We examine a two-media 2-dimensional fluid system consisting of a lower
medium bounded underneath by a flatbed and an upper medium with a free surface
with wind generated surface waves but considered bounded above by a lid by an
assumption that surface waves have negligible amplitude. An internal wave
driven by gravity which propagates in the positive $x$-direction acts as a free
common interface between the media. The current is such that it is zero at the
flatbed but a negative constant, due to an assumption that surface winds blow
in the negative $x$-direction, at the lid. We are concerned with the layers
adjacent to the internal wave in which there exists a depth dependent current
for which there is a greater underlying than overlying current. Both media are
considered incompressible and having non-zero constant vorticities. The
governing equations are written in canonical Hamiltonian form in terms of the
variables, associated to the wave (in a presence of a constant current). The
resultant equations of motion show that wave-current interaction is influenced
only by the current profile in the 'strip' adjacent to the internal wave.Comment: 13 pages, 1 figur

Compelli, Alan

Ivanov, Rossen

English

arXiv

[Ivanov Rossen; Иванов Росен]We examine a two-media 2-dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface with wind generated surface waves but considered bounded above by a lid by an assumption that surface waves have negligible amplitude. An internal wave driven by gravity which propagates in the positive x-direction acts as a free common interface between the media. The current is such that it is zero at the flatbed but a negative constant, due to an assumption that surface winds blow in the negative x-direction, at the lid. We are concerned with the layers adjacent to the internal wave in which there exists a depth dependent current for which there is a greater underlying than overlying current. Both media are considered incompressible and having non-zero constant vorticities. The governing equations are written in canonical Hamiltonian form in terms of the variables, associated to the wave (in a presence of a constant current). The resultant equations of motion show that wave-current interaction is influenced only by the current profile in the ‘strip‘ adjacent to the internal wave. 2010 Mathematics Subject Classification: 35Q35, 37K05, 74J30

Bulgarian Digital Mathematics Library at IMI-BAS

Hamiltonian Approach to Internal Wave-Current Interactions in a Two-Media Fluid with a Rigid Lid

We examine a two-media 2-dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface with wind generated surface waves but considered bounded above by a lid by an assumption that surface waves have negligible amplitude. An internal wave driven by gravity which propagates in the positive x-direction acts as a free common interface between the media. The current is such that it is zero at the flatbed but a negative constant, due to an assumption that surface winds blow in the negative x-direction, at the lid. We are concerned with the layers adjacent to the internal wave in which there exists a depth dependent current for which there is a greater underlying than overlying current. Both media are considered incompressible and having non-zero constant vorticities. The governing equations are written in canonical Hamiltonian form in terms of the variables, associated to the wave (in a presence of a constant current). The resultant equations of motion show that wave-current interaction is influenced only by the current profile in the ’strip’ adjacent to the internal wave

Arrow@TUDublin

Technological University Dublin ARROW@TU Dublin Articles School of Mathematics 2015 Hamiltonian approach to internal wave-current interactions in a two-media fluid with a rigid lid Alan Compelli Technological University Dublin, alan.compelli@tudublin.ie Rossen Ivanov Technological University Dublin, rossen.ivanov@tudublin.ie Follow this and additional works at: https://arrow.tudublin.ie/scschmatart  Part of the Fluid Dynamics Commons, and the Mathematics Commons Recommended Citation Compelli, A., Ivanov, R. (2015) Hamiltonian approach to internal wave-current interactions in a two-media fluid with a rigid lid, Pliska Stud. Math. Bulgar 25 (2015) 7–18. doi: 10.21427/9b0y-b440 This Article is brought to you for free and open access by the School of Mathematics at ARROW@TU Dublin. It has been accepted for inclusion in Articles by an authorized administrator of ARROW@TU Dublin. For more information, please contact yvonne.desmond@tudublin.ie, arrow.admin@tudublin.ie, brian.widdis@tudublin.ie. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ??????????????????????????????????????????????????????? ????????????????????????????????????????????????????????????? ??????????????????????? ????????????????????????????????????? ?????????????????????????????????????????????????????????????????????????????????????????????Pliska Stud. Math. 25 (2015), 7–18STUDIA MATHEMATICAHAMILTONIAN APPROACH TO INTERNALWAVE-CURRENT INTERACTIONS IN A TWO-MEDIAFLUID WITH A RIGID LIDAlan Compelli, Rossen IvanovWe examine a two-media 2-dimensional fluid system consisting of a lowermedium bounded underneath by a flatbed and an upper medium with a freesurface with wind generated surface waves but considered bounded aboveby a lid by an assumption that surface waves have negligible amplitude. Aninternal wave driven by gravity which propagates in the positive x-directionacts as a free common interface between the media. The current is suchthat it is zero at the flatbed but a negative constant, due to an assump-tion that surface winds blow in the negative x-direction, at the lid. We areconcerned with the layers adjacent to the internal wave in which there ex-ists a depth dependent current for which there is a greater underlying thanoverlying current. Both media are considered incompressible and havingnon-zero constant vorticities. The governing equations are written in canon-ical Hamiltonian form in terms of the variables, associated to the wave (ina presence of a constant current). The resultant equations of motion showthat wave-current interaction is influenced only by the current profile in the’strip’ adjacent to the internal wave.1. IntroductionStudies of internal waves, such as sharp temperature gradients called thermo-clines which separate oceanic bodies of water which are at different temperatures,are of significant interest to climatologists, marine biologists, coastal engineers,etc.2010 Mathematics Subject Classification: 35Q35, 37K05, 74J30.Key words: Internal waves, vorticity, current, shear flow, Hamiltonian system.8 A. Compelli, R. IvanovThe study of internal waves draws from previous single medium irrotational[1], [2], [3], [4], [5], [6] and rotational [7], [8], [9], [10], [11], [12], [13], [14] studiesand from appropriate studies of 2-media systems such as [17], [18], [19], [20].However, these studies need to be extended to include the interaction betweenwaves and currents.Recent studies include the interaction between waves that propagate acrossthe Pacific Ocean and the Equatorial Undercurrent (EUC) [21], a Hamiltonianformulation describing the 2-dimensional nonlinear interaction between coupledsurface waves, internal waves, and an underlying current with piecewise constantvorticity, in a two-layered fluid overlying a flat bed [22] and using shifted variablesto transform a non-canonical wave-current system into a canonical system whichhas zero vorticity in the layers adjacent to the internal wave [23]. This study aimsto provide a Hamiltonian formulation of a two-media bounded system which isrotational in the layers adjacent to the internal wave and hence show that wave-current interaction is influenced only by the current profile in this ’strip’.2. PreliminariesThe system under study consists of a 2-dimensional internal wave under therestorative action of gravity, which acts as a free common interface separatingtwo fluid media, and a depth dependent current as per Figure 1.The medium underneath the internal wave is defined by the domain Ω1 ={(x, y) ∈ R2 : −h1 < y < η(x, t)}. This medium is bounded at the bottom byan impermeable flatbed at a depth −h1. The medium above the internal waveis defined by the domain Ω2 = {(x, y) ∈ R2 : η(x, t) < y < h2}. This medium isregarded as being bounded on top by an impermeable lid at a height h2, but inreality is a free surface with negligible wave amplitude. Throughout the article thesubscript 1 will be used to mean evaluation for the lower medium Ω1, subscript 2means evaluation for the upper medium Ω2, subscript i = {1, 2} means evaluationfor both media and subscript c will be used to denote evaluation at the commoninterface. Non-lateral velocity flow is described by Vi(x, y, z) = (ui, vi, 0). Thearbitrary periodic function η(x, t) describes the elevation of the internal wave,i.e. y = η is the equation of the internal wave. We define the mean of η to be theshear surface at y = 0 with the centre of gravity in the negative y-direction.A depth dependent current U1(y) flows in Ω1 and, correspondingly, U2(y)flows in Ω2. Currents are described for the system under study via the continuousHamiltonian Approach to Internal Wave-Current Interactions 9xy0-h1-l1h2l2IIIIIIIV12 (x,t)U=0U=1U=2U=-3U=0windFigure 1: System setup. The current profile in layers I and IV is arbitrary as weare only concerned with layers II and III as the internal wave is a free interfacebetween these layers. Continuity of U(y) is assumed in layers I and IV.function Ui(y) as(1) Ui(y) =−σ3, y = h2 (lid)σ2, y = l2γy + κ, l2 ≥ y ≥ −l1 (layers II and III)σ1, y = −l10, y = −h1 (flatbed)for the positive constants σ1, σ2, σ3, κ, li, hi, γ and κ, where κ is the velocity ofthe time-independent current at y = 0 and γ is the non-zero constant vorticity forlayers II and III, noting that the current is arbitrary in layers I and IV (howeverrepresented by a continuous function everywhere).We consider a velocity field which is defined by:(2){ui = ϕ˜i,x + Ui(y)vi = ϕ˜i,y.We have separated the wave and current contributions to the velocity and sowe define ϕ˜i as the wave velocity potential for Ωi and in particular the velocity10 A. Compelli, R. Ivanovcomponents in layers II and III are [22](3){ui = ϕ˜i,x + γy + κvi = ϕ˜i,y.Additionally, the stream function ψi is introduced, defined by:(4){ui = ψi,yvi = −ψi,x.ρ1 and ρ2 are the respective constant densities of the lower and upper media andstability is given by the immiscibility conditionρ1 > ρ2.(5)The rotationality of the layers II and III is given by the conditionγ < 0⇔ σ1 > σ2(6)ensuring non-zero vorticity in this region. Alternatively σ2 > σ1 could also beconsidered for γ > 0.We assume that for large |x| the amplitude of η attenuates and hence makethe following assumptionslim|x|→∞η(x, t) = 0,(7)lim|x|→∞ϕ˜i(x, y, t) = 0,(8)and−l1 < η(x, t) < l2 for all x and t,(9)i.e. the wave is localised in the strip.We have the following equation (Euler’s equation)∇((ϕi,t)c +12(∇ψi)2c − γψi)= ∇(−piρ1− gη)(10)where pi is the dynamic pressure, g is the acceleration due to gravity (wherey points in the opposite direction to the center of gravity) and ∇ = (∂x, ∂y).Hamiltonian Approach to Internal Wave-Current Interactions 11The following Bernoulli condition (cf. [20]) at the interface follows from Euler’sequation and assumptions (7) and (8):ρ1((ϕ1,t)c +12(∇ψ1)2c − γχ1 + gη)= ρ2((ϕ2,t)c +12(∇ψ2)2c − γχ2 + gη)+ f(t)(11)where χi is the stream function evaluated at the interface. Since the two mediado not mix, χ1 = χ2 ≡ χ. Moreover, f(t) is an arbitrary function of time anddepends on how the potentials are defined at ±∞. Clearly such a function canbe absorbed in the definition of the wave potentials, but we will keep it separatefor further convenience. We know by comparing (3) and (4) that12(∇ψi)2c =12(ϕ˜i,x)2c +12(ϕ˜i,y)2c +12(γη + κ)2 + (ϕ˜i,x)cγη + κ(ϕ˜i,x)c(12)and hence we can express the Bernoulli condition in terms of wave and currentcomponents only as(13) (ρ1ϕ˜1,t − ρ2ϕ˜2,t)c + κ(ρ1ϕ˜1,x − ρ2ϕ˜2,x)c +ρ12|∇ϕ˜1|2c −ρ22|∇ϕ˜2|2c+12(ρ1 − ρ2)(γη + κ)2 + γη(ρ1ϕ˜1,x − ρ2ϕ˜2,x)c− (ρ1 − ρ2)γχ+ (ρ1 − ρ2)gη = f(t).The terms with γ and κ are due to the wave-current interaction. For example,the second term is due to overall translation leading to a shift ∂t → ∂t + κ∂x.The equation suggests the introduction of the variableξ := ρ1ξ1 − ρ2ξ2,(14)whereξi := (ϕ˜i)c = ϕ˜i(x, η(x, t), t).(15)We also have the following kinematic boundary conditions at the interface, usingthe velocity representations (3)(16){ηt + ηx(γη + (ϕ˜i,x)c + κ)+ (ϕ˜i,y)c = 0(ϕ˜1,y)b = (ϕ˜2,y)l = 0noting that V1(x,−h1, 0) = (u1, 0, 0) and V2(x, h2, 0) = (u2, 0, 0), where thesubscripts b and l denote evaluation at the bottom (lower boundary) and lid(upper boundary) respectively.12 A. Compelli, R. Ivanov3. Hamiltonian FormulationIf we consider the system under study as an irrotational system the Hamiltonian,H, is given by the sum of the kinetic and potential energies as:H =ρ12∫Rη∫−h1(u21+ v21)dydx+ρ22∫Rh2∫η(u22+ v22)dydx+12(ρ1 − ρ2)∫Rgη2dx.(17)The kinetic energy term for Ω1 isK1 =ρ12∫Rη∫−h1(u21+ v21)dydx(18)which we can split into layers IV and III, respectively, asK1 =ρ12∫R−l1∫−h1(u21+ v21)dydx+ρ12∫Rη∫−l1(u21+ v21)dydx.(19)For layer IV the kinetic energy is(20)ρ12∫R−l1∫−h1(u21 + v21)dydx =ρ12∫R−l1∫−h1(ϕ˜1,x)2dydx+ρ12∫R−l1∫−h1(ϕ˜1,y)2dydx+ρ12∫R−l1∫−h1γ2y2dydx+ρ12∫R−l1∫−h1U21dydx+ ρ1∫R−l1∫−h1ϕ˜1,xγydydx+ ρ1∫R−l1∫−h1γU1ydydx+ ρ1∫R−l1∫−h1U1ϕ˜1,xdydx.However, terms 3-7 combine to produce a constant which is irrelevant in terms ofdynamic considerations (does not contribute to the variations with respect to thefield variables). Moreover∫Rη(x′, t)dx′ = 0 (the mean deviation is by definitionzero) and the fields vanish at x = ±∞ so that integration of total x− derivativesproduces zero, thus(21)ρ12∫R−l1∫−h1(u21+ v21)dydx =ρ12∫R−l1∫−h1(ϕ˜1,x)2dydx +ρ12∫R−l1∫−h1(ϕ˜1,y)2dydx.Hamiltonian Approach to Internal Wave-Current Interactions 13For layer III the kinetic energy is(22)ρ12∫Rη∫−l1(u21 + v21)dydx =ρ12∫Rη∫−l1(ϕ˜1,x)2dydx+ρ12∫Rη∫−l1(ϕ˜1,y)2dydx+ρ12∫Rη∫−l1(γy + κ)2dydx+ ρ1∫Rη∫−l1ϕ˜1,xγydydx+ ρ1∫Rη∫−l1κϕ˜1,xdydx.We writeρ12∫Rη∫−l1(γy + κ)2dydx =ρ16γ∫R(γη + κ)3dx(23)noting that∫R(γη + κ)3dx can be properly re-normalised as∫R((γη + κ)3 − κ3)dxas the variation in∫Rκ3dx is zero.We introduce the Dirichlet-Neumann operator Gi(η) (see [3], [18]) given byGi(η)ξi = (∂ni ϕ˜i)√1 + (ηx)2,(24)where ∂niϕ˜i is the normal derivative of the velocity potential ϕ˜i, at the interface,for an outward normal ni, and also define [17]B := ρ1G2(η) + ρ2G1(η).(25)Thus we can determine that{ξ1 = B−1(G2(η)ξ)ξ2 = B−1(−G1(η)ξ)(26)The integral with ρ1κϕ˜1,x term, using the Leibniz integral rule with varying limits(cf. [19]), can be written asρ1∫Rη∫−l1κϕ˜1,xdydx = −ρ1κ∫Rξ1ηxdx(27)and the ρ1γyϕ˜1,x term asρ1∫Rη∫−h1γyϕ˜1,xdydx = −ρ1∫Rγξ1ηηxdx(28)14 A. Compelli, R. Ivanovand hence we write the Hamiltonian for Ω1 as(29) H1 =ρ12∫Rη∫−h1|∇ϕ˜1|2dydx+ρ12∫Rgη2 dx+ρ16γ∫R(γη + κ)3dx− ρ1∫Rγξ1ηηxdx− ρ1κ∫Rξ1ηxdx.We follow the same procedure for Ω2 to obtain the corresponding energy as(30) H2 =ρ22∫Rh2∫η|∇ϕ˜2|2dydx−ρ22∫Rgη2dydx−ρ26γ∫R(γη + κ)3dx+ ρ2∫Rγξ2ηηxdx+ ρ2κ∫Rξ2ηxdx.The total energy is therefore H = H1 +H2 or in terms of (η, ξ)(31) H(η, ξ) =12∫Rξ(G1(η)B−1G2(η))ξ dx+ρ1 − ρ22∫Rgη2 dx− κ∫Rξηxdx−∫Rγηηxξdx+(ρ1 − ρ2)6γ∫R(γη + κ)3dx.Defining the Hamiltonian which has no current or vorticity components, H0, asH0(η, ξ) =12∫Rξ(G1(η)B−1G2(η))ξ dx+ (ρ1 − ρ2)12∫Rgη2 dx(32)we can writeH(η, ξ) = H0 − κ∫Rξηxdx−∫Rγηηxξdx+(ρ1 − ρ2)6γ∫R(γη + κ)3dx.(33)The equations of motion can be written in Hamiltonian form as follows. From(16) the dynamic boundary conditionηt = −γηηx + (ϕ˜i,x)cηx − κηx − (ϕ˜i,y)c= δξH0 − κηx − γηηx = δξH.(34)Hamiltonian Approach to Internal Wave-Current Interactions 15We note that the quantities in the Bernoulli condition (13) areρ1(ϕ˜1,x)c − ρ2(ϕ˜2,x)c = ξx − (ρ1ϕ˜1,y − ρ2ϕ˜2,y)cηx(35)ρ1(ϕ˜1,t)c − ρ2(ϕ˜2,t)c = ξt − (ρ1ϕ˜1,y − ρ2ϕ˜2,y)cηt.(36)and we can write it as(37) ξt − (ρ1ϕ˜1,y − ρ2ϕ˜2,y)c(ηt + (γη + κ)ηx) +ρ12|∇ϕ˜1|2c −ρ22|∇ϕ˜2|2c+ (γη + κ)ξx +12(ρ1 − ρ2)(γη + κ)2 − (ρ1 − ρ2)γχ+ (ρ1 − ρ2)gη = f(t).or due to (34) as(38)ξt − (ρ1ϕ˜1,y − ρ2ϕ˜2,y)c(ϕ˜i,xηx − ϕ˜i,y)c +ρ12|∇ϕ˜1|2c −ρ22|∇ϕ˜2|2c + (ρ1 − ρ2)gη+ (γη + κ)ξx +12(ρ1 − ρ2)(γη + κ)2 − (ρ1 − ρ2)γχ = f(t).Noting the ’usual’, not related to the current terms,(39) ξt + δηH0 + (γη + κ)ξx +12(ρ1 − ρ2)(γη + κ)2 − (ρ1 − ρ2)γχ = f(t).and finally, from (33)(40) ξt + δηH − (ρ1 − ρ2)γχ = f(t).The equation for ξt is given up to an arbitrary function of time because theHamiltonian can always be ’renormalised’ by a term −f(t)∫Rηdx which has avariation of −f(t) with respect to η but is otherwise zero by definition. Thus, forthe renormalised Hamiltonian(41) ξt = −δηH + (ρ1 − ρ2)γχ.Since χ = −∫ x−∞ ηt(x′, t)dx′ = −∫ x−∞ δξHdx′, after a change of variables [14]via the transformation (η, ξ)→ (η, ζ)ξ → ζ = ξ −(ρ1 − ρ2)γ2∫ x−∞η(x′, t) dx′.(42)16 A. Compelli, R. Ivanovthe system acquires a canonical Hamiltonian form:(43){ηt = δζHζt = −δηHIn conclusion we have shown that the wave-current is influenced only bythe current profile in the ’strip’ (layers II and III), i.e. outside this region thecontinuous current is arbitrary.4. ConclusionThe governing equations of a system of two-media, bounded on top by a lidand on the bottom by a flatbed, with an internal wave providing a free com-mon interface and with a depth dependent current were written in a canonicalHamiltonian form in terms of the ’wave’-related variables (η, ζ).It was then shown that the wave-current interactions are influenced only bythe current profile in the ’strip’, and do not depend on the current profile in theother layers.AcknowledgementsA.C. is funded by the Fiosraigh Scholarship Programme of Dublin Institute ofTechnology. The support of the FWF Project I544-N13 “Lagrangian kinematicsof water waves” of the Austrian Science Fund is gratefully acknowledged by R.I.The authors are grateful to Prof. A. Constantin for many valuable discussions.References[1] V. Zakharov. Stability of periodic waves of finite amplitude on the surfaceof a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 9 (1968), 86–94.[2] T. Benjamin, P. Olver. Hamiltonian structure, symmetries and conser-vation laws for water waves. J. Fluid Mech. 125 (1982), 137–185.[3] W. Craig.Water waves, Hamiltonian systems and Cauchy integrals. IMAVol. Math. Appl. 30 (1991), 37–45.[4] D. Milder. A note regarding ‘On Hamilton‘s principle for water waves‘.J. Fluid Mech. 83 (1977), 159–161.Hamiltonian Approach to Internal Wave-Current Interactions 17[5] J. Miles. Hamiltonian formulations for surface waves. Appl. Sci. Res. 37(1981), 103–110.[6] J. Miles. On Hamilton‘s principle for water waves. J. Fluid Mech. 83(1977), 153–158.[7] A. Constantin. Reappraisal of the Kelvin-Helmholtz problem. Part 1.Hamiltonian structure. J. Phys. A 34 (2001), 1405–1417.[8] A. Constantin, J. Escher. Symmetry of steady periodic surface waterwaves with vorticity. J. Fluid Mech. 498 (2004), 171–181.[9] A. Constantin, D. Sattinger, W. Strauss. Variational formulationsfor steady water waves with vorticity. J. Fluid Mech. 548 (2006), 151–163.[10] A. Constantin, W. Strauss. Exact steady periodic water waves withvorticity. Comm.Pure Appl. Math. 57 (2004), 481–527.[11] A. Constantin, J. Escher. Analyticity of periodic traveling free surfacewater waves with vorticity. Ann. of Math. 173 (2011), 559–568.[12] A. Teles da Silva, D. Peregrine. Steep, steady surface waves on waterof finite depth with constant vorticity. J. Fluid Mech. 195 (1988), 281–302.[13] A. Constantin, R. Ivanov, E. Prodanov. Nearly-Hamiltonian struc-ture for water waves with constant vorticity. J. Math. Fluid Mech. 9 (2007),1–14.[14] E. Wahle´n. A Hamiltonian formulation of water waves with constant vor-ticity. Lett. Math. Phys. 79 (2007), 303–315.[17] W. Craig, P. Guyenne, H. Kalisch. Hamiltonian long wave expansionsfor free surfaces and interfaces. Comm. Pure Appl. Math. 24 (2005), 1587–1641.[18] W. Craig, P. Guyenne, C. Sulem. Coupling between internal and sur-face waves. Nat. Hazards 57 (2011), 617–642.[19] A. Compelli. Hamiltonian formulation of 2 bounded immiscible mediawith constant non-zero vorticities and a common interface. Wave Motion54 (2015), 115–124.18 A. Compelli, R. Ivanov[20] A. Compelli. Hamiltonian Approach to the Modeling of Internal Geo-physical Waves with Vorticity. Monatshefte fu¨r Mathematik TBD (2015),http://arrow.dit.ie/scschmatart/178/.[21] A. Constantin, R. Johnson. The dynamics of waves interacting withthe Equatorial Undercurrent. Geophysical & Astrophysical Fluid Dynamics109 (2015), 311-358.[22] A. Constantin, R. Ivanov. A Hamiltonian Approach to Wave-CurrentInteractions in Two-Layer Fluids. Physics of Fluids 27, 4 (2015), DOI:10.1063/1.4929457.[23] A. Compelli, R. Ivanov. On the dynamics of internal waves observed inequatorial undercurrent. J. Nonlinear Math. Phys., 22 (2015), 531–539.[24] C. Lanczos. Variational Principles of Mechanics. University of TorontoPress, Toronto, 1949.Alan CompelliSchool of Mathematical SciencesDublin Institute of TechnologyKevin Street, Dublin 8, IrelandRossen IvanovSchool of Mathematical SciencesDublin Institute of TechnologyKevin Street, Dublin 8, IrelandandFaculty of MathematicsOskar-Morgenstern-Platz 1University of Vienna1090 Vienna, Austriae-mail: rossen.ivanov@univie.ac.at

Hamiltonian approach to internal wave-current interactions in a two-media fluid with a rigid lid

Irish Universities

https://arrow.tudublin.ie/cgi/viewcontent.cgi?article=1246&amp;context=scschmatart

Hamiltonian Approach to Internal Wave-Current Interactions in a Two-Media Fluid with a Rigid Lid

Abstract

Similar works

Full text

Available Versions

Bulgarian Digital Mathematics Library at IMI-BAS

Arrow@TUDublin

Irish Universities