12. Shallow water equations with rotation Poincaré waves. Considering now motions with L<<R, we can write the equations of motion in

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1 . Shallw water equatins with rtatin Pincaré waves Cnsidering nw mtins with L<<R, we can write the equatins f mtin in Cartesian crdinate:. u t - fv = p ρ. v t + fu = p ρ 3. gρ 0 = pz ρ ρ 4. u +v +w z = 0 ρ t +wρ z = 0 cmbined: In the general case: Again, the vertical mmentum equatin 3, and the adiabatic equatin 5 can be ρ p zt + N w = 0 Cnsider first a hmgeneus fluid fr which ρ ttal = ρ + ρ = ρ the hdrstatic equatin. Then p z = -gρ is the ttal pressure. Integrating frm z t the free surface : p - p(z) = -gρ (-z) r p(z) = p atm + gρ (-z) => p(z) = gρ (-z) as p atm = 0 Then the hrizntal pressure gradient p = gρ is independent f z. The equatins f mtin can be written: u t -fv = -g v t +fu = -g (there is n adiabatic equatin as there is n perturbatin densit) u +u +w z = 0

2 Ntice that the right-hand side f the hrizntal mmentum equatins is independent f z: (u,v) are independent f z Integrate the cntinuit equatin dz D (u + v )dz + w z= w z= D = 0 D Tp and bttm b.c. are (including nn-linearities) We have: d dd w = at z = ; w = = u H D at z = D As: w zz = 0 w z = a(t) w = a(t)z + b(t) at z = -D w = 0 -Da+b = 0 b = ad w = a(t)(z+d) at z = 0 w = Da(t) = w = (z D) D + linearized versin In general w = a(,,t)(z+d) at z = a = d D w = D d (z + D) d = + u H

3 and we btain d (u + v )( + D) + + u H D = d (u + v )( + D) + + u H ( + D) = 0 r t +[u(+d)] + [v(+d)] eactl Assuming <<D i.e. linearizing, we have 0 u t -fv = -g v t +fu = -g t + (ud) + (VD) = 0 These are the linear shallw water equatins fr a hmgenus fluid with rtatin. Nw return t the equatins with stratificatin and separate variables u = U(,,t)F(z) v = V(,,t)F(z) p = P(,,t)H(z) w = W(,,t)G(z) We get - ( U t -fv)f = PH ρ (V t +fu)f = PH ρ N WG = ρ P t H z (U +V )F + WG z = 0 Chse W = P t (as we derived in LTE) 3

4 We have H = gρ F G z = F/D U t -fv = -gp Hrizntal structure equatins V t +fv = -gp P t +D n (U +V ) = 0 Cmpare with the hmgeneus laer equatins with D=cnstant. The are the same with P =. The pressure plas the part f the sea/surface elevatin. The vertical structure equatin is again: G zz + N (z) gd n G = G z - D n G = 0 at z= The same identical as fr LTE with h n = D n G= at z = -D The hdrstatic apprimatin we have made assumes w t <<g which is equivalent t assuming ω <<N -> In a flat-bttm cean stratificatin makes pssible an infinite sequence f internal replicas f the bartrpic, lng, shallw water gravit waves. We shall stud the latter first. Frm nw n we shall stud the hmgeneus ne laer prblem as it is equivalent t the hrizntal structure equatins (P=) fr the full stratified case. With D = cnstant u t fv = -g v t + fu = -g t + D(u +v ) = 0 4

5 Frm the vrticit equatin ζ = v -u crss-differentiating the hrizntal mmentum equatins r ζ = f(u + v ) = f D f (ξ- )=0 D Statement f cnservatin f ptential vrticit fr the linear, hmgeneus mdel with f = cnstant q =ζ f D relative vrti vrticit stretching Fr peridic mtins q = iwq = 0 q vanishes Nw we want an equatin fr : take the divergence f hrizntal mmentum equatins (u Frm cntinuit + v ) fζ = g + D (u + v ) = 0 (u + v ) = D fζ=-g D Frm the statement fr PV ζ = q + f D f - f(q+ )+g =0 D D gd f gd =+f g q 5

6 Nte that as q = 0; ptential vrticit is cnserved steadil. S we can separate = stead + wave = s + w. The unstead part f wave carries n ptential vrticit s w w gd f gd w = 0 hmgeneus equatin D D s = q = q particular slutin f f Stead part is in gestrphic balance with (u,v) and reflects the initial distributin f q PV as 0 = ; q = q The wave equatin is: f c c = 0 When c = gd is the phase speed fr lng gravit waves. If there were n rtatin f = 0 we wuld get the nn-dispersive wave equatin (nedimensin) c tt = 0 = F(-c t) + G(+c t); with slutin F and G determined b initial cnditin Taking a slutin f the frm: = Ae i(k+l-ωt) K= k + we btain ω = c (k + l ) + f ω=± c (k + l ) + f These are lng, shallw water gravit waves mdified b rtatin, ften called Pincaré waves. Visualize the particle mtin: 6

7 K, c, c g K, c, c g phases phases N Rtatin Figure - Rtatin Figure b MIT OpenCurseWare. All these waves have ω> f, f is the lwest pssible frequenc: Grup velcit c g ω = = c k k ω = c f k + c K c //K g c g ω = = c = c ω f + c K The hrizntal velcities are btained i) eliminating v frm mmentum eqn. eq. fr u ii) eliminating u eq. fr v 7

8 u + f u = g fg peratr LHS = 0 w = ±f inertial scillatins v + f If we align with v = g gf K then full slutin is u = cs(ft); v = sin(ft) = cs(k-ωt) ω u= cs (k-ωt) Dk f v= sin(k-ωt) Dk Pincaré wave energ cncentrated at lwest pssible frequenc, near f 8

9 MIT OpenCurseWare Wave Mtin in the Ocean and the Atmsphere Spring 008 Fr infrmatin abut citing these materials r ur Terms f Use, visit: