CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL. Yuri N. Skiba and Denis M. Filatov

Transcription

1 Universidad Nacional Autónoma de México Centro de Ciencias de la Atmósfera CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL Yuri N. Skiba and Denis M. Filatov s:

2 BASIC VARIABLES z h( x, y, t) H( x, y, t) ( uv, ) ht ( x, y) x y ( uv, ) - velocity field ht ( x, y) - topography hxy (,, t) = H ht - free surface height H ( xyt-,, ) fluid depth

3 SHALLOW WATER MODEL IN PERIODIC DOMAINS ON A PLANE a) periodic domain in both x and y u u u h + u + v fv = g t x y x v v v h + u + v + fu = g t x y y b) periodic channel in x H Hu Hv + + = t x y Initial conditions: Periodic conditions in x: Lateral boundary conditions : (,,) = (, ), where = {,, } (,, ) = (,, ), where = {,, } (,, ) = (,, ) = R xy R xy R uvh R yt R Lyt R uvh v x t v x M t

4 LOCAL CONSERVATION LAWS FOR SHALLOW-WATER MODEL Since 1 dh H dt r = divu and 1 da Adt = r divu Column volume conservation: d dt ( HA ) = Relative hight conservation: d dt z h { T } = H Potential vorticity conservation: d dt ζ + f r { } =, ζ = (rot u) z H

5 INTEGRAL CONSERVATION LAWS FOR THE SHALLOW WATER MODEL Mass conservation: M t ( ) = Hdxdy = const D Total energy conservation: 2 2 u + v g E() t = H + h h dxdy = 2 2 D 2 2 ( T ) const Potential enstrophy () conservation: v u 1 J t = + f dxdy = 2 x y D 2H 2 const The classic schemes by Arakawa & Lamb (1981), Sadourny (1975), Takano & Wurtele (1982), Kim (1984) as well as recent schemes by Ringler & Randall (22) and Salmon (24), conserve the energy and/or the potential enstrophy only if the model is still continuous in time, while the discretization in time destroys all these laws.

6 CONSERVATION LAW AND TIME DISCRETIZATION r dϕ r Let A be antisymmetric: + A() t ϕ = in (, T) dt r r r Aϕ, ϕ = ϕ r r ϕ () = g r r r ( ) ϕ() t = ϕ() = g, t, T r ϕ Discretization effect: j+ 1 r j ϕ r j+ 1 r j + A αϕ (1 α) ϕ τ + = ( α, α 1) 1 2 r ϕ j+ 1 r j ϕ r ϕ j+ 1 r j ϕ τ ( α = ) 1 j+ 1 1 j + A 2ϕ + 2ϕ = 1 2 r r r ϕ j+ 1 = r j ϕ

7 DIVERGENT FORM OF THE MODEL Using the change of variables (Skiba, 1995) z = H U = zu V = zv the equations of shallow-water model are transformed to a divergent form U 1 uu U 1 vu U h + + u + + v fv = gz t 2 x x 2 y y x V 1 uv V 1 vv V h + + u + + v + fu = gz t 2 x x 2 y y y H zu zv + + = t x y which is essential for constructing conservative difference schemes.

8 BASIC PROPERTY OF THE DIVERGENT FORM The discrete divergent forms I 1 i = i+ 1 i+ 1 i i ar l guarantee the conservation of mass, since a R a R Δl a R a R Δl i+ 1 i+ 1 i i for periodic conditions as well as for conditions at the lateral boundary of channel. Similarly, the discrete divergent forms of advective terms guarantee the conservation of total energy, since ar R a R ar R R a R ar + a + a = l l Δl Δl Δl 1 = ( ar I I ar ) = Δl = a = i+ 1 i+ 1 i i i i 1 i+ 1 i+ 1 i i 1 i ai and I 1 i = ai+ 1Ri+ 1 airi 1 1 R i a I R I R I 1 a R R 1 Δl ( ) = = Δl

9 SHALLOW WATER MODEL ON A SPHERE ( uu ) ( vu cosϕ ) U 1 1 U 1 U + + u + + vcosϕ t acosϕ 2 λ λ 2 ϕ ϕ u gz h f + tan ϕ V =, a acosϕ λ ( uv ) ( vv cosϕ ) V 1 1 V 1 V + + u + + vcosϕ + t acosϕ 2 λ λ 2 ϕ ϕ u gz h + f + tan ϕ U =, a a ϕ ( zu ) ( zv cosϕ ) h = t acosϕ λ ϕ

10 y SPLITTING METHOD Splitting steps 1 (in x) and 2 (in y): U 1 uu U h + + u = gz t 2 x x x V 1 uv V + + u = t 2 x x H t r u Δt x zu + = x Splitting step 3 (sphere rotation): U 1 vu U + + v = t 2 y y V 1 vv V h + + v = gz t 2 y y y H t zv + = y U t fv = V t + fu =

11 TWO MAPS USED IN THE SPLITTING METHOD

12 THE SPLITTING ALONG THE LONGITUDINAL DIRECTION λ

13 PROBLEM 1: THE SPLIT PROBLEM IN λ n+ 1 n Uk Uk 1 1 uk+ 1Uk+ 1 uk 1Uk 1 Uk+ 1 Uk uk = τ a cosϕl 2 2Δλ 2Δλ gzk hk+ 1 hk 1 =, a cosϕ 2Δλ l V V 1 1 u V u V V V + + = n+ 1 n k k k+ 1 k+ 1 k 1 k 1 k+ 1 k 1 uk τ a cosϕl 2 2Δλ 2Δλ n+ 1 n Hk Hk 1 zk+ 1Uk+ 1 zk 1Uk 1 + = τ a cosϕ 2Δλ l 1 ( n+ 1 n ) R = R + R 2

14 THE SPLITTING ALONG THE LATITUDINAL DIRECTION ϕ

15 PROBLEM 2: THE SPLIT PROBLEM IN ϕ n+ 1 n Ul Ul 1 1 vl+ 1Ul+ 1cosϕl+ 1 vl 1Ul 1cosϕl τ a cosϕl 2 2Δϕ U U + vl = 2Δϕ l+ 1 l 1 cosϕl, n+ 1 n Vl Vl 1 1 vl+ 1Vl+ 1cosϕl+ 1 vl 1Vl 1cosϕl τ a cosϕl 2 2Δϕ V V gz h h + vl = 2Δϕ a 2Δϕ l+ 1 l 1 l l+ 1 l 1 cos ϕl, n+ 1 n Hl Hl 1 zl+ 1Vl+ 1cosϕl+ 1 zl 1Vl 1cos ϕl 1 + = τ a cosϕl 2Δϕ 1 ( n+ 1 n R = R + R ) 2

16 PROBLEM 3: A SPHERE ROTATION n 1 n U + kl Ukl ukl fl + tanϕl Vkl = τ a n 1 n V + kl Vkl ukl + fl + tanϕl Ukl = τ a

17 MASS AND TOTAL ENERGY CONSERVATION LAWS The mass and total energy conservation laws M t ( ) Hdxdy const = = 2 2 u + v g E() t = H + h h dxdy = 2 2 D D 2 2 ( T ) const Evidently, it is sufficient to conserve only the variable part of total energy E t H h dxdy 2 2 u + v g 2 1 () = + = const 2 2 D

18 CONSERVATION LAWS FOR DISCRETE MODELS In discrete SW system, the mass and total energy conservation laws accept the forms M M n + 1 n = and E n + 1 n 1 = E1 where M M a λ ϕ cosϕ H n 1 n 2 n l kl l k + = Δ Δ ( ) E = a ΔλΔ ϕ cosϕ U + V + g h n 2 1 n n n 1 l 2 kl kl kl l k

19 EXPERIMENT 1: PLANE. THE HEIGHT h T=1,3,4,6,7,14 days 4th approximation order

20 EXPERIMENT 1: PLANE. VELOCITY T=1,3,4,5,6,7 days 4th approximation order

21 EXPERIMENT 1: PLANE. POTENTIAL VORTICITY T=1,3,4,5,6,7 days 4th approximation order

22 U EXPERIMENT 2: SPHERE. ROSSBY WAVE SPHERE h INITIAL CONDITIONS T=

23 EXPERIMENT 2: SPHERE. VELOCITY 2-order scheme Velocity at t = 3.5 days T=3.5 days 4-order scheme Velocity at t = 3.5 days ϕ ϕ λ λ T=7 days Velocity at t = 7 days Velocity at t = 7 days ϕ ϕ λ λ

24 EXPERIMENT 2: SPHERE. HEIGHT GRID 6º X 6º 4-order scheme GRID 3º X 3º T=3.5 days h T=7 days

25 EXPERIMENT 2: SPHERE. POTENTIAL ENSTROPHY 1-2 order scheme 3-4 order scheme x 14 Potential Enstrophy x 14 Potential Enstrophy J(t) MAXIMUM VARIATION OF POTENTIAL ENSTROPHY FOR DIFFERENT GRIDS in % J(t) J(t) Time (days) x 14 Potential Enstrophy order scheme Time (days) J(t) Time (days) x 14 Potential Enstrophy order scheme Time (days)

26 EXPERIMENT 3: SPHERE. TIME BEHAVIOR OF POTENTIAL ENSTROPHY WITH RANDOM INITIAL PERTURBATIONS J () t J() log1 J () 2nd order scheme 4th order scheme

27 Universidad Nacional Autónoma de México Centro de Ciencias de la Atmósfera CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL Yuri N. Skiba and Denis M. Filatov s: