Exact Solution of the Linear, 4D-Discrete, Implicit, Semi-Lagrangian Dynamic Equations with Orographic Forcing

Transcription

1 1 Exact Solution of the Linear, 4D-Discrete, Implicit, Semi-Lagrangian Dynamic Equations with Orographic Forcing Marko Zirk, Rein Rõõm Tartu University, Estonia

2 1 1 Introduction Exact solution of completely discrete linear equations has multiple outputs: Discovery of the exact structure of the solution of a non-stationary numerical model Investigation of specific details of flows for optional (non-homogenous) wind and temperature stratification Investigation of the influence of vertical discretization (variable z) on the quality of numerical models Investigation of the time step t size impact on the numerical stability and precision of numerical schemes Testing of adiabatic cores of NWP models

3 2 2 Domain of integration Geometry is plane: p s (x, y) p = const. Orography (taken into account in boundary conditions but not in geometry) h is presented by 2D Agnesi profile h(x, y) = 1 + (x/a (h height, a x, a y -half-widths): x ) 2 + (y/a y ) 2. The reference state of the atmosphere is given by wind-speed U(p) and temperature T(p). The Lagrangian time-derivative is approximated as d dt = t + U(p) x.

4 3 3 Continuous linear equations Linear form of White equations (White 1989, Rm 21). Notation is standard, except of ϕ which is the nonhydrostatic geopotential fluctuation. dω dt = R p H 2 T p2 H 2 ϕ p, du dt = ϕ f u, dt dt = θω p, (1) u + ω p =, dp s dt + da dt = ω p. Reference state parameters: H(p) = RT(p), p θ(p) = R c p T(p) p T p. Vertical boundary conditions ϕ p = gh(p ) p s p. ω p= =.

5 4 4 Vertical grid and difference operators Vertical grid is the Arakawa C-grid. Fields ϕ k, u k, pressure difference p k and non-dimensional height difference ζ k are located on full levels. Pressure p k+1/2, T k+1/2, H k+1/2, θ k+1/2, ω k+1/2 and T k+1/2 belong to the half-levels. Vertical discretization is originated by p k+1/2. Non-dimensional height ζ k+1/2 : p k+1/2 = p klev+1/2 e ζ k+1/2, ζ k+1/2 = ln(p klev+1/2 /p k+1/2 ). Full-level height and pressure: height differences ζ k = ζ k 1/2 + ζ k+1/2 2, p k = p klev+1/2 e ζ k. ζ k = ζ k 1/2 ζ k+1/2, ζ k+1/2 = ζ k ζ k+1.

6 5 The vertical difference derivatives are ( ) ψ = ψ ( ) k ψ, p p k ζ k p k k+1/2 = ψ k+1/2 p k+1/2 ζ k+1/2. 5 Horizontal grid and difference operators ψ n ijk = ψ(x i, y j, p k, t n ), ψ n 1 i l k,jk = ψ(x i U k t, y j, p k, t n 1 ), l k = U k t x, non-dimensional displacement along x-axes during t. δ t ψijk n = ψn ij,k ψn 1 i l k,j,k t δ i ψ = 1 x δ j ψ = 1 y, ψ n ijk = ψn ijk + ψn 1 i l k,jk 2 ( ) ψi+1/2 ψ i 1/2, δi+1/2 ψ = 1 x (ψ i+1 ψ i ), ( ) ψj+1/2 ψ j 1/2, δj+1/2 ψ = 1 y (ψ j+1 ψ j )

7 6 ψ x i = 1 2 ψ y j = 1 2 ( ψi 1/2 + ψ i+1/2 ), ψ x i+1/2 = 1 2 (ψ i + ψ i+1 ), ( ψj 1/2 + ψ j+1/2 ), ψ y j+1/2 = 1 2 (ψ j + ψ j+1 ) 6 Semi-Implicit, Semi-Lagrangian Equations Equations (1) become in SISL presentation ( p δ t ωijk+1/2 n = R T H )k+1/2 2 n ijk+1/2 ( p H 2 ζ ) k+1/2 ϕ n ijk+1/2, δ t u n i+1/2,jk = (δ xϕ) n i+1/2,jk + fvxyn i+1/2,jk, δ t vij+1/2,k n = (δ yϕ) n ij+1/2,k fuxyn ij+1/2,k, ( ) δ t T n θ ijk+1/2 = ω n ijk+1/2 p, (2) k+1/2 ( u) n ijk + ωn ijk (p ζ) k =,

8 7 δ t (p s) n ij,klev+1/2 + δ t(a) ij,klev+1/2 = ω n ijklev+1/2. 7 Embedding into 3D Fourier space In following, the discrete Fourier transformation is employed in the form f i = 2N 1 q= ˆf q e iη qi, kus ˆf q = 1 2N 2N 1 i= f q e iη qi ja η q = π q N. Orography in the 2D Fourier basis is a ij a(x i, y j ) = qr â qr e i(ηx q i+ηy r j), Using short notation A general presentation holds ψ n ijk = { } T ijk+1/2 n, ωn ijk+1/2, un i+1/2jk, vn ij+1/2k, ϕn ijk, (p s) n ij ψ n ijk = qrs ˆψ s qrke i(ηx q i+ηy r j ds qrk n), (3)

9 8 where d s qrk are the non-dimensional frequencies. 8 Normal mode equations: (3) (2) : iν s qrk+1/2ˆωs qr,k+1/2 = R ( p H 2 )k+1/2 ( ) p ˆT qrk+1/2 s H 2 ζ k+1/2 ˆϕ s qrk+1/2, iν s qrkû s qrk = iµ x q ˆϕ s qrk +F qrˆv s qrk, iν s qrkˆv s qrk = iµ y r ˆϕ s qrk F qrû s qrk, (4) iν s qrk+1/2 ˆT s qrk+1/2 = θ k+1/2 p k+1/2 ˆω s qrk+1/2, i[(µx q) û s qrk+(µ y r) ˆv s qrk]+ ˆωs qrk (p ζ) k =. F qr = fa x q(a y r), a x q = 1 + eiηx q 2, a y r = 1 + eiηy r 2 ν s qrk = 2 t tan[1 2 (ηx q l k d s qrk)]., µ x q = eiηx q 1 i x, µy r = eiη y r 1. i y Stationary case s = : d qrk = ν s qrk = ν qk = 2 t tan(1 2 ηx q l k ). Non-stationary case s : d s qrk = η x q l k 2 d s qr ν s qrk = ν s qr = 2 t tan(ds qr).

10 9 9 Wave equation for vertical velocity ˆω s qrk+1/2 From normal mode equations, the wave equation follows for ˆω s qrk+1/2 (Lˆω s ) qrk+1/2 + λ s qrk+1/2ˆωs qrk+1/2 =, (5) where λ s qrk+1/2 = H2 k+1/2 µ2 qr (ν s qrk+1/2 )2 N 2 k+1/2 F qr 2 (ν s qrk+1/2 )2. (Lˆω s ) qrk+1/2 = L + qrk+1/2 ˆωs qrk+1 L qrk+1/2 ˆωs qrk, L + qrk+1/2 = νs qrk+1 ν s qrk+1/2 e ζ k+1/2 ζ k+1/2 ζ k+1, L qrk+1/2 = νs qrk ν s qrk+1/2 e ζ k/2 ζ k+1/2 ζ k, ν s qrk = (νs qrk )2 F qr 2 ν s qrk.

11 1 1 Stationary solution 1.1 Main recurrence formula Solutions of Eq. (5) are sought in the form k χ k+1/2 = χ 1/2 c j, χ 1/2 = 1. (8) j= From (5), a nonlinear, two-point recurrence follows for growth factors c k L + k+1/2 (c k+1 1) + L k+1/2 (1/c k 1) + λ k+1/2 =. (9) The initial value c(), required to start this simple straightforward formula, can be established, assuming the homogeneous layer in the top in the free buoyancy-wave radiating state. The solution then presents ω qrk+1/2 = Cχ k+1/2, where C is specified from boundary condition on the surface.

12 Special case of homogeneous stratification Homogeneous atmosphere : ζ k = ζ k+1/2 = ζ, Homogeneous atmosphere : λ qrk = λ qr = H 2 µ 2 qr Consequently c k = c, χ k+1/2 = χ 1/2 c k, L ± k+1/2 = e ζ/2 ( ζ) 2, (ν qr) 2 N 2 F qr 2 (ν qr) 2. where c satisfies equation e ζ/2 (c 1) + e ζ/2 (c 1 1) + ( ζ) 2 λ =. The solution of this equation is: ( trapped wave, Q 1 : c = e ζ/2 Q + ) Q 2 1, where free wave, Q < 1 : c = e ( ζ/2 i ξ), Q Q qr = cosh( ζ/2) 1 2 ( ζ)2 λ qr. 1 Q 2 ξ = Arctan, Q

13 12 11 Non-stationary free waves Assuming ν s qrk+1/2 = νs qr, the Eq. (5) simplifies, as the coefficients L ± become independent of the frequency (and hence, of horizontal wave-numbers q, r): L + k+1/2 = p k+1/2/p k+1 ζ k+1/2 ζ k+1 = e ζk+1/2 ζ k+1/2 ζ k+1, L k+1/2 = p k+1/2/p k ζ k+1/2 ζ k = e ζ k/2 ζ k+1/2 ζ k. Quest quantities are the frequencies ν s qr in λ and the corresponding eigenfunctions. The solution presents as a combination of functions (8), found from recurrence (9). The wanted frequencies are selected by the homogeneous bottom boundary condition.

14 13 12 Examples: homogeneous stratification 3 V z : u = 5 m/s, T = 265 K, h =.2 km, a x = 2 km, zlev = 3, xlev = 124, z =.1 km, x =.4 km X, km V z : u = 1 m/s, T = 265 K, h =.2 km, a x = 2 km, zlev = 3, xlev = 124, z =.1 km, x =.4 km X, km

15 14 13 Examples: BW refraction on tropopause 3 3 Vz: D(Vz)=.1m/s,U= 5m/s,gam=4.7K/km,h=1m,a= 3km,dx=.55km,Mlev=3,dz=1m T U γ=4.7 K/km T, K 5 1 U, m/s X, km 3 3 Vz: D(Vz)=.1m/s,U= 5m/s,gam=9.5K/km,h=1m,a= 3km,dx=.5km,Mlev=3,dz=1m T U γ=9.5 K/km T, K U, m/s X, km

16 Vz: D(Vz)=.1m/s,U= 5-2m/s,gam=9.5K/km,h=1m,a= 3km,dx=.5km,Mlev=3,dz=1m T U γ=9.5 K/km T, K U, m/s X, km 3 3 Vz: D(Vz)=.1m/s,U= 1-5m/s,gam=9.5K/km,h=1m,a= 3km,dx=.5km,Mlev=3,dz=1m T U γ=8.8 K/km T, K U, m/s X, km

17 16 14 Examples: Hyperbolic reference wind profile Reference temperature and wind ω: u = m/s, γ =.65 K/m, a x = 2 km, h =.2 km, xlev = 496, zlev = 3, x =.4 km, z = 1 m 2 T U γ=6.5k/km p, hpa p,hpa T, K 2 U, m/s x, km ω: u = m/s, γ =.65 K/m, a x = 2 km, h =.2 km, xlev = 496, zlev = 3, x =.4 km, z = 1 m 2 p, hpa x, km

18 17 15 Example: Testing NH HIRLAM HIRLAM, V z : D(V z )=.5m/s,a x =3km, h=1m, MLEV=1,dx=.55km,dt=3s,6 steps Reference temperature and wind T γ=8. K/km U p, hpa γ=4.5 K/km 8 1 p,hpa T, K 2 U, m/s X, km V z : D(V z )=.5m/s; U,T-HIRLAM,h=1m,a x = 3km,dx=.55km,MLEV=2,dz=1m Reference temperature and wind T γ=8. K/km U p, hpa γ=4.5 K/km 8 1 p,hpa T, K 2 U, m/s X, km

19 18 16 References White, A.A.,1989: An extended version of nonhydrostatic, pressure coordinate model. QJRMS115, Robert, A., T. L. Yee, H. Richie, 1985: A semi-lagrangian and semiimplicit integration scheme for multi-level atmospheric models. Mon. Weather Rev., 113, Rõõm, R., 21: Nonhydrostatic adiabatic kernel for HIRLAM. Part I: Fundametals of nonhydrostatic dynamics in pressure-related coordinates. HIRLAM Technical Report, 48, 26p.

20 19 Vz: D(Vz)=.1m/s,U,T-HIRLAM,h=1m,a= 3km,dx=.55km,Mlev=22,dz=1m X, km

21 2 Linear, Vz: D(Vz)=.1m/s,U,T-HIRLAM,h=1m,a= 3km,dx=.55km,Mlev=22,dz=1m X, km

22 21 HIRLAM, V z : D(V z )=.1m/s,a x =3km, h=1m, MLEV=1,dx=.55km,dt=3s,6 steps X, km