The Basic (Physical) Problem: Experiments. Internal Gravity Waves and Hyperbolic Boundary-Value Problems. Internal Gravity Waves. Governing Equations

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1 The Basic (Phsical) Problem: Eperiments Internal Gravit Waves and Hperbolic Boundar-Value Problems P. A. Martin Colorado chool of Mines Collaborator: tefan Llewelln mith, UC an Diego D. E. Mowbra & B.. H. Rarit, J. Fluid Mech. 28 (1967) 1 16 Internal Gravit Waves 19 Governing Equations Horizontal clinder, out of screen from middle of the X. Figure 5: t Andrew's Cross in a stratied uid. In the top gure!=n = 0:9 and in the left bottom gure!=n =0:7. t. Andrew s Cross Gravit acts down. Incompressible fluid. Densit increases with depth. The clinder oscillates and radiates energ. The energ is confined to the arms of the cross: we call these wave beams Can the fluid motion be predicted? How does it depend on the shape of the oscillating object? Linear theor, Boussinesq approimation, unbounded fluid Figure 5: t Andrew's Cross in a stratied uid. In the top gure!=n = 0:9 and in the left bottom gure!=n =0:7. pressure = Re { p() e iωt} First derivatives of p give velocit v. Governing PDE [z points up, gravit g goes down] p 2 + p λ 2 z 2 = 0, λ = ω 2 ω 2 N 2 N is the Brunt Väisälä frequenc. N depends on g and the basic densit stratification. In general, N = N(z), but we take N = constant, impling eponential densit variation. Boundar condition, v n = given on oscillating surface. Far-field, radiation conditions??

2 The PDE : Preliminar Remarks The PDE : Preliminar Remarks Re { p() e iωt}, p 2 + p λ 2 z 2 = 0, λ = ω 2 ω 2 N 2 Re { p() e iωt}, p 2 + p λ 2 z 2 = 0, λ = ω 2 ω 2 N 2 N is a given positive constant: keep it fied. The frequenc ω is also constant, but it can be selected. igwf1 N is a given positive constant: keep it fied. The frequenc ω is also constant, but it can be selected. 1. uppose ω > N = λ > 0: PDE is elliptic. Dull! B scaling z, we get Laplace s equation. The PDE : Preliminar Remarks Re { p() e iωt}, p 2 + p λ 2 z 2 = 0, λ = ω 2 ω 2 N 2 N is a given positive constant: keep it fied. The frequenc ω is also constant, but it can be selected. 1. uppose ω > N = λ > 0: PDE is elliptic. Dull! B scaling z, we get Laplace s equation. 2. uppose ω < N = λ < 0. The PDE is hperbolic: it is recognised as the 2D wave equation, with z plaing the role of time. The characteristics (cones in 3D) give the beam angles. Interior Boundar-Value Problems + 2 u olve (elliptic) PDE in disc with From now on, we onl consider Case 2 (0 < ω < N). Appending boundar conditions gives a hperbolic BVP.

3 Interior Boundar-Value Problems Interior Boundar-Value Problems igwf1 + 2 u olve (elliptic) PDE in disc with Laplace s equation: uniqueness of interior Dirichlet problem olution: u(, ) u olve (elliptic) PDE in disc with Laplace s equation: uniqueness of interior Dirichlet problem olution: u(, ) 0 igwf2 2 u olve (hperbolic) PDE in disc with Interior Boundar-Value Problems Interior Hperbolic Boundar-Value Problems + 2 u olve (elliptic) PDE in disc with Laplace s equation: uniqueness of interior Dirichlet problem olution: u(, ) 0 Mathematics Few mathematical studies: earl eamples are Bourgin and Duffin, Bull. AM 45 (1939) F. John, Amer J. Math. 63 (1941) Internal wave attractors 2 u olve (hperbolic) PDE in disc with Non-trivial solution: u(, ) = a Much recent work on internal waves in closed regions. Prett picture stolen from Leo Maas s webpage: But we are interested in eterior problems...

4 Digression: Acoustic cattering Wave equation (hperbolic): 2 U = 1 2 U c 2 t 2 U = Re {u() e iωt } = ( 2 + k 2 ), k = ω/c Eterior problem: n olve ( 2 + k 2 ) outside u with = g on. n For unique solution, impose Acoustics: ommerfeld Radiation Condition Mathematicall, the radiation condition ensures that the BVP for the Helmholtz equation is uniquel solvable. Phsicall, the radiation condition ensures that waves go awa from the oscillating object. OMMERFELD RADIATION CONDITION u eikr r f (θ, φ) as r, r, θ, φ are spherical polars. f is unknown far-field pattern. Acoustics: ommerfeld Radiation Condition Acoustics: ommerfeld Radiation Condition Mathematicall, the radiation condition ensures that the BVP for the Helmholtz equation is uniquel solvable. Phsicall, the radiation condition ensures that waves go awa from the oscillating object. Actuall, we don t care about the waves, we care about energ. Fortunatel, with acoustics and with waves on the surface of the ocean, the energ goes in the same direction as the waves. Mathematicall, the radiation condition ensures that the BVP for the Helmholtz equation is uniquel solvable. Phsicall, the radiation condition ensures that waves go awa from the oscillating object. Actuall, we don t care about the waves, we care about energ. Fortunatel, with acoustics and with waves on the surface of the ocean, the energ goes in the same direction as the waves. Actuall, I lied: we don t care about energ, we care about causalit, which means working with time-dependent problems. Fortunatel, with acoustics and with waves on the surface of the ocean, causalit is ensured.

5 Acoustics: Plane Waves Acoustics: Fundamental olution, G ( 2 + k 2 ), k 2 = ω 2 /c 2 u() = ep (ik ), k = (k, k, k z ) = k 2 = k 2 ctac4 Equivalentl, ω = c k 2 + k 2 + kz 2 [Dispersion relation] Phase (wave) velocit, c, is in direction of k: c = (c 2 /ω)k, c = c = ω/k Group (energ) velocit is ( ) ω ω ω c g =, k k k z Calculation gives c = c g : energ propagates with the wave: same direction, same speed What is a Fundamental olution? ingular solution of governing PDE. Helmholtz equation, ( 2 + k 2 ), in 3D: G(P, P ) = G(, ) = eikr R, R = This G also satisfies the radiation condition. Wh is G useful? uperposition: for eample, u(p) = satisfies ( 2 + k 2 ) and the radiation condition for an µ. Appling the boundar condition gives integral equation for µ. Acoustics: Another use of G Plane Internal Gravit Waves P u n = g n Given that u and G both satisf the radiation condition, a calculation with Green s theorem gives As 0 < ω < N, define angle θ c b ω = N cos θ c In 2D, PDE is 2 cot2 θ c z 2 = 0 Characteristics are lines at angle θ c to z-ais u(p) = 1 ( u G ) 4π n Gg ds Boundar integral equations Letting P p gives 2πu(p) u(q) G (p, q) ds q = n q This is a BIE for u on BEM... g(q) G(p, q) ds q Plane waves p() = ep (ik ), k = (k, k z ) = kz 2 = k 2 tan 2 θ c Equivalentl, ω 2 = k 2 N 2 /(k 2 + kz 2 ) [Dispersion relation] Phase (wave) velocit, c, is in direction of k ( ) ω ω Group (energ) velocit is c g =, k k z Calculation gives c c g = 0: energ propagates perpendicularl to the direction of wave propagation!

6 Qualitative Comparison Fundamental olution for Internal Waves in 3D Recall: z ω = N cos θ c c g 19 Let P = (X, Y, Z ), Q = (,, z). Fi P. G(P, Q) = 1 (X ) 2 + (Y ) 2 (Z z) 2 tan 2 θ c θ c c Energ propagates along the arms of the X Waves propagate perpendicular to the arms z θ c P -plane G(P, Q) is singular at all points Q on a double cone, not just at a single point, P = Q. If P is in a conical wave beam (as shown), the (red) cone will intersect the (blue) radiator in a curve: this means trouble ahead!! Laer Potentials Given G, construct p(p) = Figure 5: t Andrew's Cross in a stratied uid. In the top gure!=n = 0:9 and in the left bottom gure!=n =0:7. This is a single-laer potential. We can also define double-laer potentials. Two issues 1. The integrand is singular for all points q along a certain curve on, when P is in a wave beam: how do we handle this? 2. How can we decide if the radiation condition is satisfied? Laer Potentials Given G, construct p(p) = This is a single-laer potential. We can also define double-laer potentials. Two issues 1. The integrand is singular for all points q along a certain curve on, when P is in a wave beam: how do we handle this? 2. How can we decide if the radiation condition is satisfied? 3. In fact, what is the radiation condition?!

7 ome Technical Bits Introduce spherical polars [P = (X, Y, Z ), Q = (,, z)] X = R sin Θ cos Φ, Y = R sin Θ sin Φ, Z z = R cos Θ ω G(P, Q) = R ω 2 N 2 cos 2 Θ with ω = N cos θ c o, G is singular at Θ = θ c. Fi P in wave beam. Q varies as we integrate over the surface. Locate Q using (Θ, Φ). Integration domain, E, is elliptical. E shrinks as P Θ Θ = θ ingularit now along straight line. c We must also define in G correctl for Θ < θ c ; this is another manifestation of the radiation condition. Uses analtic continuation in comple ω-plane, and causalit. Idea goes back to A. D. Pierce (1963) and D. G. Hurle (1972). The Far Field Recall p(p) = We can find the asmptotic behaviour of p(p) as P. p decas as with 2 outside the beams: dull. Much trickier inside the beams: p(p) r 1/2 F(σ), Φ E with r = OP, σ is a lateral coordinate across the beam. F can be calculated in terms of µ. In the beam, v is along the beam, with v r 1/2 F (σ). Knowing p and v, we can calculate energ transport. It turns out that, for some choices of µ, energ transport is not outwards all across the beam. Thus, although G radiates, µg ma not! Eplanation is clear: energ depends quadraticall on µ. The Far Field Recall p(p) = We can find the asmptotic behaviour of p(p) as P. p decas as with 2 outside the beams: dull. Much trickier inside the beams: p(p) r 1/2 F(σ), with r = OP, σ is a lateral coordinate across the beam. F can be calculated in terms of µ. In the beam, v is along the beam, with v r 1/2 F (σ). Knowing p and v, we can calculate energ transport. Discussion and Conclusions Hperbolic BVPs do arise, and the are worth of stud. A proper formulation is missing: in particular, how should one specif a radiation condition so as to (i) get a well-posed problem and (ii) retain all the interesting phsics? ome possibilities are Go back to the time domain and solve an initial-bvp We have done this for oscillating horizontal thin disc Develop analtic continuation arguments Check/calculate energ transport at the end of a calculation Formulating and solving boundar integral equations will also be challenging, because of the complicated singularit structure: ver different to linear acoustics. There is plent to do!

8 References Papers b PAM and tefan G. Llewelln mith: Internal gravit waves, boundar integral equations and radiation conditions. Wave Motion 49 (2012) Generation of internal gravit waves b an oscillating horizontal disc. Proc. Ro. oc. A 467 (2011) Generation of internal gravit waves b an oscillating horizontal elliptical plate. IAM J. Appl. Math. 72 (2012) Also, some (unpublished) work done on 2D problems: simpler geometr but more complicated due to logs.