Equations / variables Vertical coordinate Terrain representation Grid staggering

– Equations / variables – Vertical coordinate– Terrain representation– Grid staggering– Time integration scheme– Advection scheme– Boundary conditions– Map projections– Dynamics parameters

WRF Mass-Coordinate Dynamical Solver

Flux-Form Equations in Mass Coordinate

ts

t

,

Hydrostatic pressure coordinate:

,,,, wWvVuU

Conserved state variables:

hydrostatic pressure

Non-conserved state variable: gz

Flux-Form Equations in Mass Coordinate

gwdt

d

x

U

t

Qx

U

t

w

x

Uwpg

t

W

u

x

Uu

x

p

x

p

t

U

0

Inviscid, 2-Dequations without rotation:

Diagnosticrelations:

,,0p

Rp

Flux-Form Equations in Mass Coordinate

Introduce theperturbation variables:

)(,)(

;)(,)(

zpzpp

zz

w

x

Uwpg

t

W

u

x

Uup

xxxx

p

t

U

Momentum and hydrostatic equations become:

),,(),,,(

),,,()(

yxyxp

yxz Note –

likewise

Flux-Form Equations in Mass Coordinate

Acoustic mode separation:

Recast Equations in terms of perturbation about time t

'''',''''

;'''','''',''''

,'''','''',''''

tt

ttt

ttt

ppp

WWWVVVUUU

''''1

''

''''''''

2

tt

ttttscpLinearize ideal gas law

about time t

Vertical pressure gradientbecomes

tts

tt

s ccp '''''' 2

2

2

Flux-Form Equations in Mass Coordinate

ttt

twtt

s

tt

s

tt

t

tu

ttttt

RWg

Rcc

gW

R

R

Rp

xxxx

pU

''1

''

''''''''

''

''

''''''

''''

''

'V'

'V'

'V'

2

2

2

Small (acoustic) timestep equations:

Moist Equations in Mass-Coordinate Model

Moist Equations:

vdodd

lvlvdlvdlvd

d

dd

dd

p

Rp

Qq

x

qU

t

q

x

U

t

w

x

Uwpg

t

W

u

x

Uu

x

p

x

p

t

U

,

)()()(

0

,,,,

Diagnostic relations:

Mass-Coordinate Model, Terrain Representation

Vertical coordinate:

tst

,hydrostatic pressure

gwy

vx

ut

Lower boundary condition for the geopotential specifies the terrain elevation, and specifyingthe lowest coordinate surface to be the terrainresults in a terrain-following coordinate.

)( gz

Height/Mass-coordinate model, grid staggering

z

W

W

U U

x

lv qq ,,,

V

V

U U

x

y

lv qq ,,,

C-grid staggering

horizontal vertical

Height/Mass-Coordinate Model, Time Integration

3rd Order Runge-Kutta time integration

Rt

Rt

Rt

tt

t

tt

tt

3

2

3

1

1advance

Amplification factor

241;;

41 tk

AAki nnt

Time-Split Leapfrog and Runge-Kutta Integration Schemes

Phase and amplitude errors for LF, RK3

Oscillation equationanalysis

ikt

Phase and amplitude errors for LF, RK3

Advection equationanalysis

xt U

5th and 6th order upwind-biased and centered schemes.Analysis for 4x wave.

Acoustic Integration in the Mass Coordinate Model

Forward-backward scheme, first advance the horizontal momentum

Second, advance continuity equation,diagnose omega,and advance thermodynamic equation

ttt

twtt

s

tt

s

tt

t

tu

ttttt

RWg

Rcc

gW

R

R

Rp

xxxx

pU

''1

''

''''''''

''

''

''''''

''''

''

'V'

'V'

'V'

2

2

2

Finally, vertically-implicit integration of the acoustic and gravity wave terms

Advection in the Height/Mass Coordinate Model

2nd, 3rd, 4th, 5th and 6th order centered and upwind-biased schemesare available in the WRF model.

Example: 5th order scheme

UFUFxx

Uii

2

1

2

1

1

12132

32211

2

1

2

1

10560

1,1

60

1

15

2

60

37

iiiiii

iiiiiiii

Usign

UUF

where

For constant U, the 5th order flux divergence tendency becomes

TOHx

Cr

x

tU

x

Ut

x

Ut

iiiiiii

thth

..60

6152015660

1

6

6

321123

65

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.

Runge-Kutta loop (steps 1, 2, and 3) (i) advection, p-grad, buoyancy using (ii) physics if step 1, save for steps 2 and 3 (iii) mixing, other non-RK dynamics, save… (iv) assemble dynamics tendencies Acoustic step loop (i) advance U,V, then , then w, (ii) time-average U,V, End acoustic loop Advance scalars using time-averaged U,V, End Runge-Kutta loopOther physics (currently microphysics)

Begin time step

End time step

WRF Mass-Coordinate Model Integration Procedure

,,t

Mass Coordinate Model: Boundary Condition Options

1. Specified (Coarse grid, real-data applications).2. Open lateral boundaries (gravity-wave radiative).3. Symmetric lateral boundary condition (free-slip wall).4. Periodic lateral boundary conditions.5. Nested boundary conditions (not yet implemented).

Lateral boundary conditions

Top boundary conditions

1. Constant pressure.2. Gravity-wave radiative condition (not yet implemented).3. Absorbing upper layer (increased horizontal diffusion).4. Rayleigh damping upper layer (not yet implemented).

Bottom boundary conditions1. Free slip.2. Various B.L. implementations of surface drag, fluxes.

Mass/Height Coordinate Model: Coordinate Options

1. Cartesian geometry (idealized cases)2. Lambert Conformal3. Polar Stereographic4. Mercator

3rd order Runge-Kutta time step

Acoustic time step

Divergence damping coefficient: 0.1 recommended.

Vertically-implicit off-centering parameter: 0.1 recommended.Advection scheme order: 5th order horizontal, 3rd order vertical

recommended.

Mass Coordinate Model: Dynamics Parameters

Courant number limited, 1D: 73.1

x

tUCr

Generally stable using a timestep approximately twice as large as used in a leapfrog model.

2D horizontal Courant number limited: 2

1

h

CC sr

stepsacousticofnumberRKsound t

External mode damping coefficient: 0.05 recommended.