ATM 298, Spring 2013 Lecture 6 Numerical Methods: VerBcal DiscreBzaBons April 22, Paul A. Ullrich (HH 251)

Transcription

1 ATM 298, Spring 2013 Lecture 6 Numerical Methods: VerBcal DiscreBzaBons April 22, 2013 Paul A. Ullrich (HH 251) paullrich@ucdavis.edu

2 Course Projects Check out the short project descriptions! The next phase is turning towards hashing out the details for these projects. Recommended: Break up your project into a set of 3-5 steps. For each step, indicate what you need to do and what you need from me. 42 Days unbl project presentabons! Paul Ullrich ATM 298: Lecture 04 April 10,

3 Outline 1. Unique Aspects of the Vertical 2. Overview of Vertical Coordinate Systems 3. Terrain-following Variations 4. Vertical Computational Modes 5. Semi-Lagrangian Layers Slides are based on Michael Toy s talk on verhcal discrehzahons from the DCMIP workshop (2012)

4 Radius of the Earth km Atmosphere Depth 100 km Troposphere Depth 10 km Mountain Height 5 km

5 Unique Aspects of the Vertical Gravity acts in the verhcal (let s look at the verhcal momentum equahon) Typical scales (in m/s 2 ): dw dt u 2 + v 2 r g +2 u cos + r 2 w Dominant balance is between these two terms. Hydrosta)c balance

6 Unique Aspects of the Vertical For an approximately isothermal layer of the atmosphere, HydrostaBc balance: Ideal gas = p R d T 0 g 0 R d T = g Approx. SoluBon: gz p = p 0 exp R d T 0 z r a ExponenHal decay of pressure and density with height! Highly strabfied!

7 Unique Aspects of the Vertical Gradients are much stronger in the verbcal: It s much colder 10km straight up (- 50 C) and harder to breathe than 10km down the road. Paul Ullrich ATM 298: Lecture 06 April 22, 2013

8 Unique Aspects of the Vertical Boundary condihons at r = a (surface) and r = No flow through the surface Zero pressure up here

9 Unique Aspects of the Vertical Many dishnct physical processes: ConvecHon Boundary layer Viscous processes RadiaHon Waves Troposphere- Stratosphere InteracHon Some not- so physical processes: Model- top sponge layer (numerical viscosity)

10 Numerical Considerations Choice of coordinate system Height coordinates (Richardson 1922) Pressure coordinates (Eliassen 1949) Isentropic coordinates (Eliassen and Raustein 1968) Mass coordinates (Laprise 1992) Terrain- following or cut- cell? Ensure the hydrostahc relahon is sahsfied for a strahfied atmosphere Staggering of variables? How to handle boundary condihons?

11 Non-Hydrostatic Primitive Equations Five prognoshc equahons, one constraint equahon Shallow Atmosphere approximahon assumed du uv tan = +2 vsin dt a a z dv dt + u2 tan = 2 u sin a z dw dt = d dt = c v dt dt + pd dt = J r u p = R d T d + u h r z

12 Hydrostatic Primitive Equations Four prognoshc equahons, two constraint equahon du uv tan = +2 vsin dt a a z dv dt + u2 tan = 2 u sin a d dt = c v dt dt + pd dt = J r u z d + u h r z p = R d T g

13 Z-Coordinates (Shallow Atmosphere) CompuHng the verhcal velocity Non- hydrostabc (Predicted / Prognosed) dw dt = w = 1 Z g HydrostaBc (Diagnosed) r u h dz 1 Z z 0 w dz dt 1 p (B + Q)dz + 1 c p u h (u, v) HydrostaHc dynamical cores rarely use z- coordinates due to this mess Z z 0 J T dz B = 1 R z T 0 Q = u h rp Q p dz 1 c p R zt g Z zt z 0 R 1 z T 0 J T dz + R z T r u 0 h dz dz p r ( u h )dz Examples: Kasahara and Washington (1967) DeMaria (1995)

14 Vertical Coordinate Transforms z(x, y,,t)! (x, y, z, t) Z New generalized verbcal velocity: z z = = d u h r z

15 Vertical Coordinate Transforms z(x, y,,t)! (x, y, z, t) Z Prejy much any quanhty can be chosen for your verhcal coordinate, but it must be monotone: It must be either strictly increasing or decreasing as a func5on of height. Eliminates ophons such as temperature from being used as a verhcal coordinate.

16 Non-Hydrostatic Equations Six prognoshc equahons, one constraint equahon " du uv tan = dt a a m " dv dt + u2 tan 1 = + 1 a m dw dt = c v dt dt + pd dt = J dz dt = @ New Mass Variable (General Coordinates) # # +2 v sin 2 u sin + r (mu h (m ) d + u h r z p = R d T

17 Non-Hydrostatic Equations Six prognoshc equahons, one constraint equahon " du uv tan = dt a a m " dv dt + u2 tan 1 = + 1 a m dw dt = c v dt dt + pd dt = J dz dt = + r (mu h (m ) d + u h z is predicted! Two- term horizontal pressure gradient! # # +2 v sin 2 u p = R d T

18 Pressure Coordinates = p! Non- HydrostaBc EquaBons HydrostaBc EquaBons VerHcal Momentum Layer mass is constant dw dt = 1 m g m = 1 g Horizontal pressure gradient Single term HPGF = 1 m r pz HPGF = gr p z p ConHnuity EquaHon + r p (mu (m!) =0 r p (u =0 VerHcal velocity difficult to diagnose DiagnosHc conhnuity equahon! ṗ = Z p 0 r p u h dp

19 Mass Coordinates AKA HydrostaHc pressure coordinates (Laprise 1992) = (x, y, z, t) Z 1 Arises from the hydrostahc pressure equahon: z (x, y, z = g Represents the mass of air above a given height. For a hydrostahcally balanced atmosphere this is the pressure p.

20 Mass Coordinates Non- HydrostaBc EquaBons HydrostaBc EquaBons Pseudo- density m = dz d = 1 g Layer mass is constant Same as Non- Hydrosta5c Horizontal pressure gradient HPGF = 1 r (gz) Double term pressure gradient Same as =1 ConHnuity EquaHon DiagnosHc conhnuity equahon r u =0 = Z 0 r u h d Same as Non- Hydrosta5c

21 Isentropic Coordinates AKA AdiabaHc coordinates = (PotenHal temperature) = T R/cp p0 p The verhcal velocity is proporhonal to the diabahc heahng: d dt = J c p R/cp p0 p For an adiabahc atmosphere the verhcal mohon is zero and coordinate surfaces are material surfaces (a quasi- Lagrangain verhcal coordinate) This minimizes errors associated with verhcal advechon

22 Isentropic Coordinates In non- hydrostahc models at high horizontal resoluhon, stahc instabilihes and turbulence present a challenge since the coordinate loses the monotonicity property. This is typically solved by hybridizing the coordinate with something which maintains monotonicity. VerHcal cross- sechon of isentropes associated with a breaking mountain wave.

23 Vertical Coordinates Summary of four common verhcal coordinates. Coordinate Non- HydrostaBc Models HydrostaBc Models Height (z) Suitable Not preferred (difficult to diagnose w) Pressure (p) Not preferred (difficult to diagnose ω) Suitable Mass (π) Suitable Suitable (idenhcal to p coordinate) PotenHal temperature (θ) Suitable (some challenges) Suitable

24 Representation of Topography Three main ophons for represenhng topography: Step topography Sigma coordinates (terrain- following) Shaved cell Figure: Adcror et al. (1997)

25 Representation of Topography Advantages: No issues with small cells Accurate representahon of horizontal pressure gradient force Disadvantages: Poorly represents the underlying topography Hard corners create a lot of spurious noise

26 Representation of Topography Advantages: Accurate treatment of pressure gradient force Accurate treatment of topography Disadvantages: Small grid cells can affect the maximum Hmestep size (CFL condihon)

27 Representation of Topography Advantages: Topography absorbed into evoluhon equahons Very accurate for smooth topography Disadvantages: Poor representahon of horizontal pressure- gradient force near steep p x p = g z topography gr p z =0= gr z 1 r p The discrete form of these terms don t necessarily cancel!

28 Representation of Topography p p s z z z s z T z s Gal- Chen and Somerville (1975) = T S T Skamarock and Klemp (2008) = p p s + Phillips (1957) Sigma coordinate Normalized hydrostahc pressure p p 1 p s p s p p 0 Simmons and Burridge (1981) Hybrid coordinate Schar et al. (2002) Hybrid coordinate

29 Representation of Topography Second- Order AdvecBon over Topography Sigma Coordinate Simmons and Burridge (1981) Hybrid coordinate Schar (2002) Hybrid coordinate Source: Schar et al. (2002)

30 Vertical Staggering Lorenz ( L ) grid Charney- Phillips ( CP ) grid v, p v, p v, p LocaHon of levels and interfaces is idenhcal. Only difference is staggering of velocity and potenhal temperature. p v p v p v p Like an Arakawa A- grid Like an Arakawa C- grid

31 Vertical Staggering Lorenz ( L ) grid Linearized verhcal + g =0 v, p v, p v, p Density perturbahon (affects magnitude) Ignore for now DiscreHzaHon on + 1 pk+1 p k 1 z k+1 z k 1 Pressure perturbahon (affects propagahon) =0 Like an Arakawa A- grid

32 Vertical Staggering: Lorenz Lorenz ( L ) grid Linearized verhcal + g =0 v, p v, p DiscreHzaHon on Lorenz + 1 pk+1 p k 1 z k+1 z k 1 =0 v, p SeparaHon of odd/even modes: VerHcal velocity evoluhon equahon is unable to see 2Δz modes in the pressure field. Like an Arakawa A- grid Called a ComputaBonal Mode

33 Vertical Staggering: Charney-Phillips Linearized verhcal + g =0 DiscreHzaHon on Charney- Phillips + 1 pk+1/2 p k 1/2 =0 z k+1/2 z k 1/2 No separahon of odd/even modes. No computahonal mode supported. Also see: Tokioka (1978) Arakawa and Moorthi (1988) Arakawa and Konor (1996) Charney- Phillips ( CP ) grid p v p v p v p Like an Arakawa C- grid

34 Vertical Staggering There are many possible choices of verhcal staggering besides the ones chosen here: For non- hydrostahc system there are 5 prognoshc variables We can choose any two thermodynamic variables from ρ, p, T, θ, etc. Some have computahonal modes. Others do not. Accurate representahon of waves (acoushc, inerha- gravity, Rossby) best achieved by minimizing finite differences over 2Δz. Charney- Phillips ( CP ) grid p v p v p v p Analyzed by Thuburn and Woollings (2005) in three coordinate systems.

35 Semi-Lagrangian Coordinates Instead of fixing the verhcal coordinate, a common strategy is the use of semi- Lagrangian (or quasi- Lagrangian) verhcal coordinates. This is analogous to the semi- Lagrangian advechon technique discussed last Hme, except only applied to the verhcal. Remapping to a fixed Eulerian grid is typically performed every 15 or 30 minutes. Introduced by Starr (1945) In models: Skamarock (1998) He (2002) Lin (2004) Zängl (2007) Toy and Randall (2009) Toy (2011) Figure: Nair et al. (2009)

36 Semi-Lagrangian Equations Six prognoshc equahons, one constraint equahon " du uv tan = If are floahng +2 v with sin dt a a verhcal velocity, " then dv dt + u2 tan 1 = + 1 there is # through layer surfaces. 2 u sin a m dw dt = c v dt dt + pd dt = J dz dt = z is predicted! + r (mu h (m ) d + u h @ p = R d T

37 Semi-Lagrangian Coordinates Layer thickness becomes a prognoshc variable: d( p k ) dt = k+1/2 k 1/2 Before layer thicknesses can become too thin / thick, they are remapped back to the fixed Eulerian grid. The remapping procedure takes the place of verhcal advechon. Figure: Nair et al. (2009)

38 Timestep Limitations (Non-Hydrostatic) Recall that stability of an explicit numerical method requires a CFL condihon to be sahsfied. For example: u t x apple 1 Here u is the maximum wave speed of the system, Δx is the grid spacing and Δt is the Hme step size. In the atmosphere, the fastest waves are sound waves, with a maximum speed of u r p 342m/s 100m 110km Only relevant for non- hydrostahc atmospheric models. HydrostaHc models do not support verhcally propagahng sound waves!

39 Timestep Limitations CFL CondiHon: u t x apple 1 with u 342 m/s 100m for x = 110 km t apple 321? s 110km for x = 100 m t apple 0.3? s In prachce, limihng the Hme step size to be governed by the verhcal coordinate is too severe! Need a bejer approach

40 Timestep Limitations Filtered EquaBons: AnelasHc approximahon (Ogura and Phillips, 1962) Pseudo- Incompressible approximahon (Durran, 1989) Unified approximahon (Arakawa and Konor 2005) (basic premise is to derive a new set of equahons which do not contain sound waves and so maximum wave speed is limited by advechon) Numerical Methods: Implicit Hme stepping in the verhcal

41 That s all!