The Case for Using High-Order Numerical Methods in Mesoscale and Cloudscale NWP Bill Skamarock NCAR/MMM

WRF (Weather Research and Forecast) Model NCAR NOAA - NCEP NOAA - FSL Air Force Weather Agency Federal Aviation Administration NRL, Universities and other labs Collaborative development effort by Develop an advanced mesoscale forecast and assimilation system Design for 1-10 km horizontal grids Portable and efficient on parallel computers Advanced data assimilation and model physics Well suited for a broad range of applications Community model with direct path to operations

The principal objective when developing an NWP model is to maximize efficiency (1)Maximize forecast (solution) accuracy for a given computer resource, or (2) minimize computer resource needed for a given forecast (solution) accuracy.

Consider … Existing nonhydrostatic NWP models use low-order accuracy numerics (1 st /2 nd order time, 2 nd order space) MM5 ARPS COAMPS NMM GEMS LM UKMO Unified Model WRF dynamical core development projects (1) Eulerian mass and height coordinate cores - 3 rd order RK3 time int. - high order advection. (2) Semi-Lagrangian core - High order RK time int. - High order spatial operators for interp and gradient operators. Question: Is the use of higher-order methods in the WRF cores justified?

Why are low-order numerics used in most mesoscale NWP models? 1.High-order methods examined in the early development of NWP models were generally not robust. 2.Traditional verification measures will not show increased accuracy of higher order methods when phenomena at small scale are inherently unpredictable at typical mesoscale NWP timescales (1-3 days). 3.There is a widely perceived need to conserve higher moments of the model solutions where possible. Conservation of this sort usually leads to the use of lower order methods. Are these reasons (still) valid?

Are these reasons (still) valid? NO! 1.High-order methods examined in the early development of NWP models were generally not robust. Robust higher-order methods have been developed. 2.Traditional verification measures will typically not show the increased accuracy of higher order methods when phenomena at small scale are inherently unpredictable at typical mesoscale NWP timescales (1-3 days). Other error and verification measures should be used at mesoscale and cloudscale resolutions in addition to the traditional methods. 3.There is a widely perceived need to conserve higher moments of the model solutions where possible. Conservation of this sort usually leads to the use of lower order methods. Conservation is not necessary or appropriate at small scales.

Outline for the remainder of this talk: Describe the higher order numerical methods used in the WRF model. Present theoretical arguments for increased accuracy of these methods. Present examples demonstrating this increased accuracy and efficiency. Present some arguments suggesting that we should expect increased efficiency using these methods. Consider some arguments against the need for conservation of higher order moments in NWP models at small (and perhaps even large) scale. Mention some issues that arise in global modeling that do not arise in limited-area modeling.

Conservation in numerical schemes What quantities should we consider conserving? first-order quantities: mass, momentum, entropy. second-order quantities: energy, enstrophy? Historically, energy conservation was used to analyze and prove stability (Keller and Lax, 1960’s) for nonlinear systems (esp. with recognition of nonlinear instability by Phillips, 1959). Energy: Equations conserve energy (and mass, momentum, and entropy). Enstrophy: Equations do not conserve enstrophy (barotropic vorticity equations do). Arakawa Jacobian formulation is unsuitable for flows exhibiting systematic downscale energy cascade.

No, but there is no theory supporting a yes or no answer. Question: Does it make sense to conserve higher-order moments (energy) in numerical solutions when first order quantities are not conserved? Question: Does it make sense to require conservation of energy in a numerical scheme when parameterized sources/sinks and boundary fluxes are orders of magnitude larger than the conservation errors in a non-conservative but more accurate scheme. No, because this conservation would be meaningless. Energy conservation: Observation: Accurate solutions from non-conservative models should be very nearly conservative.

For short range high resolution NWP, conservation of higher order quantities is likely irrelevant to forecast skill. - frequent assimilation of new data - errors have little time to accumulate For long range forecasts and climate applications, preceding arguments are still valid. Our philosophy: First and foremost, we should conserve the first-order quantities in which the governing equations of the model are cast.

Global models must solve the equations of motion on a sphere: Pole problem: converging meridians (possibly severe stability/timestep restrictions). Solutions - spectral formulations (implicit), other implicit formulations, semi-Lagrangian formulations. No lateral boundary specification needed. Dynamical solver issues for global and regional models Regional models solve equations on some portion of the globe: Lateral boundary condition problem. No pole problem. Thus, there is more latitude in the choice of numerical schemes for use in regional models than for use in global models.

What is in the WRF model? The official core is the Eulerian mass coordinate core Hydrostatic pressure coordinate: Conserved state variables: hydrostatic pressure Non-conserved state variable:

Inviscid, 2-D equations without rotation: Diagnostic relations: WRF model flux-form mass coordinate equations

Height/Mass-Coordinate Model, Time Integration 3 rd Order Runge-Kutta time integration advance Amplification factor

Time-Split Leapfrog and Runge-Kutta Integration Schemes

Phase and amplitude errors for LF, RK3 Oscillation equation analysis

Phase and amplitude errors for LF, RK3 Advection equation analysis 5 th and 6 th order upwind-biased and centered schemes. Analysis for 4  x wave.

Advection in the Height/Mass Coordinate Model 2 nd, 3 rd, 4 th, 5 th and 6 th order centered and upwind-biased schemes are available in the WRF model. Example: 5 th order scheme where

For constant U, the 5 th order flux divergence tendency becomes Advection in the Height/Mass Coordinate Model The odd-ordered flux divergence schemes are equivalent to the next higher ordered (even) flux-divergence scheme plus a dissipation term of the higher even order with a coefficient proportional to the Courant number.

2 nd Order4 th Order6 th Order 3 rd Order5 th Order Advection of Top-Hat Profile

Phase and amplitude errors for LF, RK3 Oscillation equation analysis

Advection Dispersion Equation

Maximum Courant Number for Advection (Wicker & Skamarock, 2002) Time Scheme 3 rd 4 th 5 th 6th Leap Frog U 0.72 U 0.62 RK U 0.30 U RK Spatial Order U = unstable

Theory – Numerical Solution of PDEs solution error resolution low-order method high-order method Error versus resolution solution error cost low-order method high-order method Error versus cost

(From Wicker & Skamarock, MWR 2002)

What can we take from theory? (1)Increasing the order of the method: - increases the cost per timestep (e.g., in WRF, RK3 is twice the cost per timestep compared to leapfrog). - decrease in error is problem dependent (error reduction by small fraction to orders of magnitude are possible). (2)Increasing the resolution for a given method: - cost scales as where n is the number of spatial dimensions refined. Example: for a doubling of the horizontal resolution, the computational cost increases by a factor of 8. Consider,

For multidimensional problems, higher-order methods are usually more efficient than lower-order methods Another example: On the same grid, if a high order dynamical core takes twice as long to produce a solution compared to a low order dynamical core, the high order model will run in the same time as the low order model on a grid with horizontal resolution

Practice (or the problem with theory) In NWP models, the solutions are (1) not smooth, and (2) the solutions do not converge. Why? Higher resolution leads to more fine-scale structure in the solution, because So, we cannot rely solely on theory to guide us in choosing the most efficient methods for our models. (1) model physics depend on the resolution, and (2) the resolution of the terrain, initial conditions and boundary conditions also increases.

(1) How do we define solution error in NWP? - verification measures? - can we associate errors with a model component, such as the dynamical core? (2) What should we expect from our models as we increase resolution and resolve motions that are inherently unpredictable (e.g., convective cells for forecasts greater than O(hour)). - pointwise verification (implied determinacy) is not appropriate. - Need measures of resolution and spatial variability. How do we evaluate the efficiency of a dynamical core?

5 min 10 min15 min Comparison of Gravity Current Simulations Height Coordinate Mass Coordinate Dx = Dz = 100 m

2-D Mountain Wave Simulation a = 1 km, dx = 200 ma = 100 km, dx = 20 km Mass Coordinate Height Coordinate

Comparison of Height and Mass Coordinates

Supercell Thunderstorm Simulation Height coordinate model ( dx = 2 km, dz = 500 m, dt = 12 s, 80 x 80 x 20 km domain ) Surface temperature, surface winds and cloud field at 2 hours

Baroclinic Wave Simulation – Surface Fields Pressure (solid, c.i.= 4 mb), temperature (dashed, c.i.= 4K), cloud field (shaded) Mass Coordinate, 4 days 12 h Height Coordinate, 4 days 6 h Dx = 100 km, Dt= 10 min

From Takemi and Rotunno, 2002 WRF squall-line simulations N-S periodic, W-E open b.c. TKE tests (t = 4h, dx = 1 km, gust front dashed)

From Takemi and Rotunno, 2002

36 h Forecast Valid 12Z 1 April 02, 24 h Precip 12 km ETA22 km WRF24 h RFC Analysis

Precipitation Threat Score and Bias March 2002April 2002

The problem with precip threat scores (and other pointwise verification schemes) truth forecast 1forecast 2 Issue: the obviously poorer forecast has better skill scores From Mike Baldwin NOAA/NSSL

Observations22 km WRF10 km WRF 3 hour accumulated precip, forecasts, valid 18Z 4 June Z runs, 15-18Z precip accumulation 4 km obs analysis (radar and gages) precip (mm) From Mike Baldwin and Matt Wandishin, NOAA/NSSL

8 km NMM12 km opnl ETA precip (mm) 3 hour accumulated precip, forecasts, valid 18Z 4 June Z runs, 15-18Z precip accumulation 4 km obs analysis (radar and gages) From Mike Baldwin and Matt Wandishin, NOAA/NSSL Observations

Power spectra for precip (obs and forecasts) From Matt Wandishin and Mike Baldwin, NOAA/NSSL, Spectra code from Ron Errico, NCAR 12Z forecasts, Z accum precip, valid 4 June 2002

Power spectra for precip (obs and forecasts) From Matt Wandishin and Mike Baldwin, NOAA/NSSL, Spectra code from Ron Errico, NCAR 0Z and 12Z forecasts, 3 hourly accum precip, averaged over June 2002

Conclusion: Given the high cost of large, multi-dimensional atmospheric simulations (i.e., NWP), the use of high-order numerical methods maximizes efficiency (accuracy for a given cost) in many (perhaps most?) error measures. Power spectra for precip (obs and forecasts) 12Z forecasts, Z accum precip, valid 4 June 2002