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Analytical hydrodynamics

Contents

Idea

Analytical hydrodynamics, as it is understood here, means the investigation of fluid flows using tools from topology and differential geometry. These tools enable us to describe properties and conservation laws of fluid flows that may be important for computational fluid dynamics, especially for climate models. Climate models need to solve the Navier-Stokes equations for the atmosphere and the oceans, they need to use discrete approximation schemes for this task. Such schemes can cause artefacts like the violation of conservation laws of the partial differential equations that they approximate. This phenomenon poses a huge problem for long running models like climate models.

A climate model that violates the conservation of energy, for example, cannot be used to predict an average temperature (or, quite possible, to predict anything about the climate at all). Some currently used climate model suffer from these insufficiencies. Necessary corrections have to be added manually and heuristically and are commonly called “flow corrections”. It is a part of ongoing research to remove these kinds of corrections from climate models.

Details

Ideal Fluids

The flow of an ideal fluid, that is an incompressible, inviscid and homogenuous fluid, filling a certain domain, is from the mathematical viewpoint described by a geodesic on the group of diffeomorphisms of that domain that preserve volume elements. The geodesics are the geodesics of the Riemannian metric given by the kinetic energy.

We’ll make all of this precise in the following:

As domain we take a compact Riemannian manifold M. The diffeomorphisms of M form an infinite dimensional group D(M). D(M) is a Fréchet manifold, so that some concepts from finite dimensional differential geometry may be defined on it.

For ideal fluids we’ll actually look at the subgroup of diffeomorphisms that conserve a given volume form ω on M:

SDiff(M):={fD(M):f *ω=ω}SDiff(M) := \{ f \in D(M): f^* \omega = \omega \}

SDiff is clearly a subgroup, it is also a closed subgroup and therefore a Lie group.

The (infinite dimensional) Lie algebra of SDiff(M) is the vector space of all vector fields on M with zero divergence.

Quick Reminder of Differential Geometry

Divergence, Pullback

Let M be a smooth orientable Riemannian manifold of dimension n. A volume form ω is a n-form that vanishes nowhere. In 3 with cartesian coordinates x,y,z the canonical example would be

ω=dxdydz\omega = d x \wedge d y \wedge d z

The dual basis of dx etc. is denoted by x etc. in our example.

The divergence of a vector field X with respect to a volume form ω is the unique scalar function div(X) such that:

div(X)ω=d(ι Xω)div(X) \omega = d(\iota_X \omega)

When we use our example we’d write of course

X=f 1 x+f 2 y+f 3 zX = f_1 \partial_x + f_2 \partial_y + f_3 \partial_z

and the divergence of X would be

div(X)= xf 1+ yf 2+ zf 3div(X) = \partial_x f_1 + \partial_y f_2 + \partial_z f_3

which we get if we plug in the expression for X into the formula d(ι Xω).

Given two manifolds M,N and a differentiable map f:MN, we can pull back a differential form ω on N to one on M via

f *ω p(v 1,...,v n):=ω f(p)(df(v 1),...,df(v n))f^{*} \omega_p (v_1, ..., v_n) := \omega_{f(p)} (d f(v_1), ..., d f(v_n))

Levi-Civita Connection and Geodesics

On a Riemannian manifold M with tangential bundle TM there is a unique connection, the Levi-Civita connection, with the following properties for vector fields X,Y,ZTM:

ZX,Y= ZX,Y+X, ZY(compatibility with the metric)Z \langle X, Y \rangle = \langle \nabla_Z X, Y \rangle + \langle X, \nabla_Z Y \rangle \; \text{(compatibility with the metric)}
XY YX=[X,Y](torsion freeness)\nabla_X Y - \nabla_Y X = [X, Y] \; \text{(torsion freeness)}

If we combine both formulas we get

2 XY,Z=XY,Z+YZ,XZX,Y+[X,Y],Z[Y,Z],X+[Z,X],Y2 \langle \nabla_X Y, Z \rangle = X \langle Y, Z \rangle + Y \langle Z, X \rangle - Z \langle X, Y \rangle + \langle [X, Y], Z \rangle - \langle [Y, Z], X \rangle + \langle [Z, X], Y \rangle

If the scalar products are constant along every flow, i.e. the metric is invariant, then the first three terms on the right hand side vanish, so that we get

2 XY,Z=[X,Y],Z[Y,Z],X+[Z,X],Y2 \langle \nabla_X Y, Z \rangle = \langle [X, Y], Z \rangle - \langle [Y, Z], X \rangle + \langle [Z, X], Y \rangle

This latter formula can be written in a more succinct way if we introduce the coadjoint operator. Remeber the adjoint operator defined to be

ad XZ=[X,Z]ad_X Z = [X, Z]

With the help of the scalar product we may define

ad X *Y,Z:=Y,ad XZ=Y,[X,Z]\langle ad^*_X Y, Z \rangle := \langle Y, ad_X Z \rangle = \langle Y, [X, Z] \rangle

Then the formula above for the covariant derivative can be written as

2 XY,Z=ad XY,Zad Y *X,Zad X *Y,Z2 \langle \nabla_X Y, Z \rangle = \langle ad_X Y, Z \rangle - \langle ad^*_Y X, Z \rangle - \langle ad^*_X Y, Z \rangle

Since the inner product is nondegenerate, eliminating Z leads to

2 XY=ad XYad X *Yad Y *X2 \nabla_X Y = ad_X Y - ad^*_X Y - ad^*_Y X

A geodesic curve is one whose tangent vector X is transported parallel to itself, that is we have

XX=0\nabla_X X = 0

Using the formula for the covariant derivative for an invariant metric above we get

XX=ad X *X=0\nabla_X X = - ad^*_X X = 0

as a reformulation of the geodesic equation. For time dependent dynamical systems, we have the time axis as an additional dimension and every vector field has t as an additional summand. So, in this case we get as geodesic equation (again: for an invariant metric)

XX= tXad X *X=0\nabla_X X = \partial_t X - ad^*_X X = 0

Ideal Fluids Continued

The Euler viewpoint of fluids is that there are fluid packages or “particles”. The fluid flow is described by specifying where each package or particle is at a given time t. When we start with a time t 0=0 on a given manifold M, the flow of every fluid package is decribed by a path on M parameterized by time, and for every time t>t 0 there is a diffeomorphism g t of M defined by the requirement that it maps the initial position of every fluid package to the position at time t.

To be continued…

Examples

Burgers’ Equation as Geodesic Equation

It is possible to derive Burgers equation as the geodesic equation of the diffeomorphism group DS 1 of the circle S 1. See this blog post.

Hamiltonian Formulation

We can turn the linear space of divergence free vector fields into a Lie algebra 𝔤 by giving it the negative Jacobi-Lie bracket:

[v,w]= j=1 n(w j jvv j jw)[v, w] = \sum_{j = 1}^n \left( w_j \partial_j v - v_j \partial_j w \right)

TODO: explain why it is the negative of the usual definition (left Lie algebra bracket).

We also have a scalar product on 𝔤:

v,w Mvwdμ\langle v, w \rangle \coloneqq \int_M v \cdot w d \mu

Remember from last time that one can define the directional derivative on Fréchet spaces just like in finite dimensions:

Let F and G be Fréchet spaces, UF open and P:UG a continuous map. The derivative of P at the point fU in the direction hF is the map

DP:U×FGD P: U \times F \to G
DP(f)hlim t01t(P(f+th)P(f))D P(f) h \coloneqq \lim_{t \to 0} \frac{1}{t} ( P(f + t h) - P(f))

For a real valued function T of 𝔤, the derivative DT eats two vector fields and spits out a real number. If you have a derivative DT and a fixed vector field δv, you can determine for every vector field u another vector field δTδv(u) by requiring that

DT(u,δv)=u,δTδv(u)D T (u, \delta v) = \langle u, \frac{\delta T}{\delta v} (u) \rangle

holds. I used the funny δ’s to match the notation of

  • Jerrold E. Marsden, Tudor S. Ratiu: “Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems”, 2nd edition, Springer, New York, 1999

We can now define the ideal fluid bracket which plays the role of the Lie-Poisson bracket for real valued functions T,S of 𝔤:

{T,S}(u) Mu[δTδu,δSδu]dμ\{ T, S \} (u) \coloneqq \int_M u \cdot \left[ \frac{\delta T}{\delta u}, \frac{\delta S}{\delta u} \right] d \mu

We choose as energy function, the Hamiltonian H, simply the kinetic energy:

H(v)12 Mu 2dμH(v) \coloneqq \frac{1}{2} \int_M \| u \|^2 d \mu

Then we can show that

F˙={F,H}\dot{F} = \{F, H \}

for all functions F is equivalent to Euler’s equation.

References

  • V.I. Arnold, ; B.A. Khesin: Topological methods in hydrodynamics. (Springer 1998, ZMATH)

  • Boris Khesin, Robert Wendt: The geometry of infinite-dimensional groups. (Springer 2009, ZMATH)

  • Tsutomu Kambe: Geometrical theory of dynamical systems and fluid flows. Revised ed. (ZMATH)

A paper dedicated to the Burgers equation:

  • Boris Khesin, Gerard Misiolek: Shock waves for the Burgers equation and curvatures of diffeomorphism groups (arXiv)