# Presentation: 183771

## A Fully-Implicit Semi-Lagrangian Hydrostatic Model of Atmospheric Dynamics

- Sajal K. Kar1,2
- 1-I.M. Systems Group, Inc. (IMSG), 2-National Centers for Environmental Prediction/Environmental Modeling Center (NCEP/EMC)

- 21st Conference on Weather Analysis and Forecasting/17th Conference on Numerical Weather Prediction American Meteorological Society, Washington DC, 1-5 August 2005

**FISL Hydrostatic 3D Model**
-    Primitive equations
-    Terrain-following normalized pressure vertical coordinate
-    Horizontal coordinates: conformal map projection
-    Horizontal A grid; vertical Lorenz grid
-    Semi-implicit linearization wrt isothermal atmosphere
-    Horizontal advection: semi-Lagrangian
-    ‘Nonlinear’ terms left-over from semi-implicit linearization
-     and vertical advection, treated in a fully-implicit manner
-    3D elliptic equation for a generalized geopotential
-     vertically decoupled into a set of 2D elliptic equatios
-     solved iteratively using a generalized conjugate residual algorithm
-    Bicubic horizontal interpolation
-    Details of model formulation in AMS extended abstract P1.84

**Linear Hydrostatic Mountain Wave Simulation**
-  Witch of Agnesi: height, *h* = 1 m and  half-width, *a* = 10 km. Isothermal atmosphere:* T*0 = 250 K => *N* = 0.019 s-1.  No rotation; uniform zonal wind, *u*0 = 20 ms-1. *Na*/*u*0 = 9.5 >> 1 (Hydrostatic); *Nh/u*0 = .00095 << 1 (Linear).
- Vertical (*x*, η) slice of 3D FISL hydrostatic model. Model top at 0.01 hPa with 90 levels uniformly spaced in η. Rayleigh damping type sponge above 13 km. Bounded domain in *x* (and *y*) with *x* (= *y*) = 2 km.* nx* = 100, ny = 5. Rayleigh damping type sponge (25*x*) on two sides.
-  FISL scheme with 3 iterations and no uncentering. *t* = 30 s. *C*g = 314.47 m s-1  gravity-wave Courant number, *Cg***t*/*x* = 4.717. Grid-scale selective horizontal filter every 40 time steps. No gradual build up of zonal flow or mountain. 16 hour time integration. Analytic (Alaka 1960) vertical wavelength, 2π*u*0/*N* = 6.4 km.

**Nonlinear Hydrostatic Mountain Wave Simulation A**
- Parameters different from the linear wave run: Witch of Agnesi height, *h* = 632 m.* Nh/u*0 = 0 .6  (Nonlinear but below critical value 0.85) ∆*t* = 60 s.  gravity-wave Courant number, *C**g**∆*t*/∆*x* = 9.4. Grid-scale selective horizontal filter applied every 5  time steps. Theoretical height at which isentropes would first steepen, *z* = 3*u*0/(2*N*) = 4.83 km (~ 517 hPa).

**Nonlinear Hydrostatic Mountain Wave Simulation B**
- Parameters different from Simulation A: Witch of Agnesi height, *h* = 910 m.* Nh/u*0 = 0 .86  (Nonlinear and above critical value 0.85) Grid-scale selective horizontal filter applied every time steps. 8 hour time integration. Steepening of isentropes followed by convective overturning. Development of strong surface wind on lee side.

**Conclusions:** A fully-implicit semi-Lagrangian 3D hydrostatic model has been developed. Preliminary results indicate that the model is able to simulate 2D hydrostatic mountain waves, linear and nonlinear, over a bell-shaped mountain with reasonably large gravity-wave Courant numbers. The model has also been employed to simulate the nonlinear 3D evolution of mid-latitude disturbances on a β-plane channel. Some of those results are presented in the AMS extended abstract.

**FISL scheme**
-      Consider a generic prognostic equation for Ψ:
- 		(1)
- Then, based on a two time level, semi-Lagrangian backward trajectory:
- a fully-implicit, semi-Lagrangian (FISL) scheme for (1) can be written as
- 		(2)
- where the subscript *i* denotes an iterative index.