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Analytical hydrodynamics (Rev #3, changes)

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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 MM. The diffeomorphisms of MM form an infinite dimensional group D(M)D(M). 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\omega on MM:

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

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

The (infinite dimensional) Lie algebra of SDiff(M) SDiff(M) is the vector space of all vector fields on MM 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\omega is a n-form that vanishes nowhere. In 3\mathbb{R}^3 with cartesian coordinates x,y,zx, y, z the canonical example would be

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

The dual basis of dxd x etc. is denoted by x\partial_x etc. in our example.

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

div(X)ω=d(Xlrcornerω) div(X) \omega = d(X \lrcorner \omega)

When we use our example we’d write of course

X=f 1 x+f 2 y+f 3 z X = 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 3 div(X) = \partial_x f_1 + \partial_y f_2 + \partial_z f_3

which we get if we plug in the expression for XX into the formula d(Xlrcornerω)d(X \lrcorner \omega).

Given two manifolds M,NM, N and a differentiable map f:MNf: M \to N, we can pull back a differential form ω\omega on NN to one on MM 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))

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=0t_0 = 0 on a given manifold MM, the flow of every fluid package is decribed by a path on MM parameterized by time, and for every time t>t 0t \gt t_0 there is a diffeomorphism g tg_t of MM defined by the requirement that it maps the initial position of every fluid package to the position at time tt.

To be continued…

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)