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The Euler equations for three-dimensional incompressible inviscid fluid flow are

\displaystyle  \partial_t u + (u \cdot \nabla) u = - \nabla p \ \ \ \ \ (1)

\displaystyle \nabla \cdot u = 0

where {u: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}^3} is the velocity field, and {p: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}} is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the formal calculations here; in particular, we will apply inverse operators such as {(-\Delta)^{-1}} or {|\nabla|^{-1} := (-\Delta)^{-1/2}} formally, assuming that these inverses are well defined on the functions they are applied to.

Meanwhile, the surface quasi-geostrophic (SQG) equation is given by

\displaystyle  \partial_t \theta + (u \cdot \nabla) \theta = 0 \ \ \ \ \ (2)

\displaystyle  u = ( -\partial_y |\nabla|^{-1}, \partial_x |\nabla|^{-1} ) \theta \ \ \ \ \ (3)

where {\theta: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}} is the active scalar, and {u: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}^2} is the velocity field. The SQG equations are often used as a toy model for the 3D Euler equations, as they share many of the same features (e.g. vortex stretching); see this paper of Constantin, Majda, and Tabak for more discussion (or this previous blog post).

I recently found a more direct way to connect the two equations. We first recall that the Euler equations can be placed in vorticity-stream form by focusing on the vorticity {\omega := \nabla \times u}. Indeed, taking the curl of (1), we obtain the vorticity equation

\displaystyle  \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (4)

while the velocity {u} can be recovered from the vorticity via the Biot-Savart law

\displaystyle  u = (-\Delta)^{-1} \nabla \times \omega. \ \ \ \ \ (5)

The system (4), (5) has some features in common with the system (2), (3); in (2) it is a scalar field {\theta} that is being transported by a divergence-free vector field {u}, which is a linear function of the scalar field as per (3), whereas in (4) it is a vector field {\omega} that is being transported (in the Lie derivative sense) by a divergence-free vector field {u}, which is a linear function of the vector field as per (5). However, the system (4), (5) is in three dimensions whilst (2), (3) is in two spatial dimensions, the dynamical field is a scalar field {\theta} for SQG and a vector field {\omega} for Euler, and the relationship between the velocity field and the dynamical field is given by a zeroth order Fourier multiplier in (3) and a {-1^{th}} order operator in (5).

However, we can make the two equations more closely resemble each other as follows. We first consider the generalisation

\displaystyle  \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (6)

\displaystyle  u = T (-\Delta)^{-1} \nabla \times \omega \ \ \ \ \ (7)

where {T} is an invertible, self-adjoint, positive-definite zeroth order Fourier multiplier that maps divergence-free vector fields to divergence-free vector fields. The Euler equations then correspond to the case when {T} is the identity operator. As discussed in this previous blog post (which used {A} to denote the inverse of the operator denoted here as {T}), this generalised Euler system has many of the same features as the original Euler equation, such as a conserved Hamiltonian

\displaystyle  \frac{1}{2} \int_{{\bf R}^3} u \cdot T^{-1} u,

the Kelvin circulation theorem, and conservation of helicity

\displaystyle  \int_{{\bf R}^3} \omega \cdot T^{-1} u.

Also, if we require {\omega} to be divergence-free at time zero, it remains divergence-free at all later times.

Let us consider “two-and-a-half-dimensional” solutions to the system (6), (7), in which {u,\omega} do not depend on the vertical coordinate {z}, thus

\displaystyle  \omega(t,x,y,z) = \omega(t,x,y)

and

\displaystyle  u(t,x,y,z) = u(t,x,y)

but we allow the vertical components {u_z, \omega_z} to be non-zero. For this to be consistent, we also require {T} to commute with translations in the {z} direction. As all derivatives in the {z} direction now vanish, we can simplify (6) to

\displaystyle  D_t \omega = (\omega_x \partial_x + \omega_y \partial_y) u \ \ \ \ \ (8)

where {D_t} is the two-dimensional material derivative

\displaystyle  D_t := \partial_t + u_x \partial_x + u_y \partial_y.

Also, divergence-free nature of {\omega,u} then becomes

\displaystyle  \partial_x \omega_x + \partial_y \omega_y = 0

and

\displaystyle  \partial_x u_x + \partial_y u_y = 0. \ \ \ \ \ (9)

In particular, we may (formally, at least) write

\displaystyle  (\omega_x, \omega_y) = (\partial_y \theta, -\partial_x \theta)

for some scalar field {\theta(t,x,y,z) = \theta(t,x,y)}, so that (7) becomes

\displaystyle  u = T ( (- \Delta)^{-1} \partial_y \omega_z, - (-\Delta^{-1}) \partial_x \omega_z, \theta ). \ \ \ \ \ (10)

The first two components of (8) become

\displaystyle  D_t \partial_y \theta = \partial_y \theta \partial_x u_x - \partial_x \theta \partial_y u_x

\displaystyle - D_t \partial_x \theta = \partial_y \theta \partial_x u_y - \partial_x \theta \partial_y u_y

which rearranges using (9) to

\displaystyle  \partial_y D_t \theta = \partial_x D_t \theta = 0.

Formally, we may integrate this system to obtain the transport equation

\displaystyle  D_t \theta = 0, \ \ \ \ \ (11)

Finally, the last component of (8) is

\displaystyle  D_t \omega_z = \partial_y \theta \partial_x u_z - \partial_x \theta \partial_y u_z. \ \ \ \ \ (12)

At this point, we make the following choice for {T}:

\displaystyle  T ( U_x, U_y, \theta ) = \alpha (U_x, U_y, \theta) + (-\partial_y |\nabla|^{-1} \theta, \partial_x |\nabla|^{-1} \theta, 0) \ \ \ \ \ (13)

\displaystyle  + P( 0, 0, |\nabla|^{-1} (\partial_y U_x - \partial_x U_y) )

where {\alpha > 0} is a real constant and {Pu := (-\Delta)^{-1} (\nabla \times (\nabla \times u))} is the Leray projection onto divergence-free vector fields. One can verify that for large enough {\alpha}, {T} is a self-adjoint positive definite zeroth order Fourier multiplier from divergence free vector fields to divergence-free vector fields. With this choice, we see from (10) that

\displaystyle  u_z = \alpha \theta - |\nabla|^{-1} \omega_z

so that (12) simplifies to

\displaystyle  D_t \omega_z = - \partial_y \theta \partial_x |\nabla|^{-1} \omega_z + \partial_x \theta \partial_y |\nabla|^{-1} \omega_z.

This implies (formally at least) that if {\omega_z} vanishes at time zero, then it vanishes for all time. Setting {\omega_z=0}, we then have from (10) that

\displaystyle (u_x,u_y,u_z) = (-\partial_y |\nabla|^{-1} \theta, \partial_x |\nabla|^{-1} \theta, \alpha \theta )

and from (11) we then recover the SQG system (2), (3). To put it another way, if {\theta(t,x,y)} and {u(t,x,y)} solve the SQG system, then by setting

\displaystyle  \omega(t,x,y,z) := ( \partial_y \theta(t,x,y), -\partial_x \theta(t,x,y), 0 )

\displaystyle  \tilde u(t,x,y,z) := ( u_x(t,x,y), u_y(t,x,y), \alpha \theta(t,x,y) )

then {\omega,\tilde u} solve the modified Euler system (6), (7) with {T} given by (13).

We have {T^{-1} \tilde u = (0, 0, \theta)}, so the Hamiltonian {\frac{1}{2} \int_{{\bf R}^3} \tilde u \cdot T^{-1} \tilde u} for the modified Euler system in this case is formally a scalar multiple of the conserved quantity {\int_{{\bf R}^2} \theta^2}. The momentum {\int_{{\bf R}^3} x \cdot \tilde u} for the modified Euler system is formally a scalar multiple of the conserved quantity {\int_{{\bf R}^2} \theta}, while the vortex stream lines that are preserved by the modified Euler flow become the level sets of the active scalar that are preserved by the SQG flow. On the other hand, the helicity {\int_{{\bf R}^3} \omega \cdot T^{-1} \tilde u} vanishes, and other conserved quantities for SQG (such as the Hamiltonian {\int_{{\bf R}^2} \theta |\nabla|^{-1} \theta}) do not seem to correspond to conserved quantities of the modified Euler system. This is not terribly surprising; a low-dimensional flow may well have a richer family of conservation laws than the higher-dimensional system that it is embedded in.

As in the previous post, all computations here are at the formal level only.

In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for these equations. In a bit more detail: starting with Lagrangian space {{\cal L} = ({\bf R}^n, \hbox{vol})} and Eulerian space {{\cal E} = ({\bf R}^n, \eta, \hbox{vol})}, we let {M} be the space of volume-preserving, orientation-preserving maps {\Phi: {\cal L} \rightarrow {\cal E}} from Lagrangian space to Eulerian space. Given a curve {\Phi: {\bf R} \rightarrow M}, we can define the Lagrangian velocity field {\dot \Phi: {\bf R} \times {\cal L} \rightarrow T{\cal E}} as the time derivative of {\Phi}, and the Eulerian velocity field {u := \dot \Phi \circ \Phi^{-1}: {\bf R} \times {\cal E} \rightarrow T{\cal E}}. The volume-preserving nature of {\Phi} ensures that {u} is a divergence-free vector field:

\displaystyle  \nabla \cdot u = 0. \ \ \ \ \ (1)

If we formally define the functional

\displaystyle  J[\Phi] := \frac{1}{2} \int_{\bf R} \int_{{\cal E}} |u(t,x)|^2\ dx dt = \frac{1}{2} \int_R \int_{{\cal L}} |\dot \Phi(t,x)|^2\ dx dt

then one can show that the critical points of this functional (with appropriate boundary conditions) obey the Euler equations

\displaystyle  [\partial_t + u \cdot \nabla] u = - \nabla p

\displaystyle  \nabla \cdot u = 0

for some pressure field {p: {\bf R} \times {\cal E} \rightarrow {\bf R}}. As discussed in the previous post, the time translation symmetry of this functional yields conservation of the Hamiltonian

\displaystyle  \frac{1}{2} \int_{{\cal E}} |u(t,x)|^2\ dx = \frac{1}{2} \int_{{\cal L}} |\dot \Phi(t,x)|^2\ dx;

the rigid motion symmetries of Eulerian space give conservation of the total momentum

\displaystyle  \int_{{\cal E}} u(t,x)\ dx

and total angular momentum

\displaystyle  \int_{{\cal E}} x \wedge u(t,x)\ dx;

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

\displaystyle  \int_{\Phi(\gamma)} u^*

for any closed loop {\gamma} in {{\cal L}}, or equivalently pointwise conservation of the Lagrangian vorticity {\Phi^* \omega = \Phi^* du^*}, where {u^*} is the {1}-form associated with the vector field {u} using the Euclidean metric {\eta} on {{\cal E}}, with {\Phi^*} denoting pullback by {\Phi}.

It turns out that one can generalise the above calculations. Given any self-adjoint operator {A} on divergence-free vector fields {u: {\cal E} \rightarrow {\bf R}}, we can define the functional

\displaystyle  J_A[\Phi] := \frac{1}{2} \int_{\bf R} \int_{{\cal E}} u(t,x) \cdot A u(t,x)\ dx dt;

as we shall see below the fold, critical points of this functional (with appropriate boundary conditions) obey the generalised Euler equations

\displaystyle  [\partial_t + u \cdot \nabla] Au + (\nabla u) \cdot Au= - \nabla \tilde p \ \ \ \ \ (2)

\displaystyle  \nabla \cdot u = 0

for some pressure field {\tilde p: {\bf R} \times {\cal E} \rightarrow {\bf R}}, where {(\nabla u) \cdot Au} in coordinates is {\partial_i u_j Au_j} with the usual summation conventions. (When {A=1}, {(\nabla u) \cdot Au = \nabla(\frac{1}{2} |u|^2)}, and this term can be absorbed into the pressure {\tilde p}, and we recover the usual Euler equations.) Time translation symmetry then gives conservation of the Hamiltonian

\displaystyle  \frac{1}{2} \int_{{\cal E}} u(t,x) \cdot A u(t,x)\ dx.

If the operator {A} commutes with rigid motions on {{\cal E}}, then we have conservation of total momentum

\displaystyle  \int_{{\cal E}} Au(t,x)\ dx

and total angular momentum

\displaystyle  \int_{{\cal E}} x \wedge Au(t,x)\ dx,

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

\displaystyle  \int_{\Phi(\gamma)} (Au)^*

or pointwise conservation of the Lagrangian vorticity {\Phi^* \theta := \Phi^* d(Au)^*}. These applications of Noether’s theorem proceed exactly as the previous post; we leave the details to the interested reader.

One particular special case of interest arises in two dimensions {n=2}, when {A} is the inverse derivative {A = |\nabla|^{-1} = (-\Delta)^{-1/2}}. The vorticity {\theta = d(Au)^*} is a {2}-form, which in the two-dimensional setting may be identified with a scalar. In coordinates, if we write {u = (u_1,u_2)}, then

\displaystyle  \theta = \partial_{x_1} |\nabla|^{-1} u_2 - \partial_{x_2} |\nabla|^{-1} u_1.

Since {u} is also divergence-free, we may therefore write

\displaystyle  u = (- \partial_{x_2} \psi, \partial_{x_1} \psi )

where the stream function {\psi} is given by the formula

\displaystyle  \psi = |\nabla|^{-1} \theta.

If we take the curl of the generalised Euler equation (2), we obtain (after some computation) the surface quasi-geostrophic equation

\displaystyle  [\partial_t + u \cdot \nabla] \theta = 0 \ \ \ \ \ (3)

\displaystyle  u = (-\partial_{x_2} |\nabla|^{-1} \theta, \partial_{x_1} |\nabla|^{-1} \theta).

This equation has strong analogies with the three-dimensional incompressible Euler equations, and can be viewed as a simplified model for that system; see this paper of Constantin, Majda, and Tabak for details.

Now we can specialise the general conservation laws derived previously to this setting. The conserved Hamiltonian is

\displaystyle  \frac{1}{2} \int_{{\bf R}^2} u\cdot |\nabla|^{-1} u\ dx = \frac{1}{2} \int_{{\bf R}^2} \theta \psi\ dx = \frac{1}{2} \int_{{\bf R}^2} \theta |\nabla|^{-1} \theta\ dx

(a law previously observed for this equation in the abovementioned paper of Constantin, Majda, and Tabak). As {A} commutes with rigid motions, we also have (formally, at least) conservation of momentum

\displaystyle  \int_{{\bf R}^2} Au\ dx

(which up to trivial transformations is also expressible in impulse form as {\int_{{\bf R}^2} \theta x\ dx}, after integration by parts), and conservation of angular momentum

\displaystyle  \int_{{\bf R}^2} x \wedge Au\ dx

(which up to trivial transformations is {\int_{{\bf R}^2} \theta |x|^2\ dx}). Finally, diffeomorphism invariance gives pointwise conservation of Lagrangian vorticity {\Phi^* \theta}, thus {\theta} is transported by the flow (which is also evident from (3). In particular, all integrals of the form {\int F(\theta)\ dx} for a fixed function {F} are conserved by the flow.

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The Euler equations for incompressible inviscid fluids may be written as

\displaystyle \partial_t u + (u \cdot \nabla) u = -\nabla p

\displaystyle \nabla \cdot u = 0

where {u: [0,T] \times {\bf R}^n \rightarrow {\bf R}^n} is the velocity field, and {p: [0,T] \times {\bf R}^n \rightarrow {\bf R}} is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that {u} is bounded. We will take the dimension {n} to be at least two, with the three-dimensional case {n=3} being of course especially interesting.

The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when {n=3} is to be able to construct “computers” solving the Euler equations, which generate smaller replicas of themselves in a noise-tolerant manner (as the viscosity term in the Navier-Stokes equation is to be viewed as perturbative noise).

Perhaps the most prominent obstacles to this route are the conservation laws for the Euler equations, which limit the types of final states that a putative computer could reach from a given initial state. Most famously, we have the conservation of energy

\displaystyle \int_{{\bf R}^n} |u|^2\ dx \ \ \ \ \ (1)

 

(assuming sufficient decay of the velocity field at infinity); thus for instance it would not be possible for a computer to generate a replica of itself which had greater total energy than the initial computer. This by itself is not a fatal obstruction (in this paper of mine, I constructed such a “computer” for an averaged Euler equation that still obeyed energy conservation). However, there are other conservation laws also, for instance in three dimensions one also has conservation of helicity

\displaystyle \int_{{\bf R}^3} u \cdot (\nabla \times u)\ dx \ \ \ \ \ (2)

 

and (formally, at least) one has conservation of momentum

\displaystyle \int_{{\bf R}^3} u\ dx

and angular momentum

\displaystyle \int_{{\bf R}^3} x \times u\ dx

(although, as we shall discuss below, due to the slow decay of {u} at infinity, these integrals have to either be interpreted in a principal value sense, or else replaced with their vorticity-based formulations, namely impulse and moment of impulse). Total vorticity

\displaystyle \int_{{\bf R}^3} \nabla \times u\ dx

is also conserved, although it turns out in three dimensions that this quantity vanishes when one assumes sufficient decay at infinity. Then there are the pointwise conservation laws: the vorticity and the volume form are both transported by the fluid flow, while the velocity field (when viewed as a covector) is transported up to a gradient; among other things, this gives the transport of vortex lines as well as Kelvin’s circulation theorem, and can also be used to deduce the helicity conservation law mentioned above. In my opinion, none of these laws actually prohibits a self-replicating computer from existing within the laws of ideal fluid flow, but they do significantly complicate the task of actually designing such a computer, or of the basic “gates” that such a computer would consist of.

Below the fold I would like to record and derive all the conservation laws mentioned above, which to my knowledge essentially form the complete set of known conserved quantities for the Euler equations. The material here (although not the notation) is drawn from this text of Majda and Bertozzi.

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