Rotation & nonlinear effects in shallow water: the Rossby adjustment problem

Transcription

1 Rotation & nonlinear effects in shallow water: the Rossby adjustment problem Ross Tulloch April 29,

2 1 Shallow water equations Consider, as in [6], a homogeneous incompressible 2D fluid in a channel with negligible vertical velocity, roughly constant horizontal velocity for each vertical column and height h(x, t) shallow relative to typical wavelengths. The total mass in [x 1, x 2 ] at time t is total mass in [x 1, x 2 ] = x2 x 1 ρh(x, t)dx, where ρ is constant. The momentum at (x, t) is ρu(x, t), so integrating vertically gives the mass flux ρu(x, t)h(x, t) and therefore conservation of mass gives: t x2 x 1 ρhdx = ρuh(x 1, t) ρuh(x 2, t) h t + (uh) x = 0. (1) Conservation of momentum is the same as in the Euler equations (ρhu) t + (ρhu 2 + p) x = 0 (2) and the pressure can be eliminated using the hydrostatic balance approximation, integrating vertically from z = 0 (flat bottom) to z = h(x, t) gives p(z) = ρgz p = 1 2 ρgh2, (3) so the momentum equation becomes (hu) t + (hu ) ( ) u 2 gh2 = 0 u t gh x x = 0 (4) when equation (1) is used to cancel the h t term. If we now allow for flow v in the ŷ direction (into the plane), but enforce no y dependence we have the following nonlinear shallow water PDE system h t + uh x + hu x = 0 u t + uu x + gh x = 0 v t + uv x = 0. (5) The shallow water equations describe the motion of a barotropic fluid with free surface and depth much smaller than the horizontal length scale. This 2

3 system of equations is a simplified and tractable version of the primitive equations (Navier-Stokes with rotation and stratification) and is a very useful model in geophysical fluid dynamics. One reason the shallow water model is so important is because there is an analogy between it and descriptions using isopycnal or isentropic co-ordinates, see [11] for more details. 1.1 Adding rotation Now if the channel is also rotating, Coriolis forcing adds important source terms to (5) as follows: h t + uh x + hu x = 0 u t + uu x + gh x = where f is twice the angular velocity. fv v t + uv x = fu, (6) 1.2 Linear shallow water: gravity wave equations One always gains insight into a nonlinear problem by first linearizing about a base state and solving the linearized version of the problem. In this case, the velocity and height fields are linearized about (u, v) = (U, 0) and h = H. So decompose the fields into base and perturbed variables u(x, t) = U + u (x, t), v(x, t) = v (x, t), h(x, t) = H + h (x, t), (7) then substitute these into (6) and cancel all quadratic terms h t + (U + u )h x + (H + h )u x = 0 h t + Uh x + Hu x = 0 u t + (U + u )u x + gh x = fv u t + Uu x + gh x = fv v t + (U + u )v x = fu v t + Uv x = fu. (8) Finally, setting U = 0 (no mean wind) and dropping primes gives the equation set for the Rossby adjustment problem described in [2] and [3], called the gravity wave equations : u t fv + g h x = 0, v t + fu = 0, h t + H u x = 0. (9) 3

4 1.3 Project objectives and outline The goal of this project will be to numerically analyze the time dependent evolution of shallow water equations in various regimes given a particular initial condition. Specifically, the effects of rotation and nonlinearity on geostrophic adjustment (described in the next section) are examined as in [5]. The four regimes are: (a) nonrotating-linear, (b) rotating-linear, (c) nonrotating-nonlinear and (d) rotating-nonlinear. Case (a) is just the wave equation (u tt = ghu xx ), which we can write down the solution for immediately using d Alembert s formula. Case (b) is solved using the stencils shown in Problem 12 of Chapter 3 in [2], ie. using leapfrog on an unstaggered mesh as well as using forward-backward time-differencing on a staggered mesh. An attempt is made to reproduce the cover art on Durran s textbook, which shows the emission of gravity waves during the adjustment process. Case (c) is the classic dam break problem which is solved using conservation law methods, as described in [6], [7]. Case (d) is the most difficult because it involves a nonlinear system with source terms (coriolis forces). In [5] the emphasis is on analyzing the transient behaviour and not numerical method, so they used Leveque s CLAWPACK conservation law package (amath.washington.edu/ claw) to solve (d). While CLAWPACK would be a useful tool to learn in its own right, it is not clear to me that it will handle the source terms properly. In the steady-state, the source terms should be exactly balanced by the flux gradients. To achieve this I will experiment with a central-upwind scheme adapted from [4], which examined the shallow water system with bottom topography as the source term (instead of rotation). 2 The Rossby adjustment problem Rossby was the first to completely discuss the question of how a rotating fluid with an initial perturbation adjusts under gravity. The geostrophic equilibrium that is achieved cannot be found by solving the steady-state equations because they are degenerate in that any solution of the momentum equations satisfies the continuity equation exactly [3]. So the equilibrium state depends on the initial state. 4

5 2.1 Initial conditions The problem discussed by Gill involves the linear evolution of an initial step in the height field, with the rotating fluid initially at rest, so at t = 0 u(x, t = 0) = 0, v(x, t = 0) = 0, { hl η h(x, t = 0) = 0, x < 0 h r η 0, x > 0 (10) 2.2 Steady-state analytical solution of the linear problem The analytical solution of (9) is found by first noting that the potential vorticity is conserved, which in this case means Q t = ( vx t f h ) = 0 (11) H and (10) gives Q(t = 0) = (η 0 /H) sgn(x). Manipulating (9) gives a single second order equation h tt + Hfv x ghh xx = 0 (12) which in steady state, after substituting Q(t = 0) and h for v, becomes and the solution is ghh xx + f 2 h = f 2 η 0 sgn(x) (13) { h 1 + e x/a for x > 0 = η 0 1 e x/a for x < 0, v = gη 0 fa e x /a (14) where a = gh/f is called the Rossby radius of deformation. Figure 1, as in [3], illustrates the initial and steady states. 5

6 Figure 1: Initial height field (top) and geostrophic equilibrium height and meridional velocity (below). The horizontal is scaled by the Rossby radius a = gh/f and the vertical is scaled by the magnitude of the initial height perturbation η 0. Parameter values are g = 10m/s 2, H = 10m and f = 10 4 s Transients in the linear problem The homogenous problem for the height field, upon substituting Q as in (11) for v x in (40), is h tt ghh xx + f 2 h = 0 (15) with initial condition h(t = 0) = η 0 sgn(x) h steady = η 0 e x /a sgn(x) = 2η 0 π 0 ksinkx dk, k 2 + a 2 where k is the horizontal wave number assuming wavelike solutions of the form h e i(kx ωt). Preserving antisymmetry in h, the transient height and 6

7 zonal (u) velocity field are h = 2η 0 π 0 u x = 1 h H t u = 2η 0 g π H v t = fu v = 2fη 0 πh ksinkxcosωt dk, k 2 + a sinωtcoskx dk, k2 + a 2 cosωtcoskx dk (16) k 2 + a 2 where the dispersion relation from (15) is given by ω 2 = f 2 + k 2 gh. An equivalent expression for u in terms of Bessel functions is { η 0 ( ) g u = J π H 0 f t 2 x2 for x < ght gh 0 for x >, (17) ght, and is plotted for various times in Figure 2. Notice the emission of gravity waves from the initial discontinuity. In this study, the nature of gravity wave emission will be analyzed by comparing transients in the zonal velocity u and height h for the 4 regimes discussed in Section 1.3. Also, the adjusted (steady state) height h and meridional velocity v profiles will be compared for the four regimes. Note that the final value of u will be uniformly zero when there s not dependence in the meridional (y) direction. The best possible approximation to the initial conditions in (10) for each numerical scheme will be used. That is, standard finite difference approximations to the linear equations linear equations may require smoothing of the discontinuity over a few grid cells because features of order 2 and smaller are poorly resolved [2]. In Section 4.1 smoothed-out step is used as described in [2] and in Section 4.4 the initial height field has an arctangent profile in order to mimic the cover art of [2]. To compare transients no boundary conditions need be specified, a large periodic domain is sufficient. Nonreflecting (also known as open, radiating, or outflow ) boundary conditions are applied in order to run to steady state. 7

8 Figure 2: Transient profiles of u scaled by 1/η 0 for the initial conditions in equation (10). Parameters are g = 10m/s 2, H = 10m, f = 10 4 s 1. 3 Case (a): Solution of the nonrotating-linear problem In the nonrotating-nonlinear problem there is no meridional velocity (v) and equations (9) reduce to and decouple as u t + g h x = 0, 2 u t gh 2 u 2 x = 0, 2 h t + H u x = 0 (18) 2 h t gh 2 h 2 x = 0 (19) 2 8

9 With initial conditions in (10), the solution is which is plotted in Figure 3. h(x, t) = h(x ght, 0) + h(x + ght, 0), 2 u(x, t) = 1 x+ ght 2 h(s, 0)ds gh s x ght = h(x ght) h(x + ght) 2, (20) gh Figure 3: Transient profiles of h (left) and u (right) for nonrotating-linear adjustment, both scaled by 1/η 0 and plotted against x/a. Parameters are g = 10m/s 2, H = 10m, f = 10 4 s 1. 9

10 4 Case (b): Numerical Solution of the rotatinglinear problem 4.1 Problem 12, Chapter 3: Durran s textbook The inspiration for this project comes from an exercise in [2] which asks to compare solutions of the linearized one-dimensional Rossby adjustment problem obtained using leapfrog on an unstaggered grid versus forward-backward time differencing on a staggered grid. The problem is posed with typical parameter values (f = 10 4 s 1, g = 10m/s 2, H = 10m) and initial conditions (step function as in Figure 1). Note that the initial conditions are smoothed out with three iterative applications of the filter φ f j = 1 4 (φ j+1 + 2φ j + φ j 1 ). (21) Two common schemes are used to solve this problem, the first is leapfrog on an unstaggered mesh δ 2t u fv + gδ 2x h = 0, δ 2t v + fu = 0, δ 2t h + Hδ 2x u = 0. (22) and the second is forward-backward time-differencing on a staggered mesh δ t u n+1/2 j fv n j + gδ x h n j = 0, δ t v n+1/2 j + fu n+1 j = 0, δ t h n+1/2 j+1/2 + Hδ xu n+1 j+1/2 = 0. (23) The results of the two schemes are shown in Figure 4, forward-backward differencing appears to do better than leapfrog for both large and small Courant numbers (see Section 4.3 for explanation of the relevance of Courant numbers for these schemes). The plots show nondimensionalized height and both velocity fields at t = 2 in nondimensional time as question 12 in [2] asks for. See Section 4.5 for transient and steady state plots. Note that both schemes exhibit dispersion due to the sharp gradient ( 3) in the initial condition. 10

11 (a) Leapfrog, CFL=0.9 (b) Leapfrog, CFL=0.1 (c) FBS, CFL=0.9 (d) FBS, CFL=0.1 Figure 4: Plots of h, u, and v fields for leapfrog (top) and forward-backward differencing (bottom) with Courant numbers 0.9 (left) and 0.1 (right) at time t = 21000s ft = 2 on domain x = 600km to x = 1400km. Parameters are g = 10m/s 2, H = 10m, f = 10 4 s 1, = 10 3 km. 4.2 Consistency Consistency is quite easy to show for the above schemes but stability is somewhat more difficult. The unstaggered leapfrog is second order accurate 11

12 and therefore pointwise consistent u n+1 j u n 1 j 2 t h n+1 j fu n j + g hn j+1 h n j 1 2 v n+1 j v n 1 j 2 t h n 1 j 2 t + H un j+1 u n j 1 2 = t2 6 + fu n j = t2 6 = t2 6 3 u t + g v t h t + H h x u x 3 +. (24) The forward-backward staggered time differencing is first order accurate in time and second order accurate in space u n+1 j u n j t fu n j + g hn j+1/2 hn j 1/2 v n+1 j vj n t h n+1 j+1/2 hn j+1/2 t 4.3 Stability = t 2 u 2 2 v + fu n j = t 2 + H un+1 j+1 un+1 j = t 2 t + g t h t + H h x u x 3 +. (25) Since this is a system of equations stability is ensured (sufficient) when the norm of the amplification matrix ( v n k = An k v0 n) is A k 1. (26) A necessary condition for stability is that the spectral radius is less or equal to unity, that is ρ(a k ) 1 (which is sufficient in the case that A is diagonalizable. My attempt to show stability in this way was somewhat unsuccessful. Defining φ, ψ, and χ as past values of u, v, and h respectively gives the amplification matrix for the leapfrog scheme u v h φ ψ χ n+1 j = i2g t sin k 0 2 tf tf i2h t sin k u v h φ ψ χ n j (27) 12

13 To find the norm A k, take the square root of the largest eigenvalue of B = A k A k. After a lengthy hand calculation that Maple appeared unwilling to perform, the characteristic equation of B appears to be (1 λ) 2 [ (1 λ) 2 λ(g 2 α 2 + F 2 ) ] [ (1 λ) 2 λ(h 2 α 2 + F 2 ) ] = 0 λ 2 (2 + g 2 α 2 + F 2 )λ + 1 = 0, λ 2 (2 + H 2 α 2 + F 2 )λ + 1 = 0 (28) where α = (2 t/) sin k and F = 2 tf. Apparently ρ(b) > 1 so this condition is not satisfied, but that does not mean the scheme is necessarily unstable. Progress can be made by taking a dispersion relation approach to derive a necessary condition for stability. Seek wave like solutions of the form u n j = u 0 e i(kj ωn t), v n j = v 0 e i(kj ωn t), h n j = h 0 e i(kj ωn t). (29) In matrix form the unstaggered leapfrog equations become sin ω t g sin k if u t 0 sin ω t if 0 v t 0 = 0 (30) H sin k sin ω t 0 h t 0 which have nontrivial solutions when the determinant is zero, so the dispersion relation is [ ( ) ] 2 1/2 sin ω t sin k = ± f 2 + gh (31) t and ω = 0 is another solution. So a necessary condition for stability in the unstaggered leapfrog scheme is t f 2 + gh 1. (32) 2 Similar analysis on the forward-backward staggered scheme gives the dispersion relation [ ( ) ] 2 1/2 sin ω t sin k/2 t/2 = ± f 2 + gh (33) /2 13

14 so a necessary condition for stability is then f t gh 1. (34) 2 Given the parameters used by Durran in [2] (gh = 100, f = 10 4, = 10 4 /3) it is easy to see that rotation has almost no effect on this condition, which is approximately the same as the condition from the one way wave equation, that is require gh t/ Durran s cover art A side objective of this project is to reproduce, as exactly as possible, the cover art on [2] which shows the emission of gravity waves in the transient profile of the zonal velocity u, see Figure 5. After some tuning, I obtained a Figure 5: Cover art of Dale Durran s textbook: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics [2] showing the time evolution of u(x, t) at t = 943s, t = 1222s, and t = 1501s with initial condition h(x, t = 0) =arctan(x/20km). Parameters are g = 10m/s 2, H = 10m, f = 10 4 s 1, = 10 3 km. very similar plot (Figure 6) using forward-backward time-differencing. The preface of [2] specifies all the parameters used to obtain the plots in Figure 5 14

15 Figure 6: Transient profiles of u for rotating-linear adjustment at times t = 1062 t, t = 1375 t, and t = 1689 t where t = 200s. Parameters are g = 10m/s 2, H = 10m, f = 10 4 s 1, = 3km. except for the time step t. Apparently the times quoted in [2] to produce Figure 5 are actually time step numbers n (unless the parameters of the problem are different), in which case the time step was t 224s. 4.5 Transient and steady-state solutions Adding rotation to the linear equations causes gravity waves to disperse outwards as in Figure 7 and allows for a nonzero steady state to develop, which is shown in Figure 9. The agreement between the analytical solution described in Section 2.3 and the numerical solution is shown in Figure 8. The numerical scheme does a poor job capturing the front as expected because it is a small scale O() feature. Also notice that some dispersion is near the front. Otherwise the numerical scheme matches the analytical prediction quite well. To evolve the system to steady state, outflow boundary 15

16 Figure 7: Transient profiles, using forward-backward time-differencing, of h (left) and u (right) for rotating-linear adjustment, both scaled by 1/η 0 and plotted against x/a. Parameters are g = 10m/s 2, H = 10m, f = 10 4 s 1, = 3km, t = 200s. conditions were applied to the forward-backward scheme u n+1 N = un N t gh un N un N 1 u n+1 1 = u n 1 + t gh un 2 u n 1 h n+1 N 1/2 = hn N 1/2 t gh hn N 1/2 hn N 3/2 h n+1 1/2 = hn 1/2 + t gh hn 3/2 hn 1/2. (35) 16

17 Figure 8: Numerical (solid) versus theoretical transient (dashed) profiles of u at t = 12/f for rotating-linear adjustment. The initial height perturbation was smoothed with 10 iterations of equation (21). The steady height profile is identical to the predicted height profile in Figure 1 while the velocity v has a slightly rounded vertex at x = 0 because the initial conditions were smoothed. 5 Case (c): Solution of the nonrotating-nonlinear problem 5.1 Dam break problem The nonrotating-nonlinear shallow water equations (5) with initial conditions u(x, t = 0) = 0, v(x, t = 0) = 0, { hl η h(x, t = 0) = 0 + H, x < 0 h r η 0 + H, x > 0 (36) 17

18 Figure 9: Initial height field (top) and geostrophic equilibrium height and meridional velocity (below) for the rotating-linear equations. is known as the dam break problem, which is solved semi-analytically in [9] using conservation law theory. In this configuration, a 1-rarefaction wave and a 2-shock form, as illustrated in Figure 10. One can solve for the intermediate state (numerically) by connecting the rightward Riemann invariant with the intermediate state (u, h ) ( u u l = 2 ghl ) gh (37) and using the Rankine-Hugoniot jump conditions on the shock ( ) g h + h r u u r = (h h r ). (38) 2 h h r 18

19 One can solve for the rarefaction state (u(x, t ), h(x, t ) by matching the rightward Riemann invariant with the rarefaction state u l + 2 gh l = u(x, t ) + 2 gh(x, t ) (39) and noting that the leftwards characteristic is valid in the rarefaction giving the solution u(x, t ) gh(x, t ) = x t, (40) ( ) ghl + x u(x, t ) = 2 3 h(x, t ) = 1 9g The total solution is h l, ( 1 h(x, t) = 9g 2 ghl x t h, h r, u(x, t) = 2 3 t (2 ) 2 gh l x. (41) t x < gh l t ) 2, ghl t < x < ( u ) gh t ( u ) gh t < x < ct x > ct u l, x < ( ) gh l t ghl + x t, ghl t < x < ( u ) ( gh t u, u ) gh t < x < ct u r, x > ct (42) (43) where c = (h r u r h u )/(u r u ) is the speed of the shock. With initial conditions (36), u l = u r = 0 so c = h. The solution for h l = 11 and h r = 9 corresponding to η 0 = 1 and H = 10 is plotted in Figure 10. The intermediate state is (u, h ) = ( , ), obtained from equations (37) and (38). 5.2 Finite volume methods Typically, as in [5] and [7], the dam break problem is solved numerically using finite volume methods. Specifically, equation (5) is integrated using Godunov s method with Roe linearization to solve approximate Riemann 19

20 Figure 10: Transient profiles of h (left) and u (right) for nonrotatingnonlinear adjustment, with η 0 = 1 and H = 10. Other parameters are g = 10m/s 2, f = 10 4 s 1. problems. Then to improve accuracy flux (or slope ) limiters are added to the flux term based on local slopes and an entropy fix is applied because the linear equations cannot capture rarefaction waves. The idea behind Godunov s method is to solve piecewise constant local Riemann problems exactly. The exact solution q n (x, t) can be determined by piecing together the solutions of the Riemann problems at each cell interface if the time step t is sufficiently small. The new solution is then defined as the average of the exact solutions over each cell Q n+1 j = 1 xj+1/2 x j 1/2 q n (x, t n+1 )dx (44) which does not need to be explicitly integrated since the conservation law 20

21 Q t + F (Q) x = 0 implies where Q n+1 j = Q n j t (F n j+1/2 F n j 1/2) (45) F n j 1/2 = f(q (Q n j 1, Q n j ) (46) and q (q l, q r ) is the solution to the Riemann problem between states q l and q r. Solving the Riemann problems exactly at each cell is numerically expensive so typically the nonlinear problem q t + f(q) x = 0 is replaced by a locally defined linear problem ˆq + Âj 1/2ˆq x = 0 where  is diagonalizable with real eigenvalues and is consistent with the original conservation law:  j 1/2 f ( q)asq j 1, Q j q. For  to be Roe approximate then it must also satisfy Âj 1/2(Q j Q j 1 ) = f(q j ) = f(q j 1 ). The details of the Roe solver, entropy fix, and flux limiters become quite technical. Since we already have the solution to the dam break problem in Figure 10 let us proceed to the nonlinear-rotating problem and take a look at central upwind schemes. 6 Case (d): Solution of the rotating-nonlinear problem 6.1 Central upwind schemes The idea to solve the rotating-nonlinear problem using a central upwind scheme comes from [4], which analyzed the nonrotating-nonlinear problem with bottom topography h t + (hu) x = 0 (hu) t + (hu ) gh2 = ghb x, (47) where B(x) represents the bottom elevation (topography). The right hand side of (47) acts as a source term analogously to the coriolis terms in (6). taken for such terms to ensure that they exactly the balance the appropriate flux gradients to ensure the correct steady state develops. The authors of [4] claim that standard numerical schemes typically fail to preserve the balance between the fluxes and the source terms. 21 x

22 Central upwind schemes are supposed to be simple, efficient and robust. They are based on Godunov s method in that they are based on an exact evolution of a approximate piecewise polynomial reconstruction and do not require any Riemann solvers. However the major drawback to of central schemes is the large numerical dissipation (O( t) 2r 1 ) where r is a formal order of accuracy [4]. Apparently this dissipation also affects long time simulations such as steady-state computations (as Figure 12 attests). The central upwind scheme for a one dimensional problem given in [5] is as follows. Evolve the cell averaged conservation law with source term S(u(x, t), x, t): d dtūj(t) + f ( u(xj+1/2, t) ) f ( u(x j 1/2, t) ) and the cell average ū is defined as ū j (t) = 1 xj+1/2 Substituting numerical fluxes in (48) gives = 1 xj+1/2 Sdx (48) x j 1/2 x j 1/2 u(x, t)dx. (49) d dtūj(t) = H j+1/2(t) H j 1/2 (t) + S j (t), (50) where the fluxes H j+1/2 are given by H j+1/2 (t) = a+ j+1/2 f(u j+1/2 ) a j+1/2 f(u+ j+1/2 ) a + j+1/2 + a+ j+1/2 a [ ] j+1/2 a j+1/2 a + j+1/2 u + a j+1/2 u j+1/2. j+1/2 Note that u + j+1/2 = p j+1(x j+1/2, t) is the right value of a (conservative, nonoscillatory) piecewise polynomial interpolant at x = x j+1/2 and u j+1/2 = p j+1 (x j+1/2, t) is the corresponding left value at x = x j+1/2. The details of non-oscillatory piecewise quadratic reconstructions are given in the appendix of [4]. Also, the local speeds of propagation a ± j+1/2 eigenvalues of the system { ( ( f f a + j+1/2 = max λ N, λ N ) u (u j+1/2 ) ) a j+1/2 = min { λ 1 ( f u (u j+1/2 ) 22 are determined by the ) u (u+ j+1/2 ) ), λ 1 ( f u (u+ j+1/2 ) }, 0, }, 0, (51)

23 given λ 1 < < λ N. In the Rossby adjustment problem (6) the system eigenvalues are u and u ± gh, obtained from the Jacobian in conservation form p t + f( p) x = 0 (ignoring source terms) p = h hu hv = h r s J = f i p j = f = gh u 2 2u 0 uv v u r r 2 /h + gh 2 /2 rs/h (52). (53) In steady state of the Rossby adjustment problem must support geostrophic balance, which in the one dimensional system (6) with no y-dependence is u = 0, v = g f h x. (54) The conservation form of (6) is h t + (uh) x = 0 (hu) t + (hu ) gh2 = fvh x (hv) t + (huv) x = fuh, (55) so in the numerical discretization we want to balance g 2 (h2 ) x g h 2 j+1/2 h2 j 1/2 2 fv j h j+1/2 + h j 1/2 2 fvh (56) let v j = g f 6.2 Transient and steady state solutions h j+1/2 h j 1/2. (57) Nonlinearity has a few obvious effects on geostrophic adjustment as illustrated in the transient profiles of h and u in Figure 11. First note that the right-left symmetry of the rotating-linear problem in Figure 7 is broken. One would expect this considering the rarefaction wave that developed in 23

24 Figure 11: Transient profiles of h (left) and u (right) for rotating-nonlinear adjustment, with η 0 = 1 and H = 10. Other parameters are g = 10m/s 2, f = 10 4 s 1. the break problem (Figure 10). Notice also that the initial leftward rarefaction wave is overtaken by dispersive wave, which should form a shock but apparently my code is quite dissipative. Numerical dissipation is even more evident in Figure 12 which shows the steady profiles of h and v (after a long simulation time). Part of the dissipation could be because I had trouble with quadratic polynomial interpolants and am currently using only linear interpolants. For more complicated theoretical considerations of the Rossby adjustment problem see [1], [10], and [12]. Also see [5] for a proper numerical study and [8] for a recent experimental study which claims to have quantitatively measured potential vorticity and the flow balance after a geostrophic adjustment for 24

25 Figure 12: Initial height field (top) and geostrophic equilibrium height and meridional velocity (below) for the rotating-nonlinear equations. the first time by taking simultaneous measurements the horizontal velocity field and the vertical density field. 25

26 References [1] Bouchet, F., Submitted 2003: Frontal geostrophic adjustment and nonlinear wave phenomena in one dimensional rotating shallow water. Part 2: high-resolution numerical solutions. Journal of Fluid Mechanics, Cambridge. [2] Durran, D., 1998: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Spring-Verglag, New York. [3] Gill, A., 1982: Atmosphere-Ocean Dynamics. International geophysics series, Academic Press, San Diego. [4] Kurganov, A., D. Levy, 2002: Central-Upwind Schemes for the Saint- Venant System. Mathematical Modelling and Numerical Analysis, 36, [5] Kuo, A., L. Polvani: Time-Dependent Fully Nonlinear Geostrophic Adjustment. Journal of Physical Oceanography, 27, [6] LeVeque, R., 1990: Numerical Methods for Conservation Laws. Birkhäuser, Boston. [7] LeVeque, R., 2002: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, New York. [8] Stegner, A., P. Bouruet-Aubertot, T. Pichon, 2004: Nonlinear adjustment of density fronts. Part 1. The Rossby scenario and the experimental reality. Journal of Fluid Mechanics, 502, [9] Stoker, J., 1958: Water Waves. John Wiley, New York. [10] Reznik, G., V. Zeitlin, M. Ben Jelloul, 2001: Nonlinear theory of geostrophic adjustment. Part I : Rotating shallow water model. Journal of Fluid Mechanics, 481, [11] Vallis, G., 2003: Atmospheric and Ocean Fluid Dynamics. Online text: gkv/aofd. [12] Zeitlin, V., S. Medvedev, R. Plougonven, 2003: Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in 1d rotating shallow water. Part 1. Theory. Journal of Fluid Mechanics, 445,