Incompressible Fluids

Montreal-Rochester-Syracuse-Toronto

Conference 2005 1/27 SUNY Utica May 16-18

S. G. Rajeev Department of Physics and Astronomy University of Rochester, Rochester, NY14627 email: [email protected] JJ II J

Prepared using pdfslide developed by C. V. Radhakrishnan of River Valley Technologies,Trivandrum, India I Back Close The Euler Equations 2/27 The Euler equations of an incompressible inviscid fluid are ∂ v + (v · ∇)v = −∇p, div v = 0. ∂t We can eliminate pressure p by taking a curl. Defining the vorticity 1 2 ω = curl v and using (v · ∇)v = ω × v + 2∇v , we get, ∂ ω + curl [ω × v] = 0. ∂t We can regard ω as the basic dynamic variable of the system, since v JJ determined by it 1. II J 1 R I We assume that the circulation C v around con-contactible loops (if any) C are zero and that the boundary values of the velocity are given. Back Close Poisson Brackets 3/27 See V. I. Arnold and B. Khesin Topological Methods in Hydrodynam- ics The Poisson bracket of two functions of vorticity is defined to be Z δF δG {F,G} = ω · curl × curl d3x δω δω This is the natural Poisson bracket on the dual of the Lie algebra of incompressible vector fields. Using the identity div [a × b] = b · curl a − a · curl b this may also be written as JJ II Z δF δG {F,G} = curl ω × curl · d3x J δω δω I Back Close The Hamiltonian Formalism 4/27 The hamiltonian is 1 Z H = v · vd3x. 2 Using the identity δF δF curl = δω δv we get {H, ω} = curl [ω × v] as needed to get the Euler equations. footnotesize The analogy of this to the equations of a rigid body dL JJ = L × (I−1L) dt II J could not have escaped Euler. Vorticity is analogous to angular momen- I tum; the Laplace operator is analogous to moment of inertia I. Back Close Clebsch Variables 5/27 The Poisson brackets are rather complicated in terms of vorticity. It was noticed by Clebsch that any solution to div ω = 0 can be written as

ω = ∇λ × ∇µ for a pair of functions λ, µ : M → R. The advantage of this parametrization is that the Poisson brackets of vorticity follow from canonical commutation relations of these functions: JJ {λ(x), µ(y)} = δ(x, y), {λ(x), λ(y)} = 0 = {µ(x), µ(y)}. II J I Back Close Some Technical Remarks 6/27 We will lose some global information in this parametrization; for example the ‘kinetic helicity’ will vanish for R ω × vdx = 0. To recover this we must let (λ, µ) be co-ordinates on a two dimensional manifold more general than R2. There is a ‘gauge invariance’ to this parametrization: under canonical transformations in the pair of co-ordinates (λ, µ), vorticity is unchanged. J. Marsden and A. Wienstein Physica 7(1983) 305-323 have given a nice geometrical interpretation of these variables. The space of pairs of functions (λ, µ) is a symplectic vector space using the contraction < λ, µ >= R λµd3x given by the volume form. The group of incompressible diffeomorphisms act on this symplectically. The Clebsch bracket ω = [[λ, µ]] := ∇λ × ∇µ is the moment map of this action. This is anlogous to the formula L = r × p for angular momentum. We prefer another interpretation that allows a generalization of the Clebsch bracket to any Lie algebra. Let λ, µ : G∗ → R be pair of functions on the dual of a Lie algebra G. Then dλ and dµ can be thought of as functions : G∗ → G. Thus it makes sense to define the commutator [dλ, dµ]. This is a map V ⊗ V → V ⊗ G, where V is the space of real functions on G. As a vector space JJ we can identify V ≡ U(G), the universal envelope of G. II J I Back Close Statistical Field Theory of Turbulence 7/27 The Euler equations describe the geodesics in the group of volume preserving diffeomorphisms. Arnold showed that the sectional curvature of this metric is negative in all but a finite number of directions. This means that the solutions are very unstable: small errors in the initial conditions will grow exponentially: explaining the unpredictabiliy of fluid flow. In other words, we cannot ignore fluctuations in the external forces acting on the fluid, as their effect will grow exponentially with time. This could well be the cause of turbulence. Such systems should not really be thought of as deterministic.We are led seek a theory analogous to statistical mechanics of gases, in which JJ the velocity is a random field whose probability distribution is given by II an analogue of the Fokker-Plank equation. J I Back Close Fluctuations 8/27 We can model these turbulent fluctuations by adding a Gaussian random force field to the r.h.s. ∂ ∂ v + (v · ∇)v = −∇p + f, ⇒ ω + curl [ω × v] = curl f ∂t ∂t i What would be a good choice of covariance for this Gaussian? < f (x, t) >= 0, < fi(x, s)fj(y, t) >= δ(s − t)Gij(x, y). We recall that a system with fluctuations must always have dissipation: otherwise the energy pumped into the system through fluctuations will grow without bound and we will get infinite temperature: temperature is the ratio of fluctuation to dissipation. Although there isn’t as yet JJ a universally accepted model for statistical fluctuations in turbulence, II there is one for dissipation. We will use it along with a principle of J detailed balance to get a model for fluctuations. I Back Close Viscosity 9/27 The Navier-Stokes equations of hydrodynamics allow for dissipation of energy through internal friction (viscosity): ∂ ∂ v + (v · ∇)v = ν∇2v − ∇p ⇒ ω + curl [ω × v] = ν∇2ω. ∂t ∂t What should be the covariance of fluctuation and dissipation in order that the average energy pumped in fluctuation is dissipated away?

JJ II J I Back Close A Geometric Model of Dissipation 10/27 ij Let Ω be the Poisson tensor (inverse of a symplectic form Ωij) and H the hamiltonian of some system with a finite number of degrees of freedom, for simplicity. Then the Hamilton’s equations are dξi = Ωij∂ H. dt j A nice model for dissipation is to add the negative gradient of the hamiltonian with respect to some metric to the r.h.s.: dξi dH = Ωij∂ H − gij∂ H ⇒ = −gij∂ H∂ H ≤ 0. dt j j dt i j JJ II The Navier-Stokes equations fit this mold except that it is infinite di- ij J mensional; the role of the metric g is played by the square of the I Laplace operator. Back Close A Geometric Model of Fluctuations 11/27 Suppose we add a random force to the r.h.s. dξi = Ωij∂ H − gij∂ H + ηi dt j j with zero mean and Gaussian covariance < ηi(ξ, t)ηj(ξ0, t0) >= δ(t − t0)δ(ξ, ξ0)Gij(ξ). This stochastic equation (Langevin equation) is equivalent to the Fokker- Plank equation for the probability density W : JJ ∂W √ W II + ∂ Ωij∂ H − gij∂ H W = ∂ GGij∂ √ ∂t i j j i j G J R I Note that W dξ is conserved. Back Close The Fluctuation-Dissipation Theorem 12/27 See S. Chandrashekhar Rev. Mod. Phys. 1941 for the basic idea. Let us seek a static solution of this equation. The fluctuation and dissipation terms will balance each other out if √ G W Gij ∂ √ = −gij∂ H; W i G j

h W i jk that is, i.e., ∂i log √ = −Gijg ∂kH. The Boltzmann distribution √ G W = e−βH det G will satisfy this condition if βGij = gij JJ II √ The remaining term in the Fokker-Plank equation will also vanish if G = γPfΩ so that the volume J ij −βH defined by the metrics agree with that defined by the symplectic form :∂i Ω ∂je PfΩ = 0 is I obvious in the Darboux co-ordinates. Back Close The Covariance Metric for Turbulence 13/27 We saw earlier that the phase space of an incompressible fluid is the space of vorticities; i.e., vector fields satisfying div ω = 0. The dissipation term is δH δH ν∇2ω = ν∇2 curl v = ν∇2 curl curl = −ν∇4 . δω δω Thus the tensor gij above corresponds to the square of the Laplace operator: ∇4. In order the energy pumped into the turbulent flow by fluctuations be dissipated by viscous damping, we will need the covariance of the fluctuations must be: JJ II < ηi(x)ηj(y) >= βνδij∇4δ(x, y). J I Back Close Regularization 14/27 To make sense of such singular stochastic equations we will need a regularization, replacing the system with an infinite number of degrees of freedom by a sequence approximations by finite dimensional systems. Such discretizations are necessary anyway in the numerical solutions of hydrodynamics equations. The main mathematical difficulty is in maintaining the structure of the phase space as a Lie algebra. In the case of two dimensional incompressible flow we found a regularization by a matrix model. We will need both a cut-off in real space ( so that the system has finite volume) as well as a cut-off in momentum space (so that the kinetic JJ energies of the advected particles is finite) in order to get a system with II a finite number of degrees of freedom. J I Back Close Long Distance Cut-off 15/27 The first cut-off to finite volume is easy to do within conventional differential geometry: we can modify the metric on R3 so that any two points are at a finite distance from each other. For example, if 2 i j dx2 3 ds = h dx dx = 2 we get a sphere S . The canonical commu- ij x2 [1+ 2 ] R √ tation pairing < λ, µ >= R λ(x)µ(y) hd3x continues to make sense and so does the hamiltonian 1 Z H = ωi(x)G(x − y) ωj(y)dxdy 2 ij

i ijk ij JJ where ω = ∂jλ∂jµ and G (x, y)is the Greens function of the Laplace operator on vector densities. The equations of motion for λ, µ (or even II for ω) follow from this hamiltonian and the above canonical pairing. J I The limit R → ∞ will recover the earlier case of an unbounded fluid. Back Close Two-Dimensional Euler Equations 16/27 It is useful to look first at the much simpler example of two dimensional incompressible flow. Savitri V. Iyer and S.G. Rajeev Mod.Phys.Lett.A17:1539-1550,2002; physics/0206083 Area preserving transformations are the same as canonical transformations. The corresponding Lie algebra is the algebra of symplectic transformations on the plane:

ab [f1, f2] = ∂af1∂bf2. The above Lie bracket of functions gives the Poisson bracket for vorticity: JJ II ab {ω(x), ω(y)} = ∂bω(x)∂aδ(x − y). J I Back Close Hamiltonian Formulation 17/27 Euler equations follow from these if we postulate the hamiltonian to be the total energy of the ﬂuid, 1 Z 1 Z H = u2d2x = G(x, y)ω(x)ω(y)d2xd2y. 2 2

R k 2 The quantities Qk = ω (x)d x are conserved for any k = 1, 2 ···: these are the Casimir invariants. In spite of the infinite number of conservation laws, two dimensional fluid flow is chaotic. JJ II J I Back Close Non-Commutative Regularization 18/27 The Lie algebra of symplectic transformations can be thought of as the limit of the unitary Lie algebra as the rank goes to infinity. To see this, impose periodic boundary conditions and write in terms of Fourier coefficients as X 1 2π H = (L L )2 |ω |2, {ω , ω } = − m n ω . 1 2 2 m m n ab a b m+n m L1L2 m6=(0,0)

Using an idea of Fairlie and Zachos, we now truncate this system by imposing a discrete period- icity mod N in the Fourier index m; the structure constants must be modiﬁed to preserve peridocity and the Jacobi identity: JJ 1 2π {ω , ω } = sin[θ(m n − m n )]ω , θ = . II m n θ 1 2 2 1 m+n mod N N J This can be thought of as a ‘quantum deformation’ of the Lie algebra of symplectic transformations. This is the Lie I algebra of U(N). Back Close Regularized hamiltonian 19/27 The hamiltonian also has a periodic truncation 1 X 1 H = |ω |2, 2 λ(m) m λ(m)6=0

N 2π 2 N 2π 2 λ(m) = sin m + sin m . 2π N 1 2π N 2 This hamiltonian with the above Poisson brackets describe the geodesics on U(N) with respect to an anisotropic metric. We are writing it in a JJ basis in which the metric is diagonal. II J I Back Close The Regularized Fokker-Plank Equation 20/27 Now we can write the regularized Navier-Stokes with random sources. We will not have the problem of ‘closure’ that plagues many approaches to turbulence because our regularization perserves the symmetries (the Lie algebra structure). dωm = Ωmn∂ H − gmn∂ H + ηn dt n n where the Poisson tensor is given in terms of the structure constants mn mn of the Lie algebra Ω = cp ωp. The dissipation tensor is diagonal in our basis and has the square of the eigenvalues of the discrete Lapla- cian: gmn = νδmnλ(m)2.Also, to balance fluctuation and dissipation, JJ < ηmηn >= Gmn = β−1gmn The Fokker-Plank equation is now II ∂W = ∂ (Ωmn∂ HW + gmn∂ HW + Gmn∂ W ) . J ∂t m n n n I Back Close Entropy of Matrix Models 21/27 The constants R m 2 Qm = ω (x)d x define some infinite dimensional surface in the space of all functions on the plane. The micro-canonical entropy (a la Boltzmann) of this system will be the log of the volume of this surface. How to define this volume of an infinite dimensional manifold? We can compute it in the regularization and take the limit. It is con- P venient to regard ω as a matrix by defining ωˆ = m ωmU(m) where 2πi m1 m2 U(m) = U1 U2 with the defining relations U1U2 = e N U2U1. In this basis, the Lie bracket of vorticity is just the usual matrix commutator. 1 k Then the regularized constants of motion are Qk = N trωˆ . The set JJ of herimtean matrices with a fixed value of these constants has a fi- II Q 2 nite volume,known from random matrix theory: kFlow 22/27 The entropy is thus 1 X Z S = log |λ − λ | = P log |λ − λ0|ρ(λ)ρ(λ0) N 2 k l k6=l

0 1 P where ρ(λ)dλdλ = N k δ(λ − λk). In the continuum limit this tends to a remarkably simple formula: Z S = P log |ω(x) − ω(y)|d2xd2y JJ Even without a complete theory based on the Fokker-Plank equa- II tion (assuming that its continuum limit does exist) we can make some J predictions about two dimensional turbulence. I Back Close The Shape of a Hurricane 23/27 What is the vorticity profile that maximizes entropy for fixed value of 2 mean vorticity Q1 and enstrophy ? This should be the most probable configuration for a vortex profile. Using a simple variational argument we get the answer in parametric form 1 1 1 ω(r) = 2σ sin φ + Q¯ , r2 = [a2 + a2] ± [a2 − a2] φ + sin (2φ) 1 2 1 2 2 1 π 2 in the region a1 ≤ r ≤ a2. We should expect this to be the vorticity distribution of the tornados and hurricanes. 2 2 2 Q2 = σ + Q1 JJ II J I Back Close Vorticity 7

5 24/27

radius 2 4 6 8 10 12 14

JJ II J I Back Close The Quantum Three-Sphere 25/27 To reduce the number of degrees of freedom further and get a system with a finite number of degrees of freedom, we need to deform this further. There is a well-known finite dimensional deformation of space of 3 functions on the three-sphere, the algebra F unq(S ) of functions on the iπ ‘quantum group’ SU(2)q for q = e N . The word ‘quantum’ means here simply that this deformation is non-commutative: it does not mean that we are studying any quantum mechanical effects in the physical sense. The price we pay for having a cut-off in both real space and momentum space is this non-commutativity. See for example A. Jevicki, M. Mikailov and S. Ramgoolam hep-th/0008186; A Guide to Quantum Groups by JJ V. Chari and A. Pressley. II J I Back Close Generators and Relations 26/27 iπ 3 For integer N and q = e N , F unq(S ) is the associative algebra gen- erated by α, β, α∗, β∗ with relations αβ = qβα, αβ∗ = qβ∗α, ββ∗ = β∗β, βα∗ = qα∗β, [α, α∗] = (q−1 − q)ββ∗, αα∗ + qββ∗ = 1 α2N = β2N = β∗2N = α∗2N = 1.

This is the deformation of the algebra of functions on SU(2) = S3. In the book by Chari and Pressley (section 7.4C) there is given a quantum differential calculus on this space. JJ II J I Back Close Non-Commutative Regularization of Hy- 27/27 drodynamics Together this allows us to make sense of a generalization of the Cleb- sch bracket dλ ∧ dµ. We can expand these differential forms in terms of representations of the quantum group; only representations of ‘spin’ < N − 1 will appear. The Laplacian has the irreducible representations as eigenspaces and eigenvalues that are given by a simple deformation of the usual formula. The integral of a function (quantum analogue of the Haar measure) is simply given by the projection to the trivial representation. Thus we have all the ingredients needed to make sense of the hamiltonian and JJ the Poisson bracket: λ and µ are finite dimensional hermitean matrices II representing the above algebra. J We are working on using this for numerical calculations in hydrody- I namics; also to derive a formula for entropy. Back Close