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Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities).

It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest (particularly in geometric PDE, or PDE arising from mathematical physics) admit a variational formulation; thus, a collection ${\Phi: \Omega \rightarrow M}$ of one or more fields on a domain ${\Omega}$ taking values in a space ${M}$ will solve the differential equation of interest if and only if ${\Phi}$ is a critical point to the functional

$\displaystyle J[\Phi] := \int_\Omega L( x, \Phi(x), D\Phi(x) )\ dx \ \ \ \ \ (1)$

involving the fields ${\Phi}$ and their first derivatives ${D\Phi}$, where the Lagrangian ${L: \Sigma \rightarrow {\bf R}}$ is a function on the vector bundle ${\Sigma}$ over ${\Omega \times M}$ consisting of triples ${(x, q, \dot q)}$ with ${x \in \Omega}$, ${q \in M}$, and ${\dot q: T_x \Omega \rightarrow T_q M}$ a linear transformation; we also usually keep the boundary data of ${\Phi}$ fixed in case ${\Omega}$ has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on ${\Phi}$ and ${D\Phi}$, which require the machinery of Lagrange multipliers to deal with, but which will only serve as a distraction for the current discussion.) It is common to use local coordinates to parameterise ${\Omega}$ as ${{\bf R}^d}$ and ${M}$ as ${{\bf R}^n}$, in which case ${\Sigma}$ can be viewed locally as a function on ${{\bf R}^d \times {\bf R}^n \times {\bf R}^{dn}}$.

Example 1 (Geodesic flow) Take ${\Omega = [0,1]}$ and ${M = (M,g)}$ to be a Riemannian manifold, which we will write locally in coordinates as ${{\bf R}^n}$ with metric ${g_{ij}(q)}$ for ${i,j=1,\dots,n}$. A geodesic ${\gamma: [0,1] \rightarrow M}$ is then a critical point (keeping ${\gamma(0),\gamma(1)}$ fixed) of the energy functional

$\displaystyle J[\gamma] := \frac{1}{2} \int_0^1 g_{\gamma(t)}( D\gamma(t), D\gamma(t) )\ dt$

or in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

$\displaystyle J[\gamma] = \frac{1}{2} \int_0^1 g_{ij}(\gamma(t)) \dot \gamma^i(t) \dot \gamma^j(t)\ dt.$

As discussed in this previous post, both the Euler equations for rigid body motion, and the Euler equations for incompressible inviscid flow, can be interpreted as geodesic flow (though in the latter case, one has to work really formally, as the manifold ${M}$ is now infinite dimensional).

More generally, if ${\Omega = (\Omega,h)}$ is itself a Riemannian manifold, which we write locally in coordinates as ${{\bf R}^d}$ with metric ${h_{ab}(x)}$ for ${a,b=1,\dots,d}$, then a harmonic map ${\Phi: \Omega \rightarrow M}$ is a critical point of the energy functional

$\displaystyle J[\Phi] := \frac{1}{2} \int_\Omega h(x) \otimes g_{\gamma(x)}( D\gamma(x), D\gamma(x) )\ dh(x)$

or in coordinates (again ignoring coordinate patch issues)

$\displaystyle J[\Phi] = \frac{1}{2} \int_{{\bf R}^d} h_{ab}(x) g_{ij}(\Phi(x)) (\partial_a \Phi^i(x)) (\partial_b \Phi^j(x))\ \sqrt{\det(h(x))}\ dx.$

If we replace the Riemannian manifold ${\Omega}$ by a Lorentzian manifold, such as Minkowski space ${{\bf R}^{1+3}}$, then the notion of a harmonic map is replaced by that of a wave map, which generalises the scalar wave equation (which corresponds to the case ${M={\bf R}}$).

Example 2 (${N}$-particle interactions) Take ${\Omega = {\bf R}}$ and ${M = {\bf R}^3 \otimes {\bf R}^N}$; then a function ${\Phi: \Omega \rightarrow M}$ can be interpreted as a collection of ${N}$ trajectories ${q_1,\dots,q_N: {\bf R} \rightarrow {\bf R}^3}$ in space, which we give a physical interpretation as the trajectories of ${N}$ particles. If we assign each particle a positive mass ${m_1,\dots,m_N > 0}$, and also introduce a potential energy function ${V: M \rightarrow {\bf R}}$, then it turns out that Newton’s laws of motion ${F=ma}$ in this context (with the force ${F_i}$ on the ${i^{th}}$ particle being given by the conservative force ${-\nabla_{q_i} V}$) are equivalent to the trajectories ${q_1,\dots,q_N}$ being a critical point of the action functional

$\displaystyle J[\Phi] := \int_{\bf R} \sum_{i=1}^N \frac{1}{2} m_i |\dot q_i(t)|^2 - V( q_1(t),\dots,q_N(t) )\ dt.$

Formally, if ${\Phi = \Phi_0}$ is a critical point of a functional ${J[\Phi]}$, this means that

$\displaystyle \frac{d}{ds} J[ \Phi[s] ]|_{s=0} = 0$

whenever ${s \mapsto \Phi[s]}$ is a (smooth) deformation with ${\Phi[0]=\Phi_0}$ (and with ${\Phi[s]}$ respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

$\displaystyle \int_\Omega \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}\ dx = 0. \ \ \ \ \ (2)$

Write ${\delta \Phi := \frac{d}{ds} \Phi[s]|_{s=0}}$ for the infinitesimal deformation of ${\Phi_0}$. By the chain rule, ${\frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}}$ can be expressed in terms of ${x, \Phi_0(x), \delta \Phi(x), D\Phi_0(x), D \delta \Phi(x)}$. In coordinates, we have

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \delta \Phi^i(x) L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) \ \ \ \ \ (3)$

$\displaystyle + \partial_{x^a} \delta \Phi^i(x) L_{\partial_{x^a} q^i} (x,\Phi_0(x), D\Phi_0(x)),$

where we parameterise ${\Sigma}$ by ${x, (q^i)_{i=1,\dots,n}, (\partial_{x^a} q^i)_{a=1,\dots,d; i=1,\dots,n}}$, and we use subscripts on ${L}$ to denote partial derivatives in the various coefficients. (One can of course work in a coordinate-free manner here if one really wants to, but the notation becomes a little cumbersome due to the need to carefully split up the tangent space of ${\Sigma}$, and we will not do so here.) Thus we can view (2) as an integral identity that asserts the vanishing of a certain integral, whose integrand involves ${x, \Phi_0(x), \delta \Phi(x), D\Phi_0(x), D \delta \Phi(x)}$, where ${\delta \Phi}$ vanishes at the boundary but is otherwise unconstrained.

A general rule of thumb in PDE and calculus of variations is that whenever one has an integral identity of the form ${\int_\Omega F(x)\ dx = 0}$ for some class of functions ${F}$ that vanishes on the boundary, then there must be an associated differential identity ${F = \hbox{div} X}$ that justifies this integral identity through Stokes’ theorem. This rule of thumb helps explain why integration by parts is used so frequently in PDE to justify integral identities. The rule of thumb can fail when one is dealing with “global” or “cohomologically non-trivial” integral identities of a topological nature, such as the Gauss-Bonnet or Kazhdan-Warner identities, but is quite reliable for “local” or “cohomologically trivial” identities, such as those arising from calculus of variations.

In any case, if we apply this rule to (2), we expect that the integrand ${\frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}}$ should be expressible as a spatial divergence. This is indeed the case:

Proposition 1 (Formal) Let ${\Phi = \Phi_0}$ be a critical point of the functional ${J[\Phi]}$ defined in (1). Then for any deformation ${s \mapsto \Phi[s]}$ with ${\Phi[0] = \Phi_0}$, we have

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \hbox{div} X \ \ \ \ \ (4)$

where ${X}$ is the vector field that is expressible in coordinates as

$\displaystyle X^a := \delta \Phi^i(x) L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)). \ \ \ \ \ (5)$

Proof: Comparing (4) with (3), we see that the claim is equivalent to the Euler-Lagrange equation

$\displaystyle L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) - \partial_{x^a} L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)) = 0. \ \ \ \ \ (6)$

The same computation, together with an integration by parts, shows that (2) may be rewritten as

$\displaystyle \int_\Omega ( L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) - \partial_{x^a} L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)) ) \delta \Phi^i(x)\ dx = 0.$

Since ${\delta \Phi^i(x)}$ is unconstrained on the interior of ${\Omega}$, the claim (6) follows (at a formal level, at least). $\Box$

Many variational problems also enjoy one-parameter continuous symmetries: given any field ${\Phi_0}$ (not necessarily a critical point), one can place that field in a one-parameter family ${s \mapsto \Phi[s]}$ with ${\Phi[0] = \Phi_0}$, such that

$\displaystyle J[ \Phi[s] ] = J[ \Phi[0] ]$

for all ${s}$; in particular,

$\displaystyle \frac{d}{ds} J[ \Phi[s] ]|_{s=0} = 0,$

which can be written as (2) as before. Applying the previous rule of thumb, we thus expect another divergence identity

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \hbox{div} Y \ \ \ \ \ (7)$

whenever ${s \mapsto \Phi[s]}$ arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain ${\Omega}$ is the Euclidean space ${{\bf R}^d}$, and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable ${x}$, thus

$\displaystyle L( x, \Phi(x), D\Phi(x) ) = L( \Phi(x), D\Phi(x) ), \ \ \ \ \ (8)$

then we obtain ${d}$ translation symmetries

$\displaystyle \Phi[s](x) := \Phi(x - s e^a )$

for ${a=1,\dots,d}$, where ${e^1,\dots,e^d}$ is the standard basis for ${{\bf R}^d}$. For a fixed ${a}$, the left-hand side of (7) then becomes

$\displaystyle \frac{d}{ds} L( \Phi(x-se^a), D\Phi(x-se^a) )|_{s=0} = -\partial_{x^a} [ L( \Phi(x), D\Phi(x) ) ]$

$\displaystyle = \hbox{div} Y$

where ${Y(x) = - L(\Phi(x), D\Phi(x)) e^a}$. Another common type of symmetry is a pointwise symmetry, in which

$\displaystyle L( x, \Phi[s](x), D\Phi[s](x) ) = L( x, \Phi[0](x), D\Phi[0](x) ) \ \ \ \ \ (9)$

for all ${x}$, in which case (7) clearly holds with ${Y=0}$.

If we subtract (4) from (7), we obtain the celebrated theorem of Noether linking symmetries with conservation laws:

Theorem 2 (Noether’s theorem) Suppose that ${\Phi_0}$ is a critical point of the functional (1), and let ${\Phi[s]}$ be a one-parameter continuous symmetry with ${\Phi[0] = \Phi_0}$. Let ${X}$ be the vector field in (5), and let ${Y}$ be the vector field in (7). Then we have the pointwise conservation law

$\displaystyle \hbox{div}(X-Y) = 0.$

In particular, for one-dimensional variational problems, in which ${\Omega \subset {\bf R}}$, we have the conservation law ${(X-Y)(t) = (X-Y)(0)}$ for all ${t \in \Omega}$ (assuming of course that ${\Omega}$ is connected and contains ${0}$).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if ${\Omega \subset {\bf R}}$ and the Lagrangian has no explicit time dependence, thus

$\displaystyle L(t, \Phi(t), \dot \Phi(t)) = L(\Phi(t), \dot \Phi(t)),$

then by using the time translation symmetry ${\Phi[s](t) := \Phi(t-s)}$, we have

$\displaystyle Y(t) = - L( \Phi(t), \dot\Phi(t) )$

as discussed previously, whereas we have ${\delta \Phi(t) = - \dot \Phi(t)}$, and hence by (5)

$\displaystyle X(t) := - \dot \Phi^i(x) L_{\dot q^i}(\Phi(t), \dot \Phi(t)),$

and so Noether’s theorem gives conservation of the Hamiltonian

$\displaystyle H(t) := \dot \Phi^i(x) L_{\dot q^i}(\Phi(t), \dot \Phi(t))- L(\Phi(t), \dot \Phi(t)). \ \ \ \ \ (10)$

For instance, for geodesic flow, the Hamiltonian works out to be

$\displaystyle H(t) = \frac{1}{2} g_{ij}(\gamma(t)) \dot \gamma^i(t) \dot \gamma^j(t),$

so we see that the speed of the geodesic is conserved over time.

For pointwise symmetries (9), ${Y}$ vanishes, and so Noether’s theorem simplifies to ${\hbox{div} X = 0}$; in the one-dimensional case ${\Omega \subset {\bf R}}$, we thus see from (5) that the quantity

$\displaystyle \delta \Phi^i(t) L_{\dot q^i}(t,\Phi_0(t), \dot \Phi_0(t)) \ \ \ \ \ (11)$

is conserved in time. For instance, for the ${N}$-particle system in Example 2, if we have the translation invariance

$\displaystyle V( q_1 + h, \dots, q_N + h ) = V( q_1, \dots, q_N )$

for all ${q_1,\dots,q_N,h \in {\bf R}^3}$, then we have the pointwise translation symmetry

$\displaystyle q_i[s](t) := q_i(t) + s e^j$

for all ${i=1,\dots,N}$, ${s \in{\bf R}}$ and some ${j=1,\dots,3}$, in which case ${\dot q_i(t) = e^j}$, and the conserved quantity (11) becomes

$\displaystyle \sum_{i=1}^n m_i \dot q_i^j(t);$

as ${j=1,\dots,3}$ was arbitrary, this establishes conservation of the total momentum

$\displaystyle \sum_{i=1}^n m_i \dot q_i(t).$

Similarly, if we have the rotation invariance

$\displaystyle V( R q_1, \dots, Rq_N ) = V( q_1, \dots, q_N )$

for any ${q_1,\dots,q_N \in {\bf R}^3}$ and ${R \in SO(3)}$, then we have the pointwise rotation symmetry

$\displaystyle q_i[s](t) := \exp( s A ) q_i(t)$

for any skew-symmetric real ${3 \times 3}$ matrix ${A}$, in which case ${\dot q_i(t) = A q_i(t)}$, and the conserved quantity (11) becomes

$\displaystyle \sum_{i=1}^n m_i \langle A q_i(t), \dot q_i(t) \rangle;$

since ${A}$ is an arbitrary skew-symmetric matrix, this establishes conservation of the total angular momentum

$\displaystyle \sum_{i=1}^n m_i q_i(t) \wedge \dot q_i(t).$

Below the fold, I will describe how Noether’s theorem can be used to locate all of the conserved quantities for the Euler equations of inviscid fluid flow, discussed in this previous post, by interpreting that flow as geodesic flow in an infinite dimensional manifold.

As we are all now very much aware, tsunamis are water waves that start in the deep ocean, usually because of an underwater earthquake (though tsunamis can also be caused by underwater landslides or volcanoes), and then propagate towards shore. Initially, tsunamis have relatively small amplitude (a metre or so is typical), which would seem to render them as harmless as wind waves. And indeed, tsunamis often pass by ships in deep ocean without anyone on board even noticing.

However, being generated by an event as large as an earthquake, the wavelength of the tsunami is huge – 200 kilometres is typical (in contrast with wind waves, whose wavelengths are typically closer to 100 metres). In particular, the wavelength of the tsunami is far greater than the depth of the ocean (which is typically 2-3 kilometres). As such, even in the deep ocean, the dynamics of tsunamis are essentially governed by the shallow water equations. One consequence of these equations is that the speed of propagation ${v}$ of a tsunami can be approximated by the formula

$\displaystyle v \approx \sqrt{g b} \ \ \ \ \ (1)$

where ${b}$ is the depth of the ocean, and ${g \approx 9.8 ms^{-2}}$ is the force of gravity. As such, tsunamis in deep water move very fast – speeds such as 500 kilometres per hour (300 miles per hour) are quite typical; enough to travel from Japan to the US, for instance, in less than a day. Ultimately, this is due to the incompressibility of water (and conservation of mass); the massive net pressure (or more precisely, spatial variations in this pressure) of a very broad and deep wave of water forces the profile of the wave to move horizontally at vast speeds. (Note though that this is the phase velocity of the tsunami wave, and not the velocity of the water molecues themselves, which are far slower.)

As the tsunami approaches shore, the depth ${b}$ of course decreases, causing the tsunami to slow down, at a rate proportional to the square root of the depth, as per (1). Unfortunately, wave shoaling then forces the amplitude ${A}$ to increase at an inverse rate governed by Green’s law,

$\displaystyle A \propto \frac{1}{b^{1/4}} \ \ \ \ \ (2)$

at least until the amplitude becomes comparable to the water depth (at which point the assumptions that underlie the above approximate results break down; also, in two (horizontal) spatial dimensions there will be some decay of amplitude as the tsunami spreads outwards). If one starts with a tsunami whose initial amplitude was ${A_0}$ at depth ${b_0}$ and computes the point at which the amplitude ${A}$ and depth ${b}$ become comparable using the proportionality relationship (2), some high school algebra then reveals that at this point, amplitude of a tsunami (and the depth of the water) is about ${A_0^{4/5} b_0^{1/5}}$. Thus, for instance, a tsunami with initial amplitude of one metre at a depth of 2 kilometres can end up with a final amplitude of about 5 metres near shore, while still traveling at about ten metres per second (35 kilometres per hour, or 22 miles per hour), and we have all now seen the impact that can have when it hits shore.

While tsunamis are far too massive of an event to be able to control (at least in the deep ocean), we can at least model them mathematically, allowing one to predict their impact at various places along the coast with high accuracy. (For instance, here is a video of the NOAA’s model of the March 11 tsunami, which has matched up very well with subsequent measurements.) The full equations and numerical methods used to perform such models are somewhat sophisticated, but by making a large number of simplifying assumptions, it is relatively easy to come up with a rough model that already predicts the basic features of tsunami propagation, such as the velocity formula (1) and the amplitude proportionality law (2). I give this (standard) derivation below the fold. The argument will largely be heuristic in nature; there are very interesting analytic issues in actually justifying many of the steps below rigorously, but I will not discuss these matters here.

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