A fluid system bounded by a flat bottom and a flat surface with an internal
wave and depth-dependent current is considered. The Hamiltonian of the system
is presented and the dynamics of the system are discussed. A long-wave regime
is then considered and extended to produce a KdV approximation. Finally, a
solitary wave solution is obtained

Compelli, Alan

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arXiv

A. Compelli

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Solitary wave solution of flat surface internal geophysical waves with vorticity

A fluid system bounded by a flat bottom and a flat surface with an internal wave and depth-dependent current is con-sidered. The Hamiltonian of the system is presented and the dynamics of the system are discussed. A long-wave regime is then considered and extended to produce a KdV approximation. Finally, a solitary wave solution is obtained

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Technological University Dublin ARROW@TU Dublin Conference papers School of Mathematics 2017 Solitary Wave Solution of Flat Surface Internal Geophysical Waves With Vorticity Alan Compelli Technological University Dublin, alan.compelli@tudublin.ie Follow this and additional works at: https://arrow.tudublin.ie/scschmatcon  Part of the Applied Mathematics Commons Recommended Citation Compelli, A. (2017). Solitary wave solution of flat surface internal geophysical waves with vorticity. AIP Conference Proceedings, Home > AIP Conference Proceedings, > vol. 1895, no. 1. doi:10.1063/1.5007369 This Conference Paper 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 Conference papers 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 arXiv:1708.06653v1  [physics.flu-dyn]  22 Aug 2017Solitary Wave Solution of Flat Surface Internal GeophysicalWaves with VorticityAlan Compelli1,2,a)1School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland.2Erwin Schro¨dinger Institute for Mathematics and Physics, University of Vienna, Vienna, Austria.a)Corresponding author: alan.compelli@dit.ieURL: http://www.maths.dit.ieAbstract. A fluid system bounded by a flat bottom and a flat surface with an internal wave and depth-dependent current is con-sidered. The Hamiltonian of the system is presented and the dynamics of the system are discussed. A long-wave regime is thenconsidered and extended to produce a KdV approximation. Finally, a solitary wave solution is obtained.INTRODUCTIONSurface waves typically have heights of 1-2m with extreme wave heights greater than 20m having been observed, forexample a 20.4m surface wave was measured off the northwest coast of Ireland in December 2011 by Met E´ireann(the Irish meteorological service). Internal waves are disturbances which act as an interface between discrete fluidbodies which have distinct properties such as salinity or temperature. As internal waves are typically 10 times higherthan surface waves this heuristically suggests that extreme internal waves as high as 200m may occur. Indeed Alfordet al. [1] have measured wave heights in excess of 170m in the Luzon Strait in the South China Sea.Studies of such gigantic waves are essential in the context of tsunami prediction, marine biology, the design ofmarine vessels, etc.Some studies in an irrotational [2] and a rotational [3] setting have been completed but the most pertinent is thelong-wave and KdV approximations obtained in [4]. The presented paper extends this analysis to produce a solitarywave solution.SETUPA two-dimensional water wave system consisting of two discrete fluid domains separated by a free common interfacein the form of an internal wave, such as a pycnocline or thermocline, is presented as per Figure 1.The system is bounded at the bottom by an impermeable flatbed at a depth of h1 and is considered as beingbounded on the surface by an assumption of absence of surface motion by a lid at a height of h2. The domainsΩ1 = {(x, y) ∈ R2 : −h1 < y < η(x, t)} and Ω2 = {(x, y) ∈ R2 : η(x, t) < y < h2} are defined with values associated witheach domain using corresponding respective subscript notation 1 and 2. Propagation of the internal wave is assumed tobe in the positive x-direction which is considered to be ‘eastward’. The centre of gravity is in the negative y-direction.The function η(x, t) describes the elevation of the internal wave with the mean of η assumed to be zero,∫Rη(x, t)dx = 0.The system is considered to be incompressible with ρ1 and ρ2 being the respective constant densities of the lowerand upper media and stability is given by the immiscibility conditionρ1 > ρ2. (1)x-h01h2y21(x,t)FIGURE 1. System setup.To avoid duplication of similar expressions in the 2 domains the subscript i = {1, 2} is introduced and so thevelocity vector ui = (ui, vi, 0) is given in terms of stream functions ψ1, ψ2 asui = ψi,y, vi = −ψi,x (2)or in terms of wave-only velocity potentials ϕ˜1, ϕ˜2 and current U(y) asui = ϕ˜i,x + U(y), vi = ϕ˜i,y (3)where the current profile is depicted by the heavy line in Figure 2 consisting of 5 layers: layer I having an arbitrarycurrent U2(y), layers II and III (which we will collectively refer to as the strip, that is the layers adjacent to the wave)having linear profiles characterised by constant vorticities γi and constant currents κi, layer IV having an arbitrarycurrent U1(y) and layer V having zero current.Uy0IIIIIIIVVU (y)1U (y)2 y+1 1 y+2 2-h01h2l2-m1-l1312FIGURE 2. Current profile.It is noted at y = 0 that the current is κ1 from the perspective of Ω1 and κ2 from the perspective of Ω2 and hencefor κ1 , κ1 there exists a vortex sheet.The positive constants σ1 −σ3 are included to demonstrate that a westward surface wind may be responsible fora westward drift experienced by observers on the surface but that this drift might reduce in strength and ultimatelybecome an eastward drift for, say, a scuba diver descending in the presence of the considered current. This phenomenonwas until relatively recently referred to (mysteriously) as dead water and demonstrates how recent knowledge of suchprocesses is.We make the assumption that the functions η(x, t), ϕ˜1(x, y, t) and ϕ˜2(x, y, t) belong to the Schwartz class S(R) withrespect to the x variable (for any y and t). This reflects the localised nature of the wave disturbances. The assumptionof course implies that for large absolute values of x the internal wave attenuates and solim|x|→∞η(x, t) = 0, lim|x|→∞ϕ˜1(x, y, t) = 0, lim|x|→∞ϕ˜2(x, y, t) = 0. (4)The situation with a free surface is studied in [5, 6].GOVERNING EQUATIONSThe Bernoulli condition at the interface is [4, 5]ρ1((ϕ1,t)c +12|∇ψ1|2c − (γ1 − 2ω)χ1 + gη)= ρ2((ϕ2,t)c +12|∇ψ2|2c − (γ2 − 2ω)χ2 + gη)(5)where χi(x, t) := ψi(x, η(x, t), t)is the stream function evaluated at the interface, the subscript c means evaluation atthe common interface (i.e. on the wave), g is the acceleration due to gravity and ω is the rotational speed of Earth.There is a kinematic boundary condition at the interfacevi = ηt + uiηx (6)and boundary conditions at the bottom and topv1(x,−h1, t) = 0, v2(x, h2, t) = 0 (7)respectively.HAMILTONIAN FORMULATIONThe Hamiltonian is given in terms of decomposed (wave and current) variables by the functional H viaH(η(x), ϕ˜i(x, y))=12ρ1∫Rη(x)∫−h1|∇ϕ˜1|2dydx +12ρ2∫Rh2∫η|∇ϕ˜2|2dydx + ρ1∫Rη(x)∫−h1Uϕ˜1,xdydx+ ρ2∫Rh2∫ηUϕ˜2,xdydx +12ρ1∫Rη(x)∫−h1U2dydx +12ρ2∫Rh2∫ηU2dydx +12(ρ1 − ρ2)g∫Rη2dx. (8)By introducing the interface velocity potentialsξi(x) := ϕi(x, η(x)), (9)the overall interface velocity potentialξ(x) := ρ1ξ1 − ρ2ξ2, (10)Dirichlet-Neumann operators (where ni are outward normals)Gi(η)ξi :=∂ϕi∂ni∣∣∣∣∣y=η√1 +(ηx)2, (11)the operator BB := ρ1G2(η) + ρ2G1(η) (12)and the variable µµ(x) :=((γ1 − γ2)η + (κ1 − κ2))ηx (13)the Hamiltonian can be written in terms of conjugate variables asH(η(x), ξ(x))=12∫RξG1(η)B−1G2(η)ξ dx −12ρ1ρ2∫RµB−1µ dx −∫R(γ1η + κ1)ξηx dx+ ρ2∫RµB−1G1(η)ξ dx +ρ16γ1∫R(γ1η + κ1)3dx −ρ26γ2∫R(γ2η + κ2)3dx +12g(ρ1 − ρ2)∫Rη2dx (14)and it is noted that (14) now describes the system in terms of wave quantities only.The system has non-canonical equations of motion, analogous to those in [7]:ηt =δHδξ, ξt = −δHδη+ Γχ (15)where, by establishing that χ1 = χ2,χ := χ1 = χ2 (16)and the constant Γ is introduced asΓ := ρ1γ1 − ρ2γ2 + 2ω(ρ1 − ρ2). (17)It is noted, by the absence of U1 and U2 terms, that the dynamics of the system depends on the strip only.We can use the variable transformation [8]ξ → ζ = ξ −Γ2∫ x−∞η(x′, t) dx′, (18)as ξ is defined modulo an additive constant, and use the fact that∫Rη(x, t) dx = constant (19)(noting that the variable ζ(x, t) also belongs to the Schwartz class S(R) with respect to x) to give canonical equationsof motionηt =δHδζ, ζt = −δHδη. (20)In the case of a free surface the results can be extended as in [5, 6, 9, 10].THE LONG-WAVE APPROXIMATIONAs the vortex sheet provides a physically unrealistic representation it can be eliminated simply be letting κ = κ1 = κ2giving the Hamiltonian:H(η(x), ξ(x))=12∫RξG1(η)B−1G2(η)ξ dx −12ρ1ρ2(γ1 − γ2)2∫RηηxB−1ηηx dx −∫R(γ1η + κ)ξηx dx+ ρ2(γ1 − γ2)∫RηηxB−1G1(η)ξ dx +16(ρ1γ21 − ρ2γ22)∫Rη3dx +12(g(ρ1 − ρ2) + (ρ1γ1 − ρ2γ2)κ) ∫Rη2dx. (21)The Dirichlet-Neumann operators can be expanded in terms of orders of η asGi(η) =∞∑j=0Gi j(η). (22)As η, ξ are periodic it can be shown thatG1(η)=D tanh(h1D) + DηD − D tanh(h1D)ηD tanh(h1D)+O(η2) (23)andG2(η)=D tanh(h2D) − DηD + D tanh(h2D)ηD tanh(h2D)+O(η2) (24)where D is a Fourier multiplier equivalent to the operation −i∂/∂x. B(G(η)) can hence also be expanded.The following scaling is performed:(x, y)→ h2(x, y), (u1, u2) →√gh2(u1, u2), η→ aη (25)where (u1, u2) are the velocity vectors and a is the wave amplitude.Small arbitrary constant parameters ε, δ ∈ R are introduced:ε =ah2, δ =h2λ. (26)From the kinematic boundary condition η, ξ ∼ O(ε). λ → h2λ and so λ−1 ∼ O(δ) giving the long-wave regime.As D is equivalent to the wavenumber k = 2pi/λ then D ∼ O(δ).So the expanded, nondimensionalised and scaled Dirichlet-Neumann operators and B−1 operator can be writtenasG1(η) = δ2(h1D2 + εDηD)− δ4(13h31D4 + εh21D2ηD2)+ δ6( 215h51D6)+ O(δ8, εδ6, ε2δ4), (27)G2(η) = δ2(h2D2 − εDηD)+ δ4(−13h32D4 + εh22D2ηD2)+ δ6( 215h52D6)+ O(δ8, εδ6, ε2δ4) (28)andB−1 =1δ2(ρ2h1 + ρ1h2)D−1{1 − ερ2 − ρ1ρ2h1 + ρ1h2η + ε2(ρ2 − ρ1)2(ρ2h1 + ρ1h2)2η2+ δ2[13ρ2h31+ ρ1h32ρ2h1 + ρ1h2D2 −13ε(ρ2 − ρ1)(ρ2h31+ ρ1h32)(ρ2h1 + ρ1h2)2ηD2 −13ε(ρ2 − ρ1)(ρ2h31+ ρ1h32)(ρ2h1 + ρ1h2)2D2η + ερ2h21− ρ1h22ρ2h1 + ρ1h2DηD]− δ4[215ρ2h51+ ρ1h52ρ2h1 + ρ1h2D4 −1(ρ2h31+ ρ1h32)29(ρ2h1 + ρ1h2)2D4]+ O(δ6, εδ4, ε2δ2, ε3)}D−1 (29)respectively noting the introduction of new notation such as bars or asterisks has been avoided.The approximation to O(δ6, εδ4, ε2δ2, ε3) can now be calculated asH(η, ξ) =12δ4(h1h2ρ2h1 + ρ1h2) ∫RξD2ξdx +12εδ4( ρ1h22 − ρ2h21(ρ2h1 + ρ1h2)2) ∫RξDηDξdx−12δ6(13h21h22(ρ2h2 + ρ1h1)(ρ2h1 + ρ1h2)2) ∫RξD4ξdx − εδ2κ∫Rξηxdx +12ε2δ2(γ1ρ1h2 + ρ2γ2h1ρ2h1 + ρ1h2) ∫Rη2ξxdx+16ε3(ρ1γ21 − ρ2γ22)∫Rη3dx +12ε2(g(ρ1 − ρ2) + (ρ1γ1 − ρ2γ2)κ) ∫Rη2dx. (30)THE KDV APPROXIMATIONNow the assumption is made that O(ε) ∼ O(δ2), with a view to establishing balance between nonlinearity and disper-sion when obtaining solitary wave solutions, and we can writeH(η, ξ) =12δ4α1∫RξD2ξdx +12δ6α3∫Rη(ξx)2dx −12δ6α2∫RξD4ξdx − δ4κ∫Rξηxdx+12δ6α4∫Rη2ξxdx +16δ6α6∫Rη3dx +12δ4α5∫Rη2dx (31)where the following constants have been introduced:α1 =h1h2ρ2h1 + ρ1h2, α2 =13h21h22(ρ2h2 + ρ1h1)(ρ2h1 + ρ1h2)2, α3 =ρ1h22− ρ2h21(ρ2h1 + ρ1h2)2, (32)α4 =γ1ρ1h2 + ρ2γ2h1ρ2h1 + ρ1h2, α5 = g(ρ1 − ρ2) + (ρ1γ1 − ρ2γ2)κ, α6 = ρ1γ21 − ρ2γ22. (33)This gives the following (non-canonical) equations of motion,∂η∂t= −α1ξxx − δ2α3(ηξx)x − δ2α2ξxxxx − κηx − δ2α4ηηx (34)and∂ξ∂t= −δ22α3(ξx)2− κξx − δ2α4ηξx −12δ2α6η2 − α5η + Γχ. (35)Next, we get the partial derivative of (35) with respect to x and it is noted that a term g − 2ωκ appears which canbe simplified by takingg ≈ 9.8ms−2, ω ≈ 7.3 × 10−5s−1, κ ≈ 1ms−1 (36)and thus clearly g >> 2ωκ.We introduceu = ξx (37)and use a Galilean shift(x, t) → (x − κt, t) = (X, T ) (38)givingηT + α1uX + δ2α2uXXX + δ2α3(uη)X + δ2α4ηηX = 0 (39)anduT − Γα1uX + g(ρ1 − ρ2)ηX − δ2Γα2uXXX + δ2α3uuX + δ2(α4(uη)X − Γα3(uη)X + α6ηηX − Γα4ηηX)= 0. (40)The linearised equations are henceηT + α1uX = 0, uT − Γα1uX + g(ρ1 − ρ2)ηX = 0. (41)Periodicity of η and u means we can writeη(X, T ) = η0ei(kX−Ω(k)T ), u(X, T ) = u0ei(kX−Ω(k)T ) (42)where k is the wavenumber,Ω(k) is the angular frequency and both are related viac(k) =Ω(k)k(43)where the wavespeed c(k) is that observed by an observer moving at a speed κ along y = 0.We hence have equations−ickη + iα1ku = 0 (44)and−icku + ig(ρ1 − ρ2)kη − iΓα1ku = 0. (45)This has solutions for observers moving with the flow asc =12(− α1Γ ±√α21Γ2 + 4α1g(ρ1 − ρ2))(46)and for stationary observers asc + κ =12(− α1Γ ±√α21Γ2 + 4α1g(ρ1 − ρ2)). (47)Starting with the leading approximationu =cα1η (48)the aim is to find a KdV type equation of the formηT + cηX + δ2(C1ηηX +C2ηXXX)= 0 (49)for some constants C1,C2, that is an equation that has a nonlinear and a dispersive component.By considering a Johnson-type transformation [11]u =cα1η + δ2(µη2 + σηXX)(50)we calculateσ = −cα2(1 − α1Γc)α21(2 + α1Γc) (51)andµ =α1α4(c − α1Γ) − α3c(c + 2α1Γ) + α21α62α21(2c + Γα1)(52)giving the KdV approximation [4]ηT + cηX + δ2(α21α6 + 3α3c2 + 3α1α4cα1(2c + Γα1))ηηX + δ2(c2α2α1(2c + α1Γ))ηXXX = 0. (53)The KdV regime is one of the most important propagation regimes for water waves, where stable nonlinearsolitary waves (solitons) are formed, see also [12, 13, 14]. However, there are various propagation regimes and manyother situations are possible, including δ2 ≪ ε. This is when the wavelength is very large in comparison to hi. In suchcase the contribution from the δ2 terms is not significant and can be neglected. Instead of a KdV equation the relevantmodel is the dispersionless Burgers equation (∂τ = ∂T + c∂X)ητ + ε(α21α6 + 3α3c2 + 3α1α4cα1(2c + Γα1))ηηX = 0. (54)Such an equation does not support globally smooth solutions, i.e. the solutions always form a vertical slope andbreak. Such wave-breaking phenomenon is well known for internal waves in the ocean. This is a mechanism thatcauses mixing in the deep ocean, with implications for biological productivity and sediment transport.SOLITARY WAVE SOLUTIONWe seek to obtain a KdV equation in the standard format:ηT + 6ηηX + ηXXX = 0 (55)for which a solution, η(X, T ) = 2κ2sech2(κ(X − 4κ2T )), is already known.We transform T usingX → X + cT (56)and scale as follows [12]:η→ αη, X → βX, T → γT. (57)IntroducingA =α21α6 + 3α3c2 + 3α1α4cα1(2c + Γα1), B =c2α2α1(2c + α1Γ)(58)the KdV is hence writtenηT + δ2AηηX + δ2BηXXX = 0 (59)and the relationγβ=6δ2αA=β2δ2B(60)is noted.Choosingγ =1δ2(61)we can establish thatα =6B13A, β = B13 (62)giving the solitary wave solution of (53)η(X, T ) =12κ2B1/3Asech2{κB1/3(X + (c − 4δ2κ2B1/3)T)}(63)which describes a solitary crest of amplitude 12κ2B1/3/A moving with speed c−4δ2κ2B1/3. The correction to the speedc is −4δ2κ2B1/3, which is related to the amplitude through κ.ACKNOWLEDGMENTSThe author is grateful to Prof. A. Constantin, Dr R. Ivanov and Dr C.I. Martin for many valuable discussions andto Prof. M. Todorov for organising the Ninth Conference of the Euro-American Consortium for Promoting the Ap-plication of Mathematics in Technical and Natural Sciences. Financial support from the Erwin Schro¨dinger Interna-tional Institute for Mathematics and Physics (ESI), Vienna (Austria) for participation in the Research in Teams ProjectHamiltonian approach to modelling geophysical waves and currents with impact on natural hazards is acknowledged.The author is supported by the Fiosraigh scholarship program at Dublin Institute of Technology (Ireland).REFERENCES[1] M. Alford et al., Nature 521, 65–69 (2015).[2] W. Craig, P. Guyenne, and H. Kalisch, Comm. Pure Appl. Math. 58(12), 1587–1641 (2005).[3] A. Compelli, Wave Motion 54, 115–124 (2017).[4] A. Compelli and R. Ivanov, J. Math. Fluid Mech. 19, 329–344 (2017), arXiv:1611.06581 [physics.flu-dyn].[5] R. Ivanov, Nonlinear Analysis: Real World Applications 34, 316–334 (2017), corrigendum vol. 36 (2017)115, arXiv:1702.01441 [physics.flu-dyn].[6] A. Constantin and R. Johnson, Geophysical and Astrophysical Fluid Dynamics 109, 311–358 (2015).[7] A. Constantin, R. Ivanov, and E. Prodanov, J. Math. Fluid Mech. 10, 224–237 (2008).[8] E. Wahlen, Lett. Math. Phys. 79, 303–315 (2007).[9] A. Constantin and R. Ivanov, Phys. Fluids 27 (2015), 086603.[10] A. Constantin, R. Ivanov, and C. I. Martin, Arch. Rational Mech. Anal. 221, 1417–1447 (2016), arXiv:math-ph/0610014.[11] R. Johnson, J. Fluid Mech. 455, 63–82 (2002).[12] R. Johnson, in A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge UniversityPress, Cambridge, 1997).[13] R. Ivanov, Philos. Trans. Roy. Soc.: Ser. A 365, 2267–2280 (2007), arXiv:0707.1839 [nlin.SI].[14] D. Henry and R. Ivanov, Discrete Contin. Dyn. Syst.-A 34, 3025–3034 (2014), arXiv:1402.0537 [nlin.SI].

Solitary Wave Solution of Flat Surface Internal Geophysical Waves With Vorticity

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Solitary Wave Solution of Flat Surface Internal Geophysical Waves with Vorticity

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