7.8 Transient motion in a two-layered sea
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1 1 Lecture Notes on Fluid Dynamics (1.63J/2.21J by Ciang C. Mei, layer.tex Refs: Csandy: Circulation in te Coastal Ocean Cusman-Rosin, Intro to Geopysical Fluid Dynamics 7.8 Transient motion in a two-layered sea allow water approximation Figure 7.8.1: A two-layered sea of contant dept. Let be te mean dept and U, V be te dept-integrated fluxes in te upper layer, and, U, V be te corresponding quantities in te lower layer. Let ζ and ζ be te vertical displacements of te free surface and interface respectively. Ten (U, V = 0 (u, vdz, (U, V = For sallow layers, we ave ydrostatic pressure in te upper layer: (u, v dz, (7.8.1 p = g(ζ z (7.8.2 On te interface, z = + ζ, p = g(ζ + ζ (7.8.3
2 2 In order tat pressure is continuous on te interface we must ave in te lower layer Tus p = g(ζ + ζ + g( + ζ z (7.8.4 p ( = g p = g + g = g + ( g Te dept-integrated momentum equations in te upper layer are U t fv = 1 p = g + τ x (7.8.5 (7.8.6 (7.8.7 V t + fu = 1 p = g + τ y (7.8.8 (ζ ζ + U t + V = 0 (7.8.9 after using ( Ignoring sear stresses on te interface and at te bottom, we ave, for te lower layer, U t fv = 1 p V t + fu = 1 p = g = g + V ɛg ɛg ( ( t + U = 0 ( were (7.8.2 as been used. Te density contrast ɛ = ( is usually small in oceans. Tere six equations for six unknowns : ζ, U, V, ζ, U, and V. As in te last section we can eliminate some unknowns and reduce te number of equations. For example, by eliminating U, V in te upper layer, we can get t ( 2 t 2 + f 2 (ζ ζ g 2 t = 1 ( τ x t + τ y f ( τ y τ x ( imilarly by eliminating U, V in te lower layer we get, ( 2 t t + f 2 ζ g 2 2 t ɛg 2 t = 0 ( Tese two govern te coupled beavior of ζ and ζ, in terms of wic te no-flux boundary conditons on te velocity can also be expressed. We sall owever introduce an alternate approac wic as some advantages.
3 Transformation into normal form To avoid solving six simultaneous equations at once, we seek a linear combination so tat te governing equations look like tose for a one-layer fluid wit just tree unknowns. Tis is called te normal form. Take a( (7.8.10, we get, t (au + U f(av + V = g ( aζ + ζ + ɛζ + aτ x ( imilarly a( ( gives t (av + V + f(au + U = g ( aζ + ζ + ɛζ + aτ y ( Finally a( ( gives, from te continuity equations. Now if we take and Ten ( ( become t (aζ aζ + ζ + (au + U + (av + V = 0 ( aζ + ζ + ɛζ = β ζ ( aζ aζ + ζ = aζ + (1 aζ = ζ ( Ū t f V = gβ ζ + a τ x ( V t ζ + fū = gβ + a τ y ( ζ t + Ū + V = 0 ( and take on te appearance of te governing equations for a single layer fluid. For tis reduction to be possible, conditions ( and ( imply tat aζ + ζ + ɛζ = β (aζ aζ + ζ ( Equating te coefficients of ζ and ζ separately, we get βa = a +, (7.8.25
4 4 β( a + 1 = ɛ A quadratic equation is found for β by eliminating a ( β 2 β 1 + ɛ + + ɛ = 0 ( i.e., Tere are two solutions for β { β1 β 2 β 2 } ( = [ 1 ± β + ɛ = 0 ( ] 1 4ɛ ( + 2, ( to be refered to as two normal modes. Te solutions corresonding to β 1 and β 2 will be refered to as Mode 1 and Mode 2, respectively. Let us see wat te corresponding a s are. Using (7.8.26, we ave ince from (7.8.27, it follows tat Terefore, a = 1 ɛ β. ( β 1 β 2 = ɛ a = 1 β 1β 2 β. a 1 = 1 β 2, a 2 = 1 β 1 ( Let us summarize te normal form equations for Mode k wit k = 1, 2. Defining, Ū k = a k U + U, Vk = a k V + V ( ζ k = a k ζ + (1 a k ζ ( ( τ x k = a k τ x, ( τ y k = a k τ y. ( ten Ūk t f V k = gβ k ζ k + ( τ x k ( V k t + fūk = gβ k ζ k + ( τ y k ( ζ k t + Ūk + V k = 0 (7.8.36
5 5 Note tat for mode k te effective gravity is gβ k, and te effective forcing stresses are a k τx and a k τy. olution for te set of tree equations is obviouly a simpler task. For a general wind stress field (τx, τy, we first solve for (Ū1, V 1, ζ 1 and (Ū2 V 2, ζ 2 separately from ( to ( Afterwards (U, V, ζ and (U, V, ζ can be solved from te algebraic equations ( and ( For example, From ζ 1 = a 1 ζ + (1 a 1 ζ, ζ2 = a 2 ζ + (1 a 2 ζ ( we get for te sea surface, ζ = ζ 1 1 a 1 ζ 2 1 a 2 a 1 1 a 1 a 2 1 a 2 = (1 a 2 ζ 1 (1 a 1 ζ 2 a 1 a 2 ( and for te interface imilarly, from we get, for te upper layer and for te lower layer ζ = a 1 a 2 ζ1 ζ2 a 1 1 a 1 a 2 1 a 2 = a 2 ζ 1 + a 1 ζ2 a 1 a 2 ( Ū 1 = a 1 U + U, Ū 2 = a 2 U + U ( U = U = Ū 1 1 Ū 2 1 a 1 1 a 2 1 a 1 Ū 1 a 2 Ū 2 a 1 1 a 2 1 Te formulas for V and V are similar, wit U s replaced by V s Normal form for small density difference = Ū1 Ū2 a 1 a 2 ( = a 2Ū1 + a 1 Ū 2 a 1 a 2 ( From ere we sall only be interested in small density differences, ten β 1 = 1 + ɛ + + O(ɛ2 ( β 2 = ɛ + + O(ɛ2 (7.8.44
6 6 or a 1 = 1 ɛ + + O(ɛ2, ( Taking only te leading order terms, we ave From ( We also get from ( and ( a 2 = + ɛ + + O(ɛ2 ( a 1 = 1, a 2 =. ( β 1 = +, β 2 = ɛ + ( ζ = (1 + / ζ 1 + (1 1 ζ / = ζ 1. ( ζ = ( / ζ 1 + ζ / ( U = Ū1 Ū2 1 + /, U = ( /Ū1 + Ū2 1 + / V = V 1 V /, V = ( / V 1 + V / ( ( For Mode 1 we get from (7.8.31,( and (7.8.33, Ū 1 = U + U V1 = V + V ζ 1 = ζ, ( τ x 1 = τ x ( τ y 1 = τ y. ( Let H = + denote te total dept, te normal mode equations are Ū1 t f V 1 = gh ζ 1 + τ x ( V 1 t + fū1 = gh ζ 1 + τ y ( t + Ū1 + V 1 = 0. ( In view of (7.8.53, te two layers move togeter as if a single layer of omogeneous fluid. Tis is called te Barotropic (surface mode.
7 7 For Mode 2 we recall from (7.8.34, Ten from ( to (7.8.33, and a 2 = + O(ɛ, β 2 = ɛ + ( Ū 2 = U + U, V2 = V + V ( ζ 2 = ζ ζ ( β 2 = from ( Te normal form equations are ɛ + ( ζ 2 t + Ū2 + V 2 = 0 ( Note tat te effective gravity Ū2 t f V 2 = g ɛ + ζ 2 τ x, ( V 2 t + fū2 = g ɛ + ζ 2 τ x, ( is muc smaller tan g. Moreover, te effective wind stresses are ence are oppposite to te surface wind stresses. gβ 2 = g ɛ ( τ x, τ y ( Free modes witout wind Let us consider free waves in te absence of wind forcing ((τ x k, (τ y k = 0. It is ten possible to consider one mode and assume tat te oter modes to be zero. In particular we sall examine (eigen, or natural modes of sinusoidal wave form (ζ k, U k, V k e ikxx+ikyy iωt (7.8.65
8 8 Barotropic (surface mode: Let us focus on mode 1 alone and assume mode 2 to be absent, i.e., ten ence Ū 2 = V 2 = ζ 2 = 0 ( Ū 2 = ( /U + U = 0 V 2 = ( /V + V = 0 ( ζ 2 = ( /ζ + (1 + / ζ = 0, U = U V = V, ζ = ζ. ( Again te dept-averaged velocities in bot layers are in pase and are numerically te same. Te interface displacement is in pase wit and smaller tan te free surface displacement in proportion to te vertical distance above te seabed. Moreover, ( to ( become for te upper layer: or ( ( + + t U f V = g( + ( ( + + t V + f t + + U ( U + V It is easy to get te equations for te lower layer. = g( + ( ( = 0. ( U fv = g ( t V + fu = g ( t t + + ( U + V = 0. ( U t fv = g V t + fu = g t + + ( U + V ( ( = 0. (7.8.77
9 9 Consider sinusidodal waves and te equations for te upper layer, iωu fv = gik x ζ iωv fu = gik y ζ iωζ + + (ik xu + ik y V = 0 For nontrivbil solution t eeigenvalue condition is easily found to be ω 2 = f 2 + g( + k 2 ( wic can be written as ω + g ( 2 = ± + f + k 2 ( + 2 g 1/2 ( Tis is te dispersion relation for te barotropic (surface wave mode, and is plotted in Figure Baroclinic (internal wave mode Now consider te unforced Mode 2 and assume Mode 1 to be absent, Ū 1 = V 1 = ζ 1 = 0 ( ince we ave, from ( to ( a 1 = 1 ( U + U = 0, V + V = 0. ( ζ = ɛ + ζ ( Note tat te orizontal velocities in te two layers are oppositive in pase, so are te free surface and te interface. Te last result implies te free surface beaves like a rigid lid; te interface oscillates wit muc greater vigor. Using tese results, we get ( U 2 = U + U = 1 + U ( ( V 2 = 1 + V ( ( ζ 2 = 1 + ζ (7.8.86
10 10 Figure 7.8.2: Dispersion relation for te barotropic mode in a two-layered sea of contant dept. Putting all tis in ( to (7.8.36, we get ( 1 + U f t Terefore, for te lower layer, imilarly, Also from (7.8.31, ( 1 + = g ɛ + + ( 1 + V. ( U t fv = g ɛ +. ( V t + fu = gɛ + ( t ( ( U + V = 0. (7.8.90
11 ence, and t + U + V We can also write te equations for te upper layer, 11 = 0. ( U t fv = g ɛ +. ( V t + fu = gɛ + t U V ( = 0. ( Note te sign canges in te momentum and mass conservation equations. For sinusoical waves we ave wit iωu fv = g ik x ζ iωv fu = g ik y ζ iωζ (ik x U + ik y V = 0 g = gɛ ( Te eigenvalue condition acan be easily found to be wic can be written as ω 2 = f 2 + g k 2 ( ( 1/2 ω = ± f g g + k2 2 ( Tis dispersion relation is similar to Figure ince te reduced gravity, i svery small. Te baroclinic mode as a muc longer natural period. Mode 2 is called te barolinic (internal wave mode.
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