• B. Khesin, J. Lenells, G. Misiolek, and S. C. Preston: *Curvatures of Sobolev Metrics on Diffeomorphism Groups*, (arxiv).

The authors use a nice name for the volume preserving diffeomorphisms $\mathrm{SDiff}(M)$: They call them “volumorphisms”. They mention that for a 2-dimensional base manifold $M$ the manifold $\mathrm{SDiff}(M)$ equipped with the right invariant $L^2$ scalar product is geodesically complete. In three dimensions however it would seem that “the problem is open and challenging”. So if you were looking for a math problem to beat the boredom, here is one :-)

category: blog, mathematical methods